New computational methods for NLO and NNLO calculations in QCD Stefan Weinzierl Institut für Physik, Universität Mainz I: Techniques already encountered at LO II: Methods at NLO III: Steps towards NNLO Part I: Prelude Physics is about numbers Monte Carlo integration We are interested in multi-dimensional integrals: n I = d u f (u1,...,u ) Z n [0,1]n Evaluate the integrand at N random points ~u j =(u j,1,...,u j,n). 1 N I = ∑ f (~u j) N j=1 Error scales as 1 σ ∼ √N Simulation of scattering events pdf’s hard parton hadronisation and decay scattering shower Underlying event: Interactions of the proton remnants. Multiple interactions: more than one pair of partons undergo hard scattering Pile-up events: more than one hadron-hadron scattering within a bunch crossing Fixed-order calculations Done at the parton level with quarks and gluons. Usually only a few partons in the final state. Amplitudes calculated in perturbation theory. Method of choice to describe well-separated jets. pdf’s hard scattering The master formula for the calculation of observables 1 1 O = dx1 fa(x1) dx2 fb(x2) ∑Z Z 2K sˆ spin spin colour colour h i a,b ( ) na nb na nb pdf’s flux factor average over initial spins and colours | {z } 2 dφn O(p1,..., p|n){z }An |2 {z } ×∑ Z | + | n observable amplitude integral over phase space | {z } | {z } | {z } We will discuss: Required properties of observables • Calculation of the amplitude • Integration over the phase space • Part I Infrared-safe observables Soft and collinear particles A particle detector has a finite resolution. Finite angular resolution: A pair of particles moving in (almost) the same direction • can not be resolved below a certain angle. We call these particles a pair of collinear particles. Example: An electron and a photon not resolved by the electromagnetic calorimeter. Detection only above a certain threshold: A particle with an energy below a certain • threshold will not be detected. We call this particle a soft particle. Example: A low-energy photon. Infrared-safe observables Observables which do not depend on long-distance behaviour, are called infrared-safe observables and can reliably be calculated in perturbation theory. In particular, it is required that they do not change value, if infinitessimal soft or collinear particles are added. For example: collinear : lim On(p1,..., pi,..., p j,..., pn)= On 1(p1,..., pi j,..., pn) pi p j − || soft : lim On(p1,..., p j,..., pn)= On 1(p1,..., p j 1, p j+1,..., pn) p j 0 − − → Event shapes Event shapes are observables calculated from all particles in an event. Typical examples of infrared-safe event shapes in electron-positron annihilation are thrust, heavy jet mass, wide jet broadening, total jet broadening, C parameter, etc. Definition: Thrust ∑ ~pi nˆ i | · | T = max nˆ ∑ ~pi i | | For two particles back-to-back one has T = 1 For many particles, isotropically distributed we have 1 T = 2 Spherocity versus sphericity Spherocity: 2 2 ∑ ~pi⊥ 4 i | | S = min π nˆ ∑ ~pi i | | Sphericity: 2 2 ~p 4 ∑ i⊥ min i | | 2 π nˆ ∑ ~pi i | | Sphericity is not infrared-safe ! (Altarelli, Phys. Rept. 1981) ... however this does not stop experimentalists from measuring it ... (Aleph 1990) Jet algorithms The most fine-grained look at hadronic events consistent with infrared safety is given by classifying the particles into jets. Ingredients for a sequential recombination algorithm: a resolution variable yi j where a smaller yi j means that particles i and j are “closer”; • 2(1 cosθi j) yDurham = − min(E2,E2) i j Q2 i j a combination procedure which combines two four-momenta into one; • µ µ µ p(i j) = pi + p j. a cut-off y which provides a stopping point for the algorithm. • cut Jet algorithms In electron-positron annihilation one uses mainly exclusive jet algorithms, where • each particle in an event is assigned uniquely to one jet. In hadron-hadron collisions one uses mainly inclusive jet algorithms, where each • particle is either assigned uniquely to one jet or to no jet at all. One distinguishes further sequential recombination algorithm and cone algorithms. • Infrared-safe cone algorithm: SISCone. Once jets are defined we can look at cross sections, p distributions, rapidity ⊥ • distributions, etc. Modeling of jets In a perturbative calculation jets are modeled by only a few partons. This improves with the order to which the calculation is done. y At leading order: cut y y At next-to-leading order: cut cut y y y At next-to-next-to-leading order: cut cut cut Part I Amplitudes Quantum chromodynamics QCD describes quarks and gluons. The gauge group is the non-Abelian group SU(3). 