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