Overview of MadGraph5 [email protected] Olivier MadGraph5_aMC@NLO Plan 2 •Details of the computation •Evaluation of matrix-element •Phase-Space integration •What is MG5_aMCMattelaer

Overview of MadGraph5_aMC@NLO

Olivier MattelaerUCLouvainCP3/CISM

Mattelaer Olivier MadGraph5_aMC@NLO

Plan

2

•Details of the computation•Evaluation of matrix-element•Phase-Space integration

•What is MG5_aMC


What are my goals

• Justify why analytic computation are SLOWER than numerical computation

• Justify why adding cuts to the code are POSSIBLE but can lead to PROBLEM

• Overview of MG5_aMC➡ What does each package

• Details on small width handling

Title

3


•Determine the production mechanism

• Evaluate the matrix-element

• Phase-Space Integration

4

Matrix-ElementCalculate a given process (e.g. gluino pair)

s s~ > go go WEIGHTED=2 page 1/1

Diagrams made by MadGraph5_aMC@NLO

s

1

s~

2

g

go

3

go

4

diagram 1 QCD=2, QED=0

s

1

go

3

sl

s~2

go4


s

1

go

3

sr

s~2

go4


s

1

go

4

sl

s~2

go3


s

1

go

4

sr

s~2

go3


σ =1

2s

!|M|2dΦ(n)

σ =1

2s

!|M|2dΦ(n)


•Determine the production mechanism

• Evaluate the matrix-element

• Phase-Space Integration

4

Matrix-ElementCalculate a given process (e.g. gluino pair)

s s~ > go go WEIGHTED=2 page 1/1


s

1

s~

2

g

go

3

go

4


s

1

go

3

sl

s~2

go4


s

1

go

3

sr

s~2

go4


s

1

go

4

sl

s~2

go3


s

1

go

4

sr

s~2

go3


σ =1

2s

!|M|2dΦ(n)

σ =1

2s

!|M|2dΦ(n)

Easy

Hard

VeryHard

(in general)


Matrix Elemente+ e- > mu+ mu- WEIGHTED=4 page 1/1

Diagrams made by MadGraph5

e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5

M = e2(u�µv)gµ⌫q2

(v�⌫u)


Matrix Element

1

4

X

pol

|M|2 =1

4

X

pol

M⇤M

e+ e- > mu+ mu- WEIGHTED=4 page 1/1


e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5


(v�⌫u)


Matrix Element

X

pol

uu = 6p+m

1

4

X

pol

|M|2 =1

4

X

pol

M⇤M



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5


(v�⌫u)


Matrix Element

X

pol

uu = 6p+m

1

4

X

pol

|M|2 =1

4

X

pol

M⇤M

e4

4q4Tr[ 6p1�µ 6p2�⌫ ]Tr[ 6p3�µ 6p4�⌫ ]



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5


(v�⌫u)


Matrix Element

X

pol

uu = 6p+m

1

4

X

pol

|M|2 =1

4

X

pol

M⇤M

e4

4q4Tr[ 6p1�µ 6p2�⌫ ]Tr[ 6p3�µ 6p4�⌫ ]



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5

8e4

q4[(p1.p3)(p2.p4) + (p1.p4)(p2.p3)]


(v�⌫u)


Matrix Element

Very Efficient (few computation to perform to get that number)

X

pol

uu = 6p+m

1

4

X

pol

|M|2 =1

4

X

pol

M⇤M

e4

4q4Tr[ 6p1�µ 6p2�⌫ ]Tr[ 6p3�µ 6p4�⌫ ]



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


5

8e4

q4[(p1.p3)(p2.p4) + (p1.p4)(p2.p3)]


(v�⌫u)



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4




e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


Z



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4




e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


Z

Need to compute |Ma |2 |Mz |2 2Re(M*a Mz)



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4




e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


Z


So for M Feynman diagram we need to compute different term

M2



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4




e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


Z



M2

The number of diagram scales factorially with the number of particle



e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4




e+

1

e-

2

a

mu+

3

mu-

4


e+

1

e-

2

z

mu+

3

mu-

4


Z



M2

The number of diagram scales factorially with the number of particle

In practise possible up to 2>4


Helicity• Evaluate M for fixed helicity of external particles

➡ Multiply M with M* -> |M|^2 ➡ Loop on Helicity and average the results

e+ e- > e+ e- WEIGHTED=4 page 1/1


e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4


7

Idea

M = (u e �µv)gµ⌫q2

(v e �⌫u)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4


Numbers for given helicity and momenta

7

Idea


(v e �⌫u)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4



Lines present in the code. {

7

Idea


(v e �⌫u)

v1 = fct(~p1,m1)

u2 = fct(~p2,m2)

v3 = fct(~p3,m3)

u4 = fct(~p4,m4)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4




7

Idea


(v e �⌫u)

v1 = fct(~p1,m1)

u2 = fct(~p2,m2)

v3 = fct(~p3,m3)

u4 = fct(~p4,m4)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4




7

Idea


(v e �⌫u)

v1 = fct(~p1,m1)

u2 = fct(~p2,m2)

v3 = fct(~p3,m3)

u4 = fct(~p4,m4)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4


Numbers for given helicity and momentaCalculate propagator wavefunctions


7

Idea


(v e �⌫u)

v1 = fct(~p1,m1)

u2 = fct(~p2,m2)

v3 = fct(~p3,m3)

u4 = fct(~p4,m4)

Wa = fct(v1, u2,ma,�a)






e+

1

e-

2

a

e+

3

e-

4


e+

1

e-

2

z

e+

3

e-

4


e+

1

e+

3

a

e-2

e-4


e+

1

e+

3

z

e-2

e-4


Numbers for given helicity and momentaCalculate propagator wavefunctionsFinally evaluate amplitude (c-number)


7

Idea


(v e �⌫u)

v1 = fct(~p1,m1)

u2 = fct(~p2,m2)

v3 = fct(~p3,m3)

u4 = fct(~p4,m4)

