Fulvio PiccininiINFN, Sezione di Pavia

Monte Carlo tools for LHC

Contents

• Basics of Monte Carlo methods

• Monte Carlo programs for particle physics

1. Monte Carlo integrators

2. Parton Shower

3. Matrix Element event generators

4. Matching

Martignano, 12-18 June, 2006

Useful references• Torbjorn Sjostrand, CERN Academic Training lectures, 2005

http://www.thep.lu.se/~ torbjorn

• Mike Seymour, CTEQ Summer School, 2004

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

• Stefano Weinzierl, Introduction to Monte Carlo methods

arXiv:hep-ph/0006269

• Fabio Maltoni, Lectures at Martignano School 2004

• Gennaro Corcella, Lectures at Martignano School 2005

• Wieslaw Placzek, Monte Carlo Methods in High Energy Physics

Lectures delivered in Pavia 2004

• Carlo M. Carloni Calame, Monte Carlo at Work

Lectures delivered at Master in Complexity in Pavia 2006

http://www.pv.infn.it/~ carloni/MCmaster/

some material taken “directly” from these references

Schematic view of a collision at a hadron collider

f(x,Q2) f(x,Q2)PartonDistributions

HardSubProcess

PartonShower

Hadronization

Decay

+Minimum BiasCollisions

from M. Dobbs and J.B. Hansen, Comput. Phys. Commun. 134, (2001) 41

With a Monte Carlo tool we try to simulate what is happening in acollision according to our best Lagrangian at hand up to now

This is important to

• design the experimental apparatus (it takes many years of

construction before running)

• study detector efficiencies in presence of experimental realistic

event selection (e.g. we never have a full 4π coverage)

• help in disentangling a signal from Standard Model backgrounds

• understand signals according to Standard Model or its extensions

Monte Carlo tools are the necessary link between Lagrangian and

experiment

What is a Monte Carlo?

A Monte Carlo method is any technique using stochastic variables

to solve a problem

It is widespreadly used to simulate the evolution of intrinsecally

stocastic processes such as for instance random walks (modelling

behaviours in biology, physics, finance, etc.)

But the MC method can also be used to solve a problem with a

deterministic solution (e.g. the value of a definite integral), provided

the solution can be interpreted as a parameter of a hypothetical

population. A statistical estimate of the parameter can be

constructed using a random sequence of numbers to construct a

sample of the population

Stochastic variable: a variable that can take on more than one value(discrete or continuous range of values) and for which any value thatwill be taken cannot be predicted in advance

Generating uniform random numbers

Sequence of truly random is unpredictable and unreproducible. How

can we generate such a sequence?

• exploiting physical processes e.g. tossing of a coin, results of the

roulette, radioactive decay, etc. → tables of random numbers

These are not suitable for practical calculations

• pseudo-random numbers: strictly predictable (generated by means

of mathematical formulas) but having the appearance of

randomness (their statistical properties resemble the ones of truly

random numbers)

Mathematical algorithms for their generation are called random

number generators

Several algorithms exist for uniform random number generation

The Law of Large Numbers

Aim: calculate the integral∫ b

a f(u) du

If f is

• integrable

• piecewyse continuous

• limited

1

n

n∑

i=1

f(ui) →1

b − a

∫ b

a

f(u) du

ui are n random numbers with uniform probability density over the interval (a, b)

The Monte Carlo estimator of the integral converges to the correctanswer as the random sample becomes very large

Useful quantitiesThe Monte Carlo estimate of the integral

I =

∫

dxf(x) =

∫

dduf(u1, ..., ud)

E =1

N

N∑

n=1

f(xn)

σ2(f) =

∫

dx (f(x) − I)2

∫

dx1...

∫

dxN

(

1

N

N∑

n=1

f(xn) − I

)2

=σ2(f)

N

⇑(Squared) error estimate of the Monte Carlo estimate

According to CLT the probability that the MC estimate lies between

I − aσ(f)/√

N and I + bσ(f)/√

N given by

limN→∞

Prob

(

−aσ(f)√

N≤ 1

N

N∑

n=1

f(xn) − I ≤ bσ(f)√

N

)

=1√2π

b∫

−a

dt exp

(

−t2

2

)

.

