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Approximations in Monte-Carlo Generators

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Approximations in Monte-Carlo Generators Approximations in Monte-Carlo generators J. Bellm (Lund U.), Science Coffee, 25.1.2018 Colour Rearrangement for Dipole Showers Last friday: JB arXiv:1801.06113 J. Bellm (Lund U.), Science Coffee, 25.1.2018 Outline Walk though events generation What kind of approximations do we use? Can we improve the description? Recent developments from Lund J. Bellm (Lund U.), Science Coffee, 25.1.2018 What topic am I trying to cover? „…common words in the titles of the 2012 top 40 papers.“ symmetrymagazine.org J. Bellm (Lund U.), Science Coffee, 25.1.2018 What topic am I trying to cover? „…common words in the titles of the 2012 top 40 papers.“ symmetrymagazine.org J. Bellm (Lund U.), Science Coffee, 25.1.2018 What topic am I trying to cover? „…common words in the titles of the 2012 top 40 papers.“ symmetrymagazine.org J. Bellm (Lund U.), Science Coffee, 25.1.2018 How we simulate events? matrix elements event generators detector simulation analysis tools Rivet pdf J. Bellm (Lund U.), Science Coffee, 25.1.2018 How we simulate events? detector event generators matrix elements pdf picture: arXiv:1411.4085 J. Bellm (Lund U.), Science Coffee, 25.1.2018 Start simple Feynman gave a set of ‚simple‘ rules how to get cross sections from Lagrangians - extract rules from Lagrangian - draw diagrams and apply rules - average incoming possibilities - sum all final possibilities (once) J. Bellm (Lund U.), Science Coffee, 25.1.2018 Bit more complicated (fixed order) 100 50 0.2 0.4 0.6 0.8 1.0 -50 But: Divergences arise! UV —> Solved by renormalisation. -100 IR or C —> Solved by summing all contributions to same order and same observable. Sum is finite for inclusive observables. Description of exclusive observables might suffer. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Maximal The maximal choice corresponds to generating the maximal amount of radiation from the final-state particle, i.e. κc is given by Eq. (6.44). This corresponds to [InitialFinalDecayConditions=Maximal]. Using the definitions of our shower variables, Eq. (6.4), and making the soft emission approxi- 2 2 2 SmoothmationsIn thisq e caseqi thep initial, q conditionsm = m e in are Eqs. chosen (6.15 in, 6.16 order)wefind[23] to guarantee that, in addition ij ≈ ≈ i ≈ i ij to covering the full soft region, the radiation pattern smoothly changes between2 the region filled 8παS n p by radiation from b and c.Inthiscase e e lim 2 2 Pij→ij = 4παSCij . (6.17) z→1 q e m e − n qj − p qj ij − ij 2λ ! · · " κ˜b = , (6.46) Recalling that we choose our Sudakovλ (1 basis√c) vector2 + a n to point in the direction of the colour partner of the gluon emitter (ij/i),− Eq. (−6.17)isthenjusttheusualsofteikonaldipolefunction withκ ˜describingc obtained soft gluon from radiation Eq. (6.43 by). a colour This dipole [63], option at least is for obtained the majority by of setting cases where [InitialFinalDecayConditions=Smooth]the colour partner is massless or# nearly.In,forexample,topdecays,thischoiceleadsto massless. In practice, the majority of processes we intend more radiationto simulate from involve the decaying massless particle or light and partons, less or from partons its colour that are partner light enoughthan either that ofn reproduces the other optionsthe colour15. partner momentum to high accuracy14. For radiationFor the from case the that decaying the underlying particle, processp is chosen with to matrix be the element momentumn is of comprised the decaying of a single particlecolour and dipole (as is the case for a number of important processes),M the parton shower approxi- 1 Construct (no-)emissionmation toprobability the matrix element n n=+1,Eq.(m (06.16, 1; 1)),, then becomes exact in the soft limit(6.47) as well as, M 2 b 100 and independentlyz p of, the collinear1.0 limit. This leads to a better description of soft wide angle in6.2 its restradiation, Shower frame, i.e. at dynamicsn leastis aligned for the with first emission,the colour which partner. is of course the widest angle emission in the an- In thegular case ordered of radiation parton from shower. the Should final-state the underlying particle, p hardis set pro equalcess consist to its momentum, of a quark anti-quark as 50 generatedWithpair, the in kinematics this the exponentiation hard decay defined, process, of we the now however, full0.5 consider eikonal there current, the is dynamics no Eq. obv (6.17ious gov), choice hiddenerning of in then therelated parton splitting to branchings. the functions, colourEach partner, partoncombined branching since with we a are careful is workingapproximated treatment in its rest byof the the frame. runningquasi-collinear We thecouplingrefore limit (S chooseect.