Wa = fct(v1, u2,ma,�a)M = fct(v3, u4,Wa)


Comparison

8

M diag N particle

Analytical

Helicity

Recycling

M2

M

(N!)2

(N!) 2N

M (N − 1)! 2(N−1)


Real caseKnown

s s~ > t t~ b b~ WEIGHTED=4 page 1/2


s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2


Number of routines: Number of routines:

Number of routines for both:

0 0

0

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




01

1

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




Identical

1 1

1

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




6 6

6

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




67

7

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




7

Identical

7

7

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




Identical

8 8

8

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




89

9

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




810

10

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




10 9

11

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




10 10

12

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




10 10

12

2(N+1) 2(N+1)

9

M1 M2

|M |2 = |M1 +M2|2


Real caseKnown



s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2




s

1

s~

2

gt

3

t~4

g

b

5

b~

b~6


s

1

s~

2

gt

3

t~4

g

b~

6

b

b5


s

1

s~

2

g

t

3

t~ 4g

b5

b~

6

g


s

1

s~

2

g

b

5

b~6

g

t

3

t~ t~

4


s

1

s~

2

g

b

5

b~6

g

t~

4

t t

3


t

3

t~ 4g

b5

b~

6

g

s

1

s

s~2



Number of routines for both: 12

10 102(N+1) 2(N+1)

N!*2(N+1) N! 10


Comparison

11

M diag N particle

Analytical

Helicity

Recycling

M2

M

(N!)2

(N!) 2N

M (N − 1)! 2(N−1)

Mattelaer Olivier MadGraph5_aMC@NLO 12

M diag N particle

Analytical

Helicity

Recycling

Recursion Relation

Not used in Madgraph

M2

M

(N!)2

(N!) 2N

M (N − 1)! 2(N−1)

log(M) 2N 2(N−1)


Comparison

13

M diag N particle 2 > 6

Analytical 1.6e9

Helicity 1.0e7

Recycling 6.5e5

Recursion Relation 3.2e4

M2

M

(N!)2

(N!) 2N

M (N − 1)! 2(N−1)

log(M) 2N 2(N−1)

Mattelaer Olivier MadGraph5_aMC@NLOBrussels October 2010 Tim Stelzer

ALOHA ALOHA

UFO Helicity

30 14

Mattelaer Olivier MadGraph5_aMC@NLOBrussels October 2010 Tim Stelzer

ALOHA ALOHA

UFO Helicity

30 14

Basically, any new operator can be handle by MG5/Pythia8 out of the box!


• Numerical computation faster than analytical computation

• We are able to compute matrix-element➡ for large number of final state➡ for any BSM theory

15

To Remember


Plan

16

•Details of the computation•Evaluation of matrix-element•Phase-Space integration

•What is MG5_aMC?


Monte Carlo Integration

σ =1

2s

!|M|2dΦ(n)

Calculations of cross section or decay widths involve integrations over high-dimension phase space of very peaked functions:

17



σ =1

2s

!|M|2dΦ(n)


Dim[Φ(n)] ∼ 3n

17



σ =1

2s

!|M|2dΦ(n)


General and flexible method is needed

Dim[Φ(n)] ∼ 3n

17


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

V = VN = 0

• MonteCarlo• Trapezium• Simpson

Method of evaluation1/

pN

1/N4

1/N2


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

V = VN = 0



pN

1/N4

1/N2

simpson MC3 0,638 0,35 0,6367 0,8

20 0,63662 0,6100 0,636619 0,651000 0,636619 0,636


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC


pN

1/N4

1/N2

More Dimension 1/pN

1/N2/d

1/N4/d



IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

I =!

x2

x1

f(x)dx

V = (x2 − x1)

!x2

x1

[f(x)]2dx − I2 VN = (x2 − x1)2

1

N

N!

i=1

[f(x)]2 − I2

N

IN = (x2 − x1)1

N

N!

i=1

f(x)


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

I =!

x2

x1

f(x)dx

V = (x2 − x1)

!x2

x1

[f(x)]2dx − I2 VN = (x2 − x1)2

1

N

N!

i=1

[f(x)]2 − I2

N

IN = (x2 − x1)1

N

N!

i=1

f(x)

I = IN ±!

VN/N


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

I =!

x2

x1

f(x)dx

V = (x2 − x1)

!x2

x1

[f(x)]2dx − I2 VN = (x2 − x1)2

1

N

N!

i=1

[f(x)]2 − I2

N

IN = (x2 − x1)1

N

N!

i=1

f(x)

I = IN ±!

VN/N

V = VN = 0


IntegrationI =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

Zdq2

(q2 �M2 + iM�)2

IN = 0.637 ± 0.307/√

N

ZdxC

I =!

x2

x1

f(x)dx

V = (x2 − x1)

!x2

x1

[f(x)]2dx − I2 VN = (x2 − x1)2

1

N

N!

i=1

[f(x)]2 − I2

N

IN = (x2 − x1)1

N

N!

i=1

f(x)

I = IN ±!

VN/N

V = VN = 0

Can be minimized!


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

=

! ξ2

ξ1

dξcos π

2x[ξ]

1−x[ξ]2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

=

! ξ2

ξ1

dξcos π

2x[ξ]

1−x[ξ]2

≃ 1

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

=

! ξ2

ξ1

dξcos π

2x[ξ]

1−x[ξ]2

≃ 1

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

=

! ξ2

ξ1

dξcos π

2x[ξ]

1−x[ξ]2

IN = 0.637 ± 0.031/√

N

≃ 1

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


Importance Sampling

IN = 0.637 ± 0.307/√

N

I =

! 1

0

dx cosπ

2x

IN = 0.637 ± 0.307/√

N

=

! ξ2

ξ1

dξcos π

2x[ξ]

1−x[ξ]2

IN = 0.637 ± 0.031/√

N

≃ 1

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)

The Phase-Space parametrization is important to have an efficient computation!

I =

Z 1

0dx(1� cx2)

cos�⇡2x

�

(1� cx2)


•Generate the random point in a distribution which is close to the function to integrate.