⇑Monte Carlo error scales like 1/

√N independently of the dimensions!

differently from deterministic integration algorithms like Newton-Cotes, Gaussian,

etc., which perform better for 1-2 dimensions but become rapidly worse with

higher dimensions

Moreover, Θ functions parametrizing your event selections are not amenable with

deterministic algorithms (they require typically continuity of the first derivative),

while they don’t disturbe the convergence of Monte Carlo integration

Variance estimate from your sample

S2 =1

N − 1

N∑

n=1

(f(xn) − E)2 =1

N

N∑

n=1

(f(xn))2 − E2

How to generate random variables according to a

given probability density P (x) different from the

uniform one:

the cumulative inversion method

definition: C(x) =∫ x

−∞ P (y) dy

Theorem: C(x) is distributed uniformly in [0, 1]

If C(x) is calculable and invertible we have

ξ = C(x) x = C−1(ξ)

Generating uniformly ξ we obtain x distributed according to P (x)

Example: Cauchy (Breit-Wigner) distribution

P (x) =1

π

1

1 + x2

C(x) =1

π

∫ x

−∞

dy

1 + y2=

1

πarctan(x) +

1

2

x = tan

(

π

(

ξ − 1

2

))

1

π

∫ x

−∞

dy

1 + y2=

1

πarctan(x) +

1

2

The Gaussian distribution

If we can not calculate and/or invert analytically C(x)

as e.g. in the case of the Gaussian distribution

G(x; 0, 1) =1√2π

e−x2

2

we can do it

• numerically

• exploiting the Central Limit Theorem

– The sum of a large number of independent random variables is

always distributed normally (i.e. according to the Gaussian

distribution), independently of the distributions of the single

random variables, provided they have finite expectations and

variances

Example

xi, i = 1, . . . , n uniform in [0, 1]; Rn =∑

i xi

< xi >=1

2σ2(xi) =

1

12

⇓

< Rn >=n

2σ2(Rn) =

n

12

⇓ C.L.T.

X =Rn − n/2√

n/12→ G(x; 0, 1) as n → ∞

Hit and miss techniqueAnother technique (always valid) to generate random numbersaccording to a given distribution in an interval starting from theuniform one is to

• enclose the function p(x) in a box. In principle you should know the maximum

value T of p(x)

• generate a first random x uniformly

• generate a second random ξ in [0, T ]

• if ξ < p(x) accept the point (hit)

• else reject the point (miss)

The problem in practice is the value of T . If you don’t know it you can have anestimate by sampling the function as many times as possible and storing themaximum value

Event generation in HEP is based on this simple algorithm. It allows

you to generate events if they were coming out from a real interaction

Variance reduction techniques

The scaling of the error as 1/√

N can be quite slow for practical

purposes. Several technics have been introduced to speed up the

integration convergence

• stratified sampling

• importance sampling

• control variates

• antithetic variates

• adaptive Monte Carlo methods

• multi-channel Monte Carlo

All methods can be applied to optimize the hit or miss algorithm

Stratified sampling

1∫

0

dxf(x) =

a∫

0

dxf(x) +

1∫

a

dxf(x), 0 < a < 1

More generally we split the integration region M = [0, 1]d into k regions Mj where

j = 1, ..., k. In each region we perform a Monte Carlo integration with Nj points.

E =

k∑

j=1

vol(Mj)

Nj

Nj∑

n=1

f(xjn)

σ2(f)

N→

k∑

j=1

vol(Mj)2

Nj

σ2(f)∣

∣

Mj

with

σ2(f)∣

∣

Mj=

1

vol(Mj)

∫

Mj

dx

f(x) − 1

vol(Mj)

∫

Mj

dx f(x)

2

=

1

vol(Mj)

∫

Mj

dx f(x)2

−

1

vol(Mj)

∫

Mj

dx f(x)

2

Importance sampling∫

dx f(x) =

∫

f(x)

p(x)p(x)dx =

∫

f(x)

p(x)dP (x)

p(x) =∂d

∂x1...∂xdP (x)

p(x) ≥ 0 and∫

dx p(x) = 1 ⇒ p(x) may be interpreted as a probability density function

Generating N points x1, ..., xN according to the P (x) an estimate of the integral is given by

E =1

N

N∑

n=1

f(xn)

p(xn)

The statistical error of the Monte Carlo integration is given by σ(f/p)/√

N and an estimator for

the variance σ2(f/p) is

S2

(

f

p

)

=1

N

N∑

n=1

(

f(xn)

p(xn)

