Tutorial aMC@NLO

7/25/2019 Tutorial aMC@NLO

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Parton ShowerOlivier Mattelaer

IPPP/Durham


2/182

Fabio MaltoniFabio Maltoni

Mattelaer Olivier Monte-Carlo Lecture: Bei ing 2015

1. High-Q Scattering2 2. Parton Shower

3. Hadronization 4. Underlying Event

Sherpa artist

What are the MC for?

2


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3

Sher a artist

2. Parton Shower

where new h sics lies

rocess de endent

first rinci les descri tion

it can be s stematicall im roved

1. Hi h-Q Scatterin2

3. Hadronization 4. Underl in Event


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Sherpa artist

4

1. High-Q Scattering2 2. Parton Shower

4. Underlying Event3. Hadronization

QCD -known physics

universal/ process independent

first principles description


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We need to be able to describe an arbitrarily number ofparton branchings, i.e. we need to dress partons with radiation

Parton shower

5


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This effect should be unitary:the inclusive cross sectionshouldnt change when extra radiation is added

Parton shower

5


7/182Fabio MaltoniFabio MaltoniMattelaer Olivier Monte-Carlo Lecture: Bei ing 2015



Remember that parton-level cross sections for a hard processare inclusive in anything else. E.g. for LO Drell-Yan production allradiation is included via PDFs (apartfrom non-perturbative power corrections)

Parton shower

5





Remember that parton-level cross sections for a hard processare inclusive in anything else. E.g. for LO Drell-Yan production allradiation is included via PDFs (apartfrom non-perturbative power corrections)

And finally we want to turn partons into hadrons (hadronization)....

Parton shower

5


9/182Mattelaer Olivier Monte-Carlo Lecture: Bei ing 2015 6

First Example

e+e qqg

u

4

g

3

u

e-

2

e+

1

a

u~

5

1

2

3

e+

1

e-

2

z

g

3

u~

5

u~

u

4

1

2

3

q q



First Example

e+e qqg

u

4

g

3

u

e-

2

e+

1

a

u~

5

1

2

3

e+

1

e-

2

z

g

3

u~

5

u~

u

4

1

2

3

q q

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q

2

= 2Eq/S

x2 = 2k2q/q2

= 2Eq/Sd

dx1dx2= 0CF

s

2

x21+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2



First Example

e+e qqg

u

4

g

3

u

e-

2

e+

1

a

u~

5

1

2

3

e+

1

e-

2

z

g

3

u~

5

u~

u

4

1

2

3

q q

Divergent at and

Soft Divergencies

Collinear Divergencies

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q

2

= 2Eq/S

x2 = 2k2q/q2

= 2Eq/Sd

dx1dx2= 0CF

s

2

x21+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2

x1 = 1 x2 = 1 (1 x1) = x2x3

2 (1 cos23)

(1 x2) = x1x3

2 (1 cos13)



First Example

e+e qqg

u

4

g

3

u

e-

2

e+

1

a

u~

5

1

2

3

e+

1

e-

2

z

g

3

u~

5

u~

u

4

1

2

3

q q

Divergent at and

Soft Divergencies


x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q

2

= 2Eq/S

x2 = 2k2q/q2

= 2Eq/Sd

dx1dx2= 0CF

s

2

x21+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2

x1 = 1 x2 = 1 (1 x1) = x2x3

2 (1 cos23)

(1 x2) = x1x3

2 (1 cos13)



First Example

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q2

= 2Eq/S

x2 = 2

k2

q/q2

= 2Eq/

Sd

dx1dx2= 0CF s

2x

2

1+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2



First Example

d

dx3d cos 13= 0CF

s

2

2

sin2 13

1 (1 x3)2

x3 x3

3 cos 13

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q2

= 2Eq/S

x2 = 2

k2

q/q2

= 2Eq/

Sddx1dx2

= 0CF s2

x

2

1+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2

Change the variable to and



First Example

d

dx3d cos 13= 0CF

s

2

2

sin2 13

1 (1 x3)2

x3 x3

3 cos 13

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q2

= 2Eq/S

x2 = 2

k2

q/q2

= 2

Eq/

Sddx1dx2

= 0CF s2

x

2

1+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2




First Example

d

dx3d cos 13= 0CF

s

2

2

sin2 13

1 (1 x3)2

x3 x3

3 cos 13

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q2

= 2Eq/S

x2 = 2

k2

q/q

2= 2

Eq/

Sddx1dx2

= 0CF s2

x

2

1+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2




First Example

d

dx3d cos 13= 0CF

s

2

2

sin2 13

1 (1 x3)2

x3 x3

3 cos 13

x3 = 2k3q/q2

= 2Eg/S

x1 = 2k1q/q2

= 2Eq/S

x2 = 2

k2

q/q

2= 2

Eq/

Sddx1dx2

= 0CF s2

x

2

1+ x2

2

(1 x1)(1 x2)

x1 + x2 + x3 = 2


Collinear limit Split our integral in two

2 dcos13

sin213= dcos13

1 cos13+ dcos13

1 +cos13

d2

13

2

13

+d23

2

23

d cos 13(1 cos 13)

