CASCADE3 A Monte Carlo event generator based on TMDs
S. Baranov, A. Bermudez Martinez, L.I. Estevez Banos, F. Guzman, F. Hautmann, H. Jung, A. Lelek, J. Lidrych, A. Lipatov, M. Malyshev, M. Mendizabal, S. Taheri Monfared, A.M. van Kampen, Q. Wang, H. Yang
DDESY 21-00522 Jan 2021
CASCADE3A Monte Carlo event generator based on TMDs
S. Baranov , A. Bermudez Martinez , L.I. Estevez Banos , F. Guzman ,F. Hautmann , , H. Jung , A. Lelek , J. Lidrych , A. Lipatov ,M. Malyshev , M. Mendizabal , S. Taheri Monfared ,A.M. van Kampen , Q. Wang , H. Yang , Lebedev Physics Institute, Russia DESY, Hamburg, Germany InSTEC, Universidad de La Habana, Cuba Elementary Particle Physics, University of Antwerp, Belgium RAL and University of Oxford, UK SINP, Moscow State University, Russia School of Physics, Peking University, China
Abstract
The C
ASCADE k t -factorization, either internally implemented or from external pack-ages via LHE files, can be processed for parton showering and hadronization. The initialstate parton shower is tied to the TMD parton distribution, with all parameters fixed bythe TMD distribution. The simulation of processes for high energy hadron colliders has been improved significantlyin the past years by automation of next-to-leading order (NLO) calculations and matching ofthe hard processes to parton shower Monte Carlo event generators which also include a sim-ulation of hadronization. Among those automated tools are the M AD G RAPH MC @ NLO [1]generator based on the MC @ NLO [2–5] method or the P
OWHEG [6, 7] generator for the calcu-lation of the hard process. The results from these packages are then combined with either theH
ERWIG [8] or P
YTHIA [9] packages for parton showering and hadronization. Different jetmultiplicities can be combined at the matrix element level and then merged with special pro-cedures, like the MLM merging [10] for LO processes, the FxFx [11] or MiNLO method [12]for merging at NLO, among others. While the approaches of matching and merging ma-trix element calculations and parton showers are very successful, two ingredients important1 a r X i v : . [ h e p - ph ] J a n or high energy collisions are not (fully) treated: the matrix elements are calculated withcollinear dynamics and the inclusion of initial state parton showers results in a net trans-verse momentum of the hard process; the special treatment of high energy effects (small x )is not included.The C ASCADE
Monte Carlo event generator, developed originally for small x processesbased on high-energy factorization [13] and the CCFM [14–17] evolution equation, has beenextended to cover the full kinematic range (not only small x ) by applying the Parton Branch-ing (PB) method and the corresponding PB Transverse Momentum Dependent (TMD) partondensities [18, 19]. The initial state evolution is fully described and determined by the TMDdensity, as it was in the case of the CCFM gluon density, but now available for all flavorspecies, including quarks, gluons and photons at small and large x and any scale µ . For ageneral overview of TMD parton densities, see Ref. [20].With the advances in determination of PB TMDs [18,19], it is natural to develop a scheme,where the initial parton shower follows as close as possible the TMD parton density andwhere either collinear (on-shell) or k t -dependent (off-shell) hard process calculations canbe included at LO or NLO. In order to be flexible and to use the latest developments inautomated matrix element calculations of hard process at higher order in the strong coupling α s , events available in the Les Houches Event (LHE) file format [21], which contains all theinformation of the hard process including the color structure, can be further processed forparton shower and hadronization in C ASCADE
ASCADE x mode of C ASCADE The cross section for the scattering process of two hadrons A and B can be written in collinearfactorization as a convolution of the partonic cross section of partons a and b , a + b → X , andthe densities f a ( b ) ( x, µ ) of partons a ( b ) inside the hadrons A ( B ), σ ( A + B → Y ) = (cid:90) dx a (cid:90) dx b f a ( x a , µ ) f b ( x b , µ ) σ ( a + b → X ) , (1)where x a ( x b ) are the fractions of the longitudinal momenta of hadrons A, B carried by thepartons a ( b ) , σ ( a + b → X ) is the partonic cross section, and µ is the factorization scale of theprocess. The final state Y contains the partonic final state X and the recoils from the partonevolution and hadron remnants.In C ASCADE .1 On-shell processes
The hard processes in collinear factorization (with on-shell initial partons, withouttransverse momenta) can be calculated by standard automated methods like M AD -G RAPH MC @ NLO [1] for multileg processes at LO or NLO accuracy. The matrix elementprocesses are calculated with collinear parton densities (PDF), as provided by LHAPDF [22].We extend the factorization formula given in eq.(1) by replacing the collinear parton den-sities f ( x, µ ) by TMD densities A ( x, k t , µ ) with k t being the transverse momentum of theinteracting parton, and integrating over the transverse momenta.However, when the hard process is to be combined with a TMD parton density, as de-scribed later, the integral over k t of the TMD density must agree with the collinear ( k t -integrated) density; this feature is guaranteed by construction for the PB-TMDs (also avail-able as integrated PDFs in LHAPDF format).In a LO partonic calculation the TMD or the parton shower can be included respectingenergy momentum conservation, as described below. In an NLO calculation based on theMC@NLO method [2–5] the contribution from collinear and soft partons is subtracted, asthis is added later with the parton shower. For the use with PB TMDs, the H ERWIG
