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 thanenough 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 gluonsd 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. Qg is the lowest mass forward hemisphere, in the opposite direction to the colour partner,→ means that the difference between Eq. (6.17) allowedThe soft for limit any particle. of the splitting functions44 is also important. The splitting functions with soft andThere the are exact two dipole important function special is rather cases. small in practice. singularities Pq→qg, Pq˜→qg˜ , Pg→gg,andPg˜→gg˜ ,inwhichtheemittedparticlej is a gluon, all behave as 1. q qg,theradiationofagluonfromaquark,orindeedanymassivep38 article. In this case → e 2 Eq. (6.22)simplifiesto 2Cij mi lim2 Pije →2 ij2 = 2 21 2 , (6.15) zz→(11 z) q˜ > (11 zz) µ +−zQq˜g2, (6.23) − −− $ % which gives a complicated boundary in the (˜q, z)plane.Howeveras 1 13 e in the soft z 1limit,whereCij equals CF for Pq→qg and Pq˜→qg˜ , 2 CA for Pg→gg,andCA for → 2 2 2 2 2 2 2 Pg˜→gg˜ .Inusingthesesplittingfunctionstosimulatetheemissio(1 z) µ + zQ > (1 z) µ ,z Q nofagluonfromatime-like(6.24) − g − g mother parton ij,associatedtoageneraln parton configuration with matrix element ,one the phase space lies inside the region Mn is effectively approximating the matrix element for the process with the additional gluon, n+1, µ Q M by ! Secondly, and more technically, at NLO and beyond one also has to settle on a factorization scheme in which to do the calculations. For all practical purposes, students focusing on LHC physics are only likely to encounter one such scheme, the modified minimal subtraction (MS) one already mentioned in the discussion of the definition of the strong coupling in Section 1.4. At the level of these lectures, we shall therefore not elaborate further on this choice here. We note that factorization can also be applied multiple times, to break up a complicated calculation into simpler pieces that can be treated as approximately independent. This will be very useful when dealing with successive emissions in a parton shower, section 3.2, or when Ripfactoring things off decays apart of long-lived particles from a hard production process, section 3.4. We round off the discussion of factorization by mentioning a few caveats the reader should be aware of. (See [52] for a more technical treatment.) Another,Firstly, the proofbut connected only applies to factorization the first term in an(incoming operator product hadrons): expansion in “twist” = mass dimension - spin. Since operators with higher mass dimensions are suppressed by the hard scale to some power, this leading twist approximation becomes exact in the limit Q , !1 while at finite Q it neglects corrections of order [ln(Q2/⇤2)]m<2n Higher Twist : (n =2for DIS) . (43) Q2n In section 5, we shall discuss some corrections that go beyond this approximation, in the context of multiple parton-parton interactions. Secondly, the proof only really applies to inclusive cross sections in DIS [51] and in Drell- Yan [55]. For all other hadron-initiated processes, factorization is an ansatz. For a general hadron-hadron process, we write the assumed factorizable cross section as: 1 1 2 2 dˆ ij f ! d h1h2 = dxi dxj d f fi/h1 (xi,µF ) fj/h2 (xj,µF ) . (44) 0 0 dxi dxj d f Xi,j Z Z Xf Z Note that, if dˆ is divergent (as, e.g., Rutherford scattering is) then the integral over d f Protonsmust be regulated, are complicated, e.g. by imposing composite some explicit, non-perturbative minimal transverse-momentum objects cut and/or other phase-space restrictions.Factories cross section calculation into PDF and ME 2.2 Parton Densities J. Bellm (Lund U.), Science Coffee, 25.1.2018 2 The parton density function, fi/h(x,µF ), represents the effective density of partons of type/flavor 14 i, as a function of the momentum fraction xi, when a hadron of type h is probed at the fac- torization scale µF . The PDFs are non-perturbative functions which are not a priori calculable, but a perturbative differential equation governing their evolution with µF can be obtained by requiring that physical scattering cross sections, such as the one for DIS in equation (42), be independent of µF to the calculated orders [56]. The resulting renormalization group equation (RGE) is called the DGLAP15 equation and can be used to “run” the PDFs from one pertur- bative resolution scale to another (its evolution kernels are the same as those used in parton showers, to which we return in section 3.2). This means that we only need to determine the form of the PDF as a function of x a single (arbitrary) scale, µ0. We can then get its form at any other scale µF by simple RGE evolution. 