)2

− E2

The relevant quantity is now f(x)/p(x)

Control variates∫

dx f(x) =

∫

dx (f(x) − g(x)) +

∫

dx g(x)

If g(x) is similar to f(x) over the integration region and the integral

of g(x) is known the numerical integration of f(x) − g(x) is easier

Example:∫ 1

0

f(x)

x1−αdx =

∫ 1

0

f(x) − f(0)

x1−α+

∫ 1

0

f(0)

x1−α

with f(x) regular near x = 0 and α > 0

The approximating function g(x) needs not be inverted analytically

Antithetic variates

Usually Monte Carlo calculations use random points, which are uncorrelated. The

method of antithetic variates deliberately makes use of correlated points, taking

advantage of the fact that such a correlation may be negative.

var(f1 + f2) = var(f1) + var(f2) + 2 covar(f1, f2)

covar(f1, f2) = E[f1 − E(f1)]E[f2 − E(f2)]

If we can arrange to choose points such that f1 and f2 are negative correlated, a

substantial reduction in variance may be realized

Very simple example:∫ 1

0

x dx

by choosing xi and 1 − xi as integration points the integrand becomes flat∫ 1

0

x dx =1

2

∫ 1

0

x dx +1

2

∫ x=1

x=0

y dy with y = 1 − x

=1

2

∫ 1

0

x dx +1

2

∫ 1

0

(1 − y) dy =1

2

∫ 1

0

1 dx

Adaptive MC methods

If we don’t know anything about the integrand it is better to use an

algorithm which iteratively learns the peaking structure of the

function as it proceeds (one vey popular is VEGAS) combining in an

automated way stratified and importance sampling

After a number of iterations the phase space is subdivided into a grid

and the integration is carried out in each cell. The number of points

for each cell is proportional to the contribution to the total integral

The most efficient way of working in d− dimensions is when

p(u1, ..., ud) = p1(u1) · p2(u2) · ... · pd(ud)

i.e. factorized peaking behaviour

Routines like FOAM and VAMP try to overcome this limitation

Multi-channel Monte Carlo

If the integrand has peaks in different regions of phase space it can be difficult tomap analytically simultaneously all the peaks

Idea: make a combination of mappings

If the single mapping Pi(x) are known, with∫

Pi(x) dx = 1, puttingp(x) =

∑

αiPi(x), with αi ≥ 0,∑

αi = 1, αi is the probability ofselecting the i−th channel

I =

∫

dx f(x) =m∑

i=1

αi

∫

f(x)

p(x)dPi(x)

Monte Carlo estimate for the integral

E =1

N

m∑

i=1

Ni∑

ni=1

f(xni)

p(xni)

Expected error of integration√

W (α) − I2

N

W (α) =m∑

i=1

αi

∫ (

f(x)

p(x)

)2

dPi(x)

By adjusting the arbitrary parameters αi one may try to minimize W (α)R. Kleiss and R. Pittau, Comp. Phys. Comm. 83 (1994) 141

T. Ohl, Comp. Phys. Comm. 120 (1999) 13

Monte Carlo for particle physicsGenerally the partonic kernel of a cross section is written as

dσ

dX=∑

a,b

∫

dx1 dx2

∫

dΦn fa/h1(x1, µ

2F ) fa/h2

(x2, µ2F )

dσab(x1p1, x2p2; αs(µ2R), µ2

R, µ2F )

dΦnΘ(cuts)

see previous lecture of B. Anastasiou

• phase space point generation for 2 → n

• matrix element calculation summed over spin/polarisation and

colour (averaging on initial state degrees of freedom)

dΦn(P, p1, .., pn) =

n∏

i=1

d4pi

(2π)3Θ(p0

i )δ(p2i − m2

i )(2π)4δ4

(

P −n∑

i=1

pi

)

=n∏

i=1

d3pi

(2π)32Ei(2π)4δ4

(

P −n∑

i=1

pi

)

dΦn(P, p1, ..., pn) =1

2πdQ2dΦj(Q, p1, ..., pj)dΦn−j+1(P, Q, pj+1, ..., pn)

Q =

j∑

i+1

pi

Recursive generation of 2 → n phase space through two-body decays

dΦn =1

(2π)n−2dM 2

n−1...dM 22dΦ2(n)...dΦ2(2)

M 2i = q2

i , qi =i∑

j=1

pi, dΦ2(i) = dΦ2(qi, qi−1, pi)

(m1 + ... + mi)2 ≤ M 2

i ≤ (Mi+1 − mi+1)2.