ASCADE p a,b are recalculatedtaking into account the virtual masses Q a = k t,a and Q b = k t,b [25], E a,b = 12 √ ˆ s (cid:0) ˆ s ± ( Q b − Q a ) (cid:1) (2) p z a,b = ± √ ˆ s (cid:113) (ˆ s + Q a + Q b ) − Q a Q b (3)with ˆ s = ( p a + p b ) with p a ( p b ) being the four-momenta of the interacting partons a and b . Thepartonic system is then rotated and boosted back to the overall center-of-mass system of thecolliding particles. By this procedure, the parton-parton mass √ ˆ s is exactly conserved, whilethe rapidity of the partonic system is approximately restored, depending on the transversemomenta.In Fig. 1 a comparison of the Drell-Yan (DY) mass, transverse momentum and rapidityis shown for an NLO calculation of DY production in pp collisions at √ s = 13 TeV in the3ass range < m DY < GeV. The curve labelled NLO(LHE) is the calculation of M AD -G RAPH MC @ NLO with the subtraction terms, the curve NLO(LHE+TMD) is the predictionafter the transverse momentum is included according to the procedure described above. Inthe p T spectrum one can clearly see the effect of including transverse momenta from theTMD distribution. The DY mass distribution is not changed, and the rapidity distribution isalmost exactly reproduced, only at large rapidities small differences are observed. NLO (LHE)NLO (LHE+TMD) − − − − − − − − − − − − Drell-Yan production at √ s =
13 TeV / σ d σ / d m ( G e V − ) . . . m DY (GeV) R a t i o NLO (LHE)NLO (LHE+TMD)30 < m DY < − − − − − − Drell-Yan production at √ s =
13 TeV / σ d σ / dp T ( G e V − ) − . . . p T (GeV) R a t i o NLO (LHE)NLO (LHE+TMD)30 < m DY < − − − Drell-Yan production at √ s =
13 TeV / σ d σ / dy - - . . . y R a t i o Figure 1: Distributions of Drell-Yan mass, transverse momentum and rapidity for pp → DY + X at √ s = 13 TeV. The hard process is calculated with M AD G RAPH MC @ NLO .NLO(LHE) is the prediction including subtraction terms, NLO(LHE+TMD) includes trans-verse momenta of the interacting partons according to the description in the text.The transverse momenta k t are generated according to the TMD density A ( x, k t , µ ) , atthe original longitudinal momentum fraction x and the hard process scale µ . In a LO calcu-lation, the full range of k t is available, but in an NLO calculation via the MC@NLO method a shower scale defines the boundary between parton shower and real emissions from the matrixelement, limiting the transverse momentum k t . Technically the factorization scale µ is calcu-lated within C ASCADE lhescale ) as it is not directly accessible from theLHE file, while the shower scale is given by
SCALUP . The shower scale guarantees that the TMDand later the parton shower does not generate transverse momenta which would violate thecollinear factorization ansatz.The advantage of using TMDs for the complete process is that the kinematics are fixed,independent of simulating explicitly the radiation history from the parton shower. For in-clusive processes, for example inclusive Drell-Yan processes, the details of the hadronic finalstate generated by a parton shower do not matter, and only the net effect of the transversemomentum distribution is essential. However, for processes which involve jets, the details ofthe parton shower become also important. The parton shower, as described below, followsvery closely the transverse momentum distribution of the TMD and thus does not changeany kinematic distribution after the transverse momentum of the initial partons are included.All hard processes, which are available in M AD G RAPH MC @ NLO can be used withinC
ASCADE
3. The treatment of multijet merging is described in Section 8.4 .2 Off-shell processes
In a region of phase space, where the longitudinal momentum fractions x become very small,the transverse momentum of the partons cannot be neglected and has to be included alreadyat the matrix element level, leading to so-called off-shell processes.In off-shell processes a natural suppression at large k t [26] (with k t > µ ) is obtained,shown explicitly in Fig. 2, where the matrix element for g ∗ g ∗ → Q ¯ Q , with Q being a heavyquark, is considered. The process is integrated over the final state phase space [27], ˜ σ ( k t ) = (cid:90) dx x dφ , dLips | M E | (1 − x ) , (4)where dLips is the Lorentz-invariant phase space of the final state, ME is the matrix-elementfor the process, φ , is the azimuthal angle between the two initial partons, and a simplescale-independent and k t -independent gluon density xG ( x ) = (1 − x ) is included whichsuppresses large- x contributions. In Fig. 2 we show ˜ σ ( k t ) normalized to its on-shell value ˜ σ (0) at √ s = 13000 GeV as a function of the transverse momentum of the incoming gluon k t, for different values of x , which are chosen such that the ratio k t, / ( x s ) is kept constant.