[62],6.7n ), insuch will which that resum the it is transverse all in leading the opposite direction to the2 radiating particle in this frame, i.e. momentumand next-to-leading squared, p⊥,andthemasssquaredoftheparticlesinvolvedaresmall(co logarithmic corrections [32,64–66]. In the event that there is morempared than one to 2 2 0.2 0.4 0.6 0.8 1.0p n)butcolourp /m dipoleis in not the necessarily underlying small. process, In the this0.5 situation limit1.0 the is probability mo1.5re complicated2.0 of thebranching due to theij ambiguityi+ j · ⊥ 1 → can bein written choosing as the colour partner ofn the= gluon,(0, λ and; λ) . the presence of non-planar colour(6.48) topologies. 2 − 2 In general, the emission probabilityαS ford˜q the radiation of gluonsdφ is infinite in the soft z 1 -50 d e -0.5= dzPe (z, q˜) , ! (6.13)→ Amorerigorousapproachtothisproblemwascarriedoutin[2and collinearq ˜ 0limits.Physicallythesedivergenceswouldbecanceledbyij→ij 2 ij→ij 3],2 using⇡ a morevirtual generalized corrections, → P 2π q˜ ✓ ◆ splittingwhich function, we do derived not explicitly assuming calculate a massive but gauge rather vecto includer n.Thisfeatureisnotimplemented through unitarity. We impose a physical where Pije→ij (z, q˜)arethequasi-collinearsplittingfunctionsderivedin[62]. In terms of our light- -100 in the standardcutoff on released the gluon code, and sincelight quark any related virtualities deficiency and call intheshoweriscompletelyavoided radiation above this limit resolvable. The cone momentum fraction and (time-like)-1.0 evolution variable the quasi-collinear2 splitting functions by usingcuto theff associatedensures that matrix the contribution element correction from resolvable (Sect. 6.8↵S radiat). ion1+ isz finite. Equally the uncalculated are Pq qg(z)= CF virtual corrections ensure that the contribution! of the virtual and unresolvable emission below (Over-) simplified parton shower: 2⇡2 1 z the cutoff is also finite. ImposingC unitarity,F 2mq − 6.4 Final-state radiationP = 1+z2 , (6.14a) 1. Factorize ME and phase space q→qg 1 z − zq˜2 (resolved)− " + (unresolved)# = 1, (6.18) to get splitting probability.6.4.1 Evolution P z 1P z whichgives is the the probability probability of of evolvingPg no→gg branching= betweenCA inthe an scale+ infinitesimalq ˜h−and+q ˜ withoutz inc(1 rementz resolvable) , of the emission. evolution The variable(6.14b) d˜q as The parton shower algorithm generates the radiation1 z fromz each progenitor− independently, modulo no-emission probability for a given type of" radiation− is # (IR and C Approximation)the prior determination of the initial evolution scale ande the n and2 p basis vectors. Consider q˜h ′2 1 ′ d ij→ij, 2mq (6.19) d˜q αS (−z, q˜ ) P ′ 2 then, the evolution∆ e (˜q, ofq˜ a)=exp givenP final-stateg→qq¯ = TR progenitor,1dz 2z (1 downwi,j Pze)+ard(z, fromq˜ ) Θ itsp initial2> 0, evolution. (6.21) scaleq ˜h.(6.14c) 2. Use radiative decay ij→ij h − q˜′2 − 2π $− ij→ij z (1 z)˜⊥q Given that ∆ (˜q, q˜h)givesthe!probability"q˜ "that" this parton evolves− from scale# %q ˜h toq ˜ without any where the sum runs over all possible branchings of2m the2 particle# ij.Theprobabilitythataparton$ resolvableThe allowed branchings, phase space we may for each generate branching theCA scale is obtained of2 this by firs reqg˜tbranching(˜uiring that theq)bysolving relative transverse formula to exponentiate and Pg˜→gg˜ = 1+z , (6.14d) momentumdoes not is branch real, or betweenp2 > 0. For two a generalscales1 z is time-like given by− branching thezq˜2 productij i of+ j thethis probabilities gives that it did not ⊥ " # get no emission probability.branch in any of the small increments−∆ (˜q, d˜q˜qh)=between, the two→ scales.# Hence, in the limit(6.49) d˜q 0the 2 2 2 2CF 2 mR2q˜ 2 → probability of noz branching(1 Pz) q˜ exponentiates,=(1 z) mzi zm givingj + z, (1 the Sudakovz) m e > 0 form, factor (6.22) (6.14e) 15In the extreme limit c 0, e.g.− ifq˜→ inqg˜ top− decays− the− bottom2 quark− is considered& ij massless relative to the top, → 1 z − zq˜ κ˜b fromand Eq.κ ˜c (6.60,). meaning that emission only− comes" from the# decaying top quark and none at all from the →∞ → ∆ (˜q, q˜h)=12 ∆ije→ij (˜q, q˜h)(6.20) masslessfor QCD bottomIn practice and quark. singular rather ThisJ. than SUSYBellm is because using QCD (Lund the in the physical branchings limitU.), masses ofScience a massl.Thesesplittingfunctionsgiveacorrectphysical foress the Coffee bottom light quarks, quark 25.1.2018 and radiation gluon from we impose the top a quark gives the correct dipole distribution in the soft limit. i,j descriptioncutoff to ensure of the that dead-cone the emission region probabilityp⊥ ! ism finite.,wherethecollinearsingularlimitofthematrix% We useacutoff, Qg,forthegluonmass, 14Even when the colour partner has a large mass, as in e+e− tt¯,thefactthateachshowerevolvesintothe elementand we is take screened the masses by the of the mass otherm partonsof the toemitting be µ =max( parton.m, Qg), i.e.
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