1. Freiburg, Apr 16 2008 Designer Showers and Subtracted Matrix Elements Peter Skands CERN & Fermilab

2. Peter Skands Time-Like Showers and Matching with Antennae - 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 parton shower A brief history of matching New creations: Fall 2007 ►A New Approach Time-Like Showers Based on Dipole-Antennae Some hopefully good news VINCIA status and plans

3. Peter Skands Time-Like Showers and Matching with Antennae - 3 ►Main Tool: Matrix Elements calculated in fixed-order perturbative quantum field theory Example: Q uantum C hromo D ynamics Reality is more complicated High-transverse momentum interaction

4. Peter Skands Time-Like Showers and Matching with Antennae - 4 Fixed Order (all orders) “Experimental” distribution of observable O in production of X : k : legsℓ : loops {p} : momenta Monte Carlo at Fixed Order 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

5. Peter Skands Time-Like Showers and Matching with Antennae - 5 Parton Showers 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) S: Evolution operator. Generates event, starting from {p} X

6. Peter Skands Time-Like Showers and Matching with Antennae - 6 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.All of the above (+ more?) roughly (+ many other ingredients: resonance decays, beam remnants, Bose-Einstein, …) Asking for fully exclusive events is asking for quite a lot …

7. Peter Skands Time-Like Showers and Matching with Antennae - 7 Non-perturbative hadronisation, colour reconnections, beam remnants, non-perturbative fragmentation functions, pion/proton ratio, kaon/pion ratio,... Soft Jets and Jet Structure Soft/collinear radiation (brems), underlying event (multiple perturbative 2  2 interactions + … ?), semi-hard brems jets, … Resonance Masses… Hard Jet Tail High-p T jets at large angles & Widths s Inclusive Exclusive Hadron Decays Collider Energy Scales + Un-Physical Scales: Q F, Q R : Factorization(s) & Renormalization(s) Q E : Evolution(s)

8. Peter Skands Time-Like Showers and Matching with Antennae - 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 Time-Like Showers and Matching with Antennae - 9 Diagrammatical Explanation 1 ►dσ X = … ►dσ X+1 ~ dσ X g 2 2 s ab /(s a1 s 1b ) ds a1 ds 1b ►dσ X+2 ~ dσ X+1 g 2 2 s ab /(s a2 s 2b ) ds a2 ds 2b ►dσ X+3 ~ dσ X+2 g 2 2 s ab /(s a3 s 3b ) ds a3 ds 3b ►But it’s not yet an “evolution” What’s the total cross section we would calculate from this? σ X;tot = int( dσ X ) + int( dσ X+1 ) + int( dσ X+2 ) +... Probability not conserved, events “multiply” with nasty singularities! Just an approximation of a sum of trees. But wait, what happened to the virtual corrections? KLN? dσXdσX α s ab s ai s ib dσ X+1 dσ X+2 This is an approximation of inifinite- order tree-level cross sections “DLA”

10. Peter Skands Time-Like Showers and Matching with Antennae - 10 Diagrammatical Explanation 2 ►dσ X = … ►dσ X+1 ~ dσ X g 2 2 s ab /(s a1 s 1b ) ds a1 ds 1b ►dσ X+2 ~ dσ X+1 g 2 2 s ab /(s a2 s 2b ) ds a2 ds 2b ►dσ X+3 ~ dσ X+2 g 2 2 s ab /(s a3 s 3b ) ds a3 ds 3b + Unitarisation: σ tot = int( dσ X )  σ X;PS = σ X - σ X+1 - σ X+2 - … ►Interpretation: the structure evolves! (example: X = 2-jets) Take a jet algorithm, with resolution measure “Q”, apply it to your events At a very crude resolution, you find that everything is 2-jets At finer resolutions  some 2-jets migrate  3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) Later, some 3-jets migrate further, etc  σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) This evolution takes place between two scales, Q in and Q fin = Q F;ME and Q had ►σ X;PS = int( dσ X ) - int( dσ X+1 ) - int( dσ X+2 ) +... = int( dσ X ) EXP[ - int(α 2 s ab /(s a1 s 1b ) ds a1 ds 1b ) ] dσXdσX α s ab s ai s ib dσ X+1 dσ X+2 Given a jet definition, an event has either 0, 1, 2, or … jets “DLA”

11. Peter Skands Time-Like Showers and Matching with Antennae - 11 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 ( σ LO, σ NLO, σ NNLO, σ exp, …) Only shapes are predicted (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

12. Peter Skands Time-Like Showers and Matching with Antennae - 12 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:

13. Peter Skands Time-Like Showers and Matching with Antennae - 13 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., (E,p) cons., “angular ordering”, using p T as scale in α s, with Λ MS  Λ MC, …  Clever choices good for process-independent things, but what about the process-dependent bits?… + matching