10 [GeV] t,2 k00.20.40.60.811.21.41.61.82 ( ) s / s =0.001000 , x =1.000000 GeV t,12 k =0.010000 , x =10.000000 GeV t,12 k =0.100000 , x =100.000000 GeV t,12 k =13 TeVsm=5 GeV, pp
10 [GeV] t,2 k00.20.40.60.811.21.41.61.82 ( ) s / s =0.001000, m=5 GeV , x =1.000000 GeV t,12 k =0.001000, m=175 GeV , x =1.000000 GeV t,12 k =13 TeVspp Figure 2: The reduced cross section ˜ σ ( k t ) / ˜ σ (0) as a function of the transverse momentum k t, of the incoming gluon at √ s = 13000 GeV. (Left) for different values of k t, and x , (right) fordifferent heavy flavor masses and fixed values of k t, and x .In Fig. 2 (left) predictions are shown for bottom quarks with mass m = 5 GeV and dif-ferent k t, , in Fig. 2 (right) a comparison is made for different heavy quark masses. Usingoff-shell matrix elements a suppression at large transverse momenta of the initial partons isobtained, depending on the heavy flavor mass and the transverse momentum. In a collinearapproach, with implicit integration over transverse momenta of the initial state partons, the5ransverse momenta are limited by a theta function at the factorization scale, while off-shellmatrix elements give a smooth transition to a high k t tail.When using off-shell processes, BFKL or CCFM type parton densities should be used tocover the full available phase space in transverse momentum, which can lead to k t ’s largerthan the transverse momentum of any of the partons of the hard process [28]. Until now, onlygluon densities obtained from CCFM [14–17] or BFKL [29–31] are available, thus limiting theadvantages of using off-shell matrix elements to gluon induced processes.Several processes with off-shell matrix elements are implemented in C ASCADE A T IE [33] and P EGA - SUS [34]. The events from the hard process are then read with the C
ASCADE A T IE or P EGASUS no further corrections need tobe performed and the event can be directly passed to the showering procedure, described inthe next section.Lepto(photo)production process IPRO Reference γ ∗ g ∗ → q ¯ q
10 [35] γ ∗ g ∗ → Q ¯ Q
11 [35] γ ∗ g ∗ → J/ψg g ∗ g ∗ → q ¯ q
10 [35] g ∗ g ∗ → Q ¯ Q
11 [35] g ∗ g ∗ → J/ψg g ∗ g ∗ → Υ g g ∗ g ∗ → χ c g ∗ g ∗ → χ b g ∗ g ∗ → J/ψJ/ψ
21 [40] g ∗ g ∗ → h
102 [41] g ∗ g ∗ → ZQ ¯ Q
504 [42, 43] g ∗ g ∗ → Zq ¯ q
503 [42, 43] g ∗ g ∗ → W q i Q j
514 [42, 43] g ∗ g ∗ → W q i q j
513 [42, 43] qg ∗ → Zq
501 [44] qg ∗ → W q
511 [44] qg ∗ → qg
10 [45] gg ∗ → gg
10 [45]Table 1:
Processes included in C ASCADE . Q stands for heavy quarks, q for light quarks. Initial State Parton Shower based on TMDs
The parton shower, which is described here, follows consistently the parton evolution of theTMDs. By this we mean that the splitting functions P ab , the order and the scale in α s as wellas kinematic restrictions are identical to both the parton shower and the evolution of theparton densities (for NLO PB TMD densities, the NLO DGLAP splitting functions [46, 47]together with NLO α s is applied, while for the LO TMD densities the corresponding LOsplitting functions [48–50] and LO α s is used). The PB method describes the TMD parton density as (cf eq.(2.43) in Ref. [18]) x A a ( x, k t , µ ) = ∆ a ( µ ) x A a ( x, k t , µ ) + (cid:88) b (cid:90) dq q dφ π ∆ a ( µ )∆ a ( q ) Θ( µ − q ) Θ( q − µ ) × (cid:90) z M x dz P ( R ) ab ( α s ( f ( z, q )) , z ) xz A b (cid:16) xz , k (cid:48) t , q (cid:17) , (5)with z M < defining resolvable branchings, k ( q c ) being the transverse momentum vectorof the propagating (emitted) parton, respectively. The transverse momentum of the partonbefore branching is defined as k (cid:48) t = | k + (1 − z ) q | with q = q c / (1 − z ) being the rescaledtransverse momentum vector of the emitted parton (see Fig. 3, with the notation k t = | k | and q = | q | ) and φ being the azimuthal angle between q and k . The argument in α s is in generala function of the evolution scale q . Higher order calculations indicate the transverse momen-tum of the emitted parton as the preferred scale. The real emission branching probabilityis denoted by P ( R ) ab ( α s ( f ( z, q )) , z ) including α s as described in Ref. [18] (in the following weomit α s in the argument of P ( R ) ab for easier reading). The Sudakov form factor is given by: ∆ a ( z M , µ, µ ) = exp (cid:32) − (cid:88) b (cid:90) µ µ dq q (cid:90) z M dz z P ( R ) ba (cid:33) . (6)Dividing Eq.