Polylogs for Polygons Bootstrapping Amplitudes and Wilson Loops

Polylogs for Polygons: Bootstrapping Amplitudes and Wilson Loops in Planar N=4 Super-Yang-Mills Theory Lance Dixon New Directions in Theoretical Physics 2 Higgs Centre, U. Edinburgh 12 Jan. 2017

LHC is producing copious data There is much more to explore – Higgs boson properties, as well as searches for new particles L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 2

LHC data often more accurate than state of the art theory (NLO QCD) pp 4 jets 1509. 07335 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 3

Why? • QCD 2 -loop amplitudes all unknown for 2 3 or higher Li 2(…) ? ? ? • Can we learn more about QCD – at least the functions needed for it – by studying a toy theory, planar N=4 SYM? L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 4

QCD vs. planar N=4 SYM fundamental QCD adjoint + adjoint scalars N=4 SYM Change gauge group from SU(3) to SU(Nc), let Nc ∞ so planar Feynman diagrams dominate L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 5

QCD is asymptotically free: b < 0 Running of as is logarithmic, slow at short distances (large Q) N=4 SYM is conformally invariant, b = 0 Bethke a Q confining L. Dixon Polylogs for Polygons calculable Edinburgh - 2017/1/12 6

Remarkable properties of planar N=4 SYM • • Conformally invariant (b = 0) Scattering amplitudes have uniform transcendental weight: “ ln 2 Lx ” at L loops Perturbation theory has finite radius of convergence (no renormalons, no instantons) Amplitudes for n=4 or 5 gluons “trivial” to all loop orders Amplitudes equivalent to Wilson loops Dual (super)conformal invariance for any n Strong coupling minimal area surfaces Integrability + OPE exact, nonperturbative predictions for near-collinear limit L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 7

L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 8

1960’s Analytic S-Matrix No QCD, no Lagrangian or Feynman rules for strong interactions. Bootstrap program: Reconstruct scattering amplitudes directly from analytic properties: “on-shell” information • Poles • Branch cuts Landau; Cutkosky; Chew, Mandelstam; Frautschi; Eden, Landshoff, Olive, Polkinghorne; Veneziano; Virasoro, Shapiro; … (1960 s) Analyticity fell out of favor in 1970 s with the rise of QCD & Feynman rules Now reincarnated for computing amplitudes in perturbative QCD – as alternative to Feynman diagrams! Perturbative information now assists analyticity. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 9

Perturbation theory linearizes the bootstrap Tree amplitudes fall apart into simpler tree amplitudes in special limits – pole information Trees recycled into trees Britto, Cachazo, Feng, Witten, hep-th/0501052 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 10

Branch cut information (Generalized) Unitarity loop momentum Well actually, loop integrands Trees recycled into loops! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 11

In fact, for planar N=4 SYM, we now know the all-loop integrand Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka, 1008. 2958 Amplituhedron Arkani-Hamed, Trnka, 1312. 2007, 1312. 7878 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 12

But we don’t yet know how to integrate it • We don’t need to, if we can bootstrap the integrated amplitude. • At least for 6 gluons, we can guess what function space the integrated amplitude lies in, using the exact 2 -loop result Goncharov, Spradlin, Vergu, Volovich, 1006. 5703 • Also at 7 gluons, where cluster algebras provide strong clues. Golden, Goncharov, Paulos, Spradlin, Volovich, Vergu, 1305. 1617, 1401. 6446, 1411. 3289 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 13

Strategy • Instead of integrating this • We write down all integrals that look like they could have come from this. • Then pick the right one using some physical constraints. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 14

Hexagon function bootstrap LD, Drummond, Henn, 1108. 4461, 1111. 1704; Caron-Huot, LD, Drummond, Duhr, von Hippel, Mc. Leod, Pennington, 1308. 2276, 1402. 3300, 1408. 1505, 1509. 08127; 1609. 00669 • First “nontrivial” amplitude in planar N=4 SYM is 6 -gluon • Bootstrap works to 5 loops so far, for both MHV = (--++++) and NMHV = (---+++) • Heptagon bootstrap for 7 -gluon amplitudes at symbol level for MHV = (--+++++) through 4 loops, NMHV = (---++++) through 3 loops Drummond, Papathanasiou, Spradlin, 1412. 3763 LD, Drummond, Harrington, Mc. Leod, Papathanasiou, Spradlin, 1612. 08976 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 15

Holy grail: solving planar N=4 SYM exactly Images: A. Sever, N. Arkani-Hamed Alday, Maldacena, Gaiotto, Sever, Vieira, … Basso, Sever, Vieira L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 16

