Lance Dixon (SLAC) CERN Theory Colloquium - PowerPoint Presentation

1. Lance Dixon (SLAC)CERN Theory Colloquium 8 February 2023The DNA of Particle Scattering LD, Ö. Gürdoğan, Y.-T. Liu, A. McLeod, M. Wilhelm 2112.06243, 2204.11901, 2212.02410

2. “Don’t know much biology…”Particle physics has its Standard ModelMolecular biology has its Central DogmaL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20232instructions for life

3. Code of life carried by nucleotidesL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20233T  U for RNA

4. Codon redundancyAmino acids are coded by triplets or codons of base pairs, and the code has redundanciesL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20234There are only 20 amino acids, and stop and “start” codonsInformation content is not but only In fact, this is still a vast overcount,but the number of foldable proteinsthat do interesting things is still huge!  

5. Analogous code for particle scatteringInstruction set for quantum mechanical scattering amplitudes, building blocks for all reaction probabilities (cross sections).The code is not as universal as ATGC, it depends on the processWe have only seen 16 letter sequences so farStill, cracking the code lets us understand scattering at a deeper level than with traditional methods, like Feynman diagrams, and sometimes do more precise computationsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20235

6. LHC is QCD MachineCopious production of quarks and gluons, materialize as collimated jets of hadrons, predicted by Quantum ChromodynamicsConfrontation between experiment and theory at high precision requires higher order corrections in the strong coupling  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20236

7. Typical LHC Collision7F. KraussCalculablein principleModel or getfrom experimentL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 2023

8. Producing Higgs bosons at LHC L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20238Leading Order (LO) cross section =  Since ,interaction between gluons and Higgs is approximately local (mediated by a leading dim 5 operator ) Higgs boson dominantly produced by gluon fusion, a quantum process at “one loop”, mediated by top quark, because t couples strongly to both gluons and Higgs

9. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 20239Poor convergenceof expansion in Uncertainty bands from varyingNecessitates high orders!Perturbative Short-Distance Cross Section Higgs gluon fusioncross section at LHCvs. CM energyLO approx. is terrible!LO  NNNLO factor of 2 or 3 increase! LONLONNLOAnastasiou, Duhr, Dulat, Herzog, Mistlberger, 1503.060565082014NNNLO

10. Some NLO QCD Feynman diagramsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202310virtual gg  H real, gg  Hg

11. Some NNNLO QCD topologiesL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202311+ …+ quarks+ operator renormalization+ corrections+ parton distributions gg  Hg@ 2 loops, state of artin QCD Scattering amplitudes are the underlying building blocks

12. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202312Whole more than sum of its partsSimplicity often hidden from individual Feynman diagrams  Parke-Taylor (1986)

13. 13Granularity vs. FluidityL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 2023

14. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202314 Fluid Tree AmplitudesTree amplitude is a rational function of kinematic variables. Falls apart into simpler tree amplitudes in special limitsPicture leads directly to BCFW (on-shell) recursion relations:Reconstruct amplitude from poles in complex plane, where itfactorizes into simpler amplitudes Britto, Cachazo, Feng, Witten, hep-th/0501052Similarly determine (multi)loopintegrands from generalized unitarity

15. Beyond tree levelLoop level Feynman diagrams come with an instruction to integrate over all loop momenta L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202315 Hggg where the dilogarithm is  For example, at one loop the amplitude for gg  Hg involves the “scalar box” integral

16. One loop not too badFor any number of external particles, all one-loop integrals (even in dimensional regularization, ) can be reduced to scalar box integrals + simplerBrown-Feynman (1952), Melrose (1965), ‘t Hooft-Veltman (1974), Passarino-Veltman (1979), van Neerven-Vermaseren (1984), Bern, LD, Kosower (1992) combinations of where is (many different) functions of the kinematic variables (Mandelstam invariants),plus logarithms L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202316

17. Multi-loop much more complexL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202317At L loops, instead of just ’s, get special functions with up to 2L integrations Weight 2L “iterated integrals”Best case: generalized polylogarithms, defined iteratively by  and  Still very intricate multi-variate functions

18. Complexity tamed by “the symbol”L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202318Entrance to Northwestern Physics Department

19. Symbol ~ DNA CodeL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202319Complexity encoded in words written in an alphabet defined by iterative differentiationCode is analog of ATGC code for DNA.Characterizes answer fairly completelyWe can learn to read itWe can use that understanding to go to ever higher loop orders (at least in simple theories)