(5) by ∆ a ( µ ) and differentiating with respect to µ gives the differential formof the evolution equation describing the probability for resolving a parton with transversemomentum k (cid:48) and momentum fraction x/z into a parton with momentum fraction x andemitting another parton during a small decrease of µ , µ ddµ (cid:18) x A a ( x, k t , µ )∆ a ( µ ) (cid:19) = (cid:88) b (cid:90) z M x dz dφ π P ( R ) ab xz A b (cid:0) xz , k (cid:48) t , µ (cid:1) ∆ a ( µ ) . (7)The normalized probability is then given by ∆ a ( µ ) x A a ( x, k t , µ ) d (cid:18) x A a ( x, k t , µ )∆ a ( µ ) (cid:19) = (cid:88) b dµ µ (cid:90) z M x dz dφ π P ( R ) ab xz A b (cid:0) xz , k (cid:48) t , µ (cid:1) x A a ( x, k t , µ ) (8)7his equation can be integrated between µ i − and µ to give the no-branching probability(Sudakov form factor) for the backward evolution ∆ bw , log ∆ bw ( x, k t , µ, µ i − ) = log (cid:18) ∆ a ( µ )∆ a ( µ i − ) x A a ( x, k t , µ i − ) x A a ( x, k t , µ ) (cid:19) (9) = − (cid:88) b (cid:90) µ µ i − dq (cid:48) q (cid:48) dφ π (cid:90) z M x dz P ( R ) ab x (cid:48) A b ( x (cid:48) , k (cid:48) t , q (cid:48) ) x A a ( x, k t , q (cid:48) ) , with x (cid:48) = x/z . This Sudakov form factor is very similar to the Sudakov form factor inordinary parton shower approaches, with the difference that for the PB TMD shower the ratioof PB TMD densities [ x (cid:48) A b ( x (cid:48) , k (cid:48) t , q (cid:48) )] / [ x A a ( x, k t , q (cid:48) )] is applied, which includes a dependenceon k t .In Eq.(9) a relation between the Sudakov form factor ∆ a used in the evolution equationand the Sudakov form factor ∆ bw used for the backward evolution of the parton shower ismade explicit. A similar relation was also studied in Refs. [51, 52]. The PB approach allowsa consistent formulation of the parton shower with the PB TMDs, as in both Sudakov formfactors ∆ a and ∆ bw the same value of z M is used.The splitting functions P ( R ) ab contain the coupling, P ab ( α s , z ) = ∞ (cid:88) n =1 (cid:18) α s ( f ( z, q ))2 π (cid:19) n P ( n − ab ( z ) , (10)where the scale f ( z, q ) in the coupling depends on the ordering condition as discussed later(see Eq.(11)). A backward evolution method, as now common in Monte Carlo event generators, is ap-plied for the initial state parton shower, evolving from the large scale of the matrix-elementprocess backwards down to the scale of the incoming hadron. However, in contrast to theconventional parton shower, which generates transverse momenta of the initial state partonsduring the backward evolution, the transverse momenta of the initial partons of the hardscattering process is fixed by the TMD and the parton shower does not change the kinemat-ics. The transverse momenta during the backward cascade follow the behavior of the TMD.The hard scattering process is obtained as described in section 2. The backward evolutionof the initial state parton shower follows very closely the description in [32, 53, 54], which isbased on Ref. [25].The starting value of the evolution scale µ is calculated from the hard scattering process,as described in Section 2. In case of on-shell matrix elements at NLO, the transverse momen-tum of the hardest parton in the parton shower evolution is limited by the shower-scale , asdescribed in Section 2.1. In Eq.(9) ordering in µ is assumed. However, if angular ordering as in CCFM [14–17] is applied then the ratioof parton densities would change to [ x (cid:48) A b ( x (cid:48) , k (cid:48) t , q (cid:48) /z )] / [ x A a ( x, k t , q (cid:48) )] as discussed in [32]. t , µq ti − , µ i − k ti − , z i − k ti − , z i − k t , z q ti − , µ i − acz = x/x (cid:48) x (cid:48) p + , k (cid:48) t xp + , k t q t,c b Figure 3: Left: Schematic view of a parton branching process. Right: Branching process b → a + c .Starting at the hard scale µ = µ i , the parton shower algorithm searches for the next scale µ i − at which a resolvable branching occurs (see Fig. 3 left). This scale µ i − is selected fromthe Sudakov form factor ∆ bw as given in Eq.(9) (see also [32]). In the parton shower language,the selection of the next branching comes from solving R = ∆ bw ( x, k t , µ i , µ i − ) for µ i − usinguniformly distributed random numbers R for given x and µ i . However, to solve the integralsin Eq.(9) numerically for every branching would be too time consuming, instead the veto-algorithm [25, 55] is applied.The splitting function P ab as well as the argument f ( z, q ) in the calculation of α s is cho-sen exactly as used in the evolution of the parton density. In a parton shower one treats“resolvable” branchings, defined via a cut in z < z M in the splitting function to avoid thesingular behavior of the terms / (1 − z ) , and branchings with z > z M are regarded as “non-resolvable” and are treated similarly as virtual corrections: they are included in the Sudakovform factor ∆ bw . The splitting variable z i − is obtained from the splitting functions followingthe standard methods (see Eq.(2.37) in [18]).The calculation of the transverse momentum k t is sketched in Fig. 3 (right). The trans-verse momentum q t c can be calculated in case of angular ordering (where the scale q of eachbranching is associated with the angle of the emission) in terms of the angle Θ of the emittedparton with respect to the beam directions q t,c = (1 − z ) E b sin Θ , q c = (1 − z ) q . (11)Once the transverse momentum of the emitted parton q c is known, the transverse mo-mentum of the propagating parton can be calculated from k (cid:48) = k + q c (12)with a uniformly distributed azimuthal angle φ assumed for the vector components of k and q c . The generation of the parton momenta is performed in the center-of-mass frame ofthe collision (in contrast to conventional parton showers, which are generated in differentpartonic frames). 9he whole procedure is iterated until one reaches a scale µ i − < q with q being a cut-off parameter, which can be chosen to be the starting evolution scale of the TMD. It is ofadvantage to continue the parton shower evolution to lower scales q ∼ Λ qcd ∼ . GeV.The final transverse momentum of the propagating parton k is the sum of all transversemomenta q c (see Fig. 3 right): k = k − (cid:88) c q c . (13)with k being the intrinsic transverse momentum.The PB TMD parton shower is selected with PartonEvolution=2 (or