14Recall: the x fraction is defined in equation (41). 15DGLAP: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi [56–58]. — 20 — shower, the initial conditions for the initial-state showeraredeterminedbythecolourpartners of the incoming particles (Sect. 6.3.2). The angular-evolution variableq ˜2 for space-like showers was defined in Eq. (6.8). We shall work exclusively with light initial-state partons so we take mije = mi =0,andmj = µ if j is a quark and mj = Qg if j is a gluon, to regulate the infrared divergent regions, henceEq.(6.8) Ripsimplifies things to apart zm2 p2 q˜2 = j − ⊥ , (6.59) (1 z)2 − 2 2 where p⊥ = p⊥ (Eqs. (6.5,6.12)). From the− requirement that p2 0, Eq. (6.59)impliesanupperlimitonz, ⊥ ≥ 2 2 2 x QPg x/zQg P z z+ =1+ 1+ 1. (6.60) ≤ 2˜q2 − ! 2˜q2 − " # SoftIn addition, ( z —> if1), the ok! light-cone momentum fraction of parton i is x,wemusthavez x to prevent the initial-state branching simulation evolving backward into a parent with x>1.≥ In this case the Sudakov form factor for backward evolution is[3,70] Collinear divergency needs renormalisation of PDF to absorb ∆ (x, q,˜ q˜ )= ∆ e (x, q,˜ q˜ ) , (6.61) the remaining part (universal). h ij→ij h e $ij,j Forwhere parton the Sudakov shower form this factor means: for the backward evolution of a given parton i is x x ′ q˜h ′2 z+ ′ e d˜q αS (z, q˜ ) ′ z fij z , q˜ 2 ∆ e (x, q,˜ q˜ )=exp dz P e (z, q˜ ) Θ p > 0 (6.62), ij→ij h − q˜′2 2π ij→ij xf (x, q˜′) ⊥ % &q˜ &x i ' ( ) ' ( and the product runs over all possible branchings ij i+j capable of producing a parton of type → i.Thisissimilartotheformfactorusedforfinal-stateradiation, Eq. (6.21), with the addition J. Bellm (Lund U.), Science Coffee, 25.1.2018 of the PDF factor, which guides the backward evolution.* The backward evolution can be performed using the veto algorithm in the same way as the forward evolution. We need to solve ∆ (x, q,˜ q˜ )= , (6.63) h R to give the scale of the branching. We start by considering thebackwardevolutionofi via a particular type of branching, ij i + j.Wecanobtainanoverestimateoftheintegrandinthe → Sudakov form factor * over q˜h ′2 z+ over over d˜q αS over over ∆ e (x, q,˜ q˜ )=exp dz P e (z)PDF (z) , (6.64) ij→ij h − q˜′2 2π ij→ij + &q˜ &x , where P over (z), αover and the overestimate of the limits must have the same properties as for ije →ij S final-state branching. In addition x x f e , q˜ PDFover (z) z ij z z, q,˜ x. (6.65) ≥ xfi ('x, q˜)( ∀ 48 Parton shower J. Bellm (Lund U.), Science Coffee, 25.1.2018 Colorlines J. Bellm (Lund U.), Science Coffee, 25.1.2018 Cluster or Lund Strings One also need to care about the rest of the proton colors. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Cluster or Lund Strings Cluster decay (string breaking) to get down to energies in the hadronic energies. Form mesons and barions. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Full Picture Factorized parts of the programs describe factorized physics. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Can we improve the description? Yes we can! Correct IR/C approximation with actual matrix elements! And possibly higher orders (loops). Problem: The parton shower already approximates both. Solution: Avoid double counting of similar contributions. Matching: Expand shower to first order in coupling. Subtract from NLO configuration. Start shower. Merging: Calculate the higher multiplicities with merging scale but multiply with no emission probability of shower. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Matching and Merging Matching: MC@NLO arXiv:0204244 matched LO PS[d ]=PS0[d ] Nason arXiv:0409146 Powheg arXiv:0709.2092 V LO + PS0[d + d 1P (z)d ] MLM arXiv:0108069 R Z LO + PS1[d d 1P (z)d ] CKKW arXiv:0109231 Merging: CKKW-L arXiv:0112284 merged V 0,Q PS[d 0+3 ]=PS0 [d 0 ⇢ ] NL3 arXiv:0811.2912 V 0,Q 1,q1 + PS1 [d 1 q1 ⇢ ] MEPS@NLO arXiv:1207.5031 V 0,Q 1,q1 2,q2 FxFx arXiv:1209.6215 + PS [d 2 ] 2 q1 q2 ⇢ UNLOPS arXiv:1211.7278 0,Q 1,q 2,q 1 2 Plätzer arXiv:1211.5467 + PS3[d 3 q1 q2 q3 ] H71 arXiv:1705.06700 Remove double counting: More details in lecture on the topic next month. J. Bellm (Lund U.), Science Coffee, 25.1.2018 p i CS Dipole Shower i ij j Vij,k ! pj pk k Instead of 1—>2 splitting use 2—>3 Modified splitting functions to Figure 1: Etoff ectivepreserve diagram momentum for the splitting of a final-stateremove parton soft codoublennected counting to a final-state 2 spectator.conservation The blob denotes between the m -partonshower matrix steps. element, and the outgoing↵S lines1+ labelz the Pq qg(z)= CF ! 