•This is a change of variable, such that the function is flatter in this new variable.

•Needs to know an approximate function.

22

Importance SamplingKey Point

Adaptative Monte-Carlo•Create an approximation of the function on the flight!


1. Creates bin such that each of them have the same contribution.➡Many bins where the function is large

2. Use the approximate for the importance sampling method.

Algorithm

Adaptative Monte-Carlo•Create an approximation of the function on the flight!


•Events are generated according to our best knowledge of the function

➡Basic cut include in this “best knowledge”➡Custom cut are ignored

24

Cut Impact

No cut Run card cut Custom cut


Cut Impact


Might miss the contribution and think it is just zero.



Example: QCD 2 → 2

u u~ > g g QED=0 page 1/1


u

1

u~

2

g

3

g

4

g

diagram 1 QCD=2

u

1

g

3

u~

2

g

4

u

diagram 2 QCD=2

u

1

g

4

u~

2

g

3

u

diagram 3 QCD=2



u

1

u~

2

g

3

g

4

g

diagram 1 QCD=2

u

1

g

3

u~

2

g

4

u

diagram 2 QCD=2

u

1

g

4

u~

2

g

3

u

diagram 3 QCD=2



u

1

u~

2

g

3

g

4

g

diagram 1 QCD=2

u

1

g

3

u~

2

g

4

u

diagram 2 QCD=2

u

1

g

4

u~

2

g

3

u

diagram 3 QCD=2

/ 1

s=

1

(p1 + p2)2/ 1

t=

1

(p1 � p3)2/ 1

u=

1

(p1 � p4)2

Three very different pole structures contributing to the same matrix element.

26


Let’s cut the problem in piece

27

|MT |2 =|M1 |2 + |M2 |2

|M1 |2 + |M2 |2 |MT |2

Z|Mtot|2 =

Z Pi |Mi|2Pj |Mj |2

|Mtot|2 =X

i

Z |Mi|2Pj |Mj |2

|Mtot|2



27

|MT |2 =|M1 |2 + |M2 |2

|M1 |2 + |M2 |2 |MT |2

⇡ 1

Z|Mtot|2 =

Z Pi |Mi|2Pj |Mj |2

|Mtot|2 =X

i

Z |Mi|2Pj |Mj |2

|Mtot|2



27

|MT |2 =|M1 |2 + |M2 |2

|M1 |2 + |M2 |2 |MT |2

⇡ 1

Z|Mtot|2 =

Z Pi |Mi|2Pj |Mj |2

|Mtot|2 =X

i

Z |Mi|2Pj |Mj |2

|Mtot|2

– One diagram is manageable– All other peaks taken care of by denominator sum

Key Idea



27

|MT |2 =|M1 |2 + |M2 |2

|M1 |2 + |M2 |2 |MT |2

⇡ 1

Z|Mtot|2 =

Z Pi |Mi|2Pj |Mj |2

|Mtot|2 =X

i

Z |Mi|2Pj |Mj |2

|Mtot|2

– One diagram is manageable– All other peaks taken care of by denominator sum

Key Idea

N Integral– Errors add in quadrature so no extra cost– Parallel in nature


Z|Mtot|2 =

Z Pi |Mi|2Pj |Mj |2

|Mtot|2 =X

i

Z |Mi|2Pj |Mj |2

|Mtot|2

term of the above sum.

each term might not be gauge invariant


What’s the difference between weighted and unweighted?

Weighted:

Same # of events in areas of phase space with very different probabilities:events must have different weights

29

Event generation


# events is proportional to the probability of areas of phase space:events have all the sameweight (”unweighted”)

Events distributed as in nature

What’s the difference between weighted and unweighted?

Unweighted:

30

Event generation


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )

Number between 0 and 1 (assuming positive function)-> re-interpret as the probability to keep the events


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

P(xi)max( f )


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

P(xi)max( f )

Let’s reduce the sample size by playing the lottery.For each events throw the dice and see if we keep or reject the events


Event generation

31

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

P(xi)max( f )

Let’s reduce the sample size by playing the lottery.For each events throw the dice and see if we keep or reject the events

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

P(xi)max( f ) ≃max( f )

N

n

∑i=1

1


Event generation

f(x)

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


Event generation

1. pick xi

f(x)

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )


Event generation

1. pick xi

f(x)2. calculate

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )

f(xi)


Event generation

1. pick xi

3. pick f(x)

2. calculate

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )

y ∈ [0, max( f )]

f(xi)


Event generation

1. pick xi

3. pick f(x)

2. calculate

4. Compare:if accept event,

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )

y ∈ [0, max( f )]

y < f(xi)

f(xi)


Event generation

1. pick xi

3. pick f(x)

2. calculate

4. Compare:if accept event,

else reject it.

32

∫ f(x)dx =1N

N

∑i=1

f(xi) =1N

N

∑i=1

f(xi)max( f )

max( f )

y ∈ [0, max( f )]

y < f(xi)

f(xi)


MC integrator

33

Event generation


MC integrator

O

dσ

dO

33

Event generation


MC integrator

Acceptance-Rejection

O

dσ

dO

33

Event generation


Event generator

MC integrator


O

dσ

dO

33

Event generation


Event generator

MC integrator


O

dσ

dO

O

dσ

dO

33

Event generation


Event generator

MC integrator


☞ This is possible only if f(x)<∞ AND has definite sign!