+ d cos 23(1 cos 23)

2

Fi E l



First Example

z fraction of energy

Generic Formula

d

dx3d cos 13= 0CF

s

2

2

sin2 13

1

(1

x3)2

x3 x3

x3 cos 13 Change the variable to and

Collinear limit

Split our integral in two

2 dcos13

sin213

=dcos13

1 cos

13

+dcos13

1 +cos

13

d2

13

213

+d23

223

d cos 13(1 cos 13)

+ d cos 23(1 cos 23)

2

C lli f i i



Consider a process for which two particles are separated by a smallangle ".

In the limit of "!0the contribution is coming from a single parentparticle going on shell: therefore its branching is related to time

scales which are very long with respect to the hard subprocess.

Collinear factorization

9

2b

c"

Mn+1

C lli f t i ti



2a

b

c"

Mn+1"!0





9

2b

c"

Mn+1

C lli f t i ti


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Fabio MaltoniFabio MaltoniMattelaer Olivier Monte-Carlo Lecture: Bei ing 2015

2a

b

c"

Mn+1"!0





9

"!0

2b

c"

Mn+1

C lli f t i ti


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2a

b

c"

Mn+1





9

"!0 !b

c

a

2a

Mn

C lli f t i ti


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2a

b

c"

Mn+1




The inclusion of such a branching cannot change the picture set upby the hard process: the whole emission process must be writablein this limit as the simpler one times a branching probability.


9

"!0 !b

c

a

2a

Mn

C lli f t i ti


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The process factorizes in the collinear limit. This procedure ituniversal!


10

ab

cn+1 "!

b

c

a

a

Mn

z

1-z

Mp a

b

c

z = Eb/Ea

"

1

(pb+pc)2 '

1

2EbEc(1 cos ) =1

t

C lli f t i ti


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10

ab

cn+1 "!

b

c

a

a

Mn

soft

z

1-z

Mp a

b

c

z = Eb/Ea

"

1

(pb+pc)2 '

1

2EbEc(1 cos ) =1

t

C lli f t i ti


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10

ab

cn+1 "!

b

c

a

a

Mn

soft

z

1-zMp a

b

c

z = Eb/Ea

"

andcollinear

divergencies

1

(pb+pc)2 '

1

2EbEc(1 cos ) =1

t

C lli f t i ti


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10

ab

cn+1 "!

b

c

a

a

Mn

soft

z

1-zMp a

b

c

z = Eb/Ea

"

andcollinear

divergencies

1

(pb+pc)2 '

1

2EbEc(1 cos ) =1

t

Collinear factorization:

when "is small.

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

C lli f t i ti


29/182


tcan be called the evolution variable (will become clearer later): itcan be the virtualitym2of particleaor itspT2orE2"2

It represents the hardness of the branching and tends to 0 in thecollinear limit.

Different choice of evolution parameter in different Parton-shower code

d2/2 = dm2/m2 = dp2T/p2

T


11

m2

' z(1 z)2

E2

a

p2T ' zm2

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

Collinear factori ation


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12

zis the energy variable: it is defined to be the energy fraction taken by partonbfrom partona. It represents the energy sharing betweenbandcand tends to

1 in the soft limit (partoncgoing soft)

# is the azimuthal angle. It can be chosen to be the angle between thepolarization ofaand the plane of the branching.

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

P t Sh b i


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Mattelaer Olivier Monte-Carlo Lecture: Beijing 2015

The spin averaged (unregulated) splitting functions for the various typesof branching are (Altarelli-Parisi):

Parton Shower basics

13

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

P t Sh b i


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The spin averaged (unregulated) splitting functions for the various typesof branching are (Altarelli-Parisi):

Comments:* Gluons radiate the most* There are soft divergences in z=1 and z=0.* Pqg has no soft divergences.

Parton Shower basics

13

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

Argument of !


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Each choice of argument for $Sis equally acceptable at the leading-logarithmic accuracy.However, there is a choice that allows one to resum certain classes of subleadinglogarithms.

The higher order corrections to the partons splittings imply that the splitting kernelsshould be modified: Pabc(z)Pabc(z) + $s Pabc(z)

For gggbranchings Pabc(z)diverges as -b0log[z(1-z)] Pabc(z)

(just zor 1-zif quark is present)

Recall the one-loop running of the strong coupling:

We can therefore include the P(z)terms by choosing pT2~z(1-z)Q2as argument of $S:

Argument of !S

14

S(Q2) = S

(2

)1 + S(2)b0log

Q2

2

S(2)

1 S(2)b0logQ2

2

S(Q2)

Pabc(z) + S(Q

2)Pabc

= S(Q

2)

1 S(Q2)b log z(1 z)

Pabc(z)

S(z(1 z)Q2

)Pabc(z)

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

Argument of !


