The State of the Amplitude David A Kosower

The State of the Amplitude David A. Kosower Institut de Physique Théorique, CEA–Saclay Pascos 2014 Warsaw, Poland June 23– 27, 2014

Amplitudes • Scattering matrix elements: basic quantities in field theory • Basic building blocks for computing scattering cross sections • Using crossing MHV • Primary interest: in gauge theories; can derive all other physical quantities (e. g. anomalous dimensions) from them • In gravity, they are the only physical observables

Traditional Approach • Feynman Diagrams – – – Widely used for over 60 years Heuristic pictures Introduces idea of virtual or unphysical intermediate states Precise rules for calculating amplitudes Classic successes: electron g-2 to 1 part in 1010; discovery of asymptotic freedom • How it works Pick a process Grab a graduate student Lock him or her in a room Provide a copy of the relevant Feynman rules, or at least of Peskin & Schroeder’s Quantum Field Theory book – Supply caffeine, a modicum of nourishment, and occasional instructions – Provide a computer, a copy of Mathematica, a copy of FORM & a C++ compiler – –

A Difficulty • Huge number of diagrams in calculations of interest — factorial growth with number of legs or loops • 2 → 6 jets: 34300 tree diagrams, ~ 2. 5 ∙ 107 terms ~2. 9 ∙ 106 1 -loop diagrams, ~ 1. 9 ∙ 1010 terms

• In gravity, it’s even worse

Results Are Simple! • Parke–Taylor formula for AMHV Parke & Taylor; Mangano, Parke, & Xu

Even Simpler in N=4 Supersymmetric Theory • Nair–Parke–Taylor form for MHV-class amplitudes

Answers Are Simple At Loop Level Too One-loop in N = 4: • All-n QCD amplitudes for MHV configuration on a few Phys Rev D pages

Calculation is a Mess • Diagram insides involve unphysical states • Each diagram does not respect symmetry of theory (“not gauge-invariant”) — huge cancellations of gaugenoninvariant, redundant, parts are to blame (exacerbated by algebra) • There is almost no information in any given diagram

On-Shell Methods • Use only information from physical states • Avoid size explosion of intermediate terms due to unphysical states • Use properties of amplitudes as calculational tools – Factorization → on-shell recursion (Britto, Cachazo, Feng, Witten, …) – Unitarity → unitarity method (Bern, Dixon, Dunbar, DAK, …) Known integral basis: – Underlying quantum in field theory integral basis Integrals expressible terms of logarithms, dilogarithms, rational functions of invariants • Formalism Unitarity On-shell Recursion; D-dimensional unitarity via ∫ mass

We can now calculate large classes of amplitudes in gauge theories Gauge Theoryto infinite numbers of legs Sometimes Amplitudes A wealth of data for further study A foundation for a new subfield Integrability String Theory

N = 4 super-Yang–Mills • Theoretical laboratory at weak coupling over the years – Particle content and Lagrangian more complicated than pure Yang–Mills: add 4 Majorana fermions and 3 complex scalars in the adjoint, with carefully-tuned Yukawa and four-scalar interactions – Amplitudes are much simpler, because theory has a much larger symmetry – Connection to strong coupling via Ad. S/CFT • Superconformal symmetry • A new symmetry, not seen in the Lagrangian, was discovered a few years ago • Dual Conformal Symmetry

Dual Coordinates • x 1 k 5 x 5 k 4 x 2 x 4 k 2 x 3 k 3

Dual Conformal Invariance •

Amplitudes to All Orders BDS exponentiation conjecture for planar MHV amplitudes Anastasiou, Bern, Dixon, DAK; Bern, Dixon, & Smirnov Exponentiated structure holds for singular terms in all gauge theories — in N = 4 super-Yang–Mills the conjecture is for finite terms too True for n=4, 5 Thanks to dual conformal invariance

Higher-Loop Feynman Integrals • Basic ingredient for scattering amplitudes in more general theories • At one loop, Feynman integrals are expressible in terms of logarithms and dilogarithms • Trace back to computations of double boxes by Smirnov and Tausk in late 1990 s • At higher loops, we need a more general class of iterated -integral functions called Goncharov polylogs