In the rest frame of qi dΦ2(qi, qi−1, pi) is given by

dΦ2(qi, qi−1, pi) =1

(2π)2

√

λ(q2i , q

2i−1, m

2i )

8q2i

dϕid(cos θi)

λ(x, y, z) = x2 + y2 + z2 − 2xy − 2yz − 2zx

Example: algorithm for p1 + p2 → q1 + q2 + q3 + q4

1. generate M12 and M34

2. generate ϑ12 and φ12 and calculate the momenta q12 and q34 in the p1 + p2 rest

frame

3. in the q12 rest frame generate q1 and q2 through 2 additional random numbers

giving e.g. ϑ1 and φ1

4. in the q34 rest frame generate q3 and q4 through 2 additional random numbers

giving e.g. ϑ3 and φ3

5. boost the momenta q1 and q2 from the q12 c.m. frame to the p1 + p2 frame

6. boost the momenta q3 and q4 from the q34 c.m. frame to the p1 + p2 frame

In total 8 random numbers have been used equal to the number of independent

variables for 2 → 4 including an overall arbitrary φ (cilindrical symmetry around

the beam axis)

The weight of the generated event depends on the event and is given by

w = (2π)4−3n21−2n 1

Mn

n∏

i=2

√

λ(M 2i , M2

i−1, m2i )

Mi

Three main classes of MC programs

• MC integrators

• Parton Shower MC event generators

• Multi-parton MC event generators

Each of these classes has pros and cons

Can we combine good features from different classes?

Monte Carlo integrators

• Only partonic final states, with arbitrary event selection

• Final state hadronic particles identified with partons (parton-hadron duality)

• Events with flat distribution on phase space and weighted by the matrix

element. Events used to fill histograms for distributions

• Typically they are used to obtain accurate predictions in fixed order

perturbation theory beyond Leading Order

see lecture by B. Anastasiou

• At present some NLO programs are available for a limited set of “simple” (but

important) final states (difficulty in calculating virtual corrections)

• NLO calculations work well in describing hard radiation but fail in the region

of soft/collinear singularities

• The accuracy can be increased in certain regions of phase space implementing

resummed calculations (valid generally for one observable at a time)

• The NLO corrections give an handle to test the theoretical uncertainty of the

calculation by studying the stability with respect to variations of the

renormalisation and factorisation scales

σ(pp_ → tt

_ H + X) [fb]

√s = 2 TeV

MH = 120 GeV

µ0 = mt + MH/2

NLO

LO

µ/µ0

0.2 0.5 1 2 50

2

4

6

8

10

12

14

16

1

σ(pp → tt_ H + X) [fb]

√s = 14 TeV

MH = 120 GeV

µ0 = mt + MH/2

NLO

LO

µ/µ0

0.2 0.5 1 2 50

200

400

600

800

1000

1200

1400

1

W. Beenakker et al., Phys. Rev. Lett 87 (2001) 201805

• NLO programs can test the K-factors at the distribution level. Generally they

are defined in an inclusive way as σNLO/σLO but different bins can receive

different corrections

• NLO corrections consist of Real ⊕ Virtual contributions, which display strong

cancellations. The Virtual part can become negative in the phase space ⇒difficulty in producing unweighted events

Available processes in NLO QCD MC integrators

• N jets N ≤ 3

• V V ′ V, V ′ = W, Z

• V j

• γ + 1 jet

• γγ

• V + N jets N ≤ 2

• V + bb

• QCD production of H + 2 jets, WW/ZZ + 2 fwd jets

• heavy flavour production

Other NLO calculations without a publicly released code

• Single top production (qb → bq′ & qq′ → tb)

• QQH

Parton Shower MC event generators

• General-purpose tools

• They describe the complete history of the hadron-hadron interaction, from

ISR, hard scattering, showering, hadronization, to final state hadrons and

leptons, including the underlying event (beam remnants, collisions between

other partons in the hadrons and collisions between other hadrons in the

colliding beams)

• Essentially only the hard subprocess is process dependent

• They provide an exclusive description of the events: complete information

related to every particle is recorded

• Unweighted events are produced ⇒ events are distributed in phase space as in

the real experiment (provided the underlying theory is correct)