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Each choice of argument for $Sis equally acceptable at the leading-logarithmic accuracy.However, there is a choice that allows one to resum certain classes of subleadinglogarithms.

The higher order corrections to the partons splittings imply that the splitting kernelsshould be modified: Pabc(z)Pabc(z) + $s Pabc(z)

For gggbranchings Pabc(z)diverges as -b0log[z(1-z)] Pabc(z)

(just zor 1-zif quark is present)

Recall the one-loop running of the strong coupling:

We can therefore include the P(z)terms by choosing pT2~z(1-z)Q2as argument of $S:

Argument of !S

14

S(Q2) = S

(2

)1 + S(2)b0log

Q2

2

S(2)

1 S(2)b0logQ2

2

S(Q2)

Pabc(z) + S(Q

2)Pabc

= S(Q

2)

1 S(Q2)b log z(1 z)

Pabc(z)

S(z(1 z)Q2

)Pabc(z)

|Mn+1|2dn+1 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

t

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

Collinear Limit

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

Collinear Limit

tis the evolution parameter (control the collinear behaviour)

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

Collinear Limit

tis the evolution parameter (control the collinear behaviour) zis the energy sharing variable

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

Collinear Limit


alpha_sneed to be evaluated at the scale t

To Remember


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To Remember

15

|Mn+1|2dn+1 ' |Mn|

2dn

dt

t dz

d

2

S

2Pabc(z)

Collinear Limit


alpha_sneed to be evaluated at the scale t

Pis the splitting Kernel (control the soft behaviour)

Multiple emission


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Now consider Mn+1as the new core process and use the recipe weused for the first emission in order to get the dominant contributionto the (n+2)-body cross section: add a new branching at angle muchsmaller than the previous one:

This can be done for an arbitrary number of emissions. The recipe toget the leading collinear singularity is thus cast in the form of aniterative sequence of emissions whose probability does not depend onthe past history of the system: a Markov chain. No interference!!!

Multiple emission

16

Mn+2|2dn+2 ' |Mn|

2dn

dt

tdz

d

2

S

2Pabc(z)

dt0

t0 dz0d0

2

S

2Pbde(z

0

)

","!0

" "

2a b

c"

"

d

e!

b

c

a

2a

Mnd

e

b!Mn+2

Multiple emission


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The dominant contribution comes from the region where thesubsequently emitted partons satisfy the strong ordering requirement:"!"!"...

For the rate for multiple emission we get

where Qis a typical hard scale and Q0is a small infrared cutoff thatseparates perturbative from non perturbative regimes.

Each power of $scomes with a logarithm. The logarithm can be easilylarge, and therefore it can lead to a breakdown of perturbation theory.

Multiple emission

17

n+k kS

Z Q2Q20

dt

t

Z tQ20

dt0

t0...

Z t(k2)Q20

dt(k1)

t(k1) n

S

2

klogk(Q2/Q20)

","!0

" "

2a b

c"

"

d

e!

b

c

a

2a

Mnd

e

b!Mn+2

Sudakov Form Factor


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What is the probability of no emission?

18

Sudakov Form Factor

Sudakov Form Factor


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18

Sudakov Form Factor

Pnonbranching(ti) = 1 Pbranching(ti) = 1 t

ti

s

2

Z 1

z

dzP(z)

Sudakov Form Factor


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So the probability of no emission between

two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)

Sudakov Form Factor


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two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)

Pnobranching(Q2, t) = lim

N

NYi=0

1

t

ti

S

2

Z dzP(z)

Sudakov Form Factor


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two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)

Sudakov Form Factor


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two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)

R

Sudakov Form Factor


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two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)

(Q2, t)Sudakov form factor R

Sudakov Form Factor


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two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)


Sudakov Form Factor


51/182




two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)


Sudakov Form Factor


52/182




two scales:

18

Sudakov Form Factor


ti

s

2

Z 1

z

dzP(z)


N

NYi=0

1

t

ti

S

2

Z dzP(z)

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)

' limN

eP

N

i=0

t

ti

S

2

R dzP(z)

(Q2, t)Sudakov form factor

"Property:%(A,B) = %(A,C) %(C,B)

R

Parton shower


53/182


Parton shower

19

The Sudakov form factor is the heart of the parton shower. It gives the

probability that a parton does not branch between two scales

Using this no-emission probability thebranching tree of a par ton is generated.

