David Farhi (Harvard University) Work in progress with Ilya Feige, Marat Freytsis, Matthew Schwartz SCET Workshop, 3/27/2014

Outline  Existing SCET distributions  Numerical SCET using Monte Carlo  Results  Extensibility of calculation 2

Power of SCET at LHC  LHC measurements: full of jets  QCD calculations of these observables full of large logs  SCET allows controlled calculations  … So we should go calculate distributions! 3

Some Existing SCET Calculations 4 Jouttenus, Stewart, Tackmann, Waalewijn (arxiv ) Higgs + Jet (NNLL) Chien, Kelley, Schwartz, (arxiv ) Photon + Jet (NNLL) 2 QCD Partons: ee-thrust; Higgs + Jet Veto; … 3 QCD Partons: Becher (yesterday’s talk) Z+ Jet (N 3 LL)

5 Might also want: (full PS) Various cuts multijets 1. Phase space integrals are prohibitive. 2. Many distributions have same IR structure. 4+ QCD Partons: LHC Dasgupta et al ( ) Only at single phase space point … Phase space integrals are hard! Some Existing SCET Calculations Higgs + Jet (NNLL) Photon + Jet (NNLL) 2 QCD Partons: ee-thrust; Higgs + Jet Veto; … 3 QCD Partons: Z+ Jet (N 3 LL)

1. Phase Space Prohibitive  For NLL, need:  2-loop cusp anomalous dimension (known)  1-loop anomalous dimension of S and J (known)  Tree-level matrix element (known)  Color mixing: Hard, Soft functions are matrices  Multiparticle phase space integrals are prohibitive (analytically). 6 Factorization Formula:

2. Same SCET Structure 7 Soft and Jet Functions are universal (if observables depend only on s, k) Integral over s and k Depends on observable… But not hard process Phase space integral depends on hard process, but not on observable Doing these phase space integrals allows the same IR pieces to compute distributions for many hard processes. Factorization Formula:

Numerical Integrals  Compute IR portion analytically (at NLL):  IR function F depends on observable (A) but not hard process; reuse across hard processes.  Use existing Monte Carlo Generators to do integral for each process.  Only use MC to do the integral; don’t generate resummed events (see Geneva for that). 8

Integration with Monte Carlo  Monte Carlo generators (e.g. MadGraph) give list of phase space points Φ i (with weights w i ) to compute any integral numerically:  Put color matrix structure back in: 9

Processes 10 4-jettiness 2-(sub)jettiness Beam Thrust cut Steps neededMadgraph?Universal? Sum all channels and crossings (quark and gluon channels) ✔ Compute 5 and 6 parton matrix elements ✔ Compute 4 QCD parton jet and soft functions ✔ Integrate 5 and 6 particle phase space, account for cuts. ✔

Calculation Outline 11 MadGraph Interleave various channels Append matrix element H Process Partonic events Per-PS pt distributions Observable (2-subjettiness) SCET Convolve Jet, Beam, Soft Fncs Include Hard Fnc evolution Analytic Distribution Sum = Full Distribution

Details of MadGraph Piece 12  Wrestle with MadGraph to get it to print out the matrix element:

Details of SCET Piece 13 Focus on 4-QCD-particle observables related to jettiness: For instance, 2-subjettiness with beam thrust cut: Jet Momentum Jet Energy

14 Details of SCET Piece 2. These are matrices in color space; Evolution is given by matrix exponential 4. MadGraph incorporates PDFs; include PDFs in Hard Function. Account for that by scaling f out from B. 1. Integrate to get F for any function of jettinesses (before setting scales) 3. Different F for each channel (qqqq, qqgg, …); depends on anomalous dimension of hard/soft function Subtleties For instance, 2-subjettiness with beam thrust cut:

Results 15 1 TeV

16 2-subjettiness with beam thrust cut 8 TeV, LHC-like cuts (pT, η, etc) Results

Extensibility 17 Trivial Theoretically doable DistributionAdditional workAdditional Calculation Any 4-QCD-particle tree process: pp-jjW, pp-jjμμ None (re-run code) Observable = Different function of τ i Different (trivial)analytic integral of F 4 >4 QCD particlesMultijet channels (implement anomalous dim of hard/soft func) Different SCET ObservableCorresponding beam, jet, soft function, convolution NNLLNLO Hard function NLO Soft function

Conclusions  Machinery to numerically compute many-parton SCET distributions (NLL).  Computed:  4-jettiness in ee-4j  2-jettiness in pp-jjγ with beam thrust cut  Extensible to other hard processes and observables. 18

Thank you 19