• PSMC’s are invaluable tools for detector simulations

• For these reasons they are so widespreadly used by experimentalists

• Key theoretical ingredient: parton shower technique to generate higher order

corrections starting from a simple (2 → 1 or 2 → 2) hard scattering

f(x,Q2) f(x,Q2)PartonDistributions

HardSubProcess

PartonShower

Hadronization

Decay

+Minimum BiasCollisions

from M. Dobbs and J.B. Hansen, Comput. Phys. Commun. 134, (2001) 41

Basic principles of parton shower

Main approximation (very powerful!): factorisation of soft and

collinear singularities

Ex.: the propagator of a massless quark emitting a gluon

1

(pq + pg)2=

1

2EqEg(1 − cos ϑ)

γ∗ → qqg

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

from Corcella lectures

Available PSMC Event Generators

• HERWIG, PYTHIA, ISAJET

• SHERPA, very recent (T. Gleisberg et al.)

• HERWIG++, PYTHIA7, under development

They implement many hard processes (within and beyond SM), a realization ofparton shower and a model of hadronization

While PSMC event generators describe well radiation in the soft/collinear regions

(resumming large logs), they fail to describe hard wide angle radiation and cross

sections are correct at LO

First improvement: Matrix element corrections

HERWIG and PYTHIA have been corrected by means of the exact O(αs) real matrix

element by filling the dead-zones of phase space (due to angular ordering) and by

reweighting the PS weight of the hardest emission using the matrix element

correction

Corrected processes: top quark decay, DY, gg → H

W qT distribution compared with D0 data and calculated for LHC

G. Corcella, M.H. Seymour, Nucl. Phys. B565 (2000) 227

Normalization still at LO: virtual corrections are missing

Combining NLO calculations with PS’s

This would allow to have

• Normalizations accurate at NLO

• Hard tails of distributions as in NLO calculations

• Soft/Collinear emissions treated as with PS

• Smooth matching between soft/collinear and hard regions without

double counting

• Generate unweighted exclusive events

• Negative weight events could be generated

• several methods under study

• at present only MC@NLO (Frixione, Webber and Nason) is working

Based on NLO subtraction method. It is interfaced to HERWIG but

the method is general.

Main idea: remove the Leading Log O(αs) content of the parton

shower and replace it with the NLO calculation

A fraction of the generated events have negative weight, due to

the negative contribution of the virtual NLO correction.

S. Frixione and B.R. Webber, hep-ph/0212216

S. Frixione, P. Nason, B.R. Webber, hep-ph/0305252

Multiparton MC Event Generators

The LHC (and also Tevatron) center of mass energy is large enough to open many

high multiplicity hard final states in hadronic collisions. These multiparticle final

states can originate from

• hard QCD radiative processes

• decay of standard massive particles (W , Z gauge bosons, top quarks, Higgs

bosons)

• decay of heavy supersymmetric particles

• decay of more exotic heavy particles

In the case of accurate measurements of standard particles as well as in the search

for new physics, the knowledge of multijet QCD backgrounds is an essential part of

any experimental analysis

Examples:

• Top and Higgs studies

• gluino pair production gives rise to a final state with 8 jets plus missing energy

g → qq → qq′q′′q′′′χ0. Background: 8 jets + Z0 → νν

For such complex final states Parton Shower prediction can become inadequate

because they start from a 2 → 1 or 2 → 2 kernel process and add via showering

additional gluons, missing some important subprocesses

Example: the Wbb final state in ALPGEN

M.L. Mangano, M. Moretti, F. Piccinini, R. Pittau, A.D. Polosa, JHEP0307 (2003) 001

jp subprocess jp subprocess jp subprocess

1 qq′ → WQQ 2 qg → q′WQQ 3 gq → q′WQQ

4 gg → qq′WQQ 5 qq′ → WQQq′′q′′ 6 qq′′ → WQQq′q′′

7 q′′q → WQQq′q′′ 8 qq → WQQq′q′′ 9 qq′ → WQQqq

10 q′q → WQQqq 11 qq → WQQqq′ 12 qq → WQQq′q

13 qq → WQQqq′ 14 qq′ → WQQqq 15 qq′ → WQQq′q′

16 qg → WQQq′q′′q′′ 17 gq → WQQq′q′′q′′ 18 qg → WQQqqq′

19 qg → WQQq′qq 20 gq → WQQqqq′ 21 gq → WQQq′qq

22 gg → WQQqq′q′′q′′ 23 gg → WQQqqqq′

pit > 20 GeV, |ηi| < 2.5 ∆Rij > 0.4

Process NJ = 2 NJ = 3 NJ = 4 NJ = 5 NJ = 6

1 360(1) 68.6(4) 10.4(1) 1.46(1) 0.20(1)