Define dPkas the probability for k ordered splittings from leg a at given scales

Q02 is the hadronization scale (~1 GeV). Below this scale we do not trust theperturbative description for parton splitting anymore.

dP1(t1) = (Q2, t1) dp(t1)(t1, Q

2

0),

dP2(t1, t2) = (Q2, t1) dp(t1) (t1, t2) dp(t2) (t2, Q

2

0)(t1 t2),

... = ...

dPk(t1,...,tk) = (Q2, Q2

0)

k

l=1

dp(tl)(tl1 tl)

Unitarity


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The parton shower has to be unitary (the sum over allbranching trees should be 1). We can explicitly show this by

integrating the probability for ksplittings:

Unitarity

20

dPk(t1,...,tk) = (Q2, Q20)

k

l=1

dp(tl)(tl1

tl)

Unitarity


55/182




Unitarity

20

dPk(t1,...,tk) = (Q2, Q20)

k

l=1

dp(tl)(tl1

tl)

Pk dPk(t1,...,tk) = (Q

2, Q20

)1k!

Q

2

Q20

dp(t)k

, k= 0, 1,...

Unitarity


56/182




Summing over all number of emissions

Unitarity

20

dPk(t1,...,tk) = (Q2, Q20)

k

l=1

dp(tl)(tl1

tl)


2, Q20

)1k!

Q

2

Q20

dp(t)k

, k= 0, 1,...

Unitarity


57/182





Unitarity

20

dPk(t1,...,tk) = (Q2, Q20)

k

l=1

dp(tl)(tl1

tl)


2, Q20

)1k!

Q

2

Q20

dp(t)k

, k= 0, 1,...

k=0

Pk = (Q2

, Q2

0)

k=0

1

k! Q2

Q20

dp(t)k

= (Q2

, Q2

0)exp Q2

Q20

dp(t)

= 1

Unitarity


58/182





Hence, the total probability is conserved

Unitarity

20

dPk(t1,...,tk) = (Q2

, Q2

0)

k

l=1

dp(tl)(tl1

tl)


2, Q20

)1k!

Q

2

Q20

dp(t)k

, k= 0, 1,...

k=0

Pk = (Q2

, Q2

0)

k=0

1

k! Q2

Q20

dp(t)k

= (Q2

, Q2

0)exp Q2

Q20

dp(t)

= 1

singularities


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We have shown that the showers is unitary. However, how arethe IR divergences cancelled explicitly? Lets show this for the

first emission:Consider the contributions from (exactly) 0 and 1 emissionsfrom leg a:

Expanding to first order in $sgives

Same structure of the two latter terms, with opposite signs:cancellation of divergences between the approximate virtualand approximate real emission cross sections.

The probabilistic interpretation of the shower ensures thatinfrared divergences will cancel for each emission.

singularities

21

n= (Q2, Q2

0) +(Q2, Q2

0)

bc

dz t

t

2

S

2Pabc(z)

d

n 1

bc

Q2Q20

dt

tdz

d

2

S

2Pabc(z) +

bc

dzdt

t

d

2

S

2Pabc(z)

Final-state parton showers


60/182


p

22



61/182


p

With the Sudakov form factor, we can now implement a final-stateparton shower in a Monte Carlo event generator!

22



62/182


p


1. Start the evolution at the virtual mass scale t0(e.g. the mass of thedecaying particle) and momentum fraction z0= 1

22



63/182


p



2. Given a virtual mass scale tiand momentum fractionxiat some stage

in the evolution, generate the scale of the next emission ti+1according tothe Sudakov probability &(ti,ti+1) by solving&(ti+1,ti) = Rwhere Ris a random number (uniform on [0, 1]).

22



64/182


p





3. If ti+1< tcutit means that the shower has finished.

22



65/182


p






4. Otherwise, generate z= zi/zi+1with a distribution proportional to ($s/2')P(z), where P(z) is the appropriate splitting function.

22



66/182


p






4. Otherwise, generate z= zi/zi+1with a distribution proportional to ($s/2')P(z), where P(z) is the appropriate splitting function.

5. For each emitted particle, iterate steps 2-4 until branching stops.

22

Soft Limit


67/182


There is a lot of freedom in the choice of evolution parametert.It can be the virtuality m2of particle aor its pT2or E2"2... Forthe collinear limit they are all equivalent

However, in the soft limit (z"0,1) they behave differently Can we chose it such that we get the correct soft limit?

Soft gluon comes from the full event!

Soft Limit

23

(Q2, t) = exp

bc Q2

t

dt

tdz

d

2

S

2Pabc(z)

Quantum Interference

Angular ordering


68/182


Radiation inside cones around the original partons is allowed(and described by the eikonal approximation), outside the conesit is zero (after averaging over the azimuthal angle)

Angular ordering

24

photon+photon

To Remember


69/182


Sudakov Form-Factor: Probablility of No-emission between two scale.

Probalitity of K-emission

Ensure that the parton shower is unitary

Ensure cancelation of IR divergency

Interference effect via Angular ordering

25

' eRQ2

tdt0

t0 dz

S2

P(z) e

RQ2

t dp(t0)(Q2, t)

Pk

dPk(t1,...,tk) = (Q

2, Q20

)1

k!

