Multiloop scattering amplitudes in the LHC Era

Transcription

1 Multiloop scattering amplitudes in the LHC Era Francesco Moriello Based on arxiv: In collaboration with: R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, V. Smirnov October 26, 2016

2 Outline Scattering amplitudes - Perturbation theory (Loop) Feynman Integrals Differential equations Method (Very powerful but till recently case-by-case analysis) > Major breakthrough: Canonical basis of Feynman integrals [Henn, 2013] This framework has to be generalized to the case of Feynman integrals depending on elliptic integrals > 2-loop planar QCD corrections to p p Higgs + jet [Bonciani, Del Duca, Frellesvig, Henn, FM, Smirnov, 2016] 2

3 Scattering Amplitudes Fixed loop order (regularized) scalar Feynman integrals 3

4 Differential Equations Method (DE) Compute the derivative of N integrals [Kotikov, 1991] [Remiddi, 1997] [Gehrmann, Remiddi, 2000] N-vector N X N matrix > In this framework we compute the integrals by solving the corresponding first order linear DE 4

5 Canonical form Direct solution is not feasible (many coupled equations) > How do we decouple DE? > Henn: define a new basis of integrals (linear combinations of integrals) such that is factorized out of the DE: [Henn, 2013] Integrals Linear combinations of integrals (> VERY HARD!) (> ELEMENTARY!) We solve in power series around = 0 5

6 The solution is elementary! 6

7 Multiple Polylogarithms (MPL) In many cases Feynman integrals can be solved in terms of special functions called Multiple Polylogarithms: [Goncharov, 1998] > The number of integrations is called the weight of the polylog. At weight one they are ordinary logarithms: > Dedicated fast and precise numerical routines [Vollinga et al., 2005] 7

8 Solution - MPL In the rational case we can always reduce the DE to the form (partial fractioning) constant matrices independent of x 8

9 H + jet : 2 Loop QCD corrections with exact top mass dependence [arxiv: ] > So far known analytically at 2-loop (NLO) only in the large top mass limit (expansion around (1/m_t^2)=0 ) [Gehrmann et al,2012] > High transverse Higgs momentum (high p_t) distribution has been shown to be very sensitive to new physics effects in many New Physics models [Azatov et al,2013][grojean et al,2014]... > The large top mass approximation doesn't work for high p_t 9

10 Finite top mass effects in H+j (LO) [Harlander et al, 2012] [Frederix et al, 2016] 10

11 H+j integrals Elliptic integrals Elliptic integrals 11

12 Non-elliptic DE > The differential equations for a set of integrals h can be written in total generality as: Homogeneous equation > The problem is solved once we solve the homogeneous DE With a proper basis choice the equations can be triangularized > Only 1 st order DE have to be solved - The solution is in terms of elementary functions 12

13 DE in the elliptic case > We have to solve a 2 X 2 system of ODE 13

14 2 nd order DE > The solution of the four integrals can be reduced to the solution of one 2 nd order DE for the first integral 2 nd order DE are the bottleneck of the computation of elliptic Feynman integrals 14

15 Non-elliptic Elliptic [Adam et al. 2015] [Tancredi, talk at LoopFest 2016] and many more... [Bonciani et al. 2016] 15

16 Solving 2 nd order DE (algorithms) If P(x) and Q(x) have up to 3 singular points the equation can be algorithmically solved in terms of hypergeometric functions [B. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, London (1958)] If P(x) and Q(x) have up to 4 singular points the equation can be algorithmically solved in terms of Heun functions More than 4 singular points...? 16

17 2 nd order DE (Mandelstam var.) P(X) and Q(X) have 6 singular points when x is a Mandelstam variable 17

18 DE w.r.t. a non-physical parameter > Integrals depend on three dimensionless mandelstam invariants > We can rescale them with a non-physical parameter and define DE( ) [chain rule] After reparametrization there are 3 singular points > solution is algorithmic! see also [arxiv: ] 18

19 Complete Elliptic integrals >The homogeneous solutions are complete elliptic integrals of the first kind: Complete elliptic integral of the first kind 19

20 Useful in practice? Having an analytic expression is not the end of the story! What experiments measure are cross sections (not Feynman integrals) We need: - Analytic expression OK - Fast numeric computation (10 MIN/point) - Precise numeric computation (8 digits) 20

21 Summary and outlook Canonical basis for Feynman integrals algorithmic solution New result: analytic computation of all planar 2-loop QCD integrals for H+j production (elliptic integrals ) > solution in terms of iterated integrals over elliptic kernels > next step: non-planar topologies Perspectives: Systematize the computation of Feynman integrals that requires elliptic integrals > canonical basis > algorithmic solution of 2 nd (or higher) order DE (e.g. reparametrization) 21

22 Extra slides 22

23 Canonical form - algorithms Integrals with constant leading singularities should satisfy canonical DE (Maximal cut) [Arkani-Hamed, Bourjaily, Cachzao, Trnka, 2010] [Arkani-Hamed et al, 2012] Univariate problems with rational alphabet (algorithmic) [Lee, 2014] DE linear in (algorithmic) [Argeri et al, 2014] Block diagonal linear DE [Gehrmann et al, 2014] 23

24 The equations are not uniquely defined! We have the freedom to change the set of master integrals for which we solve the equations Under a base change the system of equations transforms according to: 24

25 Canonical differential equations New computation paradigm! Find the basis such that the solution is as simple as possible, possibly elementary! Is this always possible? What do we mean by simple? It has been conjectured [Henn, 2013] that with a proper basis choice the system can be cast into the simplified form: 25