2+3 – 37.6(2) 12.1(1) 2.63(3) 0.47(1)

4+. . . +15 – – 4.3(1) 1.66(3) 0.41(1)

16+. . . +21 – – – 0.085(2) 0.036(1)

22+23 – – – – 0.00038(2)

Total 360(1) 106.4(4) 26.8(2) 5.84(4) 1.11(2)

Contributions from different initial states for Tevatron; rates in fb

Process NJ = 2 NJ = 3 NJ = 4 NJ = 5 NJ = 6

1 2.60(1) 0.63(1) 0.144(3) 0.036(2) 0.008(1)

2+3 – 2.97(1) 2.11 (2) 1.08(2) 0.47(2)

4+. . . +15 – – 0.288(1) 0.24(1) 0.13(2)

16+. . . +21 – – – 0.030(1) 0.031(4)

22+23 – – – – 0.0010(3)

Total 2.60(1) 3.60(1) 2.54(2) 1.38(2) 0.64(3)

The same as before but for the LHC. Rates in pb

and for heavy flavours

QQQ′Q′

+ N jets N = 0 N = 1 N = 2 N = 3 N = 4

tttt, LHC (fb) 12.73(8) 17.4(2) 13.5(1) 7.55(6) 3.48(5)

ttbb, LHC (pb) 1.35(1) 1.47(2) 0.94(2) 0.457(8) 0.189(4)

ttbb, FNAL (fb) 3.44(3) 0.95(1) 0.154(1) 0.0187(2) 0.00187(5)

bbbb, LHC (pb) 477(2) 259(5) 95(1) 28.6(6) 25.0(3)

bbbb, FNAL (pb) 6.64(5) 2.25(3) 0.470(5) 0.076(1) 0.0025(5)

Multiparton MC Event Generators

The previous strategy of matching PSMC’s with NLO calculations is not feasible

now for arbitrary multiparton processes. We don’t have NLO calculations for

arbitrary external legs. But we do have techniques for computing exact LO matrix

elements for multiparton hard scattering

Recenlty several matrix element event generators have been built up, thanks to

helicity amplitudes algorithms or completely numerical algorithms (and of course

computing power)

• ACERMC, ALPGEN, CompHEP, GRACE, HELAC/PHEGAS/JETI, MADEVENT, SHERPA,

VECBOS, NJETS, . . .

• Matrix elements involving a very large number of Feynman diagrams

• Complex peaking structure in the phase space

• They can generate weighted (for cross sections and distributions) and

unweighted events

• The strategy to describe real final states with hadrons is to pass the

unweighted event samples (in LHA format) to the PSMC for further showering

and hadronization ⇒ problems . . .

Up to now available processes (in ALPGEN v2.0)

• (W → ff ′) + N jets, N ≤ 6, f = l, q

• (Z/γ∗ → ff) + N jets, N ≤ 6, f = l, ν

• (W → ff ′)QQ + N jets, (Q = b, t), N ≤ 4, f = l, q

• (Z/γ∗ → ff)QQ + N jets, (Q = c, b, t), N ≤ 4, f = l, ν

• (W → ff ′) + c + N jets, N ≤ 5, f = l, q

• n W + m Z + l H + N jets, n + m + l ≤ 8, N ≤ 3

• QQ + N jets, (Q = c, b, t), N ≤ 6

• QQQ′Q′ + N jets, (Q, Q′ = c, b, t) , N ≤ 4

• QQH + N jets, (Q = b, t), N ≤ 4

• N jets, N ≤ 6

• N γ + N jets, N ≥ 1, N + M ≤ 8, M ≤ 6

• gg → H + N jets (mt → ∞)

• single top

Tuned comparisons (very important!) during the 2003 CERN MC4LHC Workshop

X-sects (pb) Number of jets

e−νe + n QCD jets 0 1 2 3 4 5 6

ALPGEN 3904(6) 1013(2) 364(2) 136(1) 53.6(6) 21.6(2) 8.7(1)

