From Maximal Helicity to Maximal Jets David A. Kosower Institut de Physique Théorique, CEA–Saclay and Institute for Advanced Study 30 Fermilab, March 16–19, 2015

A story that starts at Columbia Fermilab

Themes Jet Physics Amplitudes Computability

Prehistory (< 1986) No-one had laptops No-one had smartphones Almost no-one had a cell phone There was no arXiv – Institutions physically mailed preprints to each other!

Prehistory (< 1986)

A Seed is Planted Parke–Taylor squared amplitude (1985–6) showed that there was something simple Simple expressions should have a simple derivation With the benefit of hindsight, simple expressions should have many simple derivations

As a young postdoc, newly arrived at Columbia, what did I start doing? – Those of you who know my advisor — Howard Georgi — and his tastes in physics, can guess… I knew nothing about QCD beyond what I’d learned in a wonderful course given by John Preskill – Because QCD, of course, was messy and at Harvard, we didn’t do messy things I started working with a newly-arrived assistant professor, Parameswaran Nair We derived the five-point QCD amplitude from open strings (along with a Columbia graduate student, B-H Lee) – Color ordering; “simple” single object also Mangano, Parke, Xu (1987) Connection to perturbative string theory still going strong Stieberger & Taylor, Green & Vanhove, others; Kawai– Lewellen–Tye connection between gravity and gauge theory

But the most fruitful thing to come out of Columbia in that period was Nair’s observation that the Parke–Taylor amplitude could be thought of as a correlator on CP 1 Nair (1988) Fifteen years later, Edward Witten would develop it into twistor-string theory, and spark the modern Amplitudes revolution But what I was doing wasn’t the right thing to be doing at the time

Refinements of Current Recursion

Onward to Loops! Thanks to a kind invitation from Jan Ambjorn, I was able to spend a month in 1987 at the Niels Bohr Institute With Zvi Bern, we started trying to derive one-loop amplitudes in QCD from heterotic string theory It was a long project! A lot of elementary objections: – How can you be sure the extra string states decouple? – How can you get a beta function??? String theory is finite! – You can’t get a consistent string theory with SU(N)!

In the Mean time, in Jet Physics

Five Gluons Fortified by having Lance Dixon join us, we set off to compute our first new result: the five-gluon amplitude The first thing we did is to understand the integral basis, and compute integrals, in D dimensions (1993–4) – Study of integrals & structures underlying them is still an on- going project at higher loops – Early use of differential equations String-based calculation (1993)

Simple results! Modern form: spinor products  polylogs Other subprocesses (2q2g, 4q1g), with friendly competition Kunszt Signer Trocsanyi (1994–5)

Impact on Jet Algorithms

One-Loop All-plus Simple result at four and five points Following in the spirit of Parke & Taylor, working with Zvi’s student Gordon Chalmers, we guessed an all-n form Demonstrated its correctness through allowed analytic structures and collinear limits Not a derivation One missing ingredient: complex momenta! – Later supplied by Witten (2003) and Britto, Cachazo, Feng & Witten (2004)

All- n MHV at One Loop in SUSY With Zvi Bern, Lance Dixon, Dave Dunbar Integral basis Restricted set of integrals in N = 4 First: guess based on collinear limits for N = 4 Then: derivation from unitarity Later: derivation for N = 1 SUSY Analytic calculation: required symbolic manipulation But a powerful new tool

W+4 partons Applying the unitarity method: the artisinal era Bern, Dixon, DAK, Weinzierl (1997–8) First application of generalized unitarity – Full picture would come later, from Britto, Cachazo, Feng Computed in parallel by Glover & Campbell: accessible to traditional methods, but the balance of power was shifting – Unitarity provided digestible analytic expressions, softer spurious singularities Matrix elements used in MCFM to compute W+2 jets Campbell & Ellis (2002)

Planar collinear limits to all orders in perturbation theory from unitarity (1999) – Proof of structure – Detailed outline of explicit calculation Technology used for the calculation of the collinear splitting amplitude in N = 4 SUSY Anastasiou, Bern, Dixon, DAK (2003) A precursor to…

The Twistor Surge 2004: a new view of tree-level Yang–Mills as a twistor- string theory Witten (2003) New people, mostly from a more formal background, coming in to look at amplitudes Maximal unitarity for the box: a purely numerical approach is possible Britto Cachazo Feng (2004) – Coefficient is a product of trees, which could be computed using Berends–Giele recursion On-shell methods for tree amplitudes: factorization as a calculational principle Applicable to more general rational functions, for example purely rational parts of one-loop QCD amplitudes

NLO Revolution Maximal unitarity picture completed – Forde (for triangles and bubbles) (2007) – by Ellis, Kunszt, Giele, & Melnikov ; and by Badger (2008) for the rational terms Compute coefficients numerically – At integrand (OPP) or integrated level – Trees are tail ingredient: simple analytic formulae; Berends– Giele recursion; simple formulae from solving N =4 SUSY on- shell recursions Dixon, Henn, Plefka, Schuster (2011) Automation of processes – B LACK H AT for loops, SHERPA for real emission & phase space

NLO Revolution Lots of revolutionaries roaming the world – B LACK H AT : Berger, Bern, Dixon, Febres Cordero, Forde, Gleisberg, Hoeche, Ita, DAK, Lo Presti, Maître – HELAC-NLO: Ossola, Papadopoulos, Pittau, Actis, Bevilacqua, Czakon, Draggiotis, Garzelli, van Hameren, Mastrolia, Worek & their clients – Rocket: Ellis, Giele, Kunszt, Lazopoulos, Melnikov, Zanderighi – GoSam/Samurai: Mastrolia, Ossola, Reiter,Tramontano,Cullen, Greiner, Heinrich, Luisoni – NJet/NGluon: Badger, Biedermann, & Uwer + Sattler & Yundin – MadLoop: Hirschi, Frederix, Frixione, Garzelli, Maltoni, & Pittau – Analytics: Anastasiou, Britto, Duhr, Feng, Henn – Loop-Subtraction: Weinzierl, Becker, Goetz, Reuschle

W+5 partons  W+3 Jets At the Tevatron… Third jet in W+3 jets [ ] Reduced scale dependence at NLO Good agreement with CDF data [ ] Shape change small compared to LO scale variation SISCone ( Salam & Soyez ) vs J ET C LU

W+6 partons  W+4 Jets

7 TeV results from CMS (March 2014) 13 years: from crazy to accomplished & compared with experiment

W+7 partons  W+5 Jets Scale-uncertainty bands shrink dramatically Last jet shape is stable, harder jets have steeper spectrum at NLO Last three jet shapes look similar, just getting steeper

ATLAS 7 TeV [4.6 fb −1 ] results (Sept 2014)

Extrapolation to W+6 Jets

Looking ahead NLO is now standard The physics focus is shifting to NNLO for a broad range of processes – Two-loop technologies: maximal unitarity at integrand level & at integrated level – Infrared-cancellation technologies still ripe for development – Computation of integrals – Beginning to understand structure in systems of integrals and in integration by parts: generating vectors which avoid doubled propagators [ ] provide geometric insight into the systems — Ita [ ] and Larsen and Zhang [ ] give the first geometric interpretation of IBP relations The pace will be faster between now and 50!