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Monte Carlo tools for the LHC

“Quarks and mesons? No border control? Mach 15 jets?

Who the hell ARE these guys?”

from “Angels and Demons” by D. Brown

Fabio Maltoni

Centro Studi e Ricerche “Enrico Fermi”, Roma

it’s US!

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Basics 30’

Available Tools 60’

On air examples 30’

Plan

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Exercises:

A few exercises will be given on the way Mostly borrowed from M. Seymour’s

Cern lessons...

... solutions can be found at

madgraph.hep.uiuc.edu/MC101.nb

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Exercises:

send questions/corrections to maltoni@fis.uniroma3.it

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Basics: references

M. Seymour, Cern lectures,

http://seymour.home.cern.ch/seymour/slides/CERNlectures.html

S. Weinzierl, Introduction to MC methods,

hep-ph/0006269

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Basics:

from integration to event generation

σ = 1 2s

!

|M|2dΦ(n)

Calculations of cross section or decay widths involve integrations over high-

dimension phase space of very complex functions

General and flexible method is needed

Dim[Φ(n)] ∼ 3n

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☞ Improvement by minimizing VN

Integrals as averages

I = IN ± !VN /N 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 − IN2 IN = (x2 − x1) 1

N

!N i=1

f(x)

☞ Convergence is slow but it can be estimated easily

☞ Error does not depend on # of dimensions!

☞ Optimal/Ideal case: f(x)=C =0

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Importance Sampling

=

! ξ2

ξ1cos

π

2 x[ξ]

1−x[ξ]2 ! 1 I =

! 1

0

dx cos π 2 x

IN = 0.637 ± 0.307/ N

I =

! 1

0

dx(1 − x2) cos π2 x 1 − x2

IN = 0.637 ± 0.031/ N

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Importance Sampling

but... you need to know too much about f(x)!

idea: learn during the run and build a step-function approximation p(x) of f(x) VEGAS

many bins where f(x) is large

p(x) = N 1

b∆xi , xi − ∆xi < x < xi

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Importance Sampling

can be generalized to n dimensions:

p(x)= p(x)•p(y)•p(z)…

but the peaks of f(x) need to be “aligned” to the axis!

This is ok...

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Importance Sampling

can be generalized to n dimensions:

p(x)= p(x)•p(y)•p(z)…

but the peaks of f(x) need to be “aligned” to the axis!

This is not ok...

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Importance Sampling

can be generalized to n dimensions:

p(x)= p(x)•p(y)•p(z)…

but the peaks of f(x) need to be “aligned” to the axis!

but it is sufficient to make a change of variables!

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Multi-channel

In this case there is no unique tranformation:

Vegas is bound to fail!

Solution: use different transformations= channels

p(x) =

!n i=1

αipi(x)

!n i=1

αi = 1 with

with each pi(x) taking care of one “peak” at the time

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Multi-channel

In this case there is no unique tranformation:

Vegas is bound to fail!

p

1

(x) p

2

(x)

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Multi-channel

In this case there is no unique tranformation:

Vegas is bound to fail!

Solution: use different transformations= channels

p(x) =

!n i=1

αipi(x)

!n i=1

αi = 1 with

Exercise: show that only VN depends on the

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Phase Space

n =

!

Πni=1

d3pi

(2π)3(2Ei)

"

(2π)4δ(4)(p0

n

#

i=1

pi)

2(M ) = 1 8π

2p M

dΩ 4π

n(M ) = 1 2π

! (M −µ)2

0

22(M )dΦn−1(µ)

n

••

= 2

n-1

••

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Exercise: top decay

Easy but non-trivial

Breit-Wigner peak to be “flattened”

b

t l

w v

1

(q2 − m2W )2 + Γ2W m2W

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Exercise: top decay

b

t l

w v

1

(q2 − m2W)2 + Γ2W m2W

after analytic transformation

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Exercise: top decay

Easy but non-trivial

Breit-Wigner peak to be “flattened :

Choose the right “channel” for the phase space

b

t l

w v

1

(q2 − m2W )2 + Γ2W m2W

l

l l

b

b b

v v

v

or or ?