2⇡ 1 z final-state partons participating in the splitting. pi + pj + pk =˜pij +˜pk considering processesp = zp without˜ + y colour-charged(1 z)˜p + initial-stk ate particles, such↵S as jet2 production in i ij k T Kq qg(z,y)= CF (1 + z) ! 2⇡ 1 z + zy lepton-lepton collisions, this is the only QCD radiation process and thus constitutes✓ the basis◆ of acorrespondingfinal-statepartonshower.However,theobspj =(1 z)˜pij + yzp˜k kT erved factorisation of the differential Catani, Seymour arXiv:9605323 cross section for producing an additional parton also holds in the presence of initial-state pk =(1 y)˜pk partons, where only the additional branching channelsDinsdale, discussed Ternick, below Weinzierl then arXiv:0709.1026 have to be taken into account as well. Schumann, Krauss arXiv:0709.1027 Plätzer, Gieseke arXiv:0909.5593 3.1.1 Massive case J. Bellm (Lund U.), Science Coffee, 25.1.2018 In the most general case all partons involved in the splittingcanhavearbitrarymasses,i.e. 2 2 2 2 2 2 2 2 2 p˜ij = mij,˜pk = pk = mk, pi = mi and pj = mj ,respectively.Inordertoavoidon-shelldecays, which should be described by their respective proper matrix element, only those situations are considered, where m2 m2 + m2. ij ≤ i j Kinematics: • Exact four-momentum conservation is ensured by the requirement p˜ +˜p = p + p + p Q. (25) ij k i j k ≡ The splitting is characterised by the dimensionless variables yij,k,˜zi andz ˜j.Theyare given by pipj pipk yij,k = , z˜i =1 z˜j = . (26) pipj + pipk + pjpk − pipk + pjpk With these definitions the invariant transverse momentum of partons i and j,definedin Eq. (19), can be written as k2 =(Q2 m2 m2 m2)y z˜ (1 z˜ ) (1 z˜ )2m2 z˜2m2 . (27) ⊥ − i − j − k ij,k i − i − − i i − i j For convenience, the rescaled parton masses mn µn = (n = i, j, k, ij) , (28) √Q2 13 Random Emission Angle JB arXiv:1801.06113 k2 =2˜p p˜ yz(1 z) T ij k After z and y are created, the algorithm needs to find k T, d with random angle. 2⇡ ✓ ◆ But: How to assign colours to the gluons? Can it make a difference at all? Can we improve this? J. Bellm (Lund U.), Science Coffee, 25.1.2018 Rearranging the Colours JB arXiv:1801.06113 How to assign colours to the gluons? 2 Johannes Bellm: Colour Rearrangement for Dipole Showers We emit from a dipole, so the new gluon splits the color line and creates two new. gluon is ’in between’ the new g10 and g20 . (b) is obtained when ga is closer to the quark qp than g1. (c) is a configu- Can it make a difference at all? ration that is not obtained by CS showers but can happen Yes! if the emitter-spectator relation is not clear as in Ariadne. Longer lines —> more emission. In the colourflow picture (a), (b) and (c) in Eq. 2 cor- Shorter lines —> less emission. respond to permutations of inner gluons and the weights of assigning the colour lines depends on the full chain. If Can we improve this? we would start the shower from the configuration received We need to find a way how to assign a after emission we would assign the colours according to Diagram 1: Simple process with three colour dipoles. This physical motivated weight. the weights in the colour representation. Here the emit- process is used to calculate the weight for the colour rear- Would be nice to have the ‚right‘ ted gluon ’feels’ the nearby gluon and the colours are ar- rangement. proportion of straight or zick-zack line. ranged accordingly. The independent dipole in a chain has no possibility to distinguish a preferred direction in terms of colour amplitudes. g and g is modified as well. Thus, an emission of a gluon 1 2 J. Bellm (Lund U.), Science Coffee, 25.1.2018 In the physical picture where most of the emission of g will a↵ect the kinematics of up to four of the resulting 1 is in the angle opened by g g or possibly but suppressed dipoles. If a gluon splits into a qq¯-pair the colour of the 1 2 closer (in this example) to the quark we can assume a dipole chain breaks and the quark carries the colour of the shielding of colours of dipoles g g g ....Thenthese split gluon and the anti-quark gets the anti-colour. Once 20 3 4 distant dipoles have little e↵ect on the emission o↵ g . the emission is performed the chain or two chains is/are 1 The colour connected quark q, however, is close to the evolved further until the shower algorithm is terminated colour line of gluon g . In order to construct a weight to by finding no emission scale above the infrared (IR) cuto↵. 