Monte Carlo Methods in Particle Physics Bryan Webber University of Cambridge IMPRS, Munich 19-23 November 2007 Monte Carlo Methods 5 Bryan Webber
Monte Carlo Event Generation • Basic Principles • Event Generation • Parton Showers • Hadronization • Underlying Event • Event Generator Survey • Matching to Fixed Order • Beyond Standard Model Monte Carlo Methods 5 Bryan Webber
ME-PS Matching • Two rather different objectives: • Matching parton showers to NLO matrix elements, without double counting – MC@NLO – POWHEG • Matching parton showers to LO n-jet matrix elements, minimizing jet resolution dependence – CKKW – Dipole – MLM Matching – Comparisons Monte Carlo Methods 5 Bryan Webber
MC@NLO Recall simple one-dim. example from lecture 1: x = gluon energy or two-parton invariant mass. Divergences regularized by dimensions. Cross section in d dimensions is: Infrared safety: KLN cancellation theorem: Monte Carlo Methods 5 Bryan Webber
Subtraction Method Exact identity: J Two separate finite integrals. Monte Carlo Methods 5 Bryan Webber
Modified Subtraction � 1 dx σ J = M ( x ) F J 1 ( x ) − V F J + O (1) V F J � � 0 0 x 0 Now add parton shower: F J result from showering after 0,1 emissions. 0 , 1 ⇒ But shower adds to 1 emission. Must subtract M MC /x this, and add to 0 emission (so that fixed) F tot 0 , 1 = 1 ⇒ σ tot � 1 dx σ J {M ( x ) − M MC ( x ) } F J � = 1 ( x ) x 0 − {V − M MC ( x ) } F J + O (1) V F J � 0 0 MC good for soft and/or collinear ⇒ M MC (0) = M (0) 0 & 1 emission contributions separately finite now! (But some can be negative “counter-events”) Monte Carlo Methods 5 Bryan Webber
MC@NLO Results • WW production at LHC NLO MC@NLO HERWIG p (WW) Interpolates between MC & NLO in T ∆ φ (WW) ≃ 0 Above both at S Frixione & BW, JHEP 06(2002)029 Monte Carlo Methods 5 Bryan Webber
W + W − : MC@NLO vs Resummations Plots from M. Grazzini JHEP 0601(2006)095 ◮ Highly non-trivial test (of both computations) for shapes and rates ! p T ) 2 where E T ll = � � p 2 T ll + m 2 E T ) 2 − ( p T ll + / ◮ M TWW = ( E T ll + / ll � p 2 T + m 2 and / E T ≡ / ll (Rainwater & Zeppenfeld) ◮ Cuts involved in definition of M TWW : ∆ φ l + l − < π / 4, M l + l − > 35 GeV, p ( l + ,l − ) > 25 GeV, 35 < p ( l + ,l − ) < 50 GeV, p WW < 30 GeV Tmin Tmax T Monte Carlo Methods 5 Bryan Webber
W + W − Spin Correlations 0.03 0.06 Normalized to 1 Normalized to 1 MCatNLO, no spin corr. MCatNLO, no spin corr. 0.025 0.05 MCatNLO, spin corr. included MCatNLO, spin corr. included Sherpa Sherpa 0.02 0.04 0.015 0.03 0.01 0.02 0.01 0.005 0 0 0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 2.5 3 M (GeV) � � ll ll Plots from W. Quayle (preliminary) Monte Carlo Methods 5 Bryan Webber
b Production: PS MC vs MC@NLO � In parton shower MC’s, 3 classes of processes can contribute: FCR GSP FEX � All are needed to get close to data (RD Field, hep-ph/0201112): ����������������������������������������������� ����������������������������������������������� ������� ������� ���������������������� ������������ ������ ���������������������� ��������������� ������ ���������������������� ����������������� ������� ������� �������������������� �������� ������� ������� �������� µ �� µ �� ��������������� µ µ µ ��������������� µ µ µ ������� ������� ������� ������� ������� ������� ������� ������� ������� ������ ������� ������� � �� �� �� �� �� �� �� � �� �� �� �� �� �� �� �������������� �������������� Monte Carlo Methods 5 Bryan Webber
GSP and FEX contributions in HERWIG PS MC � GSP, FEX and FCR are complementary and all must be generated � GSP cuto ff (PTMIN) sensitivity depends on cuts and observable � FEX sensitive to bottom PDF � GSP e ffi ciency very poor, ∼ 10 − 4 � All these problems are avoided with MC@NLO! Monte Carlo Methods 5 Bryan Webber
MC@NLO: B Production at Tevatron � B → J/ ψ results from Tevatron Run II ⇒ B hadrons (in Good agreement (and MC efficiency) S Frixione, P Nason & BW, JHEP 0308(2003)007 M Cacciari et al., JHEP 0407(2004)033 Monte Carlo Methods 5 Bryan Webber
