The planar three-gluon form factor for the chiral stress tensor operator in planar maximally supersymmetric Yang-Mills theory is an analog of the Higgs-to-three-gluon scattering amplitude in QCD. The amplitude (symbol) bootstrap program has provided a wealth of high-loop perturbative data about this form factor, with results up to eight loops available. The symbol of the form factor at $\textit{L}$ loops is given by words of length 2$\textit{L}$ in six letters with associated integer coefficients. In this paper, we analyze this data, describing patterns of zero coefficients and relations between coefficients. We find many sequences of words whose coefficients are given by closed-form expressions which we expect to be valid at any loop order. Moreover, motivated by our previous machine-learning analysis, we identify simple recursion relations that relate the coefficient of a word to the coefficients of particular lower-loop words. These results open an exciting door for understanding scattering amplitudes at all loop orders.

United States

Tianji Cai

France

François Charton

Kyle Cranmer

Lance Dixon

Garrett Merz

Denmark

Matthias Wilhelm

Recurrent features of amplitudes in planar $$ \mathcal{N} $$ = 4 super Yang-Mills theory

We show how curing an anomaly of the twistor uplift of self-dual Yang-Mills theory implies linear relations among one-loop, -gluon, color-ordered subamplitudes in QCD, when all gluon helicities are positive, or when exactly one is negative. We compute the number of linearly independent subamplitudes as determined by these relations, in terms of unsigned Stirling numbers. Then we use a momentum-twistor parametrization to show that there are no further linear dependencies.

Anthony Morales

Rational QCD loop amplitudes and quantum theories on twistor space

We study the three-point form factor of the length-three half-BPS operator (Tr ) in planar $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory, using analyticity and integrability methods. We find that the functions describing the form factor in perturbation theory live in the same restrictive space of multiple polylogarithms as the one describing the form factor of the stress-tensor operator (Tr ). Furthermore, we find that the leading-order data in the collinear limit provided by the form factor operator product expansion (FFOPE) is enough to fix the form factor uniquely, at least through six loops. We perform various tests of our results using the subleading FFOPE corrections. We also analyze the form factor in the Regge limit where two Mandelstam invariants are large; we obtain a compact representation for the form factor in this limit which is valid to all orders in the coupling.

Benjamin Basso

Alexander Tumanov

The three-point form factor of Tr

Recently, the maximally-helicity-violating four-point form factor for the chiral stress-energy tensor in planar $$ \mathcal{N} $$ = 4 super Yang-Mills was computed to three loops at the level of the symbol associated with multiple polylogarithms. It exhibits , or invariance under the combined action of a kinematic map and reversing the ordering of letters in the symbol. Here we lift the two-loop form factor from symbol level to function level. We provide an iterated representation of the function’s derivatives (coproducts). In order to do so, we find a three-parameter limit of the five-parameter phase space where the symbol’s letters are all rational. We also use function-level information about dihedral symmetries and the soft, collinear, and factorization limits, as well as limits governed by the form-factor operator product expansion (FFOPE). We provide plots of the remainder function on several kinematic slices, and show that the result is compatible with the FFOPE data. We further verify that antipodal self-duality is valid at two loops beyond the level of the symbol.

Shuo Xin

A two-loop four-point form factor at function level

We study scattering amplitudes in massless non-abelian gauge theory where all outgoing gluons have positive helicity. It has been argued recently by Costello that for a particular fermion representation (8 fundamentals plus one antisymmetric-tensor representation in SU(N)) the one-loop amplitudes vanish identically. We show that this vanishing leads to previously-observed identities among one-loop color-ordered partial amplitudes. We then turn to two loops, where Costello has computed the all-plus amplitudes for this theory, as rational functions of the kinematics for any number of gluons using the celestial chiral algebra (CCA) bootstrap. We show that in dimensional regularization, these two-loop amplitudes are not rational, and they are not even finite as ϵ → 0. However, the finite remainder for four gluons agrees with the formula by Costello. In addition, we provide a mass regulator for the infrared-divergent loop integrals; with this regulator, the CCA bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to compute the double-trace terms in the theory at two loops for an arbitrary number of gluons.

