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D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops

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Zvi Bern, John Joseph Carrasco, Lance J. Dixon, Michael R. Douglas, Matt von Hippel, Henrik Johansson

High Energy Physics - Theory · Submitted 29 Oct 2012 · 63 references

Pith's verdict: 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.

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CitationHenrik Johansson, John Joseph Carrasco, Lance J. Dixon, Matt von Hippel, Michael R. Douglas, Zvi Bern. "D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops." arXiv:1210.7709 (2012). https://pith.science/paper/E6OMOCGE
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The paper

The paper claims

Planar maximally supersymmetric Yang-Mills theory in five dimensions diverges at six loops, and the coefficient of that divergence is nonzero.

The reading

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.

What carries the argument

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.

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $D = 4$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Original abstract

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,259 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.37 · compares with prior results 0.91 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. background 1.00 0.04 0.02 speculative 0.57
s1 We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. contribution 0.96 0.93 0.07 asserted 0.82
s2 We show that the coefficient is nonvanishing. result 1.00 0.96 0.87 asserted 1.00
s3 Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. result 0.95 0.90 0.12 asserted 0.96
s4 This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. interpretation 0.54 0.51 0.06 asserted 0.89
s5 To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. method 0.99 0.95 0.02 asserted 0.99
s6 The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. method 1.00 0.84 0.02 asserted 1.00
s7 The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program. method 1.00 0.85 0.02 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from simulation or computation 0.95.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Graphs 1 through 35 for the planar six-loop four-poin
Figure 2. FIG. 2: Graphs 36 through 68 for the planar six-loop four-poi
Figure 3. FIG. 3: The six-loop cut evaluated in
Figure 4. FIG. 4: A few sample graphs with labels corresponding to the l
Figure 5. FIG. 5: The six distinct vacuum diagrams that appear in eqs. (
Figure 6. FIG. 6: The canonical vacuum integral. All contributions ca
Figure 7. FIG. 7: Two simpler vacuum integrals that appear in the UV div
Figure 8. FIG. 8: Six scalar vacuum integrals that appear at intermedi
Figure 9. FIG. 9: An extra scalar diagram used for checking integral id
Figure 10. FIG. 10: The sector integrals for diagram
Figure 11. FIG. 11: The total contribution of sectors 2
Figure 12. FIG. 12: The dots indicated the numerical coefficients in eq. (

Figures reproduced from arXiv:1210.7709 by the authors.

References
How the literature uses it

52 arXiv papers cite this in 66 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 49 sentences without a confident label

Builds on it 1 paper
Uses its data or values 1 paper
Cites it as support 1 paper
Cites it as background 13 papers

What it cites · 59 sentences without a confident label

Uses the method or tool of 12 papers
  • A long-standing challenge of numerically evaluating Euclidean Feynman integrals has been addressed by a computational technique called sector decomposition [ref] , implemented in several software packages [ref] including the FIESTA package [ref] , used here.

    SecDec: A general program for sector decomposition · Introduction · 1.00

  • A long-standing challenge of numerically evaluating Euclidean Feynman integrals has been addressed by a computational technique called sector decomposition [ref] , implemented in several software packages [ref] including the FIESTA package [ref] , used here.

    Resolution of singularities for multi-loop integrals · Introduction · 1.00

  • A long-standing challenge of numerically evaluating Euclidean Feynman integrals has been addressed by a computational technique called sector decomposition [ref] , implemented in several software packages [ref] including the FIESTA package [ref] , used here.

    Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA) · Introduction · 1.00

Uses data or values from 3 papers
Compares with 3 papers
Cites as background 6 papers
  • [ref] .) The current consensus for $$ supergravity is that a $D=4$ potential counterterm valid under all known symmetries exists at seven loops [ref] .

    The vanishing volume of D=4 superspace · Conclusions and outlook · 0.97

  • [ref] .) The current consensus for $$ supergravity is that a $D=4$ potential counterterm valid under all known symmetries exists at seven loops [ref] .

    String theory dualities and supergravity divergences · Conclusions and outlook · 0.97

  • [ref] .) The current consensus for $$ supergravity is that a $D=4$ potential counterterm valid under all known symmetries exists at seven loops [ref] .

    Progress on Ultraviolet Finiteness of Supergravity · Conclusions and outlook · 0.97

Discussion

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