20 distinguish between configurations (a) and (b) we can use Splitting a Dipole: Once a dipole in the chain is the simplest matrix element available that includes three identified to be the winning participant, a momentum dipoles namely e.g. uugg¯ , see Diagram 1. We are fraction z is chosen according to the functional form of the only interested in the distinction⇤! between colour structure splitting function. If spin-averaged splitting functions are q g g g and q g g g as the rest of the used, the radiation angle of the emission plane around 10 a 20 a 10 20 event remains unchanged. Even the identification of theu ¯ the dipole axis is chosen randomly on the interval [0, 2⇡). to represent gluon g is a good approximation as its colour For spin-dependent splitting functions, the radiation angle 20 charge vanishes in the weight ratio used to decide between can be biased by the helicity of the emitting parton. the states. After the Shower: Once the IR cuto↵ is reached the In the actual implementation, we define a phase space colour chains are interpreted as colour strings and hardonised point from three dipoles and incoming beams to de- in a Lund string model or the remaining gluons are split liver the energy needed for the dipole combination. We to break the chains in colour anti-colour qq¯-pairs which use MadGraph to generate the process e+e uugg¯ and then build the clusters of the cluster models. After form- calculate the weights of the squared colour! amplitudes ing of clusters/strings the process of colour reconnection w(1; 2; 3; 4) = jamp2[0] and w(1; 3; 2; 4) = jamp2[1] at the can rearrange the constituents of the clusters/strings and given . If a flat random number in [0, 1) is smaller than in addition cluster fissioning or string breaking happens before cluster masses/string lengths are reached that al- w(1; 3; 2; 4) Pswap = lows the conversion to hadronic states. w(1; 2; 3; 4) + w(1; 3; 2; 4) 3 Colour Rearrangement we swap the momenta of the gluons g2 and g3 which cor- responds to rearranging the colour structure. Note that If we interpret terms as in Eq. 1 as probabilities to choose the weights take into account interferences and parts of colour lines for the starting conditions of the showering o↵-shell e↵ects but neglect the non-diagonal elements in process, we now want to know what happens to the colour the colour basis. structure after emitting o↵ a dipole in a given dipole chain. As the matrix element is simple and fast to compute Emitting a gluon g from dipole g g 1 leads us to: a 1 2 we also allow swapping in the chain that was not modified by the last emission, by simply calculating the swapping q g1 = g2 g3 g4 .... gn q¯ (2) probability for all neighboring triple dipoles. If the colour chain is already in an order preferred by the matrix ele- (a) q g0 ga g0 g3 g4 .... gn q¯ ! 1 2 ment this will not change the probability of having this (b) q g g0 g0 g g .... g q¯ ! a 1 2 3 4 n colour structure, see third comment in Sec. 4, if not the (c) q g0 g0 ga g3 g4 .... gn q¯ lines will be rearranged to the ’preferred’ order. Preferred ! 1 2 is a probabilistic mixture of short or long chains which is now given byP rather than an uncontrolled function Here gluon ga should be identified as the softer gluon swap of evolution variable and emission angle. in the splitting of g1 with a spectator g2. Configuration (a) is obtained if the emission angle is such that the softer 1 The following argument holds for emissions o↵ the ends of the chains with a less complex structure. where we follow the convention of reading the fully anti-symmetric structure constant A Classindices member in counter functions clock-wise order. The color structure of any amplitude,24 tree-level or beyond, pure QCD or not, can thus be expressed in terms of theseobjectsalone.For observables we are – as QCD is confining – only interested in color summed/averaged quantities. 