• Sophisticated mathematical tools applied over the years • Several methods of attack • Dramatic advances in the last year in the differential equations approach Henn; Caola, Henn, Melnikov, Smirnov & Smirnov – Choose basis of Feynman integrals [using integration-by-parts identities (Chetyrkin & Tkachov; Laporta)] – Basis integrals satisfy coupled differential equations – Very typically evaluate to iterated integrals • Choose basis functions to be of uniform trans* weight, so that the differential equations are simple, and the iterated representation is manifest

Higher Loops for the LHC • At one loop, – Integral reduction long understood (Brown & Feynman; Passarino & Veltman) – Apply generalized unitarity to computing amplitudes: put internal lines on shell, either at integrand level (Ossola, Papadopoulos, Pittau), or in solving for coefficients of integrals (“integral level”) (Bern, Dixon, Dunbar, DAK; Britto, Cachazo, Feng) – Solutions to on-shell conditions are in general complex

• At higher loops, – One encounters irreducible numerators – Need to add integration-by-parts identities (Chetyrkin & Tkachov; Laporta) to the reduction toolkit – Need more sophisticated tools from computational algebraic geometry: Gröbner bases; primary ideals; algebraic varieties – Need notions from multivariate complex analysis: global poles; multivariate contour integration Badger, Frellesvig, & Zhang; Huang, Søgaard, & Zhang Caron-Huot, DAK, Larsen, Johansson, Søgaard Mastrolia, Mirabella, Ossola, Peraro • First explicit calculations A 5(1+, 2+, 3+, 4+, 5+) Badger, Frellesvig, & Zhang • Understanding the ABDK/BDS relation at the level of cuts Caron-Huot, DAK, Larsen, Johansson, Søgaard

Supergravity • For many years, people believed that all supergravity theories were UV divergent • Belief was based on an inability to rule out counterterms • The original papers (Howe, Stelle) were careful, but many people weren’t: Þ counterterms there with non-vanishing coefficient, at three loops Þ supergravity diverges Þ only string theory remains as a candidate theory of gravity • Pure Einstein gravity known to diverge at two loops • Until recently, no other explicit divergence has been seen in any pure supergravity theory

• Explicit calculation shows that N=8 supergravity is finite through four loops Bern, Carrasco, Dixon, Johansson, & Roiban • And further study that the lowest-order counterterm appears at seven loops Bossard, Howe, Stelle; Elvang, Freedman, Kiermaier; Green, Russo, Vanhove; Green and Bjornsson ; Bossard , Hillmann & Nicolai; Ramond & Kallosh; Broedel & Dixon; Elvang & Kiermaier; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger What about lower supergravities?

• Calculate UV divergences in amplitudes • Generalized unitarity — break apart loops into trees • Gravity ≅ (Yang–Mills)2 – Kawai–Lewellen–Tye relation – Bern–Carrasco–Johansson representation • BCJ representation color factor kinematic numerator propagators where ni satisfy same Jacobi identities as ci

• Gravity amplitudes (or integrands) • N = 8 SUGRA ≅ N = 4 s. YM • Construction now understood for a wide variety of supergavity theories, with and without matter Johansson; Roiban

• N = 4 SUGRA ≅ N = 4 s. YM N = 0 s. YM • At three loops, theory is finite! Bern, Davies, Dennen, & Huang • In spite of possible counterterm Bossard, Howe, Stelle; Bossard, Howe, Stelle, Vanhove • At four loops, a divergence appears Bern, Davies, Dennen, Smirnov • But may be specific to this theory because of an anomaly Carrasco, Kallosh Roiban, Tseytlin • Next target: N = 5 SUGRA ≅ N = 4 s. YM N = 1 s. YM

Summary • Explicit calculations of scattering amplitudes have led to – New insights – New techniques – Many new results in QCD, super-Yang Mills, and supergravity New Techniques Generalize Calculate New Insights Interpret New Results • We can look forward to more rapid progress in coming years