A three-point form factor through five loops

Abstract

We bootstrap the three-point form factor of the chiral part of the stress­tensor supermultiplet in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, “extended-Steinmann-like” conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as gg → Hg and γ* → $$ q\overline{q}g $$ q q ¯ g . We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar $$ \mathcal{N} $$ N = 4 three-point form factor can be placed in an even smaller space of functions, with no independent ζ values at weights two and three.

Operator Product Expansion for Form Factors

“…To our surprise, we observe that the conjectures obtained in this way indeed match with the one-loop data at order e −4τ . Moreover, with the known corrections to the energy of the flux-tube excitations, these conjectures correctly reproduce the terms of order e −4τ τ −1 up to seven loops [10,11,26]. Taking into account the all-loop two-particle transitions we derive in [25], we also find perfect agreement with the terms of order e −4τ τ −2 and e −4τ τ −3 up to seven loops [10,11,26].…”

“…In section 4, we use the resulting transitions together with (1.2) to produce OPE predictions for the three-particle FF at order e −2τ , providing explicit results up to sevenloop order. These match with the independent available data up to two-loop order [26], and have already been used for a perturbative bootstrap of the three-point form factor up to seven-loop order [10,11].…”

“…In addition to the leading FFOPE contribution, these transitions also provide us with the terms of order e −2τ τ −1 at loops; we attach these predictions up to seven-loop order in the auxiliary file OPE-predictions.txt. These predictions have already been used, combined with other data, for a perturbative bootstrap of the three-point form factor at any value of τ and σ up to seven-loop order [10,11]. In [25], we will use a bootstrap procedure to determine the two-particle form factor transitions at finite coupling, thus lifting the Born-level results of this paper to any value of the coupling.…”

“…Our three-, four-, five-, six-, and seven-loop predictions have already provided valuable input for the perturbative form factor bootstrap up to seven-loop order [10,11]. 8…”

“…In this paper, we implement in detail the FFOPE at leading order in perturbation theory for the operators in the chiral half of the stress-tensor supermultiplet and use it to generate new predictions for FFs at all loop orders. These predictions have already played a crucial role in the perturbative bootstrapping of these form factors [10,11].…”