1Introduction Letting c1 denote the color structure of the amplitude under consideration, we are thus 2 3 The descriptioninterested of QCD in |c color1| where structure the in scalar the presence product of manis givenyexternalcoloredpartonsis by summing over all external color afieldofincreasedimportance.Somemethodsforperformingindices, i.e., automatic color summations of fully contracted vacuum bubbles, for example as implemen∗teda1 a2 in... FeynCalca1 a2... [1], in the C ⟨c1|c2⟩ = c1 c2 (2.3) program COLOR [2], or as presented in [3], have beena around1,a2,... for a while, and recently a more flexible Mathematica package, ColorMath [4], allowing! color structures with any number of2 with ai =1,...,Nc if parton i is a quark or an anti-quark and ai =1,...,N − 1ifpartoni open indices, has been published. Yet other general purpose event generator codes, suchc is a gluon. as MadGraph [5], have separate built in routines for dealing with the color structure. In the presentClearly, paper in any a stand-alone QCD calculation, C++ code, theColorFull color amplitudes,,designedfordealingwithc1 and c2,maybekeptasthey color contractionare, with using color color structure bases is read presented off from1. ColorFull the contributingis written Feyn withman interfacing diagrams. to Alternatively event generators–andthisislikelytobethepreferredsolutionsformoretha in mind, and is currently interfaced to Herwig++ (2.7) [6, 7],nafewpartons–theymay but can also be usedbe as a decomposed stand-alone package into a color for investigations basis (spanning in color set),space. such as a trace basis [8–16], a color flow ColorFullbasis [19is]oramultipletbasis[ based on trace bases [8–1716,],18 where]. the basis vectors are given by (prod- ucts of) closed and open quark-lines, but the code also offers functionality for reading in and treating any (orthogonal) basis for color space, such as multiplet bases [17, 18]. 3Tracebaseswhere we follow the convention of reading the fully anti-symmetric structure constant The intent of this paper is to convey the underlying idea of ColorFull.Forfull indices in counter clock-wise order. The2 color structure of any amplitude, tree-level or technicalOne details way we of refer organizing tobeyond, the online calculations pure Doxygen QCD or in not, documentat color can thus spaceion be is expressed.Tosetthestage,abrief to use in tr termsace bases of thes [eobjectsalone.For8–16]. To see that introductionthis is to always the QCD possible, colorobservables space we note is we given are that – in as the section QCD triple-gluon is2 confiningand the – verte trace onlyxcanbeexpressedas interested basis approach in color is summed/averaged presented in section 3.Followingthis,someremarksaboutthecomputationalstraquantities. tegy are a a a made in section 4 and the key designLetting featuresc1 denote are the presented color structure in section of the amplitude5,whereasexamples under consideration, we are thus abc 12 3 if = interested in=|c1| where the scalar product− Herwig++is given by summing over all external color(3.1) of stand-alone usage are given in sectionTR6,andtheinterfaceto⎡ ⎤is commented indices,c i.e., b c b c upon in section 7.Thefollowingsection,sectionb 8,describestheclassesof∗a1 a2... ColorFulla1 a2... and ⎢ ⟨c1|c2⟩ = c1 c⎥2 (2.3) code validation is discussed1 ina sectioni b j 9.Finallysomeconcludingremarksaremadeinc ⎣k b i a j a1,ac 2k,... 1 ⎦ a b c b a c = (t ) j(t ) k(t ) i − (t ) j(t ) k(t!) i = tr[t t t ] − tr[2t t t ] . section 10. TR with ai =1,...,Nc if parton i is a quark or an anti-quarkTR and ai =1,...,Nc − 1ifpartoni (is a gluon. ) a (b ab ) where TR is theover normalization generators, meaning of the SU( thatNc a)generators,tr( general tree-levelt t gluon)=TR amδ plitude,commonlytaken can be decomposed as 2Colorspace Clearly, in any QCD calculation, the color amplitudes, c1 and c2,maybekeptasthey to be 1/2or1. are, with color structure read off from the contributing Feynman diagrams. Alternatively g1 gσ2 gσn Apart from four-gluonUsing thisTrace vertices relation–andthisislikelytobethepreferredsolutionsformoretha forBasis on which every the triple-gluonq color q +n structur vertexgecanberewrittenintermsof in any amplitudenafewpartons–theymay results in general in a be decomposed into a color basis (spanning set), such as a trace basis [8–16], a