Monte Carlo tools for LHC

Transcription

1 Fulvio Piccinini INFN, 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, June, 2006

2 Useful references Torbjorn Sjostrand, CERN Academic Training lectures, 2005 Mike Seymour, CTEQ Summer School, torbjorn Stefano Weinzierl, Introduction to Monte Carlo methods arxiv:hep-ph/ Fabio Maltoni, Lectures at Martignano School 2004 Gennaro Corcella, Lectures at Martignano School 2005 Wieslaw Placzek, Monte Carlo Methods in High Energy Physics Carlo M. Carloni Calame, Monte Carlo at Work Lectures delivered in Pavia 2004 Lectures delivered at Master in Complexity in Pavia carloni/mcmaster/ some material taken directly from these references

3 Schematic view of a collision at a hadron collider Decay Hadronization Parton Shower + Minimum Bias Collisions Hard SubProcess Parton Distributions f(x,q 2 ) f(x,q 2 ) from M. Dobbs and J.B. Hansen, Comput. Phys. Commun. 134, (2001) 41

4 With a Monte Carlo tool we try to simulate what is happening in a collision 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

5 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 that will be taken cannot be predicted in advance

6 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

7 If f is The Law of Large Numbers Aim: calculate the integral b a f(u) du integrable piecewyse continuous limited 1 n n i=1 f(u i ) 1 b a b a f(u) du u i are n random numbers with uniform probability density over the interval (a, b) The Monte Carlo estimator of the integral converges to the correct answer as the random sample becomes very large

8 Useful quantities The Monte Carlo estimate of the integral I = dxf(x) = d d uf(u 1,..., u d ) E = 1 N f(x n ) N n=1 σ 2 (f) = dx (f(x) I) 2 ( 1 dx 1... dx N N ) 2 N f(x n ) I = σ2 (f) N n=1 (Squared) error estimate of the Monte Carlo estimate

9 According to CLT the probability that the MC estimate lies between I aσ(f)/ N and I + bσ(f)/ N given by lim N Prob ( a σ(f) N 1 N N n=1 ) f(x n ) I b σ(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 S 2 = 1 N 1 N (f(x n ) E) 2 = 1 N n=1 N (f(x n )) 2 E 2 n=1

10 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)

11 Example: Cauchy (Breit-Wigner) distribution P (x) = 1 π x 2 C(x) = 1 π x dy 1 + y 2 = 1 π arctan(x) x = tan ( ( π ξ 1 )) 1 2 π x dy 1 + y 2 = 1 π arctan(x) + 1 2

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13 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 e x2 2 2π 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

14 Example x i, i = 1,..., n uniform in [0, 1]; < x i >= 1 2 R n = i x i σ 2 (x i ) = 1 12 < R n >= n 2 σ 2 (R n ) = n 12 C.L.T. X = R n n/2 n/12 G(x; 0, 1) as n

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16 Hit and miss technique Another technique (always valid) to generate random numbers according to a given distribution in an interval starting from the uniform 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 an estimate by sampling the function as many times as possible and storing the maximum 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

17 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

18 Stratified sampling 1 dxf(x) = a dxf(x) + 1 dxf(x), 0 < a < More generally we split the integration region M = [0, 1] d into k regions M j where j = 1,..., k. In each region we perform a Monte Carlo integration with N j points. a with σ 2 (f) Mj = = E = k j=1 σ 2 (f) N k 1 vol(m j ) M j vol(m j ) N j j=1 dx N j n=1 vol(m j ) 2 N j f(x) f(x jn ) σ 2 (f) Mj 1 vol(m j ) M j 1 dx f(x) 2 1 vol(m j ) vol(m j ) M j dx f(x) M j 2 dx f(x) 2

19 Importance sampling dx f(x) = f(x) p(x) p(x)dx = p(x) = d x 1... x d P (x) f(x) dp (x) p(x) p(x) 0 and dx p(x) = 1 p(x) may be interpreted as a probability density function Generating N points x 1,..., x N according to the P (x) an estimate of the integral is given by E = 1 N N n=1 f(x n ) p(x n ) The statistical error of the Monte Carlo integration is given by σ(f/p)/ N and an estimator for the variance σ 2 (f/p) is ( ) f S 2 = 1 N ( ) 2 f(xn ) E 2 p N p(x n ) n=1 The relevant quantity is now f(x)/p(x)