SHERPA 3905(4) 1014(3) 370(2)

CompHEP 3947.4(3) 1022.4(5) 364.4(4)

GR@PPA 3906.37 (4) 1046.85 (5)

JetI 3786(81) 1021(8) 361(4) 157(1) 46(1)

MadEvent 3902(5) 1012(2) 361(1) 135.5(3) 53.6(2)

X-sects (pb) Number of jets

tt + n QCD jets 0 1 2 3 4 5 6

ALPGEN 755.4(8) 748(2) 518(2) 310.9(8) 170.9(5) 87.6(3) 45.0(5)

SHERPA 754.2(7) 747(2)

CompHEP 757.8(8) 752(1) 519(1)

JetI 745(5) 711(7) 515(5) 24.2(5)

MadEvent 754(2) 749(2) 516(1) 306(1)

From partons to jets

To obtain realistic results the generated partonic events need to be

given as initial condition to the PSMC. However, two problems arise

• Double counting: configurations with n final state partons can be

obtained starting from (n − m) partonic configurations, with m

partons provided by the PSMC. The same n-jet configuration can

be generated starting with different (n − m) configurations

• Results depend on the unphysical partonic set of cuts, while they

should not

Example: W + 3 jets at Tevatron

Two different sources for the increasing ratio when decreasing ∆Rpart:

• collinear divergence of the matrix element

• increasing double counting for smaller ∆Rpart

Towards matching of ME & PS

For e+e− physics a solution has been proposed

S. Catani et al., JHEP 0111 (2001) 063

L. Lonnblad, JHEP 0205 (2002) 046

which avoids double counting and shifts the dependence on the

resolution parameter beyon NLL accuracy

The method consists in separating arbitrarily the phase-space regions

covered by ME and PS, and use vetoed parton showers together with

reweighted tree-level matrix elements for all parton multiplicities

Proposal to extend the procedure to hadronic collisions even if

formal proof doesn’t exist up to now

F. Krauss, JHEP 0208 (2002) 015

Necessary steps for CKKW procedure

• select the jet multiplicity n according to the jet rates obtained with matrix

elements with resolution yij > ycut, defined according to the kT -algorithm

• generate n parton momenta according to the matrix element with fixed αs(ycut)

and reweight the event with the probability of no further branching by means

of Sudakov form factors

• build a “PS history” by clustering the partons to determine the values at which

1,2,...n jets are resolved. In so doing a tree of branchings is constructed and the

nodal scales characteristic of each branching are used to reweight the event

with running αs

• apply a coupling constant reweighting factor αs(y1) αs(y2) ... αs(yn) /

αs(ycut)n ≤ 1, where yi are the nodal scales

• after successful unweighting, use the n-parton kinematics as initial condition

for the shower, vetoing all branchings such that yij > ycut

The CKKW procedure has been successfully tested on LEP data

e.g. S. Catani et al., JHEP 0111 (2001) 063

R. Kuhn et al., hep-ph/0012025

F. Krauss, R. Kuhn and G. Soff, J. Phys. G26 (2000) L11

very recent work for hadronic collisions

• HERWIG (P. Richardson), PYTHIA (S. Mrenna)

• SHERPA with APACIC++/AMEGIC++ (F. Krauss et al.)

• ALPGEN v2.0, implementation according to the proposal by

M.L. Mangano (see later)

• ARIADNE (Lavesson and Lonnblad))

Several parameters need to be tuned to the data in order to havesmooth interpolation between the regions below and above theresolution. Missing virtual corrections ⇒ still a residual cutoffdependence

Some results for W+ jets at Tevatron and LHC

/ GeV Wp0 20 40 60 80 100 120 140 160 180 200

[ p

b/G

eV

] W

/d

pσ

d

-210

-110

1

10

210

SHERPA

Wp

W + 0jetW + 1jetW + 2jetW + 3jetW + 4jetD0 Data

/ GeV Zp0 50 100 150 200 250 300 350 400 450 500

/ G

eV

Z/d

pσ

dσ

1/

-610

-510

-410

-310

-210

SHERPA

=15 GeVcutQZ + XZ + 0jetZ + 1jetZ + 2jetZ + 3jetreference

/ GeV Zp0 50 100 150 200 250 300 350 400 450 500

/ G

eV

Z/d

pσ

dσ

1/

-610

-510

-410

-310

-210

SHERPA

=100 GeVcutQZ + XZ + 0jetZ + 1jetZ + 2jetZ + 3jetreference

F. Krauss et al., hep-ph/0409106; hep-ph/0503280

S. Mrenna and P. Richardson, hep-ph/0312274

Systematic of O(30%) for cross sections

Why not use a simpler recipe (always at LL order)?

