Gluon Scattering Feynman Spinor CHY tree amplitudes C

Gluon Scattering Feynman, (Spinor), & CHY tree amplitudes C. S. Lam （藍志成） Mc. Gill & UBC, Canada CSL, York-Peng Yao ar. Xiv 1602. 06419=PRD 93, 105008 (2016) ar. Xiv 1512. 05387=PRD 93, 105004 (2016) ar. Xiv 1511. 05050=NPB (2016)

60 Feynma n CHY 3

Feynman Amplitude

color stripped amplitudes: 1234567 (cyclic ordering of external legs) Many many terms Complicated and irregular numerators Simple denominators Many helicity configurations Gauge (non-)invariance

gauge invariance

Feynman numerators for n=4, 6

n=4

n=6 + 4 g vertex diagrams

message Denominator simple: Propagators Numerator very complicated Many terms No discernable regularity Each diagram NOT gauge invariant

CHY Amplitude Cachazo, He, Yuan 1306. 6575, 1307. 2199, 1309. 0885. Significance of the CHY amplitude

Particle Physics Pendulum 1940’s 1920’s Heisenberg Born Jordan Dirac +

In the 1940’s Pauli to Dirac: Dec. 21, 1943 the S-matrix theory of Heisenberg is a picture frame without a picture

electroweak theory YM GR String (2 d conformal) (QFT symmetries) In the 1960’s divergence hadronic resonances directly observable picture frame without a picture (analyticity) (fctn of many complex variables) CHY SYMMETRY + (symmetry in a complex plane) [any spacetime dimension]

2 1 3 5 4

2 1 3 5 4

"Parke-Taylor factor" Scattering Function CHY Mobius invariant Weight 4

Scalar theory Gauge theory treats n particles symmetrically

symmetric in particle labels

Many (Mobius) gauge parameters:

Two Major Tasks 1. Expansion of the reduced Pfaffian 2. Evaluation of the (n-3)-dimensional integral

Feynman & CHY integrations (scalar amplitudes)

a Feynman a b b c c d d e CHY e sometimes this gives rise to a sum of several Feynman diagrams

a b c d e

Feynman a b d a c e c d CHY b building a house without a blue print e True for all spacetime dimensions stem cell DNA methylation kinematics dynamics blueprint provided

message There is a systematic way to do the integrations one variable at a time

Rules for CHY gauge amplitude

expansion of Pf’ U U U gauge invariant

gauge invariant expansion of Pf’

Feynman CHY

expansion of Pf’ gauge invariant for 3, 4 (double pole)

message There is a set of rules to expand the reduced Pfaffian, to do the integrations, and to compute the numerator and denominator factors of the CHY amplitude

Comparison between Feynman & CHY amplitudes

CHY Feynman One formula for all n Different assembly for diff n QM & local interaction "emergent" QM & local interaction put in Regular numerator: Denominator: several terms from Irregular numerator Denominator: 1 term Mobius parameters ----- Gauge invariant `subdivisions’ Gauge invariant `city’

Conclusion 60 Feynma n CHY 3

all roads lead to Rome