Rich theoretical “data” mine • Rare to have perturbative results to 5 loops. • Usually they are single numbers • Here we have analytic functions of 3 variables (6 variables in 7 -point case) • Many limits to study • Also some conjectures based on the Amplituhedron to test (“positivity” after integration). L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 17

Planar N=4 amplitudes = polygonal Wilson loops external momenta = sides of light-like polygons Alday, Maldacena, 0705. 0303 Drummond, Korchemsky, Sokatchev, 0707. 0243 Brandhuber, Heslop, Travaglini, 0707. 1153 Drummond, Henn, Korchemsky, Sokatchev, 0709. 2368, 0712. 1223, 0803. 1466; Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, 0803. 1465 L. Dixon Polylogs for Polygons Hodges, 0905. 1473 Arkani-Hamed et al, 0907. 5418, 1008. 2958, 1212. 5605 Adamo, Bullimore, Mason, Skinner, 1104. 2890 Edinburgh - 2017/1/12 18

(Near) collinear limit L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 19

Flux tubes at finite coupling Alday, Gaiotto, Maldacena, Sever, Vieira, 1006. 2788; Basso, Sever, Vieira, 1303. 1396, 1306. 2058, 1402. 3307, 1407. 1736, 1508. 03045 BSV+Caetano+Cordova, 1412. 1132, 1508. 02987 • Tile n-gon with pentagon transitions. • Quantum integrability compute pentagons exactly in ’t Hooft coupling • 4 d S-matrix as expansion (OPE) in number of flux-tube excitations = expansion around near collinear limit L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 20

Multi-regge limit = = • Amplitude factorizes in Fourier-Mellin space Bartels, Lipatov, Sabio Vera, 0802. 2065, Fadin, Lipatov, 1111. 0782; LD, Duhr, Pennington, 1207. 0186; Pennington, 1209. 5357; Basso, Caron-Huot, Sever, 1407. 3766 (analytic continuation from OPE limit); Broedel, Sprenger, 1512. 04963, Lipatov, Prygarin, Schnitzer, 1205. 0186; LD, von Hippel, 1408. 1505, Del Duca, Druc, Drummond, Duhr, Dulat, Marzucca, Papathanasiou, Verbeek, 1606. 08807 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 21

Sudakov regions • Electron always comes with a radiation field, or cloud of soft photons Bloch-Nordsieck (1937) • Probability of high energy electron with no photons radiated with E < Ec and q < qc is exp[ - ln(Ec /E) ln(qc /q) ] Sudakov (1954) • Similar effects very important for QCD @ LHC • There also “virtual” Sudakov regions, where the external kinematics limits virtuality of internal collections of lines. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 22

Multi-particle factorization limit Bern, Chalmers, hep-ph/9503236; LD, von Hippel, 1408. 1505; Basso, Sever, Vieira (Sever talk at Amplitudes 2015) • Virtual Sudakov region, A ~ exp[- ln 2 d ], d ~ s 345 • Can study to very high accuracy in planar N=4 SYM L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 23

Double-parton-scattering-like limit Georgiou, 0904. 4675; LD, Esterlis, 1602. 02107 = • • Self-crossing limit of Wilson loop Overlaps MRK limit Another Sudakov region Singularities ~ Wilson line RGE Korchemsky and Korchemskaya hep-ph/9409446 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 24

Amplitudes (Wilson loops) are IR (UV) divergent Bern, LD, Smirnov, hep-th/0505205 • BDS ansatz captures all IR divergences of amplitude • Accounts for anomaly in dual conformal invariance due to IR divergences • Fails to describe finite part for n = 6, 7, . . . • Failure (remainder function) is dual conformally invariant constants, indep. of kinematics all kinematic dependence from 1 -loop amplitude L. Dixon Polylogs for Polygons l = g 2 Nc Edinburgh - 2017/1/12 25

Dual conformal invariance • Amplitude fixed, up to functions of dual conformally invariant cross ratios: • no such variables for n = 4, 5 n = 6 precisely 3 ratios: Remainder function, starts at 2 loops L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 26

BDS-like – better than BDS Consider where Alday, Gaiotto, Maldacena, 0911. 4708 Contains all IR poles, but no 3 -particle invariants. Dual conformally invariant part of the one-loop amplitude: L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 27

BDS-like normalized amplitude Define where `t Hooft coupling cusp anomalous dimension – known exactly No 3 -particle invariants in denominator of simpler analytic behavior L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 28

Kinematical playground self-crossing Multi-particle factorization u, w ∞ L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 29