1 L = Fa Faµν + ψ¯ (i∂/ + gγµT aAa m )ψ QCD 4 µν ∑ µ q − quarks − Field strength: a a a abc b c F = ∂µA ∂νA + gf A A µν ν − µ µ ν SU(3) matrices: T a,T b = if abcT c If we neglect quark masses, then QCD depends on one parameter g2 α = s 4π Perturbation theory Due to the smallness of the coupling constants αs at high energies, we may compute the amplitude reliable in perturbation theory, n 2 (0) 2 (1) 4 (2) 6 (3) An = g − An + g An + g An + g An + ... (l) An : amplitude with n external particles and l loops. Some examples of diagrams: (0) (1) (2) A4 A4 A4 | {z } | {z } | {z } Perturbation theory We need the amplitude squared: At leading order (LO) only Born amplitudes contribute: ∗ g4 ∼ At next-to-leading order (NLO): One-loop amplitudes and Born amplitudes with an additional parton. ∗ ∗ 2 Re + g6, virtual part g6, real part ∼ ∼ | {z } | {z } Real part contributes whenever the additional parton is not resolved. Perturbation theory Perturbative expansion of the amplitude squared (LO, NLO, NNLO): 2 2 2 2n 4 (0) 2n 2 (0) (1) 2n (1) (0) (2) An = g − A + g − 2 Re A ∗ A + g A + 2 Re A ∗ A | | n n n n n n virtual Born one-loop squared and two-loop | {z } | {z } | {z } 2 2 2n 2 (0) 2n (0) ∗ (1) An 1 = g − A + g 2 Re A A | + | n+1 n+1 n+1 loop+unresolved real | {z } | {z } 2 2 2n (0) An 2 = g A | + | n+2 double unresolved | {z } Feynman rules Each piece of a Feynman diagram corresponds to a mathematical expression: External edge: µ,a a = εµ(k) Internal edge: i k k µ,a µ ν ab ν,b = gµν +(1 ξ) δ k2 − − k2 Vertex: µ k1,a abc µν λ λ νλ µ µ λµ ν ν = gf g k1 k2 + g (k2 k3)+ g (k3 k1) λ kν,b − − − k3,c 2 h i Inconveniences we know to handle Loop amplitudes may have ultraviolet and infrared (soft and collinear) divergences. • Dimensional regularisation is the method of choice for the regularisation of loop • integrals. Ultraviolet divergences are removed by renormalisation. • Phase space integration for the real emission diverges in the soft or collinear region. • Unitarity requires the same regularisation (i.e. dimensional regularisation) for these • divergences. Infrared divergences cancel between real and virtual contribution, or with an • additional collinear counterterm in the case of initial-state partons. The textbook method The amplitude is given as a sum of Feynman diagrams. • Squaring the amplitude implies summing over spins and colour. • One-loop tensor integrals can always be reduced to scalar integrals • (Passarino-Veltman). All one-loop scalar integrals are known. • Phase space slicing or subtraction method to handle infrared divergences. • Works in principle, but not in practice ... An analogy: Testing prime numbers To check if an integer N is prime, For 2 j √N check if j divides N. • ≤ ≤ If such a j is found, N is not prime. • Otherwise N is prime. • Works in principle, but not in practice ... Brute force Number of Feynman Feynman rules: diagrams contributing to gg ng at tree level: → abc = gf (k2 k3)µg +(k3 k1)νg − νλ − λµ 2 4 +(k1 k2)λgµν] 3 25 − 4 220 5 2485 2 abe ecd 6 34300 = ig f f g gνρ gµρg − µλ − νλ 7 559405 ace ebd + f f gµνgλρ gµρgλν 8 10525900 − ade ebc + f f gµνgλρ gµλgνρ − Feynman diagrams are not the method of choice ! Helicity amplitudes Suppose that an amplitude is given as the sum of N Feynman diagrams. To calculate the amplitude squared à la Bjorken-Drell: Sum over all spins and use kµnν + nµkν ∑εµ∗(k,λ)εν(k,λ) = gµν + , λ − kn ∑u(p,λ)u¯(p,λ) = p/ + m, λ ∑v(p,λ)v¯(p,λ) = p/ m. λ − This gives of the order N2 terms. Better: For each spin configuration evaluate the amplitude to a complex number. Taking the norm of a complex number is a cheap operation. Spinors Spinors are solutions of the Dirac equation. For massless particles two-component Weyl spinors are a convenient choice: 1 p 1 p+ p+ = − ⊥∗ p = | i p p+ | −i p p | +| | +| ⊥ p 1 p1 p+ = ( p , p+) p = (p+, p ) h | p − ⊥ h −| p ⊥∗ | +| | +| p p Light-cone coordinates: p+ = p0 + p3, p = p0 p3, p = p1 + ip2, p = p1 ip2 − − ⊥ ⊥∗ − Spinor products: pq = p q+ , [qp]= q + p . h i h −| i h | −i The spinor products are anti-symmetric. Bra-ket notation versus dotted-undotted indices Two different notations for the same thing: p+ = pB p + = p ˙ | i h | A ˙ p = pB p = pA | −i h −| The spinor helicity method Gluon polarisation vectors: + k + γµ q+ k γµ q εµ (k,q)= h | | i , εµ−(k,q)= h −| | −i √2 q k+ √2 k + q h −| i h | −i q is an arbitrary light-like reference momentum.