O

dσ

dO

O

dσ

dO

33

Event generation


Monte-Carlo Summary

• Slow Convergence (especially in low number of Dimension)

• Need to know the function • Impact on cut

Bad Point


Monte-Carlo Summary

• Slow Convergence (especially in low number of Dimension)

• Need to know the function • Impact on cut

Bad Point

Good Point

•Complex area of Integration•Easy Error estimate•quick estimation of the integral•Possibility to have unweighted events


Tree (B)SM

NLO(QCD)(SM)

NLO(QCD)(BSM)

NLO(EW)(SM)

LoopInduced (B)SM

Fix Order

+Parton Shower

Merged Sample

Type of generation

!"#$%&'()$((&*&+#,&-./."%&"0&12345!"#$%&'()$((&*&+#,&-./."%&"0&12345 67

!"#$%&'()'*+&,-./%0

s~ c > w+ g WEIGHTED=3 QED=1 QCD=1 [ all= QCD ] page 1/2


c2 w+ 3

s~

s~1

g4

s~

sg

loop diagram 1 QCD=3, QED=1

s~

1

g4

s~

c2

g

c~w+

3

s~


c2

w+ 3

s~

s~

1

s~g

g

4

g


c2

w+ 3

s~

s~

1

gs~

g

4

s~


c2

g4

c~

s~

1

gs~

w+

3

c~


s~

1

s~

g

c2

c

g

4

c

w+

3


NLO corrections

NNLO corrections

N3LO or NNNLO corrections

LO predictions

⇤ = ⇤Born

⇤1 +

�s

2⇥⇤(1) +

��s

2⇥

⇥2⇤(2) +

��s

2⇥

⇥3⇤(3) + . . .

⌅


Tree (B)SM

NLO(QCD)(SM)

NLO(QCD)(BSM)

NLO(EW)(SM)

LoopInduced (B)SM

Fix Order

+Parton Shower

Merged Sample

Type of generation

!"#$%&'()$((&*&+#,&-./."%&"0&12345!"#$%&'()$((&*&+#,&-./."%&"0&12345 67

!"#$%&'()'*+&,-./%0



c2 w+ 3

s~

s~1

g4

s~

sg


s~

1

g4

s~

c2

g

c~w+

3

s~


c2

w+ 3

s~

s~

1

s~g

g

4

g


c2

w+ 3

s~

s~

1

gs~

g

4

s~


c2

g4

c~

s~

1

gs~

w+

3

c~


s~

1

s~

g

c2

c

g

4

c

w+

3



Tree (B)SM

NLO(QCD)(SM)

NLO(QCD)(BSM)

NLO(EW)(SM)

LoopInduced (B)SM

Fix Order

+Parton Shower

Merged Sample

Type of generation

!"#$%&'()$((&*&+#,&-./."%&"0&12345!"#$%&'()$((&*&+#,&-./."%&"0&12345 67

!"#$%&'()'*+&,-./%0



c2 w+ 3

s~

s~1

g4

s~

sg


s~

1

g4

s~

c2

g

c~w+

3

s~


c2

w+ 3

s~

s~

1

s~g

g

4

g


c2

w+ 3

s~

s~

1

gs~

g

4

s~


c2

g4

c~

s~

1

gs~

w+

3

c~


s~

1

s~

g

c2

c

g

4

c

w+

3


ME ↓

PS →


LO Feature

36


LO Feature

36

Γ = ?Auto-Width

Parameter scan

~ 20% agreement

unmatched with ME/PS matching

Importance of the MC

very bad close to deg.!region

Recast of efficiencies in gluino simplified mod’

with K. Sakurai, M. Papucci, L. Zeune

ATOM/experiment ATOM/experiment


LO Feature

36

Γ = ?Auto-Width

Parameter scan

~ 20% agreement







normalized to one

Systematics

BSM re-weighting|Mnew |2 / |Mold |2


LO Feature

36

Interference

Mattelaer Olivier Looping up to be MAD 8

Some New FunctionInterference

NP^2==1 NP^2<=1

2

O(⇤2)

+

O(⇤)

2

O(⇤0)

+

NP^2>0

Γ = ?Auto-Width

Parameter scan

~ 20% agreement







normalized to one

Systematics



LO Feature

36

Interference



NP^2==1 NP^2<=1

2

O(⇤2)

+

O(⇤)

2

O(⇤0)

+

NP^2>0

Γ = ?Auto-Width

Parameter scan

~ 20% agreement







normalized to one

Systematics


Plugin

Interfacez >

e+ e

-

page

1/1

Diag

ram

s m

ade

by M

adG

raph

5_aM

C@NL

O

e+

2

e-

3

z1

dia

gram

1

QCD

=0, Q

ED=1


LO Feature

36

Interference



NP^2==1 NP^2<=1

2

O(⇤2)

+

O(⇤)

2

O(⇤0)

+

NP^2>0

Narrow-width

Γ = ?Auto-Width

Parameter scan

~ 20% agreement







normalized to one

Systematics


Plugin

Interfacez >

e+ e

-

page

1/1

Diag

ram

s m

ade

by M

adG

raph

5_aM

C@NL

O

e+

2

e-

3

z1

dia

gram

1

QCD

=0, Q

ED=1


Decaypage 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+

diagram 1 HIG=0, HIW=0, NP=0, QCD=0, QED=6

u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+

diagram 2 HIG=0, HIW=0, NP=2, QCD=0, QED

c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


• Process complicated to have the full process➡Including off-shell contribution

37

DecayNon Resonant Diagram

Problem

c u > c u e+ ve e- ve~ NP=2 WEIGHTED=20 HIW=1 HIG=1 page 3/12


c

1

c3

g e+5

ve 6w+

u2

ue- 7ve

u

4

z

ve~ 8


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ae- 7


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ze- 7


c1

c3

g

e+5

ve 6w+

u4

u~e- 7ve

u2

zve~ 8


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ae- 7


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ze- 7


page 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


• Process complicated to have the full process➡Including off-shell contribution

37

Decay

Solution• Only keep on-shell contribution

Non Resonant Diagram

Problem



c

1

c3

g e+5

ve 6w+

u2

ue- 7ve

u

4

z

ve~ 8


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ae- 7


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ze- 7


c1

c3

g

e+5

ve 6w+

u4

u~e- 7ve

u2

zve~ 8


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ae- 7


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ze- 7


page 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


Theory

�full = �prod ⇤ (BR+O(�

M))

Comment

Narrow-Width Approx.Z

dq2��

1

q2 �M2 � iM�

��2

⇡ ⇡

M��(q2 �M2)


• This is an Approximation!• This force the particle to be on-shell!