20. Planar N=4 SYM, “hydrogen atom” of amplitudesQCD’s maximally supersymmetric cousin, N=4 super-Yang-Mills theory (SYM), gauge group SU(Nc), in large Nc (planar) limitStructure very rigid: Amplitudes = For planar N=4 SYM, rational structure well understood, focus on transcendental functions.Furthermore, at least three dualities hold:AdS/CFTAmplitudes dual to Wilson loopsNew “antipodal” duality between amplitudes and form factors (or among form factors?)  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202320

21. Transcendental StructureN=4 SYM amplitudes have “uniform weight” (transcendentality) 2L at loop order LWeight ~ number of integrations, e.g. 1 2 nQCD amps typically all weights from 0 to 2L L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202321

22. “Goldilocks Process”: gg HgL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202322HtgggJust “knowing the code”, can get to eight loops – in planar N=4 SYMLD, Ö. Gürdoğan, A. McLeod, M. Wilhelm 2012.12286, 2112.06243, 2204.11901gggHtQCD state of art is two loops (not counting top quark loop)Gehrmann, Jaquier, Glover, Koukoutsakis, 1112.3554 Model system, like “bacteriophage ”for DNA in 1950s Recent progress on integrals for 3 loopsDiVita, Mastrolia, Schubert, Yundin, 1408.3107;Canko, Syrrakos, 2112.14275

23. Defining the symbolL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202323Suppose is a linear combination of generalized polylogarithms of weight Then the partial derivatives of with respect to underlying coordinates are given by where are in same space, but have weight and are letters in the symbol alphabet  

24. Now iterateL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202324Define via derivatives of :  Iterating, n times for weight n function, gives symbol , where now are just rational numbersand is short for Goncharov, Spradlin, Vergu, Volovich, 1006.5703 

25. Example: Classical polylogarithms L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202325 Regular at branch cut starts at . Iterated differentiation gives the symbol: Branch cut discontinuities displayed in first entry of symbol, e.g. clip off leading to compute discontinuity at . Derivatives computed from symbol by clipping last entry, multiplying by that . Alphabet  

26. Our one-loop gg  Hg exampleL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202326 Hggg where and are only 2 dimensionless variables () symbol Six letter alphabet:  

27. Symbol as scaffolding for functionL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202327Taking derivatives drops constants like Restore by tracking values at some boundary point(s) In (crude) DNA analogy, “beyond-the-symbol” information corresponds to epigenetics:stuff close to DNA that influencesits expressione.g. methylation

28. Better alphabet for gg  Hg L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202328These 6 letters are equivalent but “diagonalize” things better:, }Symbols of ggHg amplitude simplify (remarkably) at L = 1 and 2 loops, to just 6 and 12 terms: (really only 1 and 2 terms, plus images under dihedral symmetry)3 loops: 636 terms, … Brandhuber, Travaglini, Yang, 1201.4170

29. ggHg symbol terms per loopL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202329Not advisable to look at full symbol directly without eye protection! However, there is alot of data to be minedfrom it!Ripe potential area for machine learning (Cranmer, Charton, LD, Wilhelm,Cai, Merz, in progress)

30. Examples of patternsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202330Every term in the symbol starts with ; never Analogous to “start codon” in DNA.Physical reason related to causality, which dictates where branch cuts can appear: only for 12 pairs of adjacent letters are forbidden:Resemble constraints from causality called the Steinmann relations Steinmann, Helv. Phys. Acta (1960)But not quite, which mystified us for a while… ~ DNA codonredundancy

31. Symbol alphabets forn-gluon amplitudes in planar N=4 SYML. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202331n = 6 has 9 letters: n = 7 has 42 lettersn = 8 has at least 222 letters, could even be infinite as  Goncharov, Spradlin, Vergu, Volovich, 1006.5703Golden, Goncharov, Paulos, Spradlin, Volovich, Vergu, 1305.1617, 1401.6446, 1411.3289Arkani-Hamed, Lam, Spradlin, 1912.08222; Drummond, Foster, Gürdoğan, Kalousios, 1912.08217 Henke, Papathanasiou 1912.08254; Z. Li, C. Zhang, 2110.00350parity-odd letters, algebraic in   trivial in this theory (3 kinematic variables)(6 var’s)(9 var’s)Structure closely tied to cluster algebras Fomin, Zelevinsky (2001)tropical geometry & polytopes

32. We know MHV amplitude to 7 loops L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202332Caron-Huot, LD, Dulat, von Hippel, McLeod, Papathanasiou, 1903.10890+ ~109 more=There’s a “parity-preserving”surface, , whereany word containing them is 0,9 letters drops to 6 letters Q: Where have we seen these numbers before? A: In ggHg !!!

