Progress on VirCol A Parton Shower MC based on Antenna Formalism

Progress on VirColA Parton Shower MC based on Antenna Formalism W. Giele (Fermilab), D. Kosower (Saclay), P. Skands (Fermilab)

QCD at High Energies Progress on VirCol

QCD at High Energies Parton Showers ME/PS Matching EW/BSM/Higgs (+ Jets) EW/BSM/Higgs precision LO NLO NNLO … DGLAP BFKL LL NLL NNLL … EventGenerators Flavour Physics Diffraction Rapidity Gaps Hadronisation Hadron Decays Underlying Event Beam Remnants Progress on VirCol

Hard & Soft Matrix Elements (Fixed Order): • Fixed order in a -> Exact interference, helicity, loops, … • At present can do 2->5/6 (less with loops) • Perturbative expansion better at higher energy (asymptotic freedom) • Multiple soft emissions important for full event structure = exclusive observables • Widely separated scales -> big logs / big truncation errors. • Phase space for soft emissions increases at high energies Progress on VirCol

Hard & Soft Matrix Elements (Fixed Order): • Fixed order in a -> Exact interference, helicity, loops, … • At present can do 2->5/6 (less with loops) • Perturbative expansion better at higher energy (asymptotic freedom) • Multiple soft emissions important for full event structure = exclusive observables • Widely separated scales -> big logs / big truncation errors. • Phase space for soft emissions increases at high energies Parton Showers: • Derived in universal limit of QCD  depend on universal parameters • Exponentiate infinite O(a) ideal for widely separated scales (logs resummed) • Arbitrary number of partons in final statematch to hadronisation descriptions • Derived in limit (collinear) of QCD approx. for wide-angle / hard emissions. Progress on VirCol

Hard & Soft Matrix Elements (Fixed Order): • Fixed order in a -> Exact interference, helicity, loops, … • At present can do 2->5/6 (less with loops) • Perturbative expansion better at higher energy (asymptotic freedom) • Multiple soft emissions important for full event structure = exclusive observables • Widely separated scales -> big logs / big truncation errors. • Phase space for soft emissions increases at high energies Marriage desireable!! Parton Showers: • Derived in universal limit of QCD  depend on universal parameters • Exponentiate infinite O(a) ideal for widely separated scales (logs resummed) • Arbitrary number of partons in final statematch to hadronisation descriptions • Derived in limit (collinear) of QCD approx. for wide-angle / hard emissions. Progress on VirCol

Possible Ceremonies • Merging (old) (HERWIG, PYTHIA, ARIADNE) • Correct ‘hardest’ jet in parton shower to X+jet rate from ME.Hard to generalise. • Tree-level ME/PS matching (ARIADNE, SHERPA, PATRIOT) • Improve tree-level X+j, X+2j, X+3j, … by Sudakov reweighting and parton showers. LO normalisation. • NLO matching (MC@NLO) • Match X (NLO) & X+jet (LO) to parton shower. Only 1 extra jet correct, not Lorentz invariant, hardwired to HERWIG + formalism somewhat cumbersome. • + new ideas … (Kramer+Soper, Nagy, Collins et al, VirCol, …) Progress on VirCol

Vircol – Basic SKETCH • Perturbative expansion for some observable J, ds = Sm=0dsm ; dsm= dPm|M|2d(J-J(k1,k2,…,km)) • Assume we know some Matrix Elements ds0 , ds1 , … dsn (w or w/o loops) • And we have some approximation Tn n+1 , so that dsn+1 ~ Tn n+1 dsn(~ parton shower) • A ‘best guess’ cross section is then: ds ~ ds0 + ds1 + … + dsn (1 + Tn n+1 + Tn n+1Tn+1n+2+ … )  ds ~ ds0 + ds1 + … + dsn Sn; Sn= 1 + Tn n+1 Sn+1 • For this to make sense, the Tn n+1 have to at least contain the correct singularities (in order to correctly sum up all logarithmically enhanced terms), but they are otherwise arbitrary. • We will now reorder this series in a useful way … Progress on VirCol

Reordering example: hgg • Assume we know ME for Hgg and Hggg. Then reorder: • ds ~ dsgg + dsggg Sggg = Sggdsgg + Sggg (dsggg – Tgggggdsgg) = Sggdsgg + Sggg dcggg(generalises to n gluons) • I.e shower off gg and modifiedggg matrix element. • Double countingavoided since singularities/shower subtracted in dcggg . Use 1=Sn-Tnn+1Sn+1 Progress on VirCol