20 dx f(x) = Control variates 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) x 1 αdx = 1 0 f(x) f(0) x 1 α + with f(x) regular near x = 0 and α > f(0) x 1 α The approximating function g(x) needs not be inverted analytically

21 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(f 1 + f 2 ) = var(f 1 ) + var(f 2 ) + 2 covar(f 1, f 2 ) covar(f 1, f 2 ) = E[f 1 E(f 1 )]E[f 2 E(f 2 )] If we can arrange to choose points such that f 1 and f 2 are negative correlated, a substantial reduction in variance may be realized Very simple example: 1 0 x dx by choosing x i and 1 x i as integration points the integrand becomes flat 1 0 x dx = 1 2 = x dx x dx x=1 x=0 1 0 y dy with y = 1 x (1 y) dy = dx

22 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(u 1,..., u d ) = p 1 (u 1 ) p 2 (u 2 )... p d (u d ) i.e. factorized peaking behaviour Routines like FOAM and VAMP try to overcome this limitation

23 Multi-channel Monte Carlo If the integrand has peaks in different regions of phase space it can be difficult to map analytically simultaneously all the peaks Idea: make a combination of mappings

24 If the single mapping P i (x) are known, with P i (x) dx = 1, putting p(x) = α i P i (x), with α i 0, α i = 1, α i is the probability of selecting the i th channel I = dx f(x) = Monte Carlo estimate for the integral m i=1 α i f(x) p(x) dp i(x) Expected error of integration E = 1 N m i=1 N i n i =1 W (α) I 2 f(x ni ) p(x ni ) W (α) = m i=1 N α i ( f(x) p(x) ) 2 dp i (x) By adjusting the arbitrary parameters α i 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

25 Monte Carlo for particle physics Generally the partonic kernel of a cross section is written as dσ dx = a,b dx 1 dx 2 dφ n f a/h1 (x 1, µ 2 F ) f a/h 2 (x 2, µ 2 F ) dˆσ ab(x 1 p 1, x 2 p 2 ; α s (µ 2 R ), µ2 R, µ2 F ) Θ(cuts) dφ n phase space point generation for 2 n see previous lecture of B. Anastasiou matrix element calculation summed over spin/polarisation and colour (averaging on initial state degrees of freedom) dφ n (P, p 1,.., p n ) = = n i=1 n i=1 d 4 p i (2π) 3 Θ(p0 i )δ(p 2 i m 2 i )(2π) 4 δ 4 d 3 p i (2π) 3 2E i (2π) 4 δ 4 ( P ( ) n p i i=1 P ) n p i dφ n (P, p 1,..., p n ) = 1 2π dq2 dφ j (Q, p 1,..., p j )dφ n j+1 (P, Q, p j+1,..., p n ) j Q = i+1 p i i=1

26 Recursive generation of 2 n phase space through two-body decays dφ n = 1 (2π) n 2dM 2 n 1...dM 2 2 dφ 2 (n)...dφ 2 (2) M 2 i = q 2 i, q i = i p i, dφ 2 (i) = dφ 2 (q i, q i 1, p i ) j=1 (m m i ) 2 M 2 i (M i+1 m i+1 ) 2. In the rest frame of q i dφ 2 (q i, q i 1, p i ) is given by 1 λ(qi 2, q2 i 1, m2 i ) dφ 2 (q i, q i 1, p i ) = dϕ (2π) 2 i d(cos θ i ) 8q 2 i λ(x, y, z) = x 2 + y 2 + z 2 2xy 2yz 2zx

27 Example: algorithm for p 1 + p 2 q 1 + q 2 + q 3 + q 4 1. generate M 12 and M generate ϑ 12 and φ 12 and calculate the momenta q 12 and q 34 in the p 1 + p 2 rest frame 3. in the q 12 rest frame generate q 1 and q 2 through 2 additional random numbers giving e.g. ϑ 1 and φ 1 4. in the q 34 rest frame generate q 3 and q 4 through 2 additional random numbers giving e.g. ϑ 3 and φ 3 5. boost the momenta q 1 and q 2 from the q 12 c.m. frame to the p 1 + p 2 frame 6. boost the momenta q 3 and q 4 from the q 34 c.m. frame to the p 1 + p 2 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 3n 2 1 2n 1 n λ(m 2 i, Mi 1 2, m2 i ) M n i=2 M i

28 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?