33. Strange new “antipodal” dualityL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202333 “Antipode map” S reverses order of all entries in symbol!!! Relates two seemingly unrelated processes!LD, Ö. Gürdoğan, A. McLeod, M. Wilhelm, 2112.06243ggHg(MHV) gggggg8 loops now also checked; LD, Liu, 2302.nnnnn

34. Antipodal dualityAntipode map S, at symbol level, reverses order of all letters:Kinematic map isMaps to parity-preserving surface  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202334weak-weak dualityLD, Ö. Gürdoğan, A. McLeod, M. Wilhelm, 2112.06243

35. Map covers entire phase space for 3-gluon form factorSoft is dual to collinear; collinear is dual to softWhite regions in map to some of  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202335

36. “OPE coordinates” simplify kinematic mapAmplitude:Form factor:Kinematic map L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202336   

37. Not clear why antipodal duality holdsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202337We have some clues, but far from a physical understandingBut it does explain the mystery of the “Steinmann-like”adjacency constraints:They are actual Steinmann constraints for the 6 gluon amplitude!Also, many other previously empirical “final entry relations” become part of the causal construction of the 6 gluon function space.Important to try to uncover the physical reason, besides just “it’s in their genes!”New clues provided by four-gluon form factor!

38. Four-gluon form factorL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202338Depends on 5 kinematical variables instead of 2.Even just at two loops, contains state-of-the art loop integrals  113 possible symbol letters!Abreu, Ita, Moriello, Page, Tschernow, 2005.04195; Abreu, Ita, Page, Tschernow, 2107.14180; Abreu, Chicherin, Ita, Page, Sotnikov, Tschernow, Zoia, to appear

39. [Z. Bern et al., 0803.1465, or FFOPE]Antipodal Self DualityL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202339LD, Ö. Gürdoğan, Y.-T. Liu, A. McLeod, M. Wilhelm, 2212.02410Bootstrapped symbol of 4-gluon form factor F4 at 2 loopsLight-like limit matches Y. Guo, L. Wang, G. Yang, 2209.06816 It has an antipodal self-dualitywhich subsumes the F3 – A6 duality!   

40. Amplitudes virtuous circleafter JJ CarrascoL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202340computehigher ordersfind new patternsor principlesinvent betteralgorithms

41. Summary & OutlookThere is a code based on the symbol of generalized polylogs, which is analogous to the code of DNAThe symbol of amplitudes is a linear combination of code words of maximal length 2L at L loops, and the coefficients are highly correlated, ~ “quantum DNA”The symbol is a general tool for disentangling the structure of scattering amplitudes in gauge theory, not just planar N=4 SYMComparing 3-gluon form factor to 6-gluon amplitude, a strange new antipodal duality swaps the role of branch cuts and derivatives.“Explained” by antipodal self duality of 4 gluon form factorBut who ordered that?How much more can we exploit the symbol, to learn more about why and how particles scatter, in gauge theory and in (quantum) gravity?L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202341

42. Hggg kinematics is two-dimensionalL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202342       dihedral symmetry generated by:cycle: , orb. flip:  N=4 amplitude is invariant 

43. Symbology enables plots to 8 loopsFor ratio at is within 3% of cusp anomalous dimension ratio,  same finite radius of convergence L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202343

44. Amplitudes = Wilson loopsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202344Alday, Maldacena, 0705.0303Drummond, Korchemsky, Sokatchev, 0707.0243Brandhuber, Heslop, Travaglini, 0707.1153Drummond, Henn, Korchemsky, Sokatchev, 0709.2368, 0712.1223, 0803.1466;Bern, LD, Kosower, Roiban, Spradlin, Vergu, Volovich, 0803.1465Polygon vertices xi are not positions but dual momenta, xi – xi+1 = ki Transform like positions under dual conformal symmetryDuality holds at both strong and weak coupling=weak-weak duality,holds order-by-order

45. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202345Dual conformal invarianceWilson n-gon invariant under inversion:  no such variables for n = 4,5 n = 6  precisely 3 ratios: n = 7  6 ratios. In general, 3n-15 ratios.Fixed, up to functions of invariant cross ratios:123456

