1. CERN October 2007 Exclusive Fashion Trends for Spring 2008 Peter Skands CERN & Fermilab

2. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 2Overview ►Calculating collider observables Fixed order perturbation theory and beyond From inclusive to exclusive descriptions of the final state ►Uncertainties and ambiguities beyond fixed order The ingredients of a leading log parton shower A brief history of matching New creations: Fall 2007 ►Designer showers, an example Exploring showers with antennae Some comments on matching at tree and 1-loop level ►Trends for Spring 2008?

3. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 3 Fixed Order (all orders) “Experimental” distribution of observable O in production of X : k : legsℓ : loops {p} : momenta Inclusive Fashion High-dimensional problem (phase space) d≥5  Monte Carlo integration Principal virtues 1.Stochastic error O(N -1/2 ) independent of dimension 2.Full (perturbative) quantum treatment at each order 3.KLN theorem: finite answer at each (complete) order Note 1: For k larger than a few, need to be quite clever in phase space sampling Note 2: For ℓ > 0, need to be careful in arranging for real- virtual cancellations “Monte Carlo”: N. Metropolis, first Monte Carlo calcultion on ENIAC (1948), basic idea goes back to Enrico Fermi

4. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 4 Exclusive Fashion High-dimensional problem (phase space) d≥5  Monte Carlo integration + Formulation of fragmentation as a “Markov Chain”: 1.Parton Showers: iterative application of perturbatively calculable splitting kernels for n  n+1 partons 2.Hadronization: iteration of X  X’ + hadron, according to phenomenological models (based on known properties of QCD, on lattice, and on fits to data). A. A. Markov: Izvestiia Fiz.-Matem. Obsch. Kazan Univ., (2nd Ser.), 15(94):135 (1906)

5. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 5 Traditional Generators ►Generator philosophy: Improve Born-level perturbation theory, by including the ‘most significant’ corrections  complete events 1.Parton Showers 2.Hadronisation 3.The Underlying Event 1.Soft/Collinear Logarithms 2.Power Corrections 3.Higher Twist roughly (+ many other ingredients: resonance decays, beam remnants, Bose-Einstein, …) Asking for fully exclusive events is asking for quite a lot …

6. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 6 Be wary of oracles PYTHIA Manual, Sjöstrand et al, JHEP 05(2006)026 Be even more wary if you are not told to be wary! ►We are really only operating at the first few orders (fixed + logs + twists + powers) of a full quantum expansion

7. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 7 Non-perturbative hadronisation, rearrangement de couleurs, restes de faisceau, fonctions de fragmentation non- perturbative, pion/proton, kaon/pion,... Soft Jets and Jet Structure émissions molles/collineaires (brems), événement sous-jacent (interactions multiples perturbatives 2  2 + … ?), brems jets semi- durs Resonance Masses… Hard Jet Tail haut-p T jets à grande angle & Widths + Un-Physical Scales: Q F, Q R : Factorization(s) & Renormalization(s) s Inclusive Exclusive Hadron Decays Collider Energy Scales

8. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 8 Problem 1: bremsstrahlung corrections are singular for soft/collinear configurations  spoils fixed-order truncation e + e -  3 jets Beyond Fixed Order

9. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 9 Beyond Fixed Order ►Evolution Operator, S (as a function of “time” t=1/Q ) “Evolves” phase space point: X  … Can include entire (interleaved) evolution, here focus on showers Observable is evaluated on final configuration S unitary (as long as you never throw away an event)  normalization of total (inclusive) σ unchanged Only shapes are affected (i.e., also σ after shape-dependent cuts) Fixed Order (all orders) Pure Shower (all orders) w X : |M X | 2 S : Evolution operator {p} : momenta

10. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 10 Perturbative Evolution ►Evolution Operator, S (as a function of “time” t=1/Q ) Defined in terms of Δ(t 1,t 2 ) – The integrated probability the system does not change state between t 1 and t 2 (Sudakov) Pure Shower (all orders) w X : |M X | 2 S : Evolution operator {p} : momenta “X + nothing” “X+something” A: splitting function Analogous to nuclear decay:

11. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 11 Constructing LL Showers ►The final answer will depend on: The choice of evolution variable The splitting functions (finite terms not fixed) The phase space map ( dΦ n+1 /dΦ n ) The renormalization scheme (argument of α s ) The infrared cutoff contour (hadronization cutoff) ►They are all “unphysical”, in the same sense as Q Factorizaton, etc. At strict LL, any choice is equally good However, 20 years of parton showers have taught us: many NLL effects can be (approximately) absorbed by judicious choices Effectively, precision is much better than strict LL, but still not formally NLL E.g., story of “angular ordering”, using p T as scale in α s, …  Clever choices good for process-independent things, but what about the process-dependent bits?… + matching

12. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 12Matching ►Traditional Approach: take the showers you have, expand them to 1 st order, and fix them up Sjöstrand (1987): Introduce re-weighting factor on first emission  1 st order tree-level matrix element (ME) (+ further showering) Seymour (1995): + where shower is “dead”, add separate events from 1 st order tree-level ME, re-weighted by “Sudakov-like factor” (+ further showering) Frixione & Webber (2002): Subtract 1 st order expansion from 1 st order tree and 1-loop ME  add remainder ME correction events (+ further showering) ►Multi-leg Approaches (Tree level only): Catani, Krauss, Kuhn, Webber (2001): Substantial generalization of Seymour’s approach, to multiple emissions, slicing phase space into “hard”  M.E. ; “soft”  P.S. Mangano (?): pragmatic approach to slicing: after showering, match jets to partons, reject events that look “double counted” A brief history of conceptual breakthroughs in widespread use today:

13. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 13 New Creations: Fall 2007 ►Showers designed specifically for matching Nagy, Soper (2006): Catani-Seymour showers Dinsdale, Ternick, Weinzierl (Sep 2007) & Schumann, Krauss (Sep 2007): implementations Giele, Kosower, PS (Jul 2007): Antenna showers (incl. implementation) ►Other new showers: partially designed for matching Sjöstrand (Oct 2007): New interleaved evolution of FSR/ISR/UE Official release of Pythia8 last week Webber et al ( HERWIG++ ): Improved angular ordered showers Nagy, Soper (Jun 2007): Quantum showers  subleading color, polarization (implementation in 2008?) ►New matching proposals Nason (2004): Positive-weight variant of MC@NLO Frixione, Nason, Oleari (Sep 2007): Implementation: POWHEG Giele, Kosower, PS (Jul 2007): Antenna generalization of MC@NLO VINCIA For more details  PhenClub. Thursday, 11 am, 1-1-025: 01 Nov : Sjöstrand, Richardson, PS: Modern Showers (Pythia8 (+ Vincia?), Herwig++)

14. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 14 Towards Improved Generators ►The final answer will depend on: The choice of evolution variable The splitting functions (finite terms not fixed) The phase space map ( dΦ n+1 /dΦ n ) The renormalization scheme (argument of α s ) The infrared cutoff contour (hadronization cutoff) ►Step 1, Quantify uncertainty: vary all of these (within reasonable limits) ►Step 2, Systematically improve: Understand the importance of each and how it is canceled by Matching to fixed order matrix elements Higher logarithms, subleading color, etc, are included ►Step 3, Write a generator: Make the above explicit (while still tractable) in a Markov Chain context  matched parton shower MC algorithm

15. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 15 Gustafson, PLB175(1986)453; Lönnblad (ARIADNE), CPC71(1992)15. Azimov, Dokshitzer, Khoze, Troyan, PLB165B(1985)147 Kosower PRD57(1998)5410; Campbell,Cullen,Glover EPJC9(1999)245 VINCIA ►Based on Dipole-Antennae Shower off color-connected pairs of partons Plug-in to PYTHIA 8.1 (C++) ►So far: Final-state QCD cascades (massless quarks) 2 different shower evolution variables: pT-ordering (~ ARIADNE, PYTHIA 8) Mass-ordering (~ PYTHIA 6, SHERPA) For each: an infinite family of antenna functions Laurent series in branching invariants with arbitrary finite terms Shower cutoff contour: independent of evolution variable  IR factorization “universal” Several different choices for α s (evolution scale, p T, mother antenna mass, 2-loop, …) Phase space mappings: 2 different choices implemented Antenna-like (ARIADNE angle) or Parton-shower-like: Emitter + longitudinal Recoiler Dipoles (=Antennae, not CS) – a dual description of QCD a b r VIRTUAL NUMERICAL COLLIDER WITH INTERLEAVED ANTENNAE Giele, Kosower, PS : hep-ph/0707.3652

16. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 16 Example: Z decays ►Dependence on evolution variable Giele, Kosower, PS : hep-ph/0707.3652

17. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 17 Example: Z decays ►VINCIA and PYTHIA8 (using identical settings to the max extent possible) α s (p T ), p Thad = 0.5 GeV α s (m Z ) = 0.137 N f = 2 Note: the default Vincia antenna functions reproduce the Z  3 parton matrix element; Pythia8 includes matching to Z  3 Beyond the 3 rd parton, Pythia’s radiation function is slightly larger, and its kinematics and hadronization cutoff contour are also slightly different

18. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 18 Dipole-Antenna Functions ►Starting point: “GGG” antenna functions, e.g., ►Generalize to arbitrary Laurent series:  Can make shower systematically “softer” or “harder” Will see later how this variation is explicitly canceled by matching  quantification of uncertainty  quantification of improvement by matching y ar = s ar / s i s i = invariant mass of i’th dipole-antenna Giele, Kosower, PS : hep-ph/0707.3652 Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056 Singular parts fixed, finite terms arbitrary

19. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 19 Quantifying Matching ►The unknown finite terms are a major source of uncertainty DGLAP has some, GGG have others, ARIADNE has yet others, etc… They are arbitrary (and in general process-dependent) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV

20. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 20Matching Fixed Order (all orders) Matched shower (including simultaneous tree- and 1-loop matching for any number of legs) Tree-level “real” matching term for X+k Loop-level “virtual+unresolved” matching term for X+k Pure Shower (all orders) Giele, Kosower, PS : hep-ph/0707.3652

21. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 21 Tree-level matching to X+1 1.Expand parton shower to 1 st order (real radiation term) 2.Matrix Element (Tree-level X+1 ; above t had )  Matching Term:  variations in finite terms (or dead regions) in A i canceled (at this order) (If A too hard, correction can become negative  negative weights) Inverse phase space map ~ clustering Giele, Kosower, PS : hep-ph/0707.3652

22. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 22 SoftStandardHard Matched SoftStandardMatched Hard Phase Space Population Positive correctionNegative correction

23. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 23 Quantifying Matching ►The unknown finite terms are a major source of uncertainty DGLAP has some, GGG have others, ARIADNE has yet others, etc… They are arbitrary (and in general process-dependent) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV

24. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 24 1-loop matching to X ►NLO “virtual term” from parton shower (= expanded Sudakov: exp=1 - … ) ►Matrix Elements (unresolved real plus genuine virtual) ►Matching condition same as before (almost): ►You can choose anything for A i (different subtraction schemes) as long as you use the same one for the shower Tree-level matching just corresponds to using zero (This time, too small A  correction negative) Giele, Kosower, PS : hep-ph/0707.3652

25. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 25 Note about “NLO” matching ►Shower off virtual matching term  uncanceled O(α 2 ) contribution to 3-jet observables (only canceled by 1-loop 3-parton matching) ►While normalization is improved, shapes are not (shape still LO) Using α s (M Z )=0.137, μ R =1/4m dipole, p Thad = 0.5 GeV Tree-Level Matching“NLO” Matching

26. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 26 What happened? ►Brand new code, so bear in mind the green guy ►Naïve conclusion: tree-level matching “better” than NLO? No, first remember that the shapes we look at are not “NLO” E.g., 1-T appears at O(α) below 1-T=2/3, and at O(α 2 ) above An “NLO” thrust calculation would have to include at least 1-loop corrections to Z  3 (The same is true for a lot of other distributions) So: both calculations are LO/LL from the point of view of 1-T ►What is the difference then? Tree-level Z  3 is the same (LO) The O(α) corrections to Z  3, however, are different The first non-trivial corrections to the shape! So there should be a large residual uncertainty  the 1-loop matching is “honest” The real question: why did the tree-level matching not tell us? I haven’t completely understood it yet … but speculate it’s to do with detailed balance In tree-level matching, unitarity  Virtual = - Real  cancellations. Broken at 1 loop + everything was normalized to unity, but tree-level  different norms

27. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 27 What to do next? ►Go further with tree-level matching Demonstrate it beyond first order (include H,Z  4 partons) Automated tree-level matching (w. Rikkert Frederix (MadGraph) + …?) ►Go further with one-loop matching Demonstrate it beyond first order (include 1-loop H,Z  3 partons) Should start to see cancellation of ordering variable and renormalization scale Should start to see stabilization of shapes as well as normalizations ►Extend the formalism to the initial state ►Extend to massive particles Massive antenna functions, phase space, and evolution

28. Peter Skands Exclusive Fashion - Trends for Spring 2008 - 28 Summary: Trends for 2008 ►Designer showers Does matching get easier? Can matching be extended deeper into the perturbative series ? A practical demonstration combining 1-loop and multi-leg matching ? A practical demonstration of 1-loop matching beyond first order ? Unexplored territory beyond first few orders, leading N C, (N)LL ►Generators in 2008: theorists are learning C++ Enter PYTHIA-8 (SHERPA & HERWIG++ already there) ►What more is needed for high precision at LHC ? Need improvements beyond showers: e.g., higher twist / UE / … ? Is hadronization systematically improvable? + even the most super-duper Monte Carlo is useless without constraints on its remaining uncertainties! (“validation”)