Q2Q20

dp(t)

k, k= 0, 1,...

Initial-state


70/182


So far, we have looked at final-state (time-like) splittings. For

initial state, the splitting functions are the same

However, there is another ingredient: the parton density (ordistribution) functions (PDFs). Naively: Probability to find a

given parton in a hadron at a given momentum fraction x = pz/

Pzand scale t.

t a state

26

x0 t0

Q2

x1 t1

xn1 tn1

xn tn

p

Initial-state


71/182






Pzand scale t.

How do the PDFs evolve with increasing t?

26

x0 t0

Q2

x1 t1

xn1 tn1

xn tn

p

Initial-state


72/182






Pzand scale t.

How do the PDFs evolve with increasing t?

26

t

tfi(x, t) =

Z 1

x

dz

z

s

2Pij(z)fj

xz, t

DGLAP

x0 t0

Q2

x1 t1

xn1 tn1

xn tn

p

Initial-state parton splittings


73/182


Start with a quark PDF f0(x)at scale t0. After a singleparton emission, the probability to find the quark atvirtuality t > t0is

After a second emission, we have

p p g

x0 t0

Q2

x1 t1

xn1 tn1

xn tn

p

27

f(x, t) =f0(x) +

t

t0

dt

ts

2

1

x

dz

zP(z)f0

xz

f(x/z, t)f(x, t) =f0(x) +

t

t0

dt

ts

2

1

x

dz

zP(z)

f0

xz

+

t

t0

dt

ts

2

1

x/z

dz

zP(z)f0

x

zz

The DGLAP equation


74/182


So for multiple parton splittings, we arrive at an integral-differential equation:

This is the famous DGLAP equation (where we have taken intoaccount the multiple parton species i,j). The boundarycondition for the equation is the initial PDFs fi0(x)at a startingscalet0(around 2 GeV).

These starting PDFs are fitted to experimental data.

q

28

x0 t0

Q2

x1 t1

xn1 tn1

xn tn

p

t

tfi(x, t) =

Z 1

x

dz

z

s

2Pij(z)fj

xz, t

parton showers


75/182


To simulate parton radiation from the initial state, we start withthe hard scattering, and then deconstruct the DGLAP

evolution to get back to the original hadron: backwardsevolution!

i.e. we undo the analytic resummation and replace it withexplicit partons (e.g. in Drell-Yan this gives non-zero pTto

the vector boson)

In backwards evolution, the Sudakovs include also the PDFs --this follows from the DGLAP equation and ensuresconservation of probability:

This represents the probability that parton iwill stay at thesame x(no splittings) when evolving from t1to t2.

The shower simulation is now done as in a final state shower!

p

29

Ii(x, t1, t2) = exp

t2

t1

dtj

1

x

dx

x

s(t)

2Pij x

x fi(x

, t)

fj(x, t

)

Hadronization


76/182


The shower stops if all partons are characterized by a scale atthe IR cut-off: Q0~ 1 GeV.

Physically, we observe hadrons, not (colored) partons.

We need a non-perturbative model in passing from partons tocolorless hadrons.

There are two models (string and cluster), based on physicaland phenomenological considerations.

30

Parton Shower MC event generators


77/182

Fabio MaltoniFabio MaltoniMattelaer Olivier Monte-Carlo Lecture: Bei ing 2015 31

A parton shower program associates one of the possible histories (and pre-histories in case of pp) of an hard event in an explicit and fully detailed way,

such that the sum of the probabilities of all possible histories is unity.



78/182


General-purpose tools

Complete exclusive description of the events: hard scattering,showering & hadronization (and underlying event)

Reliable and well-tuned tools

Significant and intense progress in the development of newshowering algorithms with the final aim to go at NLO in QCD

31





79/182






Shower MC Generators: PYTHIA, HERWIG, SHERPA

31





80/182






Shower MC Generators: PYTHIA, HERWIG, SHERPA

31



"Note that a banching tree is not a Feynman diagram: it

represents the coherent sum of many real and virtual diagrams

which are summed by the branching algorithm" (HERWIG

manual)

To Remember


81/182


The parton shower dresses partons with radiation. This makesthe inclusive parton-level predictions (i.e. inclusive over extraradiation) completely exclusive

In the soft and collinear limitsthe partons showers areexact, but in practice they are used outside this limit as well.

