1. 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 Jan. 2017

2. 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 /1/12

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

5. 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
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6. 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
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7. 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 /1/12

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

9. 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
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10. 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/
L. Dixon Polylogs for Polygons
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11. Branch cut information  (Generalized) Unitarity
loop momentum 
Well actually,
loop integrands
Trees recycled into loops!
L. Dixon Polylogs for Polygons
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12. In fact, for planar N=4 SYM, we now know the all-loop integrand
Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka,
Amplituhedron
Arkani-Hamed, Trnka, ,
L. Dixon Polylogs for Polygons
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13. 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,
Golden, Goncharov, Paulos, Spradlin, Volovich, Vergu, , ,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

14. 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 /1/12

15. Hexagon function bootstrap
LD, Drummond, Henn, , ;
Caron-Huot, LD, Drummond, Duhr, von Hippel, McLeod, Pennington, , , , ;
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,
LD, Drummond, Harrington, McLeod, Papathanasiou, Spradlin,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

16. 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 /1/12

17. 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 /1/12

18. Planar N=4 amplitudes = polygonal Wilson loops
external momenta = sides of light-like polygons
Alday, Maldacena,
Drummond, Korchemsky, Sokatchev,
Brandhuber, Heslop, Travaglini,
Drummond, Henn, Korchemsky, Sokatchev,
, , ;
Bern, LD, Kosower, Roiban, Spradlin,
Vergu, Volovich,
Hodges,
Arkani-Hamed et al,
, ,
Adamo, Bullimore, Mason,
Skinner,
L. Dixon Polylogs for Polygons
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19. (Near) collinear limit
L. Dixon Polylogs for Polygons
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20. Flux tubes at finite coupling
Alday, Gaiotto, Maldacena, Sever, Vieira, ;
Basso, Sever, Vieira, , , , ,
BSV+Caetano+Cordova, ,
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
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21. = = Multi-regge limit Amplitude factorizes in Fourier-Mellin space
Bartels, Lipatov, Sabio Vera, , Fadin, Lipatov, ;
LD, Duhr, Pennington, ; Pennington, ;
Basso, Caron-Huot, Sever, (analytic continuation from OPE limit);
Broedel, Sprenger, , Lipatov, Prygarin, Schnitzer, ;
LD, von Hippel, , Del Duca, Druc, Drummond, Duhr, Dulat, Marzucca, Papathanasiou, Verbeek,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

22. 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 LHC
There are also “virtual” Sudakov regions, where the external kinematics limits virtuality of internal collections of lines.
L. Dixon Polylogs for Polygons
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23. Multi-particle factorization limit
Bern, Chalmers, hep-ph/ ; LD, von Hippel, ; 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 /1/12

24. Double-parton-scattering-like limit
Georgiou, ; LD, Esterlis, =
Self-crossing limit of Wilson loop
Overlaps MRK limit
Another Sudakov region
Singularities ~ Wilson line RGE
Korchemsky and Korchemskaya hep-ph/
L. Dixon Polylogs for Polygons
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25. Amplitudes (Wilson loops) are IR (UV) divergent
Bern, LD, Smirnov, hep-th/
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 /1/12

26. 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 /1/12

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

28. 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 /1/12

29. Kinematical playground self-crossing Multi-particle
factorization u,w  ∞
L. Dixon Polylogs for Polygons
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30. 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, ;
LD, von Hippel, McLeod,
Grassmann-containing
dual superconformal
invariants, (a) = [bcdef]
E, = hexagon functions E ~
L. Dixon Polylogs for Polygons
Edinburgh /1/12

31. 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,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

32. {n-1,1} coproduct representation of functions is iterative, nested, compact
L. Dixon Polylogs for Polygons
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33. Harmonic Polylogarithms of one variable (HPLs {0,1})
Remiddi, Vermaseren, hep-ph/
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 /1/12

34. 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 /1/12

35. 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,
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36. Steinmann relations
Steinmann, Helv. Phys. Acta (1960), Bartels, Lipatov, Sabio Vera,
Caron-Huot, LD, McLeod, von Hippel,
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 /1/12

37. 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 /1/12

38. 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 /1/12

39. 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 /1/12

40. Two of many limits L. Dixon Polylogs for Polygons
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41. NMHV Multi-Particle Factorization+
Bern, Chalmers, hep-ph/ ; LD, von Hippel,
More interesting for NMHV: MHV tree has no pole
L. Dixon Polylogs for Polygons
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42. 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 /1/12

43. 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 /1/12

44. Positivity
LD, von Hippel, McLeod, Trnka,
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 /1/12

45. 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 /1/12

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

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

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

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

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

51. 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 /1/12

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

53. At (u,v,w) = (1,1,1), multiple zeta values
First irreducible MZV
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54. 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, ; Bullimore, Skinner, ,
Caron-Huot, He,
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 /1/12

55. 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,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

56. 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 /1/12

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

58. 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 /1/12

59. 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,
L. Dixon Polylogs for Polygons
Edinburgh /1/12

60. 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 /1/12

61. 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 /1/12

62. 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 /1/12

63. 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 /1/12

64. 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,
Can now check at five loops.
Resembles a second order differential equation.
L. Dixon Polylogs for Polygons
Edinburgh /1/12