This document discusses methods for generating multi-jet events with MadGraph. It describes limitations of current matrix element generators in simulating processes with more than 5 jets. The document then outlines strategies used in MadGraph to overcome its limitations and allow for multi-jet event generation. These include dividing amplitudes, speeding up evaluation with off-shell recursive relations, and reorganizing color summation using a 1/N_c expansion.

1. Generating multi-jet events
with MadGraph
Yoshitaro Takaesu (KEK)
October 27, 2016 1

2. Overview: Simulation for LHC
October 27, 2016 2
Parton Shower Generator
Matrix Element Generator
Event Generator
Detector Simulator

3. Importance of multi-jet at the LHC
Multi-jet signature
appears in many
New Physics models.
Event Generator
should be able to
generate > 4 jets.
October 27, 2016 3

4. Status of ME Generators
Model Alpgen HELAC Sherpa MadGraph
SM 6 jets 10 jets ? 6,7 jets 4,5 jets
MSSM ✕ ？ 5 jets 4,5 jets
Others ✕ ✕ 5 jets 4,5 jets
There is no ME generator
which can simulate New Physics with > 5 jets.
October 27, 2016 4

5. What is MadGraph?
5
Matrix-Element & Event Generator
October 27, 2016 5
Input
Event generator (MadEvent)
Output
Matrix-Element Generator (“MadGraph”)
+ Fortran Code
Model + process
Events

6. October 27, 2016 6
MG
MSSM,
ADD,
2HDM,
etc
Feyn-
Rules
Higher
Dim. Op.
Spin-3/2
Spin-2
NLO
Interface
PYTHIA
Detector Simulator
ROOT
Lagrangian

7. Limitation of MG
Generated Codes (> 8MB) cannot be compiled in usual PC.
October 27, 2016 7
8 MB
The file size of codes for QCD processes

8. Structure of QCD Amp
October 27, 2016 8
Color flow
Color Factor
color-ordered amp.
i: Color flow

9. October 27, 2016 9
Color Flow

10. Structure of QCD Amp
October 27, 2016 10
Color flow
Color Factor
color-ordered amp.
i: Color flow

11. How to overcome
the Limitation
October 27, 2016 11

12. October 27, 2016 12
STRATEGY
Divide
Speed up• Amp Evaluation
Off• -shell Recursive Relations
• Re-organize Color Summation
1/Nc expansion

13. Speed up Amp Evaluation
October 27, 2016 13

14. October 27, 2016 14
Off-shell Recursive Relations
Step #

15. # of vertex evaluations
October 27, 2016 15

16. Time performance
October 27, 2016 16

17. October 27, 2016 17
Recursive Relations
for gluonic sub-amplitude

18. Re-organize Color Sum
October 27, 2016 18

19. October 27, 2016 19
1/Nc expansion
Polynomial of Nc:

20. Multi-jet event generation
Generate events with• Leading Color
Approximation
For each event, include• higher order
corrections into its weight.
October 27, 2016 20

21. Leading color summation
Color-flow sampling
( ~ 40,000 for gg -> 7g )
October 27, 2016 21
Too LARGE
Event is a set of momenta,helicities and a color flow

22. Higher order corrections
• For each event with a color flow
– Specify needed color flows for the higher order
corrections
– Evaluate higher order corrections for the phase
space point and reweight
– Re-unweight
October 27, 2016 22

23. Sample Results
October 27, 2016 23

24. October 27, 2016 24

25. October 27, 2016 25

26. Total cross sections
October 27, 2016 26

27. October 27, 2016 27
Distributions

28. Conclusion
We proposed a method to generate multi• -jet
events with MadGraph.
Implemented gluonic off– -shell recursive relations
and generate color-ordered amps.
Generated LO events by sampling color flows–
Shown results for gluonic processes for LO event–
generation.
October 27, 2016 28

29. OutLook
Include higher order corrections in• 1/Nc exp.
Generate events with quarks•
Detailed study for many distributions of Multi• -
jet events
Apply the framework to New Physics• Pheno.
October 27, 2016 29

30. October 27, 2016 30

31. In color decomposition, we usually use Gell-Man matrices or
modified matrices for .
Gell-Man:
Modified:

32. Color-Flow decomposition uses another set of matrices for
.
Here are the generators of .
By using this basis, we can simplify the color factors as
much as we can: 0 or 1.

33. Color-Flow basis
33
QCD Lagrangian can be written as

34. 34
We rewrite this by introducing a U(1) gauge boson

35. 35
9 U(3) gluons are independent and their
generator matrices are
The color of a gluon can be expressed by a set of color indeces:
(i, j).
There is also the abelian gluon in this theory.

36. Feynman Rules 1
36
ColorFlowdiagrams
Color Flow

37. Feynman Rules 2
37
The abelian gluon is decoupled from U(3) gluons.

38. This is the color-flow decomposition of a n-gluon amplitude.
From these rules, we evaluate a color-fixed scattering
amplitude.
n-gluon amplitude
: (n-1)! non-cyclic permutaions
For n-gluon case, are the same as
the Color-Ordered ones.
colorfactor: 0 or1 partialamplitude

39. We add abelian gluon amplitudes to the U(3) gluon
ones.
e.g.)
1 quark line and amplitude

40. There are other contributions from propagating abelian gluons
e.g.)
2 quark lines and amplitude

41. October 27, 2016 41
Amplitude
Color
Basis
Helicity
&Color
Sum
Recursive Relations
Color-Flow Basis

42. – Compilable
– Symmetric
– Gauge invariant
October 27, 2016 42
Off-shell Recursive Relations
Divide MG code by C.O. Amp
are