Partons showers are universal(i.e. independent from theprocess)

Building block of the parton shower is the Sudakov

There is a cut-off in the shower (below which we dont trustperturbative QCD) at which a hadronization modeltakes over

32


82/182

Matching/MergingOlivier Mattelaer

IPPP/Durham

PS alone vs matched samples


83/182


GeV

0 50 100 150 200 250 300 350 400

(pb/bin)

T

/dP

!d

-310

-210

-110

1

10

(wimpy)2Q

(power)2Q

(wimpy)2TP

(power)2TP

of the 2-nd extra jetTP

(a la Pythia)tt

In the soft-collinear approximation of Parton Shower MCs, parameters are used totune the resultLarge variation in results (small prediction power)

(Pythia only)

34

Matrix Elements vs. Parton Showers


84/182

Mattelaer Olivier Monte-Carlo Lecture: Beijing 2015 35



85/182


ME

1. Fixed order calculation2. Computationally expensive3. Limited number of particles4. Valid when partons are hard and

well separated5. Quantum interference correct

6. Needed for multi-jet description

35



86/182


ME




Shower MC

1. Resums logs to all orders2. Computationally cheap3. No limit on particle multiplicity4. Valid when partons are collinear

and/or soft5. Partial interference through

angular ordering6. Needed for hadronization

35



87/182


Approaches are complementary: merge them!

ME




Shower MC




35



88/182


Difficulty: avoid double counting, ensure smooth distributions

Approaches are complementary: merge them!

ME




Shower MC




35

Goal for ME-PS merging/matching


89/182


Matrix element

Parton shower 2nd QCD radiation jet intop pair production at

the LHC, usingMadGraph + Pythia

36



90/182


Regularization of matrix element divergence

Matrix element



36



91/182


Regularization of matrix element divergence Correction of the parton shower for large momenta

Matrix element



36



92/182


Regularization of matrix element divergence Correction of the parton shower for large momenta Smooth jet distributions

Matrix element



36



93/182



Matrix element

Parton shower

Desired curve

2nd QCD radiation jet intop pair production at


36

Merging ME with PS


94/182


...

...

PS (

ME

)

[Mangano][Catani, Krauss, Kuhn, Webber][Lnnblad]

37

Merging ME with PS


95/182


...

...

PS (

ME

)


DC DC

DC

37

Merging ME with PS


96/182


...

...

PS (

ME

)


kT< Qc

kT> Qc

kT> Qc

kT> Qc

kT< Qc

kT< Qc

kT> Qc

kT< Qc

37

Merging ME with PS


97/182


...

...

PS (

ME

)

Double counting between ME and PS easily avoided using phase space cutbetween the two: PS below cutoff, ME above cutoff.


kT< Qc

kT> Qc

kT> Qc

kT> Qc

kT< Qc

kT< Qc

kT> Qc

kT< Qc

37

Merging ME with PS


98/182


So double counting problem easily solved, butwhat about getting smooth distributions that areindependent of the precise value of Qc?

Below cutoff, distribution is given by PS- need to make ME look like PS near cutoff

Lets take another look at the PS!

38

Merging ME with PS


99/182


t0

t1

t2

tcuttcut

tcut

tcut

39

Merging ME with PS


100/182


How does the PS generate the configuration above?

t0

t1

t2

tcuttcut

tcut

tcut

39

Merging ME with PS


101/182



Probability for the splitting at t1is given by

t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)

39

Merging ME with PS


102/182




t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)

39

Merging ME with PS


103/182




t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)

39

Merging ME with PS


104/182




and for the whole tree

t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)

(q(tcut, t0))2g(t2, t1)(q(cut, t2))

2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


105/182





t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


106/182





t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


107/182





t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


108/182





t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


109/182





t0

t1

t2

tcuttcut

tcut

tcut

(q(t1, t0))2s(t1)

2

Pgq(z)


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

39

Merging ME with PS


110/182


t0

t1

t2

tcuttcut

tcut

tcut


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

40

Merging ME with PS


111/182


t0

t1

t2

tcuttcut

tcut

tcut

Corresponds to the matrix element

BUT with $sevaluated at the scale of each splitting


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

40

Merging ME with PS


112/182


t0

t1

t2

tcuttcut

tcut

tcut

Corresponds to the matrix element

BUT with $sevaluated at the scale of each splitting

Sudakov suppression due to not allowing additional radiationabove the scale tcut


2s(t1)

2Pgq(z)

s(t2)

2Pqg(z

0)

40

Merging ME with PS


113/182


|M|2(s, p3, p4, ...)

41

Merging ME with PS


114/182


To get an equivalent treatment of the correspondingmatrix element, do as follows:

|M|2(s, p3, p4, ...)

41

Merging ME with PS


115/182


To get an equivalent treatment of the correspondingmatrix element, do as follows:1. Cluster the event using some clustering algorithm

- this gives us a corresponding parton shower history

|M|2(s, p3, p4, ...)

41

Merging ME with PS


116/182




t0

t1

t2 |M|2(s, p3, p4, ...)

41

Merging ME with PS


117/182




2. Reweight $sin each clustering vertex with the clusteringscale

t0

t1

t2

|M|2 |M|2s(t1)

s(t0)

s(t2)

s(t0)

|M|2(s, p3, p4, ...)