46. Hexagon function bootstrapL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202346Use analytical properties of perturbative (six point) amplitudes in planar N=4 SYM to determine them directly, without ever peeking inside the loopsStep toward doing this nonperturbatively (no loops to peek inside) for general kinematicsSame method used for “Higgs” form factor; see belowLD, 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;Caron-Huot, LD, Dulat, von Hippel, McLeod, Papathanasiou, 1903.10890, 1906.07116; LD, Dulat, 22mm.nnnnn (NMHV 7 loop)34,56,7Loops+ ~109 more=

47. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 2023476-gluon kinematics:many different regionsspurious pole u = 1self-crossingMulti-particlefactorization u,w  ∞origin

48. Two-loop storyL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202348 Hggg computed in QCD at 2 loops Gehrmann, Jaquier, Glover, Koukoutsakis, 1112.3554 Stress tensor 3-point form factor in N=4 SYM computed next (QMUL, a decade ago) Brandhuber, Travaglini, Yang, 1201.4170 Highest weight part of QCD result was same as N=4 result!!“Principle of maximal transcendentality” Kotikov, Lipatov, Velizhanin, hep-ph/0301021, hep-ph/0611204 Does it hold here beyond two loops?Other operators: Ahmed et al., 1905.12770; Guo et al., 2205.12969 

49. 2d HPLsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202349Gehrmann, Remiddi, hep-ph/0008287 Space graded by weight n. Every function F obeys: where are weight n-1 2d HPLs.To bootstrap Hggg amplitude beyond 2 loops, find as small asubspace of 2d HPLs as possible, construct it to high weight   

50. Example: Harmonic Polylogarithms in one variable (HPL{0,1})Generalize classical polylogsDefine HPLs by iterated integration: , Or by derivatives: Symbol alphabet: Weight n = length of binary string Number of functions at weight n = 2L is number of binary strings: 22LBranch cuts dictated by first integration/entry in symbolDerivatives dictated by last integration/entry in symbol L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202350Remiddi, Vermaseren, hep-ph/9905237

51. Heuristic view of function spaceL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 2023511lnulnvlnw Li2(1-1/ui) ln2ui lnui lnui+1 - z2Li3(1-1/ui), true 2D HPLs, …weight14320…   derivatives 

52. Back to 3-gluon form factorL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202352Motivated by 6 gluon case, switch to equivalent alphabet, }Symbols of form factor at 1 and 2 loops: just 1 and 2 terms, plus dihedral images(!!!): dihedral cycle: dihedral flip: 

53. Simplest analytic form is for  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202353 Harmonic polylogarithms  8 loop result has terms , }  } 

54. 6-gluon amplitude is simplest for  Let L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202354Exact map at symbol level, with , if you also reverse order of symbol entries / HPL indices!!!Works to 7 loops, where terms agree 

55. Many special dual pointsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202355There is an“f ” alphabetat all thesepoints: a way ofwriting multiplezeta values(MZV’s) sothat coactionis manifest.F. Brown, 1102.1310;O. Schnetz,HyperlogProcedures

56. Simplest pointAt this point, Reversing ordering of letters in f-alphabet, blue values show that antipodal duality holds beyond symbol level, modulo modulo is best we can get from mathematical antipode map L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202356

57. Bootstrap boundary data:Flux tubes at finite couplingL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202357Alday, Gaiotto, Maldacena, Sever, Vieira, 1006.2788; Basso, Sever, Vieira, 1303.1396, 1306.2058, 1402.3307, 1407.1736, 1508.03045BSV+Caetano+Cordova, 1412.1132, 1508.02987Tile n-gon with pentagon transitions.Quantum integrability  compute pentagons exactly in ’t Hooft coupling4d S-matrix as expansion (OPE) in number of flux-tube excitations = expansion around near collinear limit

58. A New Form Factor OPEForm factors are Wilson loops in a periodic space, due to injection of operator momentum Alday, Maldacena, 0710.1060; Maldacena, Zhiboedov, 1009.1139; Brandhuber, Spence, Travaglini, Yang, 1011.1899Besides pentagon transitions , this program needs an additional ingredient, the form factor transition Sever, Tumanov, Wilhelm, 2009.11297, 2105.13367, 2112.10569 L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202358   

59. OPE representation 3-gluon form factor: L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 2023596-gluon amplitude: weak-coupling,  expansion in    weak-coupling  expansion in (no azimuthal angle ) 

60. Exploit/test antipodal duality at 8 loopsGiven form factor, antipodal duality determines symbol of MHV 6 gluon amplitude at 8 loops on surface.Lift symbol into bulk. Only 3 free parameters!2 killed at origin, last killed in process of lifting to full function levelNeed one OPE data point to kill one beyond-symbol ambiguity  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202360LD, Y.-T. Liu, 2302.nnnnn