• Recover by re-introducing the Breit-wigner up-to a cut-off

38

Theory


M))

Comment


dq2��

1

q2 �M2 � iM�

��2

⇡ ⇡

M��(q2 �M2)


Decay chain

KIAS MadGrace school, Oct 24-29 2011 MadGraph 5 Olivier Mattelaer

Decay chains

• p p > t t~ w+, (t > w+ b, w+ > l+ vl), \ (t~ > w- b~, w- > j j), \ w+ > l+ vl

• Separately generate core process and each decay- Decays generated with the decaying particle as resulting wavefunction

• Iteratively combine decays and core processes

• Difficulty: Multiple diagrams in decays

mardi 25 octobre 2011

39

very long decay chains possible to simulate

directly in MadGraph!


Decay chain


Decay chains






39

very long decay chains possible to simulate

directly in MadGraph!

• Full spin-correlation•Off-shell effects (up to cut-off)•NWA not used for the cross-section


MadSpin

0.0001

0.001

0.01

0 25 50 75 100 125 150 175 200

1/σ

dσ

/dp T

(l+ ) [1/

GeV

]

pT(l+) [GeV]

Scalar Higgs

NLO Spin correlations onLO Spin correlations on

NLO Spin correlations offLO Spin correlations off

0.4

0.45

0.5

0.55

0.6

-1 -0.5 0 0.5 1

1/σ

dσ

/dco

s(φ)

cos(φ)

Scalar Higgs



Figure 5: Next-to-leading-order cross sections differential in pT (l+) (left pane) and in cosφ (rightpane) for ttH events with or without spin correlation effects. For comparison, also the leading-order results are shown. Events were generated with aMC@NLO, then decayed with MadSpin,and finally passed to Herwig for shower and hadronization.

0.0001

0.001

0.01

0 25 50 75 100 125 150 175 200

1/σ

dσ

/dp T

(l+ ) [1/

GeV

]

pT(l+) [GeV]

Pseudoscalar Higgs

NLO Spin correlations onNLO Spin correlations off

0.4

0.45

0.5

0.55

0.6

-1 -0.5 0 0.5 1

1/σ

dσ

/dco

s(φ)

cos(φ)

Pseudoscalar Higgs


Figure 6: Next-to-leading-order cross sections differential in pT (l+) (left pane) and in cosφ(right pane) for ttA events with or without spin correlation effects. Events were generated withaMC@NLO, then decayed with MadSpin, and finally passed to Herwig for shower and hadroniza-tion.

that preserving spin correlations is more important than including NLO corrections for this

observable. However, we observe that the inclusion of both, as it is done here, is necessary

for an accurate prediction of the distribution of events with respect to cos(φ). In general, a

scheme including both spin correlation effects and QCD corrections is preferred: it retains

the good features of a NLO calculation, i.e. reduced uncertainties due to scale dependence

(not shown), while keeping the correlations between the top decay products.

The results for the pseudo-scalar Higgs boson are shown in Figure 6. The effects of the

spin correlations on the transverse momentum of the charged lepton are similar as in the

case of a scalar Higgs boson: about 10% at small pT , increasing to about 40% at pT = 200

GeV. On the other hand, the cos(φ) does not show any significant effect from the spin-

correlations. Therefore this observable could possibly help in determining the CP nature of

the Higgs boson, underlining the importance of the inclusion of the spin correlation effects.

– 14 –

Decay as post-processing Independently of event generationBut same accuracy (spin-correlation)Use NWA for cross-section

t t > w+ b


Very small width

• Slows down the code• Can lead to numerical instability

41

Γ < 10−8M


Very small width

• Slows down the code• Can lead to numerical instability

41

Γ < 10−8M

Solution

• Use a Fake-Width for the evaluation of the matrix-element

• Correct cross-section according to NWA formula Γfake

Γtrue


What to remember•Analytical computation can be slower than numerical method

•Any BSM model are supported (at LO) •Phase Space integration are slow

•need knowledge of the function•cuts can be problematic

•Event generation are from free.•All this are automated in MG5_aMC@NLO

• Important to know the physical hypothesis

M

ADGRAPH5

aMC@NLO


Plan

43

•What is MG5_aMC? •Details of the computation

•Evaluation of matrix-element•Phase-Space integration

•Tools/functionality of MG5_aMC


List of package

44

• SysCalc (computation of systematics)• MadWidth (computation of width in NWA)• MadSpin (decay with full spin-correlation)• Re-Weighting (change of the weight of an event)• Shower / Detector Interface• MadWeight (Matrix-Element Method)• Interference• MadAnalysis5• Tau Decay• MadDM• GPU


2-body decayh > b b~ page 1/1


b

2

b~

3

h1


2 body decay

� =1

2MS

Zd�2|M|2


•By Lorentz Invariance the matrix element is constant over the phase-space.

45



b

2

b~

3

h1


2 body decay

� =1

2MS

Zd�2|M|2

� =

p�(M2,m2

1,m22)|M|2

16⇡SM3

�(M2,m21,m

22) =

�M2 �m2

1 �m22

�2 � 4m21m

22


•By Lorentz Invariance the matrix element is constant over the phase-space.

45



b

2

b~

3

h1


2 body decay

� =1

2MS

Zd�2|M|2

� =

p�(M2,m2

1,m22)|M|2

16⇡SM3

�(M2,m21,m

22) =

�M2 �m2

1 �m22

�2 � 4m21m

22

•Calculable analytically by FeynRules


•Use FeynRules formula (instateneous)

46

MadWidth hep-ph/1402.1178

2-body



46


2-body

Fast-Estimation of 3-body•Only use 2-body decay and PS factor



46


2-body


Relevant?



46


2-body


Relevant?

DONE

No



46


2-body


Relevant?

Channel Generation•Remove Sequence of 2-body/radiation diagram

DONE

NoMaybe



46


2-body


Relevant?


Estimation of 3-body•Based on the diagram. Approx. PS/Matrix-Element

DONE

Relevant?

NoMaybe



46


2-body


Relevant?