ICCF=2 ). The CCFM parton evolution and corresponding parton shower follows a similar approach asdescribed in the previous section and in detail also in Refs. [32,53,54,56]. The main differenceto the PB-TMD shower are the splitting functions with the non-Sudakov form factor ∆ ns andthe allowed phase space for emission. The original CCFM splitting function ˜ P g ( z, q, k t ) forbranching g → gg is given by ˜ P g ( z, q, k t ) = ¯ α s ( q (1 − z ))1 − z + ¯ α s ( k t ) z ∆ ns ( z, q, k t ) , (14)where the non-Sudakov form factor ∆ ns is defined as log ∆ ns = − ¯ α s ( k t ) (cid:90) dz (cid:48) z (cid:48) (cid:90) dq q Θ( k t − q )Θ( q − z (cid:48) q t ) , (15)with q t = (cid:112) q t being the magnitude of the transverse vector defined in Eq.(11) and k t themagnitude of the transverse vector in Eq.(12).The CCFM parton shower is selected with ICCF=1 ( PartonEvolution=1 ). In the previous versions of C
ASCADE the TMD densities were part of the program. With thedevelopment of TMD
LIB [57, 58] there is easy access to all available TMDs, including partondensities for photons (as well as Z, W and H densities, if available).These parton densities can be selected via
PartonDensity with a value > .For example the TMDs from the parton branching method [18, 19] are selected via PartonDensity=102100 (102200) for PB-NLO-HERAI+II-2018-set1 (set2).Note that the features of the TMD parton shower are only fully available for the PB-TMDsets and the CCFM shower clearly needs CCFM parton densities (like for instance [59]). Finite terms are neglected as they are not obtained in CCFM at the leading infrared accuracy (cf p.72 in [16]). A one loop parton shower (DGLAP like) with ∆ ns = 1 , one loop α s and strict ordering in q can be selectedwith ICCF=0 . Final state parton showers
The final state parton shower uses the parton shower routine
PYSHOW of P
YTHIA . Leptonsin the final state, coming for example from Drell-Yan decays, can radiate photons, whichare also treated in the final state parton shower. Here the method from
PYADSH of P
YTHIA is applied, with the scale for the QED shower being fixed at the virtuality of the decayingparticle (for example the mass of the Z-boson).The default scale for the QCD final state shower is µ = 2 · ( m ⊥ + m ⊥ ) ( ScaleTimeShower=1 ), with m ⊥ being the transverse mass of the hard par-ton 1(2). Other choices are possible: µ = ˆ s ( ScaleTimeShower=2 ) and µ =2 · ( m + m ) ( ScaleTimeShower=3 ). In addition a scale factor can be applied:
ScaleFactorFinalShower × µ (default: ScaleFactorFinalShower=1 ). The hadronization (fragmentation of the partons in colorless systems) is done exclusivelyby P
YTHIA . Hadronization (fragmentation) is switched off by
Hadronization = 0 (or
NFRA = 0 for the older steering cards). All parameters of the hadronization model can bechanged via the steering cards.
Uncertainties of QCD calculations mainly arise from missing higher order corrections, whichare estimated by varying the factorization and renormalization scales up and down by typi-cally a factor of 2. The scale variations are performed when calculating the matrix elementsand are stored as additional weights in the LHE file, which are then passed directly via C AS - CADE
Uncertainty_TMD=1 . Showered multijet LO matrix element calculations can be merged using the prescription dis-cussed in Ref. [62]. The merging performance is controlled by the three parameters
Rclus , Etclus , Etaclmax . Final-state partons with pseudorapidity η <
Etaclmax present in theevent record after the shower step but before hadronization are passed to the merging ma-chinery if
Imerge = 1 . Partons are clustered using the kt-jet algorithm with a cone radius
Rclus and matched to the PB evolved matrix element partons if the distance between the11arton and the jet is
R < . × Rclus . The hardness of the reconstructed jets is controlled byits minimum transverse energy
Etclus (merging scale).The number of light flavor partons is defined by the
NqmaxMerge parameter. Heavyflavor partons and their corresponding radiation are not passed to the merging algo-rithm. All jet multiplicities are treated in exclusive mode except for the highest multiplicity
MaxJetsMerge which is treated in inclusive mode.