color... flow (one-gluonsum contracted) of products triple-gluon of (connected) vertices, traces the QCD over Lag SU(rangiangN1 cg)generatorsandopenquark-lines.Moreσ2 containsgσn Mbasis(g1 [,g192]oramultipletbasis[,...,gn)= 17tr(, 18t]. t ...t )A(σ)= A(σ). (3.7) specifically, there is one open quark-lineσ∈SN for−1 every incomingquark/outgoinganti-quarkandσ∈SN −1 !g a !g pq 3Tracebases outgoing quark/incoming anti-quark.i (Notej that, from a color perspective outgoing anti- pg a i 1 quark-gluon vertices, =(t ) , (2.1) pg2 quarks are equivalentOne toway incoming of organizing quarks; calculations we will in color here spacej refer is toto use them trace collectively bases [8–16]. To as see quarks. that That only fully connected color structures enter in tree-level gluon amplitudes can eas- and Similarly outgoingthis quarks is always are possible, equivalent we note to that incoming the triple-gluon anti-quarks, vertexcanbeexpressedas and will be referred to ily be understooda froma the decomposition of Feynman diagramsintobasisvectors;upon as anti-quarks.) a abc a triple-gluonapplication vertices,abc of eq. (3.1)allexternalgluonsremainattachedtoaquark-line,and–w1 c = if , (2.2) p hile To further simplifyif the= color structure,= ⎡ we may contract− every⎤ internal gluon propaga-(3.1) q¯ b TR contracting internalb gluonsc usingb the Fierzc identity,b c eq. (3.2)–theyremainconnectedtopq tor (which after application of eq. (3.1)connectsquark-lines)usingtheFierz(completeness)⎢ ⎥ 1 ⎣ ⎦ ColorFull can be downloadedthe same from quark-line,http://colorfull.hepforge.org/1 a asi theb j colorc k suppressedb i . a j c termsk 1 cancela b eac ch otherb a c out. (This canpg1 be 2The automaticallyrelation generated Doxygen documentation= (t ) j(t is) avaik(t lable) i − at(t ) j(t ) k(t ) i = tr[t t t ] − tr[t t t ] . p proved by a shortTR calculation.) The same cancellationTR appears for gluon exchange betweeng2 http://colorfull.hepforge.org/doxygen. ( ) a (b ab ) where TR is the normalization of the SU(Nc)generators,tr(1 t t )=TRδ ,commonlytaken aquarkandagluon,meaningthatalsotree-levelcolorstructures for one qq-pair and Ng to be 1/2or1. = TR − . (3.2) gluons must be of the “fully* connected” formNc of a trace+ that hasbeencutopen,anopen Using this relation on every triple-gluon vertex in any amplitude results in general in a pq¯ 3 quark-line, It is not hard to provesum of that products this actuallyof (connected) is a scalar traces product. over SU(Nc)generatorsandopenquark-lines.More specifically, there–2– is one open quark-line for every incomingquark/outgoinganti-quarkand outgoing quark/incoming anti-quark. (Note that, from a color perspective outgoinggσ1 gσ2 anti- gσn quarks are equivalent to incominggσ quarks;gσ wegσ willq here1 refer to them collectively as quarks. M(q1,g3,...,g , q2)= (t 1 t 2 ...t n ) A(σ)= A(σ), Similarly outgoingn quarks are equivalent to incomingq2 anti-quarks, and will be referred... to σ∈S σ∈S as anti-quarks.) !Ng –3– !Ng To further simplify the color structure, we may contract every internalq1 gluon propaga- q2 (3.8) tor (which after application of eq. (3.1)connectsquark-lines)usingtheFierz(completeness) Equations from: M. Sjöldahl arXiv:1412.3967 relation i.e., only the first two basis vectors in eq. (3.41)areneeded.However,whentheFierz = TR − . (3.2) identity is applied directly to a gluon* exchange betweenJ.Nc Bellm qua (Lund+ rks, U.), as in Science eq. (3.2 Coffee), both, terms25.1.2018 do appear,3 and color structures with up to N disconnected quark-lines may appear even It is not hard to prove that this actually is a scalar product.q at tree-level. Starting from a trace basis tree-level color structure, for example a single trace over gluons, and exchanging a gluon between two partons may split off adisconnectedcolor –3– structure, such as in g1 g2 g3 g4 g1 g2 g3 g4 g1 g4 g2 g3 = −TR − TR . (3.9) Thus, counting to lg additional gluon exchanges (on top of a tree-level diagram),thecolor structures can not consist of more than max(1,Nq)+lg open and closed quark-lines, two in the above case. In general, when any Feynman diagram is decomposed into a trace basis, there can be at most Nq + ⌊Ng/2⌋ quark-lines, since all gluons may be disconnected from the quarks, but no gluon can stand alone in a trace, giving the factor ⌊Ng/2⌋. For NLO color structures having a quark-loop in the Feynman diagram, the