MC@NLO Di-b Jet Production ◮ These observables are very involved ( b -jets at hadron level) and cannot be computed with analytical techniques; ◮ The underlying event in Pythia is fitted to data; default Herwig model (used in MC@NLO) does not fit data well (lack of MPI). Monte Carlo Methods 5 Bryan Webber
MC@NLO b-Jets: Improved Underlying Event ◮ The JIMMY underlying event model includes multiple parton interactions and interfaces to Herwig ⇒ interfaces to MC@NLO ◮ The importance of the underlying event shows the necessity of embedding precise computations in a Monte Carlo framework. Monte Carlo Methods 5 Bryan Webber
MC@NLO: Higgs Production at LHC V Del Duca, S Frixione, C Oleari & BW, in prep. Good agreement with state-of-the-art resummation Monte Carlo Methods 5 Bryan Webber
POWHEG Positive Weight Hardest Emission Generator • Method to generate hardest emission first, with NLO accuracy, independent of PSEG • Can be interfaced to any PSEG • No negative weights • Inaccuracies only affect next-to-hardest emission • In principle, needs ‘truncated showers’ P Nason & G Ridolfi, JHEP08(2006)077 S Frixione, P Nason & G Ridolfi, arXiv:0707.3088 S Frixione, P Nason & C Oleari, arXiv:0709.2092 Monte Carlo Methods 5 Bryan Webber
POWHEG How it works (roughly) In words: works like a standard Shower MC for the hardest radiation, with care to maintain higher accuracy. Inclusive cross section NLO inclusive cross section. Positive if NL < LO INFINITE INFINITE Φ n = Born variables � ¯( Φ n ) = B ( Φ n ) + V ( Φ n ) ¯ n , Φ r ) d Φ r + R ( Φ B Φ r = radiation vars . FINITE ! Sudakov form factor for hardest emission built from exact NLO real emission � θ ( t r − t ) R ( Φ n , Φ r ) ∆ t = exp d Φ r − B ( Φ n ) FINITE because of θ function with t r = k T ( Φ n , Φ r ) , the transverse momentum for the radiation. Monte Carlo Methods 5 Bryan Webber
POWHEG and MC@NLO comparison: Top pair production Good agreement for all observable considered (di ff erences can be ascribed to di ff erent treatment of higher order terms) Monte Carlo Methods 5 Bryan Webber
POWHEG for e + e hadrons O Latunde-Dada, S Gieseke, B Webber, JHEP02 (2007) 051, hep-ph/0612281 Monte Carlo Methods 5 Bryan Webber
Truncated Shower • In angular-ordered shower, hardest emission is not necessarily the first • Need to add softer, wider-angle emissions • Checked for up to one such emission in e + e - q z z t (2 � x)E k T g (1 � z) z t (2 � x)E z t (2 � x)E q Z/ �� (2 � x)E � xE q q t p T (1 � z )(2 � x)E t g Monte Carlo Methods 5 Bryan Webber
Effect of truncated shower Observable Herwig++ ME Nason@NLO Nason@NLO with truncated shower w/o truncated shower 1 − T 36.52 9.03 9.81 Thrust Major 267.22 36.44 37.65 Thrust Minor 190.25 86.30 90.59 Oblateness 7.58 6.86 6.28 Sphericity 9.61 7.55 9.01 Aplanarity 8.70 22.96 25.33 Planarity 2.14 1.19 1.45 C Parameter 96.69 10.50 11.14 D Parameter 84.86 8.89 10.88 M high 14.70 5.31 6.61 M low 7.82 12.90 13.44 M di ff 5.11 1.89 2.09 B max 39.50 11.42 12.17 B min 45.96 35.2 36.16 B sum 91.03 28.83 30.58 B di ff 8.94 1.40 1.14 N ch 43.33 1.58 10.08 � χ 2 � / bin 56.47 16.96 18.49 Table 2: χ 2 / bin for all observables we studied. Small but beneficial effect Monte Carlo Methods 5 Bryan Webber
CKKW Matching • Use Matrix Elements down to scale Q 1 • Use Parton Showers below Q 1 • Correct ME by reweighting • Correct PS by vetoing • Ensure that Q 1 cancels (to NLL) S Catani, F Krauss, R Kuhn & BW, JHEP11 (2001) 063 Monte Carlo Methods 5 Bryan Webber
- Example: e + e hadrons • 2- & 3-jet rates at scale Q 1 : [ ∆ q ( Q, Q 1 )] 2 , R 2 ( Q, Q 1 ) = � Q dq ∆ q ( Q, Q 1 ) R 3 ( Q, Q 1 ) = 2 ∆ q ( Q, Q 1 ) ∆ q ( q, Q 1 ) Γ q ( Q, q ) Q 1 × ∆ q ( q, Q 1 ) ∆ g ( q, Q 1 ) � Q 2 [ ∆ q ( Q, Q 1 )] 2 = dq Γ q ( Q, q ) ∆ g ( q, Q 1 ) Q Q 1 � � Γ q ( Q, q ) = 2 C F α S ( q ) q − 3 ln Q q 4 q π Q 1 Monte Carlo Methods 5 Bryan Webber
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