In C
ASCADE
Double Precision . With C
ASCADE
YTHIA C ASCADE
RANLUX random number generator, with luxory level
LUX = 4 . Therandom number seed can be set via the environment variable
CASEED , the default value is
CASEED=12345 . When
HEPMC is included, generated events are written out in HEPMC [60] format for furtherprocessing. The environment variable
HEPMCOUT is used to specify the file name, by defaultthis variable is set to
HEPMCOUT=/dev/null .The HEPMC events can be further processed, for example with Rivet [63].
The input parameters are steered via steering files. The new format of steering is discussedin Section 9.3.1 and should be used when reading LHE files, while the other format, which isappropriate for the internal off-shell processes, is discussed in Section 9.3.2.
Examples for steering files are under $install_path/share/cascade/LHE . &CASCADE_inputNrEvents = -1 ! Nr of events to processProcess_Id = -1 ! Read LHE fileHadronisation = 0 ! Hadronisation (on =1, off = 0)SpaceShower = 1 ! Space-like Parton ShowerSpaceShowerOrderAlphas=2 ! Order alphas in Space ShowerTimeShower = 1 ! Time-like Parton ShowerScaleTimeShower = 4 ! Scale choice for Time-like Shower! 1: 2(mˆ2_1t+mˆ2_2t)! 2: shat! 3: 2(mˆ2_1+mˆ2_2)
4: 2*scalup (from lhe file)!ScaleFactorFinalShower = 1. ! scale factor for Final State Parton ShowerPartonEvolution = 2 ! type of parton evolution in Space-like Shower! 1: CCFM! 2: full all flavor TMD evolution! EnergyShareRemnant = 4 ! energy sharing in proton remnant! 1: (a+1)(1-z)**a,
Examples for steering files are under $install_path/share/cascade/HERA and $install_path/share/cascade/PP . 13
OLD STEERING FOR CASCADE** number of events to be generated*NEVENT 100** +++++++++++++++++ Kinematic parameters +++++++++++++++*’PBE1’ 1 0 -7000. ! Beam energy’KBE1’ 1 0 2212 ! -11: positron, 22: photon 2212: proton’IRE1’ 1 0 1 ! 0: beam 1 has no structure* ! 1: beam 1 has structure’PBE2’ 1 0 7000. ! Beam energy’KBE2’ 1 0 2212 ! 11: electron, 22: photon 2212: proton’IRE2’ 1 0 1 ! 0: beam 3 has no structure* ! 1: beam 2 has structure’NFLA’ 1 0 4 ! (D=5) nr of flavours used in str.fct* +++++++++++++++ Hard subprocess selection ++++++++++++++++++’IPRO’ 1 0 2 ! (D=1)* ! 2: J/psi g* ! 3: chi_c’I23S’ 1 0 0 ! (D=0) select 2S or 3S state’IPOL’ 1 0 0 ! (D=0) VM->ll (polarization study)’IHFL’ 1 0 4 ! (D=4) produced flavour for IPRO=11* ! 4: charm* ! 5: bottom’PTCU’ 1 0 1. ! (D=0) p_t **2 cut for process* ++++++++++++ Parton shower and fragmentation ++++++++++++’NFRA’ 1 0 1 ! (D=1) Fragmentation on=1 off=0’IFPS’ 1 0 3 ! (D=3) Parton shower* ! 0: off* ! 1: initial state PS* ! 2: final state PS* ! 3: initial and final state PS’IFIN’ 1 0 1 ! (D=1) scale switch for FPS* ! 1: 2(mˆ2_1t+mˆ2_2t)* ! 2: shat* ! 3: 2(mˆ2_1+mˆ2_2)’SCAF’ 1 0 1. ! (D=1) scale factor for FPS’ITIM’ 1 0 0 ! 0: timelike partons may not shower* ! 1: timelike partons may shower’ICCF’ 1 0 1 ! (D=1) Evolution equation* ! 0: DGLAP* ! 1: CCFM* ! 2: PB TMD evolution* +++++++++++++ Structure functions and scales +++++++++++++’IRAM’ 1 0 0 ! (D=0) Running of alpha_em(Q2)* ! 0: fixed* ! 1: running’IRAS’ 1 0 1 ! (D=1) Running of alpha_s(MU2)* ! 0: fixed alpha_s=0.3 ! 1: running’IQ2S’ 1 0 3 ! (D=1) Scale MU2 of alpha_s* ! 1: MU2= 4*m**2 (only for heavy quarks)* ! 2: MU2 = shat(only for heavy quarks)* ! 3: MU2= 4*m**2 + pt**2* ! 4: MU2 = Q2* ! 5: MU2 = Q2 + pt**2* ! 6: MU2 = k_t**2’SCAL’ 1 0 1.0 ! scale factor for renormalisation scale’SCAF’ 1 0 1.0 ! scale factor for factorisation scale**’IGLU’ 1 0 1201 ! (D=1010)Unintegrated gluon density* ! > 10000 use TMDlib (i.e. 101201 for JH-2013-set1)* ! 1201: CCFM set JH-2013-set1 (1201 - 1213)* ! 1301: CCFM set JH-2013-set2 (1301 - 1313)* ! 1001: CCFM J2003 set 1* ! 1002: CCFM J2003 set 2* ! 1003: CCFM J2003 set 3* ! 1010: CCFM set A0* ! 1011: CCFM set A0+* ! 