quark-loop is necessarily connected to the remaining color structure via at least one gluon exchange, –5– Pictorial representation pq pq pq pg1 pg1 pg1 pg2 pg2 pg2 N 3 N 3 N ⇠ C ⇠ C ⇠ C pq¯ pq¯ pq¯ pq pq pq pg1 pg1 pg1 pg2 pg2 pg2 pq¯ pq¯ pq¯ J. Bellm (Lund U.), Science Coffee, 25.1.2018 Pictorial representation pq pq p g1 pg1 pg2 p g2 A(q, 1, 2, q¯) 3 NC N ⇠ ⇠ C pq¯ pq¯ A(q, 1, 2, q¯)⇤ A(q, 2, 1, q¯)⇤ pq pq p pg1 ⇣ ⌘ g1 p pg2 g2 A(q, 2, 1, q¯) 3 N NC ⇠ C ⇠ pq¯ pq¯ matrix elements w1 = jamp2[0] = A(q, 1, 2, q¯) A(q, 1, 2, q¯)⇤ jamp2[1] A(q, 2, 1, q¯) A(q, 2, 1, q¯)⇤⌘ w =⇣ = ⌘⇣ 2 J. Bellm (Lund U.), Science Coffee, 25.1.2018 Rearranging the Colours JB arXiv:1801.06113 q q 1. Interpret gluons in LCA g g 2. For all triple dipoles: ⇣ ⇣ w2 If < g g R w + w ⇣ ⇣ 1 2 g g2 swap gluon momenta. Johannes Bellm: Colour Rearrangement for Dipole Showers ⇣ ⇣ gluon is ’in between’ the new g10 and g20 . (b) is obtained g g when ga is closer to the quark qp than g1. (c) is a configu- ⇣ ⇣ ration that is not obtained by CS showers but can happen if the emitter-spectator relation is not clear as in Ariadne. g g In the colourflow picture (a), (b) and (c) in Eq. 2 cor- ⇣ ⇣ respond to permutations of inner gluons and the weights of assigning the colour lines depends on the full chain. If q¯ q¯ we would start the shower from the configuration received ⇣ ⇣ after emission we would assign the colours according to Diagram 1: Simple process with three colour dipoles. This the weights in the colour representation. Here the emit- process is used to calculate the weight for the colour rear- J. Bellm (Lund U.), Science Coffee, 25.1.2018 ted gluon ’feels’ the nearby gluon and the colours are ar- rangement. ranged accordingly. The independent dipole in a chain has no possibility to distinguish a preferred direction in terms of colour amplitudes. g and g is modified as well. Thus, an emission of a gluon 1 2 In the physical picture where most of the emission of g will a↵ect the kinematics of up to four of the resulting 1 is in the angle opened by g g or possibly but suppressed dipoles. If a gluon splits into a qq¯-pair the colour of the 1 2 closer (in this example) to the quark we can assume a dipole chain breaks and the quark carries the colour of the shielding of colours of dipoles g g g ....Thenthese split gluon and the anti-quark gets the anti-colour. Once 20 3 4 distant dipoles have little e↵ect on the emission o↵ g . the emission is performed the chain or two chains is/are 1 The colour connected quark q, however, is close to the evolved further until the shower algorithm is terminated colour line of gluon g . In order to construct a weight to by finding no emission scale above the infrared (IR) cuto↵. 20 distinguish between configurations (a) and (b) we can use Splitting a Dipole: Once a dipole in the chain is the simplest matrix element available that includes three identified to be the winning participant, a momentum dipoles namely e.g. uugg¯ , see Diagram 1. We are fraction z is chosen according to the functional form of the only interested in the distinction⇤! between colour structure splitting function. If spin-averaged splitting functions are q g g g and q g g g as the rest of the used, the radiation angle of the emission plane around 10 a 20 a 10 20 event remains unchanged. Even the identification of theu ¯ the dipole axis is chosen randomly on the interval [0, 2⇡). to represent gluon g is a good approximation as its colour For spin-dependent splitting functions, the radiation angle 20 charge vanishes in the weight ratio used to decide between can be biased by the helicity of the emitting parton. the states. After the Shower: Once the IR cuto↵ is reached the In the actual implementation, we define a phase space colour chains are interpreted as colour strings and hardonised point from three dipoles and incoming beams to de- in a Lund string model or the remaining gluons are split liver the energy needed for the dipole combination. We to break the chains in colour anti-colour qq¯-pairs which use MadGraph to generate the process e+e uugg¯ and then build the clusters of the cluster models. After form- calculate the weights of the squared colour! amplitudes ing of clusters/strings the process of colour reconnection w(1; 2; 3; 4) = jamp2[0] and w(1; 3; 2; 4) = jamp2[1] at the can rearrange the constituents of the clusters/strings and given . If a