DONE

Relevant?

NoMaybe

No



46


2-body


Relevant?



DONE

Relevant?

Numerical Integration

NoMaybe

No

Yes?



46


2-body


Relevant?



DONE

Relevant?

Numerical Integration

4

4

NoMaybe

No

Yes?


MadWidth

Limitation• Only LO• No Loop Induced Decay• Valid in Narrow-width approximation• No hadronization effect

Lesson• Be aware of code limitation• Quite often those are not checked.


Decaypage 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


• Process too complicated to compute all the Feynman Diagram

48

DecayNon Resonant Diagram

Problem



c

1

c3

g e+5

ve 6w+

u2

ue- 7ve

u

4

z

ve~ 8


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ae- 7


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ze- 7


c1

c3

g

e+5

ve 6w+

u4

u~e- 7ve

u2

zve~ 8


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ae- 7


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ze- 7


page 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


• Process too complicated to compute all the Feynman Diagram

48

Decay

Solution• Only keep on-shell contribution

Non Resonant Diagram

Problem



c

1

c3

g e+5

ve 6w+

u2

ue- 7ve

u

4

z

ve~ 8


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ae- 7


c

1

c3

g e+5

ve 6w+

u2

u ve~ 8e+u

4

ze- 7


c1

c3

g

e+5

ve 6w+

u4

u~e- 7ve

u2

zve~ 8


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ae- 7


c1

c3

g

e+5

ve 6w+

u4

u~ve~ 8e+

u2

ze- 7


page 1/10


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


u1

u3

a

c2

c4

a

ve~

8

e- 7w-

e+ 5

ve6

w+


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

ave~ 8

e-7

w-

u1

u3

ae+

5

ve 6w+w-


c2c 4

a

u1

u3

ave~

8

e- 7w-w+

e+ 5

ve6

w+


Resonant Diagram


Theory


M))

Comment


dq2��

1

q2 �M2 � iM�

��2

⇡ ⇡

M��(q2 �M2)


• This is an Approximation!• This force the particle to be on-shell!

• Recover by re-introducing the Breit-wigner up-to a cut-off

49

Theory


M))

Comment


dq2��

1

q2 �M2 � iM�

��2

⇡ ⇡

M��(q2 �M2)



Decay chains






50


•Decay chains retain full matrix element for the diagrams compatible with the decay

• Full spin correlations (within and between decays)

•Full width effects•However, no interference with non-resonant diagrams ➡ Description only valid close to pole

mass➡ Cutoff at |m ± nΓ| where n is set in

run_card.

51

Decay chains


Decay chains

Thanks to developments in MadEvent, also (very) long decay chains possible to simulate directly in MadGraph!

52


MadSpin

One Event

offshell spin unweighted

No No YES

YES No No

YES YES No

YES YES YES

[Frixione, Leanen, Motylinski,Webber (2007)]

[Artoisenet, OM et al. 1212.3460]


MadSpin

One Event

Decay Events

- smear the mass- flat decay


No No YES

YES No No

YES YES No

YES YES YES




MadSpin

One Event

Decay Events



No No YES

YES No No

YES YES No

YES YES YES




MadSpin

One Event

Decay Events



No No YES

YES No No

YES YES No

YES YES YES

|MP+D

LO|2/|MPLO|2- re-weight by

Decay Events II




MadSpin

One Event

Decay Events



No No YES

YES No No

YES YES No

YES YES YES

|MP+D


Decay Events II




MadSpin

One Event

Decay Events



No No YES

YES No No

YES YES No

YES YES YES

|MP+D


Decay Events II

- accept/reject method- reject the decay not the event

Final Sample




MadSpin

One Event

Decay Events



No No YES

YES No No

YES YES No

YES YES YES

|MP+D


Decay Events II

- accept/reject method- reject the decay not the event

Final Sample




TTH Example

0.0001

0.001

0.01

0 25 50 75 100 125 150 175 200

1/σ

dσ

/dp T

(l+ ) [1/

GeV

]

pT(l+) [GeV]

Scalar Higgs



0.4

0.45

0.5

0.55

0.6

-1 -0.5 0 0.5 11/σ

dσ

/dco

s(φ)

cos(φ)

Scalar Higgs



Figure 5: Next-to-leading-order cross sections differential in pT (l+) (left pane) and in cosφ (rightpane) for ttH events with or without spin correlation effects. For comparison, also the leading-order results are shown. Events were generated with aMC@NLO, then decayed with MadSpin,and finally passed to Herwig for shower and hadronization.

0.0001

0.001

0.01

0 25 50 75 100 125 150 175 200

1/σ

dσ

/dp T

(l+ ) [1/

GeV

]

pT(l+) [GeV]

Pseudoscalar Higgs


0.4

0.45

0.5

0.55

0.6

-1 -0.5 0 0.5 1

1/σ

dσ

/dco

s(φ)

cos(φ)

Pseudoscalar Higgs


Figure 6: Next-to-leading-order cross sections differential in pT (l+) (left pane) and in cosφ(right pane) for ttA events with or without spin correlation effects. Events were generated withaMC@NLO, then decayed with MadSpin, and finally passed to Herwig for shower and hadroniza-tion.

that preserving spin correlations is more important than including NLO corrections for this

observable. However, we observe that the inclusion of both, as it is done here, is necessary

for an accurate prediction of the distribution of events with respect to cos(φ). In general, a

scheme including both spin correlation effects and QCD corrections is preferred: it retains

the good features of a NLO calculation, i.e. reduced uncertainties due to scale dependence

(not shown), while keeping the correlations between the top decay products.

The results for the pseudo-scalar Higgs boson are shown in Figure 6. The effects of the

spin correlations on the transverse momentum of the charged lepton are similar as in the

case of a scalar Higgs boson: about 10% at small pT , increasing to about 40% at pT = 200

GeV. On the other hand, the cos(φ) does not show any significant effect from the spin-

correlations. Therefore this observable could possibly help in determining the CP nature of

the Higgs boson, underlining the importance of the inclusion of the spin correlation effects.