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Event generation

Alternative way 1. pick x

3. pick 0<y<fmax

f(x) 2. calculate f(x)

4. Compare:

if f(x)>y accept event, else reject it.

I=

total tries accepted

= efficiency

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Event generation

What’s the difference?

before:

same # of events in areas of phase space with very different probabilities:

events must have different weights

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Event generation

What’s the difference?

after:

# events is proportional to the probability of

areas of phase space:

events have all the same weight (”unweighted”)

Events distributed as in Nature

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Event generation

Improved

1. pick x distributed as p(x) 2. calculate f(x) and p(x)

3. pick 0<y<1

f(x)

4. Compare:

if f(x)>y p(x) accept event, else reject it.

much better efficiency!!!

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Basics:

from integration to event generation MC integrator

Event generator

Acceptance-Rejection

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

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Basics:

from integration to event generation

1. Integrate is hard

2. Integration+ unweighting = Event generation 3. EFFICIENT event generator =

need to know how to integrate the x-section VERY well

to take home

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Basics: Final Project

1. Consider qq→tt 2. Build a MC for it

3. Include your MC for top decays

4. Make plots of the angular correlation between the charged leptons.

5. Calculate (or find ) the amplitude for the full process qq→tt→b b e+ e- v v

and compare with the results of point 4.

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Available Tools: references

Les Houches Guide Book to MC generators for Hadron Collider Physics, hep-ph/0403045

Links and descriptions of the codes at http://www.ippp.dur.ac.uk/HEPCODE/

Recent talks by Frixione, Mrenna, Schumann, Webber @ KEK 04 and Piccinini @ IFAE 04.

Several talks by M.L.Mangano (@ FNAL, CERN, MC4LHC, IFAE, KEK)

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Available Tools

Significant progress in the last few years, R&D still going on!

Many different codes available

Result: (not only) users are confused!

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Available Tools

There is no

PERFECT-FOR-ALL-PURPOSES MONTECARLO!

You’ll certainly find one suitable to your needs!

but

You must know what you need!!

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Sherpa Collaboration

1. High-Q Scattering2 2. Parton Shower

3. Hadronization 4. Underlying Event

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Main classes of MC’s

MC integrators

Parton Shower MC event generators Matrix-element based MC event

generators

MC@NLO

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MC’s integrators

Now used only for at least NLO calculations or analytically resummed results

Provide essential information on the normalization of the cross section

Produce distributions of any quantity of interest but not events (due to negative weights)

The“cleanest” tools theorists’ 1

st

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MC’s integrators

Inclusive approach (NO EVENTS)

Predictions are at parton level only. No

showering, hadronization or detector effects.

Jets contain at most two partons

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MC’s integrators

They need a lot of manual work progress is slow with only few codes available for “simple processes”

In some cases special treatment of particular areas of phase space gives an “improved”

prediction (e.g. ResBos)

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J. Campbell

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J. Campbell

+ several other results

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MC’s integrators

NLOJET++ : for jets and photons

DIPHOX family : photons w/ fragmentation ResBos family : resummed results

MCFM : many processes V,VV,VQQ...

A useful list at the HEPDATA web site

http://www.ippp.dur.ac.uk/HEPCODE/

Here some examples:

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When should they ...

...be used?

...not be used?

1. When the most precise knowledge of the cross section is needed

2. The measurement is inclusive enough for hadronization effects not to be important

3. To study the “theoretical” errors of a measurement 1. For evaluation of the tails of the distributions

2. As “blind” k-factor estimators for LO distributions

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Do we need a better “esclusive” tool?

Example: ttH @ NLO

2. Very difficult calculation: 2→3 process 1. Discovery channel for a light Higgs

4. Pt distribution of the Higgs are very similar at LO and NLO

5. In absence of a MC@NLO (very hard), ME+PS seems a very reasonable option

3. NLO result improves our prediction of the cross section dramatically

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