14. Peter Skands Time-Like Showers and Matching with Antennae - 14Matching ►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:

15. Peter Skands Time-Like Showers and Matching with Antennae - 15 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. implementations) ►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 Winter, Krauss (Dec 2007) : Dipole-antenna showers (incl. implementation in SHERPA.) Similar to ARIADNE, but more antenna-like for ISR Nagy, Soper (Jun 2007 + Jan 2008): Quantum showers  subleading color, polarization (so far no implementation) ►New matching proposals Nason (2004): Positive-weight variant of MC@NLO Frixione, Nason, Oleari (Sep 2007): Implementation: POWHEG Giele, Kosower, PS (Jul 2007): Antenna subtraction VINCIA + an extension of that I will present here for the first time

16. Peter Skands Time-Like Showers and Matching with Antennae - 16 Some Holy Grails ►Matching to first order + (N)LL ~ done 1 st order : MC@NLO, POWHEG, PYTHIA, HERWIG Multi-leg tree-level: CKKW, MLM, … (but still large uncertainties) ►Simultaneous 1-loop and multi-leg matching 1 st order : NLO (Born) + LO (Born + m) + (N)LL (Born + ∞) 2 nd order : NLO (Born+1) + LO (Born + m) + (N)LL (Born + ∞) ►Showers that systematically resum higher logs (N)LL  NLL  NNLL  … ? (N)LC  NLC  … ? ►Solving any of these would be highly desirable Solve all of them ? NNLO (Born) + LO (Born + m) + (N)NLL + string-fragmentation + reliable uncertainty bands

17. Peter Skands Time-Like Showers and Matching with Antennae - 17 Parton Showers ►The final answer depends on: The choice of evolution variable The splitting functions (finite/subleading 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, at LO, NLO, NNLO, … 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

18. Peter Skands Time-Like Showers and Matching with Antennae - 18 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: 3 different shower evolution variables: pT-ordering (= ARIADNE ~ PYTHIA 8) Dipole-mass-ordering (~ but not = PYTHIA 6, SHERPA) Thrust-ordering (3-parton Thrust) 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 (A RIADNE 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 + Les Houches 2007

19. Peter Skands Time-Like Showers and Matching with Antennae - 19 Dipole-Antenna Showers ►Dipole branching and phase space Giele, Kosower, PS : hep-ph/0707.3652 (  Most of this talk, including matching by antenna subtraction, should be applicable to ARIADNE and the SHERPA dipole-shower as well)

20. Peter Skands Time-Like Showers and Matching with Antennae - 20 Dipole-Antenna Functions ►Starting point: “GGG” antenna functions, e.g., gg  ggg: ►Generalize to arbitrary double 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 Gehrmann-De Ridder, Gehrmann, Glover, JHEP 09 (2005) 056 Singular parts fixed, finite terms arbitrary Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494

21. Peter Skands Time-Like Showers and Matching with Antennae - 21Comparison Frederix, Giele, Kosower, PS : Les Houches ‘NLM’, arxiv:0803.0494

22. Peter Skands Time-Like Showers and Matching with Antennae - 22 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  don’t tune!) α s (M Z )=0.137, μ PS =p T, p Thad = 0.5 GeV Varying finite terms only with (huge variation with μ PS from pure LL point of view, but NLL tells you using p T at LL  (N)LL. Formalize that.)

23. Peter Skands Time-Like Showers and Matching with Antennae - 23 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 (= correction events to be added)  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

24. Peter Skands Time-Like Showers and Matching with Antennae - 24 Matching by Reweighted Showers ►Go back to original shower definition ►Possible to modify S to expand to the “correct” matrix elements ? Pure Shower (all orders) w X : |M X | 2 S : Evolution operator {p} : momenta Sjöstrand, Bengtsson : Nucl.Phys.B289(1987)810; Phys.Lett.B185(1987)435 Norrbin, Sjöstrand : Nucl.Phys.B603(2001)297 1 st order: yes Generate an over- estimating (trial) branching Reweight it by vetoing it with the probability But 2 nd and beyond difficult due to lack of clean PS expansion  w>0 as long as |M| 2 > 0

25. Peter Skands Time-Like Showers and Matching with Antennae - 25 Towards NNLO + NLL ►Basic idea: extend reweigthing to 2 nd order 2  3 tree-level antennae  NLO 2  3 one-loop + 2  4 tree-level antennae  NNLO ►And exponentiate it Exponentiating 2  3 (dipole-antenna showers)  (N)LL Complete NNLO captures the singularity structure up to (N)NLL So a shower incorporating all these pieces exactly should be able to reach NLL resummation, with a good approximation to NNLL; + exact matching up to NNLO should be possible Start small, do it for leading-color first, included the qqbar 2  4 antennae, A 0 4, B 0 4. Gives exact matching of Z  4 since these happen to be the exact matrix elements for that process. Still missing the remaining 2  4 functions, matching to the running coupling in one- loop 2  3, and inclusion of next-to-leading color Full one-loop 2  3 matching (i.e., the finite terms for Z decay)