29 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)

30 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 M H = 120 GeV µ 0 = m t + M H /2 LO NLO σ(pp tt _ H + X) [fb] s = 14 TeV M H = 120 GeV µ 0 = m t + M H /2 LO NLO µ/µ µ/µ 0 W. Beenakker et al., Phys. Rev. Lett 87 (2001) NLO programs can test the K-factors at the distribution level. Generally they are defined in an inclusive way as σ NLO /σ LO 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

31 N jets N 3 Available processes in NLO QCD MC integrators V V V, V = W, Z V j γ + 1 jet γγ V + N jets N 2 V + b b QCD production of H + 2 jets, W W/ZZ + 2 fwd jets heavy flavour production Other NLO calculations without a publicly released code Single top production (qb bq & q q t b) Q QH

32 General-purpose tools Parton Shower MC event generators 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

33 Decay Hadronization Parton Shower + Minimum Bias Collisions Hard SubProcess Parton Distributions f(x,q 2 ) f(x,q 2 ) from M. Dobbs and J.B. Hansen, Comput. Phys. Commun. 134, (2001) 41

34 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 (p q + p g ) 2 = 1 2E q E g (1 cos ϑ)

35 from Corcella lectures γ q qg

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46 HERWIG, PYTHIA, ISAJET SHERPA, very recent (T. Gleisberg et al.) Available PSMC Event Generators HERWIG++, PYTHIA7, under development They implement many hard processes (within and beyond SM), a realization of parton shower and a model of hadronization

47 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(α s ) 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 q T distribution compared with D0 data and calculated for LHC Normalization still at LO: virtual corrections are missing G. Corcella, M.H. Seymour, Nucl. Phys. B565 (2000) 227

48 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

49 several methods under study at present only (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(α s ) 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.

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

51 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 q q qq q q χ 0. Background: 8 jets + Z 0 ν ν

52 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 W b b 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 q q W QQ 2 qg q W QQ 3 gq q W QQ 4 gg q q W QQ 5 q q W QQq q 6 qq W QQq q 7 q q W QQq q 8 q q W QQq q 9 q q W QQq q 10 q q W QQq q 11 q q W QQq q 12 q q W QQq q 13 qq W QQqq 14 qq W QQqq 15 qq W QQq q 16 qg W QQq q q 17 gq W QQq q q 18 qg W QQqq q 19 qg W QQq q q 20 gq W QQqq q 21 gq W QQq q q 22 gg W QQq q q q 23 gg W QQq qq q

53 p i t > 20 GeV, η i < 2.5 R ij > 0.4 Process N J = 2 N J = 3 N J = 4 N J = 5 N J = (1) 68.6(4) 10.4(1) 1.46(1) 0.20(1) (2) 12.1(1) 2.63(3) 0.47(1) (1) 1.66(3) 0.41(1) (2) 0.036(1) (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 N J = 2 N J = 3 N J = 4 N J = 5 N J = (1) 0.63(1) 0.144(3) 0.036(2) 0.008(1) (1) 2.11 (2) 1.08(2) 0.47(2) (1) 0.24(1) 0.13(2) (1) 0.031(4) (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

54 and for heavy flavours QQQ Q + N jets N = 0 N = 1 N = 2 N = 3 N = 4 t tt t, LHC (fb) 12.73(8) 17.4(2) 13.5(1) 7.55(6) 3.48(5) t tb b, LHC (pb) 1.35(1) 1.47(2) 0.94(2) 0.457(8) 0.189(4) t tb b, FNAL (fb) 3.44(3) 0.95(1) 0.154(1) (2) (5) b bb b, LHC (pb) 477(2) 259(5) 95(1) 28.6(6) 25.0(3) b bb b, FNAL (pb) 6.64(5) 2.25(3) 0.470(5) 0.076(1) (5)

55 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...