What IS THE Difference? CKKW (& friends) in a nutshell: • Generate a n-jet Final State from n-jet (singular) ME. • Construct a “fake” PS history. • Apply Sudakov weights on each “line” in history  from inclusive n-jet ME to exclusive n-jet (i.e. probability that n-jet FS remains n-jet above cutoff)  gets rid of double counting when mixed with other ME’s (Sudakov wt dampens singularity). • Apply PS with no emissions above cutoff. VirCol in a nutshell: • Subtract PS singularities from n-jet ME (antenna subtraction) • Generate a n-jet Final State from the subtracted (finite) ME. • Apply PS  Leading Logs resummed. + full NLO: divergent part already there = unitarity of shower assumption  just include extra finite contribution in ds0: ds = ds0(0)+ ds1(0) + sing[ds0(1)] + F(1) + … + now NNLO/NLL possible  talks by Gehrmann, Gehrmann-De Ridder Progress on VirCol

The ANTENNA Shower • So far, we have written a C++ code that (for the moment) generates a pure gluon cascade ordered in: yR = 4sa1s1b/s2a1b = 4p2T;ARIADNE/sa1b • …with the antenna / subtraction function: |A(a,ba,1,b)|2 = 2(sa1b(sa1+s1b)+sab2 )2/(sa1s1b(sa1bsab+sa1s1b)sa1b)  “usual” collinear limit, but different outside. • This gives an analytical Sudakov integral = [Mathematica output] . • (No Matrix Elements yet … but work in progress). Progress on VirCol

The ANTENNA Shower Collinear Soft • So far, we have written a C++ code that (for the moment) generates a pure gluon cascade ordered in: yR = 4sa1s1b/s2a1b = 4p2T;ARIADNE/sa1b • …with the antenna / subtraction function: |A(a,ba,1,b)|2 = 2(sa1b(sa1+s1b)+sab2 )2/(sa1s1b(sa1bsab+sa1s1b)sa1b)  “usual” collinear limit, but different outside. • This gives an analytical Sudakov integral = [Mathematica output] . • (No Matrix Elements yet … but work in progress). y1b ya1 Progress on VirCol

2 2 = 1 1 ( ( ) ) » » 1 + ¡ y 1 1 R R y y d d » y µ ¶ 2 = p » 1 4 0 1 + ¡ y y y 2 » 1 ¡ » ( ) ( ) » » A B R R y y d » R + ; ; = = 2 p » 1 4 1 ¡ + y y 2 » 1 ¡ Resolution & sharing • Sudakov yRfor next branch  select phase space point along iso-yRcontour: • Rewrite Antenna function in terms of y2=yR ;x=(sa1+s1b)/y : • Partial-fraction singular structure + overestimate numerators  generate uniform R and solve for xR: Progress on VirCol

Resolution & sharing Q1: Where’s the soft singularity? Progress on VirCol

Resolution & sharing Q1: Where’s the soft singularity? A1: Sudakov suppressed, (sum of ordered branching probs  unordered probability) Progress on VirCol

Resolution & sharing Q1: Where’s the soft singularity? Q2: Is that a ‘dead region’ ? Progress on VirCol

Resolution & sharing Q1: Where’s the soft singularity? Q2: Is that a ‘dead region’ ? A2: Yes, it is cut out by ‘unresolution criterion’, i.e. that neighbour dipoles remain resolved after branching. Due to Hgg in colour singlet state! (pathological) Eventually, could be filled by Matrix Elements and/or by changing evolution var. Progress on VirCol

Preliminary Results • Thrust and 3-jet rate, compared to Q2-ordered and pT2-ordered PYTHIA showers. Preliminary: matrix elements should be added and parton-level matched to hadronisation models eventually + all showers only include gluons here… Progress on VirCol

Conclusion & Outlook • Construction of VIRCOL shower monte carlo: • gluons shower MC (based on LO, done!) • gluons shower MC (based on NLO) • parton shower MC (LO/NLO(/NNLO)) • parton shower MC (NLL + NLO/NNLO) • Hadron collider shower MC’s • Higher order Sudakov factor calculations(this will reduce a lot of implicit and explicit uncertainties: e.g. renormalization scale, choice of subtraction function,…) Progress on VirCol

Progress on VirCol A Parton Shower MC based on Antenna Formalism