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

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

Why? Li2(…) ??? QCD 2-loop amplitudes all unknown for 2 3 or higher Can we learn more about QCD – at least the functions needed for it – by studying a toy theory, planar N=4 SYM? Li2(…) ??? L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

QCD vs. planar N=4 SYM QCD N=4 SYM fundamental QCD N=4 SYM adjoint + adjoint scalars 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

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

Remarkable properties of planar N=4 SYM Conformally invariant (b = 0) Scattering amplitudes have uniform transcendental weight: “ ln2Lx ” 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

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

1960’s Analytic S-Matrix Poles Branch cuts No QCD, no Lagrangian or Feynman rules for strong interactions. Bootstrap program: Reconstruct scattering amplitudes directly from analytic properties: “on-shell” information Landau; Cutkosky; Chew, Mandelstam; Frautschi; Eden, Landshoff, Olive, Polkinghorne; Veneziano; Virasoro, Shapiro; … (1960s) Poles Branch cuts Analyticity fell out of favor in 1970s 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

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

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

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

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 Also at 7 gluons, where cluster algebras provide strong clues. Goncharov, Spradlin, Vergu, Volovich, 1006.5703 Golden, Goncharov, Paulos, Spradlin, Volovich, Vergu, 1305.1617, 1401.6446, 1411.3289 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

Hexagon function bootstrap LD, Drummond, Henn, 1108.4461, 1111.1704; Caron-Huot, LD, Drummond, Duhr, von Hippel, McLeod, 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, McLeod, Papathanasiou, Spradlin, 1612.08976 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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

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 Hodges, 0905.1473 Arkani-Hamed et al, 0907.5418, 1008.2958, 1212.5605 Adamo, Bullimore, Mason, Skinner, 1104.2890 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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 4d 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

= = 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

Sudakov regions exp[ - ln(Ec /E) ln(qc /q) ] Sudakov (1954) 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 are also “virtual” Sudakov regions, where the external kinematics limits virtuality of internal collections of lines. L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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[- ln2d ], d ~ s345 Can study to very high accuracy in planar N=4 SYM L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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 = g2 Nc L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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

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

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

Basic bootstrap assumption MHV: is a linear combination of weight 2L hexagon functions at any loop order L NMHV: BDS-like normalized super-amplitude has expansion Drummond, Henn, Korchemsky, Sokatchev, 0807.1095; LD, von Hippel, McLeod, 1509.08127 Grassmann-containing dual superconformal invariants, (a) = [bcdef] E, = hexagon functions E ~ L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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} component (maximally iterated) Goncharov, Spradlin, Vergu, Volovich, 1006.5703 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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 2L: 22L L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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 GMSV, 1102.0062 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

Steinmann relations Steinmann, Helv. Phys. Acta (1960), Bartels, Lipatov, Sabio Vera, 0802.2065 Caron-Huot, LD, McLeod, 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 s234 discontinuity L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

Steinmann relations (cont.) + cyclic conditions NO OK First two entries restricted to 7 out of 9: Analogous constraints for n=7 using A7BDS-like plus L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

Iterative Construction of Steinmann hexagon functions {n-1,1} coproduct Fx characterizes first derivatives, defines F up to overall constant (a multiple zeta value). Insert general linear combinations for weight n-1 Fx Solve “integrability” constraint that mixed-partial derivatives are equal 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

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

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

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

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

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

Positivity LD, von Hippel, McLeod, 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

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

“Double scaling limit” NMHV positive kinematics collinear limit MHV positive kinematics L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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

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, McLeod, Harrington, Papathanasiou, Spradlin, 1612.08976 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

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

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

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

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

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, McLeod, 1509.08127 L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

Hexagon symbol letters Momentum twistors ZiA, 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

Space of Steinmann hexagon functions is not a ring 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 s234) * (branch cut in s345) = [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

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} (secretly Li3’s) First parity odd function, so: 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

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

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

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, go missing L. Dixon Polylogs for Polygons Edinburgh - 2017/1/12

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

Another mystery A particular linear combination of {2L-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