M.L. Mangano

• Generate partonic events for different jet multiplicities (pT > pminT , ∆Rjj > Rmin)

• Shower the events with default PSMC

• Before hadronization, process the showered events with a cone jet algorithm

• Require partons-jets matching

– require for each hard parton a jet within ∆Rmatch ' Rjet

– reject the event if two partons match to the same jet or if one parton has nomatch

– keep the event if all partons are matched

• The above procedure defines the inclusive sample

• For exclusive samples rejects events where there is an extra jet not matchedto any ME parton. Cross section = σ partonic · matching efficiency

• Inclusive sample containing events with all multiplicities obtainedcombining exclusive samples

• Physics analysis with inclusive samples should be as much as possibleindependent of generation cuts

Figure 1: pWT spectrum. The points represent run I CDF data. The curves correspond to the subsequent

inclusion of samples with higher multiplicity, form the W +0 jet, up to the W +4 jets case. The right plot

is the same as the left one, with an enhanced low-pT scale.

Figure 2: Effect of different generation cuts on the integrated pWT spectrum. Uncertainty of the order

of ± 15%. The right panel shows the ratios of the samples generated with PT20, PT30 and PT10R07,

divided by PT10. The right panel shows all four samples divided by a plain HERWIG W sample.

Last but not least... for precision measurements also EW corrections

are important

α2s ∼ αem

U. Baur, hep-ph/0511064

Sudakov EW logarithms ∼ α/π log2(s/M 2W ) important at high MT

The effect of O(α) EW corrections on W mass determination is of

the order of 50 MeV (W → eν) and 150 MeV (W → µν)

With Horace the effect of exponentiation on W -mass determination

has been studied

W → µ νW → e ν

∆MW (MeV)

∆χ2

exponentiation

0

10

20

30

40

50

60

-12 -10 -8 -6 -4 -2 0 2

C.M. Carloni Calame et al., Eur. Phys. J. C33 (2004) 665

The interplay between QCD and electroweak corrections has been

studied in the approximation of soft initial state multi-gluon

emission and final state QED corrections with Resbos modified to

include f.s. QED corrections Resbos-A

Q.-H. Cao and C.-P. Yuan, Phys. Rev. Lett. 93 (2004) 042001

While the two corrections factorize for M lνT , their relation is more

involved for the lepton p⊥ distribution

25 30 35 40 45 50pT

e+

(GeV)

0

10

20

30

40

50

60 RES + NLO QEDRES + LO QEDLO + NLO QEDLO + LO QED

25 30 35 40 45 50pT

e+

(GeV)

0.6

0.8

1

2

4

6

8

10dσdpT

e δ

[pb/GeV] RES + NLO QEDLO + LO QED

LO + NLO QEDLO + LO QED

no detector effects included

Hard QCD radiation important at LHC

σ(`ν + N jets) with Alpgen

N = 0 N = 1 N = 2 N = 3 N = 4 N = 5 N = 6

LHC (pb) 18068(4) 3412(4) 1130(2) 342.9(1.4) 100.6(1.4) 27.6(4) 7.14(15)

FNAL (pb) 2087.0(6) 225.8(2) 37.3(2) 5.66(6) 0.745(4) 0.0864(15) 0.0086(2)

M.L. Mangano, M. Moretti, F. Piccinini., R. Pittau, A.D. Polosa, JHEP07(2003)001

Under study the interface of EW corrections provided by Horace

with up-to-date QCD matrix-element based Monte Carlos, such as

for instance Alpgen, in order to have a firmer estimate of the

interplay of QCD/EW corrections at LHC for all interesting

observables

Summary

• Impressive progress in recent years in developing new MC tools

• Standards have been fixed to allow for use of different MC

outputs without problems of compatiblity (Les Houches Accords)

• It is worth emphasizing the development of techniques aimed at

exploiting good features from different Monte Carlos in different

phase space regions (e.g. NLO with Parton Shower, CKKW, . . .)

• Waiting for LHC, we can test/tune these MC tools on data from

Tevatron run II and HERA