– 14 –

TutorialOlivier Mattelaer

IPPP/Durham


• follow the built-in tutorial

• cards meaning• meaning of QCD/QED• details of syntax ($/)• script• width computation • decay chain

56

Tutorial mapLearning MG5 BSM CASE

• check the model• width computation• signal generation

• decay chain• merging sample generation

• background/NLO generation

Learning MG5_aMC


Where to find help?

• Ask me

• Use the command “help” / “help XXX”➡ “help” tell you the next command that you need to do.

• Launchpad:➡ https://answers.launchpad.net/madgraph5➡ FAQ: https://answers.launchpad.net/madgraph5/+faqs

58

https://answers.launchpad.net/madgraph5

https://answers.launchpad.net/madgraph5/+faqs


What are those cards?

• Read the Cards and identify what they do➡ param_card: model parameters➡ run_card: beam/run parameters and cuts

• https://answers.launchpad.net/madgraph5/+faq/2014

59

https://answers.launchpad.net/madgraph5/+faq/2014


Exercise II: Cards Meaning

• How do you change➡ top mass➡ top width➡ W mass➡ beam energy➡ pt cut on the lepton

60


Exercise II : Syntax

• What’s the meaning of the order QED/QCD

• What’s the difference between➡ p p > t t~ ➡ p p > t t~ QED=2➡ p p > t t~ QED=0

• Compute the cross-section for each of those and check the diagram

• Generate VBF process

61

d d > w+ w- d d QCD=0 page 4/10


d

1

d5

a

d2

w- 4

u~

w+

3

d6

u


d

1

d5

z

d2

w- 4

u~

w+

3

d6

u


d

1

d

5

a

d2

d6

a

w+

3

w-4


d1

d5

a

d2

d6

a

w+3

w-

w- 4


d1

d5

a

d2

d6

a

w-4

w+

w+ 3


d

1

d

5

a

d2

d6

z

w+

3

w-4


➡ p p > t t~ QCD=0➡ p p > t t~ QED<=2➡ p p > t t~ QCD^2==2


Exercise III: Syntax

• Generate the cross-section and the distribution (invariant mass) for ➡ p p > e+ e-➡ p p > z, z > e+ e-➡ p p > e+ e- $ z➡ p p > e+ e- / z

• Use the invariant mass distribution to determine the

Hint :To plot automatically distributions:mg5> install MadAnalysis

62


Exercise IV: Automation/Width

• Compute the cross-section for the top pair production for 3 different mass points.➡ Do NOT use the interactive interface

• hint: you can edit the param_card/run_card via the “set” command [After the launch]

• hint: All command [including answer to question] can be put in a file. (run ./bin/mg5 PATH_TO_FILE)

➡ Remember to change the value of the width

• “set width 6 Auto” works

• cross-check that it indeed returns the correct width

63

Examples

File:

import model EWDim6generate p p > z zouput TUTO_DIM6launch set nevents 5000 set MZ 100

How to Run: ./bin/mg5_amc PATH


Exercise V: Decay Chain

• Generate p p > t t~ h, fully decayed (fully leptonic decay for the top)➡ Using the decay-chain formalism➡ Using MadSpin

• Compare cross-section➡ which one is the correct one?➡ Why are they different?

• Compare the shape.

64

BSM Tutorial

MadGraph Tutorial. Beijing 2015

Exercise I: Check the model validity

• Check the model validity:➡ check p p > uv uv~➡ check p p > ev ev~➡ check p p > t t~ p1 p2

66

• This checks ➡ gauge invariance➡ lorentz invariance➡ that various way to compute the matrix element

provides the same answer


Exercise II: Width computation

• Check with MG the width computed with FR:➡ generate uv > all all; output; launch➡ generate ev > all all; output; launch➡ generate p1 > all all; output; launch➡ generate p2 > all all; output; launch

• Check with MadWidth➡ compute_widths uv ev p1 p2➡ (or Auto in the param_card)

67

FR Number

0.00497 GeV

0.0706 GeV

0 GeV

0.0224 GeV

• Muv = 400 GeV Mev = 50 GeV λ=0.1 • m1 = 1GeV m2 = 100GeV m12 = 0.5 GeV


Exercise III:

• Compute cross-section and distribution ➡ uv pair production with decay in top and / (semi leptonic

decay for the top

• Hint: The width of the new physics particles has to be set correctly in the param_card. ➡ You can either use “Auto” ➡ or use the value computed in exercise 1

• Hint: For sub-decay, you have to put parenthesis:➡ example:

p p > t t~ w+, ( t > w+ b, w+ >e+ ve), (t~ > b~ w-, w- > j j), w+ > l+ vl

68

�1 �2

arXiv:1402.1178


Too Slow?

• Use MadSpin!➡ Use Narrow Width Approximation to factorize

production and decay

• instead of ➡ p p > t t~ w+, ( t > w+ b, w+ >e+ ve), (t~ > b~ w-, w- > j j),

w+ > l+ vl

• Do

➡ p p > t t~ w+

• At the question:

• At the next question edit the madspin_card and define the decay

69

arXiv:1212.3460


Exercise IV: generate multiple multiplicity sample for pythia8

• We will do MLM matching➡ in the run_card.dat ickkw=1➡ the matching scale (Qcut) will be define in pythia

• in madgraph we use xqcut which should be smaller than Qcut (but at least 10-20 GeV)

70


Exercise V: Have Fun

• Simulate Background

• Go to NLO (ask me the model)

• …

71


Solution Learning MG5_aMC

72


Exercise II: Cards Meaning

• How do you change➡ top mass➡ top width➡ W mass➡ beam energy➡ pt cut on the lepton

73

Param_card

Run_card


• top mass

74


• W mass

75

W Mass is an internal parameter! MG5 didn’t use this value!