Basic bootstrap assumption • MHV: is a linear combination of weight 2 L hexagon functions at any loop order L • NMHV: BDS-like normalized super-amplitude has expansion Grassmann-containing dual superconformal invariants, (a) = [bcdef] L. Dixon Polylogs for Polygons Drummond, Henn, Korchemsky, Sokatchev, 0807. 1095; LD, von Hippel, Mc. Leod, 1509. 08127 ~ E, E = hexagon functions Edinburgh - 2017/1/12 30

Polylogs for polygons (also for QCD) Chen; Goncharov; Brown; … • Generalized polylogarithms, or n-fold iterated integrals, or weight n pure transcendental functions f. • Define by derivatives: S = finite set of rational expressions, “symbol letters”, and are also pure functions, weight n-1 • Iterate: • Symbol = {1, 1, …, 1} L. Dixon Polylogs for Polygons Goncharov, Spradlin, Vergu, Volovich, 1006. 5703(maximally iterated) component Edinburgh - 2017/1/12 31

{n-1, 1} coproduct representation of functions is iterative, nested, compact L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 32

Harmonic Polylogarithms of one variable (HPLs {0, 1}) Remiddi, Vermaseren, hep-ph/9905237 • Subsector of hexagon functions. • Generalize classical polylogs, • Define by iterated integration: • Or by derivatives • Symbol letters: • Weight w = length of binary string • Number of functions at weight 2 L: 22 L L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 33

9 hexagon symbol letters • yi not independent of ui : , … where • Function space graded by parity: • Also by dihedral symmetry: L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 34

Branch cut condition • All massless particles all branch cuts start at origin in Branch cuts all start from 0 or ∞ in or v or w First symbol entry L. Dixon Polylogs for Polygons GMSV, 1102. 0062 Edinburgh - 2017/1/12 35

Steinmann relations Steinmann, Helv. Phys. Acta (1960), Bartels, Lipatov, Sabio Vera, 0802. 2065 Caron-Huot, LD, Mc. Leod, von Hippel, 1609. 00669 • Amplitudes should not have overlapping branch cuts. • Cuts in 2 -particle invariants subtle in generic kinematics • Easiest to understand for cuts in 3 -particle invariants using 3 3 scattering: Intermediate particle flow in wrong direction for s 234 discontinuity L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 36

Steinmann relations (cont. ) + cyclic conditions NO OK First two entries restricted to 7 out of 9: Analogous constraints for n=7 plus L. Dixon Polylogs for Polygons using A 7 BDS-like Edinburgh - 2017/1/12 37

Iterative Construction of Steinmann hexagon functions {n-1, 1} coproduct Fx characterizes first derivatives, defines F up to overall constant (a multiple zeta value). 1. Insert general linear combinations for weight n-1 Fx 2. Solve “integrability” constraint that mixed-partial derivatives are equal 3. Stay in space of functions with good branch cuts and obeying Steinmann by imposing a few more “z -valued” conditions in each iteration weight n basis L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 38

Impose constraints: (MHV, NMHV) parameters left (0, 0) amplitude uniquely determined! # of functions grows quite slowly, about a factor of 2 per weight, same as HPL’s {0, 1} of one variable! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 39

Two of many limits L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 40

NMHV Multi-Particle Factorization Bern, Chalmers, hep-ph/9503236; LD, von Hippel, 1408. 1505 - +- + + + More interesting for NMHV: MHV tree has no pole L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 41

Multi-Particle Factorization (cont. ) look at E(u, v, w) Or rather at U(u, v, w) = ln E(u, v, w) L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 42

Factorization limit of U • Simple polynomial in ln(uw/v) ! • Sudakov logs due to on-shell intermediate state • All orders resummation possible using flux tube picture Basso, Sever, Vieira (Sever talk at Amplitudes 2015) L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 43

Positivity LD, von Hippel, Mc. Leod, Trnka, 1611. 08325 • Amplituhedron picture implies a positive integrand for certain external kinematics. • Need to continue loop momenta out of positive region • Could positivity survive anyway? • For what IR finite quantities? • “Positivity” actually means sign alternation L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 44

Positivity (cont. ) • We tested NMHV ratio function (a natural IR finite quantity) and for positivity in the appropriate regions of positive kinematics. • All test results positive through 5 loops. • MHV remainder function, other conceivable MHV IR finite quantities fail starting at 4 loops. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 45

“Double scaling limit” NMHV positive kinematics co llin ea it im rl MHV positive kinematics L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 46

[Bosonized] NMHV Ratio Function all positive – even monotonic! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 47

BDS-like normalized MHV Amplitude all positive – even monotonic! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 48