flat random number in [0, 1) is smaller than in addition cluster fissioning or string breaking happens before cluster masses/string lengths are reached that al- w(1; 3; 2; 4) Pswap = lows the conversion to hadronic states. w(1; 2; 3; 4) + w(1; 3; 2; 4) 3 Colour Rearrangement we swap the momenta of the gluons g2 and g3 which cor- responds to rearranging the colour structure. Note that If we interpret terms as in Eq. 1 as probabilities to choose the weights take into account interferences and parts of colour lines for the starting conditions of the showering o↵-shell e↵ects but neglect the non-diagonal elements in process, we now want to know what happens to the colour the colour basis. structure after emitting o↵ a dipole in a given dipole chain. As the matrix element is simple and fast to compute Emitting a gluon g from dipole g g 1 leads us to: a 1 2 we also allow swapping in the chain that was not modified by the last emission, by simply calculating the swapping q g1 = g2 g3 g4 .... gn q¯ (2) probability for all neighboring triple dipoles. If the colour chain is already in an order preferred by the matrix ele- (a) q g0 ga g0 g3 g4 .... gn q¯ ! 1 2 ment this will not change the probability of having this (b) q g g0 g0 g g .... g q¯ ! a 1 2 3 4 n colour structure, see third comment in Sec. 4, if not the (c) q g0 g0 ga g3 g4 .... gn q¯ lines will be rearranged to the ’preferred’ order. Preferred ! 1 2 is a probabilistic mixture of short or long chains which is now given byP rather than an uncontrolled function Here gluon ga should be identified as the softer gluon swap of evolution variable and emission angle. in the splitting of g1 with a spectator g2. Configuration (a) is obtained if the emission angle is such that the softer 1 The following argument holds for emissions o↵ the ends of the chains with a less complex structure. Colour Rearrangement for Dipole Showers JB arXiv:1801.06113 Total charged multiplicity Heavy jet mass (charged) − ) ch 1 s n 1 10 / 2 h 10 /d − 2 M σ 10 ALEPH Data ( d 1 ALEPH Data /d σ = −3 no swap, µ 1.0 GeV swap, µ = 1.0 GeV 10 N 2/ = no swap, µ 0.6 GeV d −1 −4 10 swap, µ = 0.6 GeV = N 10 no swap, µ 0.8 GeV swap, µ = 0.8 GeV 1/ −2 1.4 101.4 1.2 1.2 1 1 0.8 0.8 MC/Data 0.6 MC/Data 0.6 20 40 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 n M2 s ch h/ 1. Use tune for default shower. 2. CMW scheme + Cutoff variation. Not tuned results to see the effect turned out to describe data really nice. J. Bellm (Lund U.), Science Coffee, 25.1.2018 Colour Rearrangement for Dipole Showers JB arXiv:1801.06113 Differential 5-jet rate with Durham algorithm (91.2 GeV) 56 3 y 10 2 /d 10 σ 1 d 10 OPAL Data Total charged multiplicity 1 no swap, µ = 1.0 GeV − ch 1 = n − 10 swap, µ 1.0 GeV 10 1 −2 /d 10 σ d 1.4 σ 2/ − 1.2 10 2 Data 1 swap 33 0.8 swap 43 MC/Data 0.6 −3 swap 53 − − − − 10 10 5 10 4 10 3 10 2 swap 44 yDurham C parameter 56 −4 C 10 New and preliminary /d σ 1 d DELPHI Data 1.4 N no swap, µ = 1.0 GeV − 1.2 10 1 no swap, µ = 0.6 GeV 1 no swap, µ = 0.8 GeV 1.4 MC/Data 0.8 1.2 0.6 1 20 40 0.8 n MC/Data ch 0.6 0 0.2 0.4 0.6 0.8 C J. Bellm (Lund U.), Science Coffee, 25.1.2018 Next Steps 1. Extent method to LHC: - incoming colors - gluon-only chains 2. Validate eeuuggg and eeuugggg 3. Tune shower to LEP and LHC data. J. Bellm (Lund U.), Science Coffee, 25.1.2018 The End Thank you! Additional Reading: Color Flows arXiv:0209271 Trace Basis arXiv:9910563 Mangano: Introduction to QCD CERN:454171 Skands: Introduction to QCD arXiv:1207.2389 Höche: Introduction to parton shower event generators arXiv:1411.4085 General-purpose event generators for LHC physics arXiv:1101.2599 J. Bellm (Lund U.), Science Coffee, 25.1.2018 PS[d\sigma^{matched}]&=PS_0[d\sigma^ {LO}]\\ &+PS_0[d\sigma^{V}+\int d\phi_1 P(z)d\sigma^{LO}]\\ &+PS_1[d\sigma^{R}- d\phi_1 P(z)d\sigma^{LO}] K_{q\to qg}(z,y)= \frac{\alpha_S}{2\pi}C_F\left(\frac{2}{1 - z + z y} - (1+z)\right) p_i&+p_j+p_k=\tilde{p}_{ij} + \tilde{p}_{k} \\ p_i &= z \tilde{p}_{ij} + y (1-z) \tilde{p}_{k} + k_T \\ p_j &= (1-z) \tilde{p}_{ij} + y z \tilde{p}_{k} - k_T \\ p_k &= (1-y) \tilde{p}_{k} PS[d\sigma_{0+3}^{merged}]&=PS^V_0[d\sigma_0 \Delta^{0,Q}_{\rho}]\\ &+PS^V_1[d\sigma_1 \Delta^{0,Q}_{q_1}\Delta^{1,q_1}_{\rho}]\\ &+PS^V_2[d\sigma_2 \Delta^{0,Q}_{q_1}\Delta^{1,q_1}_{q_2}\Delta^{2,q_2}_{\rho}]\\ &+PS_3[d\sigma_3 \Delta^{0,Q}_{q_1}\Delta^{1,q_1}_{q_2}\Delta^{2,q_2}_{q_3}]