41

Merging ME with PS


118/182




2. Reweight $sin each clustering vertex with the clusteringscale

3. Use some algorithm to apply the equivalent Sudakovsuppression

t0

t1

t2


2

|M|2 |M|2s(t1)

s(t0)

s(t2)

s(t0)

|M|2(s, p3, p4, ...)

41

Matching for initial state radiation


119/182


x1

x2

42



120/182


We are of course not interested in e+e-but p-p(bar)- what happens for initial state radiation?

x1

x2

42



121/182



Lets do the same exercise as before:

x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)

P = (Iq(tcut, t0))2g(t2, t1)(q(tcut, t2))

2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



122/182




x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



123/182




x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



124/182




x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



125/182




x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



126/182




x1tcut

t1 t2

tcut

tcut

tcutt0

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s(t1)

2

Pgq(z)z

fq(x1, t1)fq(x01, t1)

s(t2)2

Pqg(z0)

42



127/182


tcut

t1 t2

tcut

tcut

tcutt0

x1

x1

x2

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)

(Iq(tcut, t0))2g(t2, t1)(q(tcut, t2))

2s t1

2

Pgq z

z

fq x1, t1

fq(x01, t1)

s t2

2Pqg(z

0)

43



128/182


tcut

t1 t2

tcut

tcut

tcutt0

x1

x1

x2

ME with$

sevaluated at the scale of each splitting

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s t1

2

Pgq z

z

fq x1, t1

fq(x01, t1)

s t2

2Pqg(z

0)

43



129/182


tcut

t1 t2

tcut

tcut

tcutt0

x1

x1

x2

ME with$

sevaluated at the scale of each splittingPDF reweighting

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s t1

2

Pgq z

z

fq x1, t1

fq(x01, t1)

s t2

2Pqg(z

0)

43



130/182


tcut

t1 t2

tcut

tcut

tcutt0

x1

x1

x2

ME with$

sevaluated at the scale of each splittingPDF reweighting

Sudakov suppression due to non-branching above scale tcut

qqe(s, ...)fq(x0

1, t0)fq(x2, t0)


2s t1

2

Pgq z

z

fq x1, t1

fq(x01, t1)

s t2

2Pqg(z

0)

43



131/182


x1

x2

44



132/182


x1

x2

Again, use a clustering scheme to get a parton shower history

44



133/182


x1

t1 t2

t0x1

x2

Again, use a clustering scheme to get a parton shower history

44



134/182


x1

t1 t2

t0x1

x2

Again, use a clustering scheme to get a parton shower history Now, reweight both due to $sand PDF

|M|2 |M|2

s(t1)

s(t0)

s(t2)

s(t0)

fq(x0

1, t0)

fq(x01, t1)

44



135/182


x1

t1 t2

t0x1

x2

Again, use a clustering scheme to get a parton shower history Now, reweight both due to $sand PDF

Remember to use first clustering scale on each side for PDF scale:

|M|2 |M|2s(t1)

s(t0)

s(t2)

s(t0)

fq(x0

1, t0)

fq(x01, t1)

Pevent= (x1, x2, p3, p4, . . .)fq(x1, t1)fq(x2, t0)

44

Matching schemes


136/182


We still havent specified how to apply the Sudakovreweighting to the matrix element

Three general schemes available in the literature:" CKKW scheme [Catani,Krauss,Kuhn,Webber 2001; Krauss 2002]

" Lnnblad scheme (or CKKW-L) [Lnnblad 2002]

" MLM scheme [Mangano unpublished2002; Mangano et al. 2007]

45

CKKW matching


137/182


[Catani, Krauss, Kuhn, Webber 2001][Krauss 2002]

46

CKKW matching


138/182


Apply the required Sudakov suppression

analytically, using the best available (NLL) Sudakovs.

(Iq(tcut, t0)) g(t2, t1)(q(tcut, t2))


46

CKKW matching


139/182


t0



Perform truncated showering: Run the parton shower starting att0, but forbid any showers above the cutoff scale tcut.



46

CKKW matching


140/182


t0






kT1

kT2

kT3

46

CKKW matching


141/182


t0






kT3

kT1

kT2

x

x

46

CKKW matching


142/182


t0






kT3

kT1

kT2

x

x

* Best theoretical treatment of matrix element

- Requires dedicated PS implementation- Mismatch between analytical Sudakov and (non-NLL) shower Implemented in Sherpa (v. 1.1)

46

CKKW-L matching


143/182


t0

[Lnnblad 2002][Hoeche et al. 2009]