On gauge amplitudes first appearing at two loops

Enrico Herrmann

China

Germany

Kai Yan

Hua Zhu

Erratum to: Soft gluon emission at two loops in full color

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.

Claude Duhr

Albrecht Klemm

Florian Loebbert

Christoph Nega

Franziska Porkert

The Basso-Dixon formula and Calabi-Yau geometry

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra  $$\mathbb {C}\otimes \mathbb {H}\otimes \mathbb {O}$$  <math> <mrow> <mi>C</mi> <mo>⊗</mo> <mi>H</mi> <mo>⊗</mo> <mi>O</mi> </mrow> </math>  . Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal–Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within  $$\mathbb {C}\otimes \mathbb {H}\otimes J_{2}(\mathbb {O})$$  <math> <mrow> <mi>C</mi> <mo>⊗</mo> <mi>H</mi> <mo>⊗</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>O</mi> <mo>)</mo> </mrow> </mrow> </math>  ,  $$\mathbb {O}\otimes J_{2}(\mathbb {C}\otimes \mathbb {H})$$  <math> <mrow> <mi>O</mi> <mo>⊗</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>C</mi> <mo>⊗</mo> <mi>H</mi> <mo>)</mo> </mrow> </mrow> </math>  , and  $$\mathbb {C}\otimes \mathbb {O}\otimes J_{2}(\mathbb {H})$$  <math> <mrow> <mi>C</mi> <mo>⊗</mo> <mi>O</mi> <mo>⊗</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>H</mi> <mo>)</mo> </mrow> </mrow> </math>  .

David Chester

Spain

Alessio Marrani

Portugal

Daniele Corradetti

Raymond Aschheim

Klee Irwin

Dixon-Rosenfeld lines and the Standard Model

We compute the six-particle maximally-helicity-violating (MHV) amplitude in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills theory at eight loops, using antipodal duality and the recently computed eight-loop three-point form factor for the chiral stress energy tensor multiplet. Antipodal duality maps the form factor symbol to the amplitude symbol on a two-dimensional parity-preserving surface in the three-dimensional amplitude kinematics. There are remarkably few ambiguities in lifting from two to three dimensions, nor in promoting the symbol to a function. The amplitude passes many tests, including near-collinear, multi-Regge, factorization, self-crossing and origin limits. These checks also constitute a validation of antipodal duality at eight loops.

Yu-Ting Liu

An eight loop amplitude via antipodal duality

A necessary ingredient for extending the BFKL equation to next-to-next-to-leading logarithmic (NNLL) accuracy is the one-loop central emission vertex (CEV) for two gluons which are not strongly ordered in rapidity. Here we consider the one-loop six-gluon amplitude in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills (SYM) theory in a central next-to-multi-Regge kinematic (NMRK) limit, we show that its dispersive part factorises in terms of the two-gluon CEV, and we use it to extract the one-loop two-gluon CEV for any helicity configuration within this theory. This is a component of the two-gluon CEV in QCD. Although computed in the NMRK limit, both the colour structure and the kinematic dependence of the two-gluon CEV capture much of the complexity of the six-gluon amplitudes in general kinematics. In fact, the transcendental functions of the latter can be conveniently written in terms of impact factors, trajectories, single-emission CEVs and a remainder, which is a function of the conformally invariant cross ratios which characterise the six-gluon amplitudes in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. Finally, as expected, in the MRK limit the two-gluon CEV neatly factorises in terms of two single-emission CEVs.

United Kingdom

Emmet Byrne

Switzerland

Vittorio Duca

Einan Gardi

Jennifer Smillie

One-loop central-emission vertex for two gluons in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills theory