1012: CCFM set A0-* ! 1013: CCFM set A1* ! 1020: CCFM set B0* ! 1021: CCFM set B0+* ! 1022: CCFM set B0-* ! 1023: CCFM set B1* ! 1: CCFM old set JS2001* ! 2: derivative of collinear gluon (GRV)* ! 3: Bluemlein* ! 4: KMS* ! 5: GBW (saturation model)* ! 6: KMR* ! 7: Ryskin,Shabelski* ++++++++++++ BASES/SPRING Integration procedure ++++++++++++’NCAL’ 1 0 50000 ! (D=20000) Nr of calls per iteration for bases’ACC1’ 1 0 1.0 ! (D=1) relative prec.(%) for grid optimisation’ACC2’ 1 0 0.5 ! (0.5) relative prec.(%) for integration* ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++*’INTE’ 1 0 0 ! Interaction type (D=0)* ! = 0 electromagnetic interaction*’KT1 ’ 1 0 0.44 ! (D=0.0) intrinsic kt for beam 1*’KT2 ’ 1 0 0.44 ! (D=0.0) intrinsic kt for beam 2*’KTRE’ 1 0 0.35 ! (D=0.35) primordial kt when non-trivial* ! target remnant is split into two particles* Les Houches Accord Interface’ILHA’ 1 0 0 ! (D=10) Les Houches Accord* ! = 0 use internal CASCADE* ! = 1 write event file* ! = 10 call PYTHIA for final state PS and remnant frag* path for updf files* ’UPDF’ ’./share’ C ASCADE
Acknowledgments.
FG acknowledges the support and hospitality of DESY, Hamburg, where part of this workstarted. FH acknowledges the hospitality and support of DESY, Hamburg and of CERN,Theory Division while parts of this work were being done. STM thanks the Humboldt Foun-dation for the Georg Forster research fellowship and gratefully acknowledges support fromIPM. QW and HY acknowledge the support by the Ministry of Science and Technology un-der grant No. 2018YFA040390 and by the National Natural Science Foundation of Chinaunder grant No. 11661141008. 16
Title of Program: C ASCADE
Computer for which the program is designed and others on which it is operable: any with stan-dard Fortran 77 (gfortran)
Programming Language used:
FORTRAN 77
High-speed storage required: No Separate documentation available: No Keywords:
QCD, TMD parton distributions.
Method of solution:
Since measurements involve complex cuts and multi-particle final states,the ideal tool for any theoretical description of the data is a Monte Carlo event generatorwhich generates initial state parton showers according to Transverse Momentum Depen-dent (TMD) parton densities, in a backward evolution, which follows the evolution equationas used for the determination of the TMD.
Restrictions on the complexity of the problem:
Any LHE file (with on-shell or off-shell) initialstate partons can be processed.
Other Program used: P YTHIA ( version > ) for final state parton shower and hadronization,B ASES /S PRING
LIB as a library for TMD parton densities.
Download of the program:
Unusual features of the program:
None 17 eferences [1] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, et al. , JHEP , 079 (2014). .[2] S. Frixione and B. R. Webber (2006). hep-ph/0612272 .[3] S. Frixione, P. Nason, and B. R. Webber, JHEP , 007 (2003). hep-ph/0305252 .[4] S. Frixione and B. R. Webber (2002). hep-ph/0207182 .[5] S. Frixione and B. R. Webber, JHEP , 029 (2002). hep-ph/0204244 .[6] S. Alioli, K. Hamilton, P. Nason, C. Oleari, and E. Re, JHEP , 081 (2011). .[7] S. Frixione, P. Nason, and C. Oleari, JHEP , 070 (2007). .[8] M. Bahr, S. Gieseke, M. Gigg, D. Grellscheid, K. Hamilton, et al. , Eur. Phys. J. C , 639 (2008). .[9] T. Sj ¨ostrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel,C. O. Rasmussen, and P. Z. Skands, Comput. Phys. Commun. , 159 (2015). .[10] J. Alwall, S. Hoche, F. Krauss, N. Lavesson, L. Lonnblad, et al. , Eur. Phys. J. C , 473 (2008). .[11] R. Frederix and S. Frixione, JHEP , 061 (2012). .[12] K. Hamilton, P. Nason, and G. Zanderighi, JHEP , 155 (2012). .[13] S. Catani, M. Ciafaloni, and F. Hautmann, Phys. Lett. B , 97 (1990).[14] M. Ciafaloni, Nucl. Phys. B , 49 (1988).[15] S. Catani, F. Fiorani, and G. Marchesini, Phys. Lett. B , 339 (1990).