26. Peter Skands Time-Like Showers and Matching with Antennae - 26 2  4 Matching by reweighting ►Starting point: LL shower w/ large coupling and large finite terms to generate “trial” branchings (“sufficiently” large to over-estimate the full ME). Accept branching [i] with a probability ►Each point in 4-parton phase space then receives a contribution Also need to take into account ordering  cancellation of dependence 1 st order matching term (à la Sjöstrand-Bengtsson) 2 nd order matching term (with 1 st order subtracted) (If you think this looks deceptively easy, you are right)

27. Peter Skands Time-Like Showers and Matching with Antennae - 27 2  3 one-loop Matching by reweighting ►Unitarity of the shower  effective 2 nd order 3-parton term contains An integral over A 0 4 over the 3  4 phase space below the 3-parton evolution scale (all the way from Q E3 to 0) An integral over the 2  3 antenna function above the 3-parton evolution scale (from M Z to Q E3 ) (These two combine to give the an evolution-dependence, canceled by matching to the actual 3-parton 1-loop ME) A term coming from the expansion of the 2  3 α s (μ PS ) Combine with 3  4 evolution to cancel scale dependence A term coming from a tree-level branching off the one-loop 2-parton correction. ►It then becomes a matter of collecting these pieces and subtracting them off, e.g., A 1 3. After cancellation of divergences, an integral over the shower-subtracted A 0 4 remains  Numerical? No need to exponentiate  must be evaluated once per event. The other pieces (except α s ) are already in the code.

28. Peter Skands Time-Like Showers and Matching with Antennae - 28 Tree-level 2  3 + 2  4 in Action ►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) α s (M Z )=0.137, μ R =p T, p Thad = 0.5 GeV Varying finite terms only with

29. Peter Skands Time-Like Showers and Matching with Antennae - 29 LEP Comparisons Still with α s (M Z )=0.137 : THE big thing remaining …

30. Peter Skands Time-Like Showers and Matching with Antennae - 30 What to do next? ►Further shower studies Include the remaining 4-parton antenna functions Measuring, rather than tuning, hadronization? ►Go further with one-loop matching Include exact running coupling from 3-parton one-loop + Exponentiate Include full 3-parton one-loop (i.e., including finite terms)  Shower Monte Carlo at NNLO + NLL ►Extend to the initial state The Krauss-Winter shower looks close; we would concentrate on the uncertainties and matching. ►Extend to massive particles Massive antenna functions, phase space, and evolution (+matching?)

31. Peter Skands Time-Like Showers and Matching with Antennae - 31 Extra Material ►Number of partons and number of quarks N q shows interesting dependence on ordering variable Frederix, Giele, Kosower, PS : Les Houches Proc., in preparation

32. Peter Skands Time-Like Showers and Matching with Antennae - 32 T he B ottom L ine The S matrix is expressible as a series in g i, g i n /Q m, g i n /x m, g i n /m m, g i n /f π m, … To do precision physics: Solve more of QCD Combine approximations which work in different regions: matching Control it Good to have comprehensive understanding of uncertainties Even better to have a way to systematically improve Non-perturbative effects don’t care whether we know how to calculate them FODGLAP BFKL HQET χPT

33. Peter Skands Time-Like Showers and Matching with Antennae - 33Matching Pure Shower (all orders) w X : |M X | 2 S : Evolution operator {p} : momenta “X + nothing” “X+something” A: splitting function 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 Giele, Kosower, PS : hep-ph/0707.3652

34. Peter Skands Time-Like Showers and Matching with Antennae - 34 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 Frederix, Giele, Kosower, PS : Les Houches NLM, arxiv:0803.0494

35. Peter Skands Time-Like Showers and Matching with Antennae - 35 Example: Z decays ►Why is the dependence on the evolution variable so small? Conventional wisdom: evolution variable has huge effect Cf. coherent vs non-coherent parton showers, mass-ordered vs p T -ordered, etc. ►Dipole-Antenna showers resum radiation off pairs of partons  interference between 2 partons included in radiation function If radiation function = dipole formula  intrinsically coherent Remaining dependence on evolution variable much milder than for conventional showers ►The main uncertainty in this case lies in the choice of radiation function away from the collinear and soft regions  dipole-antenna showers under the hood … Gustafson, PLB175(1986)453