61. 8 loop MHV 6-gluon amplitude at  Blue values successfully predicted by antipodal dualityResult consistent with coaction principle at weight 16.L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202361LD, Y.-T. Liu, to appear

62. Antipodal “symmetry”There’s a letter map for -gluon MHV amplitudes, at least for , on their parity preserving surfaces, which maps the symbol into its own antipode:Not a map of the underlying variables. Doesn’t currently work past 2 loops.But it’s the first evidence for some kind of antipodal action beyond one loop and  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202362Y.-T. Liu, 2207.11815

63. Antipodal Duality ? Palindromic DNA ?DNA & RNA normally read in only one directionComplementary strand normally carries no more informationHowever, if RNA has (complementary) palindromic sequences, it can form hairpinsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202363Acdx, https://commons.wikimedia.org/w/index.php?curid=11947007

64. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202364On-shell amplitudes IR divergent due to long-range gluonsPolygonal Wilson loops UV divergent at cusps, anomalous dimension – known to all orders in planar N=4 SYM: Beisert, Eden, Staudacher, hep-th/0610251Both removed by dividing by BDS-like ansatz Bern, LD, Smirnov, hep-th/0505205, Alday, Gaiotto, Maldacena, 0911.4708Normalized [MHV] amplitude is finite, dual conformal invariant, also uniquely (up to constant) maintains important symbol adjacency relations due to causality (Steinmann relations for 3-particle invariants): Removing Amplitude (or Form Factor) Infrared Divergences remainder function

65. BDS & BDS-like normalization for  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202365BDS ansatz remainder function only afunction of vanishes in all collinear limits,but no adjacency constraints split 1-loop amplitude judiciously: Now divide by:obeys “adjacency constraints” 

66. Finite radius of convergencePlanar N=4 SYM has no renormalons () and no instantons (Perturbative expansion can have finite radius of convergence, unlike QCD, QED, whose perturbative series are asymptotic.For cusp anomalous dimension, using coupling = , radius is Beisert, Eden, Staudacher (BES), 0610251Ratio of successive loop orders Find same radius of convergence in high-loop-order behavior of amplitudes and form factors, in most kinematic regions. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202366

67. Euclidean Region numericsFor ratio at is within 3% of cusp anomalous dimension ratio,  same finite radius of convergence… L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202367 

68. Values of HPLs {0,1} at u = 1Classical polylogsevaluate to Riemann zeta values HPL’s evaluate to nested sums called multiple zeta values (MZVs):Weight n = n1 + n1 + … + nm MZV’s obey many identities, e.g. stuffleAll reducible to Riemann zeta values until weight 8. Irreducible MZVs: L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202368

69. Many “empirical” adjacency constraintsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202369 Hold for 2 loop QCD amplitudes too, planar and nonplanar! LD, Mcleod, Wilhelm, 2012.12286 Latter NEW: Hold for planar N=4 SYM to 8 loops!Mnemonic for dihedral symmetry; 6 dashed lines indicate 12 forbidden pairs.

70. Number of (symbol-level) linearly independent coproducts ( derivatives) Properly normalized loop N=4 form factors belong to a small space , dimension saturates on left also obeys multiple-final-entry relations, saturation on right L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202370

71. Multi-final entry relations (plus dihedral images)2. (plus …)3. 4. … L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202371

72. Number of remaining parameters in form-factor ansatz after imposing constraintsL. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202372

73. The [Dual] Conformal Group SO(4,2) ⸧ SO(3,1) [rotations+boosts] + translations+dilatations + special-conformal 15 = 3 + 3 + 4 + 1 + 4 The nontrivial generators are special conformal KmCorrespond to inversion · translation · inversionTo obtain a [dual] conformally invariant function ) just have to check invariance under inversion, /  L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202373

74. Beyond n = 8L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202374sum all planar Feynman graphs withL loops and n external lines=+…>221

75. Numerical implications of antipodal duality? L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202375

76. L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202376Example: MHV finite remainder on (u,u,u) strong coupling resultAmazing proportionality of each perturbative coefficient at small u, and also with the strong coupling resultAlday, Gaiotto, Maldacena, 0911.4708

77. Remarkably, MHV remainder are quadratic in logarithms through 7 loops CDDvHMP, 1903.10890Coefficients involve same BES kernel as for cusp, but “tilted” by angle ,  Origin at weak coupling L. Dixon The DNA of Particle ScatteringCERN - 8 Feb. 202377B. Basso, LD, G. Papathanasiou, 2001.05460