So you need to change MZ or Gf or alpha_EW


Exercise III: Syntax

• What’s the meaning of the order QED/QCD

• What’s the difference between➡ p p > t t~ ➡ p p > t t~ QED=2➡ p p > t t~ QED=0➡ p p > t t~ QCD^2==2

76


Solution I : Syntax

• What’s the meaning of the order QED/QCD➡ By default MG5 takes the lowest order in QED!➡ p p > t t~ => p p > t t~ QED=0➡ p p > t t~ QED=2

• additional diagrams (photon/z exchange)

p p > t t~ QED=2

No significant QED contribution

p p > t t~

77


• QED<=2 is the SAME as QED=2➡ quite often source of confusion since most of the people use

the = syntax

• QCD^2==2 ➡ returns the interference between the QCD and the QED

diagram

78


Solution I Syntax

• generate p p > w+ w- j j➡ 76 processes➡ 1432 diagrams➡ None of them are VBF

79

• generate p p > w+ w- j j QED = 4➡ 76 processes➡ 5332 diagrams➡ VBF present! + those not VBF

• generate p p > w+ w- j j QED = 2➡ 76 processes➡ 1432 diagrams➡ None of them are VBF

• generate p p > w+ w- j j QCD = 0➡ 60 processes➡ 3900 diagrams➡ VBF present!

• generate p p > w+ w- j j QCD = 2➡ 76 processes➡ 5332 diagrams

• generate p p > w+ w- j j QCD = 4➡ 76 processes➡ 5332 diagrams


Exercise IV: Syntax

• Generate the cross-section and the distribution (invariant mass) for ➡ p p > e+ e-➡ p p > z, z > e+ e-➡ p p > e+ e- $ z➡ p p > e+ e- / z

Hint :To have automatic distributions:mg5> install MadAnalysis

80


p p > e+ e-(16 diagrams)

p p >z , z > e+ e-

p p > e+ e- $ z

(8 diagrams)

(16 diagrams)

p p > e+ e- /z(8 diagrams)

Z- onshell vetoNo Z 81



p p >z , z > e+ e-

p p > e+ e- $ z

(8 diagrams)

(16 diagrams)


Z- onshell vetoNo Z

Correct Distribution

81



p p >z , z > e+ e-

p p > e+ e- $ z

(8 diagrams)

(16 diagrams)


Z- onshell vetoNo Z


Z Peak

NO Z Peak

81



p p >z , z > e+ e-

p p > e+ e- $ z

(8 diagrams)

(16 diagrams)


Z- onshell vetoNo Z


Z Peak

NO Z PeakNo z/a interference z/a interference

81



p p >z , z > e+ e-

p p > e+ e- $ z

(8 diagrams)

(16 diagrams)


Z- onshell vetoNo Z


Z Peak

NO Z Peak

Wrong tail Correct tailNo z/a interference z/a interference

81


|M⇤ �M | < BWcut ⇤ �

p p > e+ e- p p >z , z > e+ e- p p > e+ e- $ z

= +Onshell cut: BW_cut

(16 diagrams) (8 diagrams) (16 diagrams)

• The Physical distribution is (very close to) exact sum of the two other one.

• The “$” forbids the Z to be onshell but the photon invariant mass can be at MZ (i.e. on shell substraction).

• The “/” is to be avoid if possible since this leads to violation of gauge invariance.

82


WARNING

• NEXT SLIDE is generated with bw_cut =5

• This is TOO SMALL to have a physical meaning (15 the default value used in previous plot is better)

• This was done to illustrate more in detail how the “$” syntax works.

83


$ explanationp p > e+ e- / Z

See previous slide warning

84

(red curve) (blue curve)


$ explanationp p > e+ e- / Z adding p p > e+ e- $ Z


84





• Z onshell veto

5 times width area

84





• Z onshell veto

• In veto area only photon contribution

5 times width area

84





• Z onshell veto


• area sensitive to z-peak

5 times width area

15 times width area

84





• Z onshell veto



• very off-shell Z, the difference between the curve is due to interference which are need to be KEPT in simulation.

5 times width area

15 times width area>15 times width area

84




The “$” can be use to split the sample in BG/SG area


• Z onshell veto



• very off-shell Z, the difference between the curve is due to interference which are need to be KEPT in simulation.

5 times width area

15 times width area>15 times width area

84



• Syntax Like ➡ p p > z > e+ e- (ask one S-channel z)➡ p p > e+ e- / z (forbids any z)➡ p p > e+ e- $$ z (forbids any z in s-channel)

• ARE NOT GAUGE INVARIANT !

• forgets diagram interference.

• can provides un-physical distributions.

85





• can provides un-physical distributions.Avoid Those as much as possible!

85





• can provides un-physical distributions.Avoid Those as much as possible!check physical meaning and gauge/Lorentz invariance if you do.

85


• Syntax like

• p p > z, z > e+ e- (on-shell z decaying)

• p p > e+ e- $ z (forbids s-channel z to be on-shell)

• Are linked to cut

• Are more safer to use

• Prefer those syntax to the previous slides one

|M⇤ �M | < BWcut ⇤ �

86


Exercise V: Automation

• Look at the cross-section for the previous processfor 3 different mass points.➡ hint: you can edit the param_card/run_card via the “set”

command [After the launch]➡ hint: All command [including answer to question] can be put

in a file.

87


Exercise V: Automation

• File content:

88

• Run it by:

• ./bin/mg5 PATH

• (smarter than ./bin/mg5 < PATH)

• If an answer to a question is not present: Default is taken automatically


• generate p p > t t~ h

➡ decay t > w+ b, w+ > e+ ve➡ decay t~ >w- b~, w- > e- ve~➡ decay h > b b~

89

Exercise VI: Decay

MadSpin

MadSpin Card

MadGraph• generate p p > t t~ h, (t > w+ b, w+ > e+ ve), (t~

>w- b~, w- > e- ve~), h > b b~

2m18.214s

0.004707

0.0030149m30.806s

Different here because of cut (not cut should be applied since 2.3.0)

Documents

Overview of MadGraph5 [email protected] Olivier MadGraph5_aMC@NLO Plan 2 •Details of the computation •Evaluation of matrix-element •Phase-Space integration •What is MG5_aMCMattelaer