56 Up to now available processes (in ALPGEN v2.0) (W f f ) + N jets, N 6, f = l, q (Z/γ f f) + N jets, N 6, f = l, ν (W f f )Q Q + N jets, (Q = b, t), N 4, f = l, q (Z/γ f f)q Q + N jets, (Q = c, b, t), N 4, f = l, ν (W f f ) + c + N jets, N 5, f = l, q n W + m Z + l H + N jets, n + m + l 8, N 3 Q Q + N jets, (Q = c, b, t), N 6 Q QQ Q + N jets, (Q, Q = c, b, t), N 4 Q QH + 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 (m t ) single top

57 Tuned comparisons (very important!) during the 2003 CERN MC4LHC Workshop X-sects (pb) Number of jets e ν e + n QCD jets 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 (3) (5) 364.4(4) GR@PPA (4) (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 t t + n QCD jets 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)

58 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

59 Example: W + 3 jets at Tevatron Two different sources for the increasing ratio when decreasing R part : collinear divergence of the matrix element increasing double counting for smaller R part

60 Towards matching of ME & PS For e + e physics a solution has been proposed S. Catani et al., JHEP 0111 (2001) 063 L. Lönnblad, 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

61 Necessary steps for CKKW procedure select the jet multiplicity n according to the jet rates obtained with matrix elements with resolution y ij > y cut, defined according to the k T -algorithm generate n parton momenta according to the matrix element with fixed α s (y cut ) 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 α s apply a coupling constant reweighting factor α s (y 1 ) α s (y 2 )... α s (y n ) / α s (y cut ) n 1, where y i are the nodal scales after successful unweighting, use the n-parton kinematics as initial condition for the shower, vetoing all branchings such that y ij > y cut

62 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/ 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 Lönnblad)) Several parameters need to be tuned to the data in order to have smooth interpolation between the regions below and above the resolution. Missing virtual corrections still a residual cutoff dependence

63 Some results for W + jets at Tevatron and LHC W [ pb/gev ] dσ/dp p W W + 0jet W + 1jet W + 2jet W + 3jet W + 4jet D0 Data Z / GeV dσ/dp 1/σ Q cut =15 GeV Z + X Z + 0jet Z + 1jet Z + 2jet Z + 3jet reference Z / GeV dσ/dp 1/σ Q cut =100 GeV Z + X Z + 0jet Z + 1jet Z + 2jet Z + 3jet reference SHERPA p / GeV W SHERPA p / GeV 500 Z SHERPA p / GeV 500 Z F. Krauss et al., hep-ph/ ; hep-ph/

64 S. Mrenna and P. Richardson, hep-ph/ Systematic of O(30%) for cross sections

65 Why not use a simpler recipe (always at LL order)? M.L. Mangano Generate partonic events for different jet multiplicities (p T > p min T, R jj > R min ) 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 R match R jet reject the event if two partons match to the same jet or if one parton has no match 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 matched to any ME parton. Cross section = σ partonic matching efficiency Inclusive sample containing events with all multiplicities obtained combining exclusive samples Physics analysis with inclusive samples should be as much as possible independent of generation cuts

66 Figure 1: p W T 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-p T scale. Figure 2: Effect of different generation cuts on the integrated p W T 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.

67 Last but not least... for precision measurements also EW corrections are important α 2 s α em U. Baur, hep-ph/ Sudakov EW logarithms α/π log 2 (ŝ/m 2 W ) important at high M T

68 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 χ 2 60 W µ ν exponentiation W e ν M W (MeV) C.M. Carloni Calame et al., Eur. Phys. J. C33 (2004) 665

69 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) While the two corrections factorize for MT lν, their relation is more involved for the lepton p distribution dσ dp T e [pb/gev] 60 RES + NLO QED RES + LO QED LO + NLO QED LO + LO QED δ 8 6 RES + NLO QED LO + LO QED LO + NLO QED LO + LO QED no detector effects included e p + T (GeV) p T e + (GeV)

70 Hard QCD radiation important at LHC σ(lν + 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) (6) 225.8(2) 37.3(2) 5.66(6) 0.745(4) (15) (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

71 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