47

CKKW-L matching


144/182


Cluster back to parton shower history

t0


47

CKKW-L matching


145/182


Cluster back to parton shower history

t0


47

CKKW-L matching


146/182


Cluster back to parton shower history Perform showering step-by-step for each step in the parton shower

history, starting from the clustering scale for that step

t0


47

CKKW-L matching


147/182




t0


kT1

47

CKKW-L matching


148/182




t0


kT1

47

CKKW-L matching


149/182




Veto the event if any shower is harder than the clustering scale forthe next step (or tcut, if last step)

t0


kT1

47

CKKW-L matching


150/182





Keep any shower emissions that are softer than the clustering scalefor the next step

t0


kT1

47

CKKW-L matching


151/182






t0


kT1

kT2

47

CKKW-L matching


152/182






t0


kT1

kT2

47

CKKW-L matching


153/182






t0


kT1

kT2

kT4

47

CKKW-L matching


154/182






t0


kT1

kT2

kT4

47

* Automatic agreement between Sudakov and shower

- Requires dedicated PS implementation" Need multiple implementations to compare between showers

Implemented in Ariadne, Sherpa (v. 1.2), and Pythia 8

MLM matching

[M.L. Mangano, ~2002, 2007] [J A l 2007 2008]


155/182


t0

[J.A. et al 2007, 2008]

48

MLM matching

[M.L. Mangano, ~2002, 2007] [J A l 2007 2008]


156/182


t0

[J.A. et al 2007, 2008]

The simplest way to do the Sudakov suppression is to run the

shower on the event, starting from t0!

48

MLM matching

[M.L. Mangano, ~2002, 2007] [J A l 2007 2008]


157/182


kT1

kT2kT3

kT4

t0

[J.A. et al 2007, 2008]

The simplest way to do the Sudakov suppression is to run the

shower on the event, starting from t0!

48

MLM matching

[M.L. Mangano, ~2002, 2007] [J A l 2007 2008]


158/182


kT1

kT2kT3

kT4

t0

[J.A. et al 2007, 2008]

The simplest way to do the Sudakov suppression is to run theshower on the event, starting from t0!

Perform jet clustering after PS - if hardest jet kT1> tcutor there arejets not matched to partons, reject the event

48

MLM matching

[M.L. Mangano, ~2002, 2007] [J A t l 2007 2008]


159/182


kT1

kT2kT3

kT4


(Iq(tcut, t0))2(q(tcut, t0))

2

t0

[J.A. et al 2007, 2008]



The resulting Sudakov suppression from the procedure is

which turns out to be a good enough approximation of the correctexpression

48

MLM matching

[M.L. Mangano, ~2002, 2007] [J A et al 2007 2008]


160/182


kT1

kT2kT3

kT4


(Iq(tcut, t0))2(q(tcut, t0))

2

t0

[J.A. et al 2007, 2008]



The resulting Sudakov suppression from the procedure is

which turns out to be a good enough approximation of the correctexpression

48

* Simplest available scheme

* Allows matching with any shower, without modification

" Sudakov suppression not exact, minor mismatch with shower

Implemented in AlpGen, HELAC, MadGraph+Pythia 6


161/182

matching schemes


162/182


We have a number of choices to make in the aboveprocedure. The most important are:

1. The clustering scheme used to determine the parton

shower history of the ME event

2. What to use for the scale Q2(factorization scale)

3. How to divide the phase space between parton showersand matrix elements

50

Back to the matching goal


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Matrix element

Parton shower

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Matrix element

Parton shower

Matching scale

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Parton shower

Matching scale

Emissions from PS

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Parton shower

Matching scale

Emissions from ME

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Parton shower

Matching scale

Sudakov suppression

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Parton shower

Matching scale51



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Parton shower

Matching scale

Desired curve

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Summary of Matching Procedure

1. Generate ME events (with different parton multiplicities) using


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( p p ) gparton-level cuts (pTME/%R or kTME)

2. Cluster each event and reweight $sand PDFs based on thescales in the clustering vertices

3. Apply Sudakov factors to account for the required non-radiation above clustering cutoff scale and generate partonshower emissions below clustering cutoff:

a. (CKKW) Analytical Sudakovs + truncated showers

b. (CKKW-L) Sudakovs from truncated showers

c. (MLM) Sudakovs from reclustered shower emissions

4. Apply separation cut

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Comparing to experiment: W+jets


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) [GeV]min

TJet Transverse Energy (E

0 50 100 150 200

[pb]

T

)dE

T

/dE

(d

min

TE

-210

-110

1

10

210

CDF Run II Preliminaryn jets) +e(W

CDF Data -1dL = 320 pbW kin: 1.1|

e20[GeV]; |eTE

30[GeV]

T]; E2

20[GeV/cWTM

Jets: |


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mg5> generate p p > w+, w+ > l+ vl @0

mg5> add process p p > w+ j, w+ > l+ vl @1

mg5> add process p p > w+ j j, w+ > l+ vl @2

mg5> output

In run_card.dat:

1 = ickkw

0 = ptj

15 = xqcut

kTmatching scale

Matching on

Matching automatically done when run through

MadEvent and Pythia!

No cone matching

Example: Simulation of pp(W with 0, 1, 2 jets

(comfortable on a laptop)

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matching in Mad

Tutorial aMC@NLO

Documents

Transcript of Tutorial aMC@NLO