[16] S. Catani, F. Fiorani, and G. Marchesini, Nucl. Phys. B , 18 (1990).[17] G. Marchesini, Nucl. Phys. B , 49 (1995). hep-ph/9412327 .[18] F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik, JHEP , 070 (2018). .[19] F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik, Phys. Lett. B , 446 (2017). .[20] R. Angeles-Martinez et al. , Acta Phys. Polon. B , 2501 (2015). .1821] J. Alwall et al. , Comput. Phys. Commun. , 300 (2007). hep-ph/0609017 .[22] A. Buckley, J. Ferrando, S. Lloyd, K. Nordstr ¨om, B. Page, M. R ¨ufenacht, M. Sch ¨onherr,and G. Watt, Eur. Phys. J. C , 132 (2015). .[23] S. Dooling, P. Gunnellini, F. Hautmann, and H. Jung, Phys.Rev.D , 094009 (2013). .[24] F. Hautmann and H. Jung, Eur. Phys. J. C , 2254 (2012). .[25] M. Bengtsson, T. Sjostrand, and M. van Zijl, Z. Phys. C , 67 (1986).[26] S. Catani, M. Ciafaloni, and F. Hautmann, Nucl. Phys. B Proc. Suppl. , 182 (1992).[27] G. Marchesini and B. R. Webber, Nucl. Phys. B , 215 (1992).[28] F. Hautmann, H. Jung, and S. T. Monfared, Eur. Phys. J. C , 3082 (2014). .[29] E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP , 443 (1976).[30] E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JETP , 199 (1977).[31] I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. , 822 (1978).[32] H. Jung, S. Baranov, M. Deak, A. Grebenyuk, F. Hautmann, et al. , Eur. Phys. J. C , 1237 (2010). .[33] A. van Hameren, Comput. Phys. Commun. , 371 (2018). .[34] A. Lipatov, M. Malyshev, and S. Baranov, Eur. Phys. J. C , 330 (2020). .[35] S. Catani, M. Ciafaloni, and F. Hautmann, Nucl. Phys. B , 135 (1991).[36] V. Saleev and N. Zotov, Mod. Phys. Lett. A , 151 (1994). [Erratum: Mod.Phys.Lett.A 9,1517–1518 (1994)].[37] A. Lipatov and N. Zotov, Eur. Phys. J. C , 87 (2003). hep-ph/0210310 .[38] S. Baranov and N. Zotov, J. Phys. G , 1395 (2003). hep-ph/0302022 .[39] S. P. Baranov, Phys. Rev. D , 114003 (2002).[40] S. Baranov, Phys. Rev. D , 054012 (2011).[41] F. Hautmann, Phys. Lett. B , 159 (2002). hep-ph/0203140 .[42] S. P. Baranov, A. V. Lipatov, and N. P. Zotov, Phys. Rev. D , 014025 (2008). .[43] M. Deak and F. Schwennsen, JHEP , 035 (2008). .1944] S. Marzani and R. D. Ball, Nucl. Phys. B , 246 (2009). .[45] M. Deak, F. Hautmann, H. Jung, and K. Kutak, JHEP , 121 (2009). .[46] W. Furmanski and R. Petronzio, Z. Phys. C , 293 (1982).[47] G. Curci, W. Furmanski, and R. Petronzio, Nucl. Phys. B175 , 27 (1980).[48] Y. L. Dokshitzer, Sov. Phys. JETP , 641 (1977). [Zh. Eksp. Teor. Fiz.73,1216(1977)].[49] G. Altarelli and G. Parisi, Nucl. Phys. B , 298 (1977).[50] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. , 438 (1972). [Yad.Fiz.15,781(1972)].[51] Z. Nagy and D. E. Soper, Phys. Rev. D , 014025 (2020). .[52] L. Gellersen, D. Napoletano, S. Prestel, Monte Carlo studies , in , p. 131. 2020. Also in preprint 2003.01700.[53] H. Jung, Comput. Phys. Commun. , 100 (2002). hep-ph/0109102 .[54] H. Jung and G. P. Salam, Eur. Phys. J. C , 351 (2001). hep-ph/0012143 .[55] S. Platzer and M. Sjodahl, Eur. Phys. J. Plus , 26 (2012). .[56] F. Hautmann and H. Jung, JHEP , 113 (2008). .[57] F. Hautmann, H. Jung, M. Kr¨amer, P. Mulders, E. Nocera, et al. , Eur. Phys. J. C74 , 3220 (2014). .[58] H. Jung et al.,
The library for transverse momentum dependent parton desities: TMDlib2 andTMDplotter . To be published, 2021.[59] F. Hautmann and H. Jung, Nuclear Physics B , 1 (2014). .[60] M. Dobbs and J. B. Hansen, Comput. Phys. Commun. , 41 (2001).[61] A. Bermudez Martinez, P. Connor, F. Hautmann, H. Jung, A. Lelek, V. Radescu, andR. Zlebcik, Phys. Rev. D , 074008 (2019). .[62] A. Bermudez Martinez et al., Jet merging with TMD parton branching . To be published,2021.[63] A. Buckley, J. Butterworth, L. Lonnblad, D. Grellscheid, H. Hoeth, J. Monk, H. Schulz,and F. Siegert, Comput. Phys. Commun. , 2803 (2013).1003.0694