Nine-Loop Amplitude

A scattering amplitude is a formula that gives the chance of each outcome when particles collide. Physicists rarely know it exactly. They write it as a sum in powers of a small number $g^{2}$ , and they compute one term at a time.

E = 1 + g^{2} E^{(1)} + g^{4} E^{(2)} + g^{6} E^{(3)} + \dots

Here $E$ is the amplitude after a known factor is divided out. Each power of $g^{2}$ counts one loop. A Feynman diagram draws a collision as lines, one line for each particle. Particles can split and rejoin on the way, and that closes a cell in the diagram. The term $E^{(L)}$ needs diagrams with $L$ closed cells.

One diagram with 6 outer lines and 9 closed cells. One cell is shaded. No lines cross.

Most amplitudes are known to 2 loops, and a few to 3. A well-known prediction for the electron uses 5. The new result is $E^{(9)}$ , the nine-loop term for six particles, in a theory called planar $𝒩 = 4$ super Yang-Mills. Theories of this type describe 3 of the 4 known forces. This one gives each particle extra partner particles, so it does not describe our world, but its formulas are simpler. Planar means that only diagrams with no crossing lines count.

A function is a rule that gives an output for each input. Each $E^{(L)}$ is a function of three numbers $u$ , $v$ and $w$ , which describe the shape of the collision. At $u = v = w = 1$ , $E^{(8)}$ is about $- 47.7$ million and $E^{(9)}$ is about 674 million. If $g^{2} = 1 / 100$ , the nine-loop term is about 0.00000000067. So at that size of $g^{2}$ , the ninth loop changes the sum in the tenth decimal place.

The bootstrap method adds no diagrams. It guesses the form of the answer: a sum of words, with a fraction in front of each word. At $L$ loops, a word has $2 L$ characters from an alphabet of 9. Six characters are $a = u / (v w)$ , $b = v / (w u)$ , $c = w / (u v)$ , $d = (1 - u) / u$ , $e = (1 - v) / v$ and $f = (1 - w) / w$ . The other three also use a square root. Each word stands for one function, built from its characters in order.

At nine loops a word has 18 characters, so there are $9^{18}$ possible words, about 150 quadrillion. One word in the answer is bbfddedfbdfaafefde, and the fraction in front of it is $- 105,757 / 65,536$ .

Rules remove most words. Only $a$ , $b$ or $c$ can come first. Two different characters among $a$ , $b$ , $c$ cannot stand side by side, and neither can $a$ and $d$ , $b$ and $e$ , or $c$ and $f$ . Such rules leave only 5,431 independent sums for the first 13 characters, out of about 2.5 trillion possible words, and 424 for the last 5. So the answer needs only $424 \times 5,431$ fractions, or 2,302,744. Known behavior in limiting cases, and a link to a simpler quantity called a form factor, then fix them all.

Anthropic reports that its AI model, Claude, computed the full answer. Two Anthropic physicists asked for the amplitude in one sentence, then kept telling it to continue. Claude wrote the code, and the cost was about $1,000 to $2,000 for each of its two routes. One route is the direct bootstrap, and the other goes through the form factor. The two routes agree on all 107,053 nonzero fractions compared.

Lance Dixon, who found the eight-loop term with Yu-Ting Liu, checked the result. Song He, Jirong Jing and Xiang Li posted their own nine-loop data the next day, from work with some help from GPT-6. Claude used known methods with more computing, and it found no new method. The answer files are public, but the programs are not. This is a calculation, not a proof. It assumes, for example, that the same 9 characters are enough at nine loops.