Beyond 6 gluons • Cluster Algebras provide strong clues to the right functions: 6 variables, 42 letter symbol alphabet Golden, Goncharov, Paulos, Spradlin, Volovich, Vergu, 1305. 1617, 1401. 6446, 1411. 3289, Spradlin talk at Amplitudes 2016 • Power seen in symbol of 3 -loop MHV 7 -point amplitude Drummond, Papathanasiou, Spradlin 1412. 3763 • With Steinmann relations, can go to 4 -loop MHV and 3 -loop non-MHV LD, Drummond, Mc. Leod, Harrington, Papathanasiou, Spradlin, 1612. 08976 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 49

What about quantum gravity? L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 50

Summary & Outlook • Hexagon function polylogarthmic ansatz planar N=4 SYM amplitudes over full kinematical phase space, for 6 gluons, both MHV and NMHV, through 5 loops • For 7 gluons, heptagon ansatz symbol of MHV (NMHV) amplitude to 4 loops (3 loops) • Function space is extremely rigid. • Need very little additional information besides collinear limits (+ multi-Regge-limits for 6 gluons) Finite coupling results for generic kinematics? ? ? Lessons for QCD? ? ? L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 51

Extra Slides L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 52

At (u, v, w) = (1, 1, 1), multiple zeta values First irreducible MZV L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 53

Dual superconformal invariance _ Dual superconformal generator Q has anomaly due to virtual collinear singularities. • Structure of anomaly constrains_ first derivatives of amplitudes Q equation • Caron-Huot, 1105. 5606; Bullimore, Skinner, 1112. 1056, Caron-Huot, He, 1112. 1060 • General derivative leads to “source term” from (n+1)-point amplitude • For certain derivatives, source term vanishes, leading to homogeneous constraints, good to any loop order L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 54

_ Q equation for MHV • Constraint on first derivative of has simple form • In terms of the final entry of symbol, restricts to 6 of 9 possible letters: • In terms of {n-1, 1} coproducts, equivalent to: • Similar (but more intricate) constraints for NMHV [Caron-Huot], LD, von Hippel, Mc. Leod, 1509. 08127 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 55

A subclass of Steinmann functions Logarithmic seeds: • Similar to definition of HPLs. • u = ∞ base point preserves Steinmann condition • cj constants chosen so functions vanish at u=1, no u=1 branch cuts generated in next step. • K functions exhaust non-y Steinmann hexagon functions L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 56

Hexagon symbol letters • Momentum twistors Zi. A, i=1, 2, …, 6 transform simply under dual conformal transformations. Hodges, 0905. 1473 • Construct 4 -brackets • 15 projectively invariant combinations of 4 -brackets can be factored into 9 basic ones: + cyclic L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 57

Very important: Space of Steinmann hexagon functions is not a ring The original hexagon function space was a ring: (good branch cuts) * (good branch cuts) = (good branch cuts) • But: (branch cut in s 234) * (branch cut in s 345) = [not Steinmann] • This fact accounts for the relative paucity of Steinmann functions – very good for bootstrapping! • In a ring, (crap) * (crap) = (more crap) L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 58

Scalar hexagon integral in D=6: First true (y-containing) hexagon function A real integral so it must be Steinmann • Weight 3, totally symmetric in {u, v, w} • First parity odd function, so: (secretly Li 3’s) • Only independent {2, 1} coproduct: • Encapsulates first order differential equation found earlier LD, Drummond, Henn, 1104. 2787 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 59

How close is Steinmann space to “optimal”? • Want to describe, not only to a given loop order, but also derivatives ({w, 1, 1, …, 1} coproducts) of even higher loop answers. • How many functions are we likely to need? • Take multiple derivatives/coproducts of the answers, and ask how much of the Steinmann space they span at each weight. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 60

Empirically Trimmed Steinmann space • First surprise is already at weight 2 • Many, many {2, 1, 1, …, 1} coproducts of the weight 10 functions span only a 6 dimensional subspace of the 7 dimensional Steinmann space, with basis: plus cyclic is not an independent element! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 61

Empirically Trimmed Steinmann space (cont. ) • At weight 3, drop out, but this is not “new” • But also is not there! • At weight 4, nothing “new” (apparently) • At weight 5, go missing (can be absorbed into other functions) • At weight 7, L. Dixon Polylogs for Polygons go missing Edinburgh - 2017/1/12 62

Empirically Trimmed Steinmann space (cont. ) 191 382 59 Almost a factor of 2 smaller at high weights Up to the mystery of the missing zeta’s, the Steinmann hexagon space appears to be “just right” for the problem of 6 point scattering in planar N=4 super-Yang-Mills theory! L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 63

Another mystery A particular linear combination of {2 L-2, 1, 1} MHV coproducts gives 2 * NMHV – MHV at one lower loop order: • First found at four loops LD, von Hippel, 1408. 1505 • Can now check at five loops. • Resembles a second order differential equation. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12 64