Calculations with off-shell matrix elements, TMD parton densities and TMD parton showers
Marcin Bury, Andreas van Hameren, Hannes Jung, Krzysztof Kutak, Sebastian Sapeta, Mirko Serino
DDESY 17-222IFJPAN-IV-2017-28version: 15 Dec 2017
Calculations with off-shell matrix elements, TMD partondensities and TMD parton showers
Marcin Bury , Andreas van Hameren , Hannes Jung , ,Krzysztof Kutak , Sebastian Sapeta , Mirko Serino , , Institute of Nuclear Physics, Polish Academy of Sciences,Cracow, Poland DESY, Hamburg, FRG Department of Physics, Ben Gurion University of the Negev,Be’er Sheva, Israel
Abstract
A new calculation using off-shell matrix elements with TMD parton densities sup-plemented with a newly developed initial state TMD parton shower is described. Thecalculation is based on the K A T IE package for an automated calculation of the partonicprocess in high-energy factorization, making use of TMD parton densities implementedin TMDlib. The partonic events are stored in an LHE file, similar to the conventional LHEfiles, but now containing the transverse momenta of the initial partons. The LHE files areread in by the C ASCADE package for the full TMD parton shower, final state shower andhadronization from P
YTHIA where events in HEPMC format are produced.We have determined a full set of TMD parton densities and developed an initial stateTMD parton shower, including all flavors following the TMD distribution.As an example of application we have calculated the azimuthal de-correlation of high p t dijets as measured at the LHC and found very good agreement with the measurementwhen including initial state TMD parton showers together with conventional final stateparton showers and hadronization. Measurements in today’s high-energy experiments have reached a new level of precision of afew percent in experimental uncertainty. In many cases in strong interactions the theoreticalpredictions have larger uncertainties, mainly coming from the unknown higher order cor-rections which can be estimated by variation of the factorization and renormalization scales.While calculations in fixed order perturbation theory in Quantum Chromodynamics (QCD)even at next-to-leading (or even next-to-next-to-leading) order expansion in the strong cou-pling α s are often not sufficient, the predictions can be improved when parton showers areincluded to simulate even higher order corrections, as done for example with the P OWHEG [1, 2] or M C @ NLO [3–6] methods. However, when supplementing a calculation of collinearinitial partons with parton showers, the kinematics of the hard process are changed due to1 a r X i v : . [ h e p - ph ] D ec he transverse momentum generated in the initial state shower [7]. This effect can be signifi-cant even at large transverse momenta, as has been discussed and shown explicitly in [8–10].With the development of transverse momentum dependent (TMD) parton distributions,this problem can be overcome, since the transverse momentum of the initial partons can beobtained from the TMD parton distributions. The great advantage of using TMD parton den-sities is that a parton shower will not change the kinematics of the matrix element process, incontrast to the conventional approach of collinear hard process calculations supplementedwith parton showers, and that the main parameters of the TMD parton shower are fixed withthe determination of the TMD.Already some time ago a TMD parton shower has been developed for the case of initialstate gluons within the frame of the CCFM evolution equation [11–14] and implemented inthe C ASCADE package [15–19]. However, TMD parton densities defined over a large range in x , k t and scale µ for all different flavors including quarks and gluons were not available untilrecently. In [20, 21] a new method for determination of TMD parton densities is described,another method to obtain TMD parton densities from collinear parton densities has beenproposed in [22], which we apply in the present study. In order to fully account for thepotential of a TMD parton shower, the initial state kinematics for the hard process calculationshould include the transverse momenta. With the development of an automated calculationof multi-leg matrix elements with off-shell initial states [23] the full potential of TMD partondensities and parton showers can be explored.In this article we will describe how the TMD parton densities can be obtained from theKMRW approach [22] and how they can be used in calculations using off-shell matrix ele-ments obtained from K A T IE [23]. We then describe how this matrix element calculation issupplemented with a newly developed TMD parton shower, which makes use of the TMDparton densities without changing the kinematics of the matrix element process. We illus-trate the advantage of using TMD densities with off-shell matrix element calculations in anapplication to azimuthal de-correlations of high p t dijet measurements at the LHC.In section 2 we briefly describe the main features of the automated calculation of off-shellmatrix elements with K A T IE and section 3 describes the procedure to obtain the TMD partondensities with the KMRW method. In section 4 we describe a new development of the TMDparton shower which can be combined with the matrix element calculation via LHE files,similar to what is being used in standard methods. In section 5 we present a case-study ofazimuthal correlations of dijets at large transverse momenta as obtained at the LHC. K A T IE is a parton-level event generator for arbitrary processes within the Standard Model,with the special feature that it can generate events with space-like initial-state momenta thathave non-vanishing transverse components. It produces weighted parton-level event filesin the Les Houches format [24], or in a custom format. For the latter, K A T IE also provides2he tools to produce distributions for arbitrary observables. It relies on LHAPDF [25] forcollinear PDFs and the running coupling constant, and on TMDlib [26] for transverse mo-mentum dependent PDFs. Alternatively, the latter can be provided as hyper-rectangulargrids which K A T IE itself interpolates. The hard matrix elements are calculated as the summedsquares of helicity amplitudes, defined following the approach of [27, 28] which guaranteesgauge invariance. The amplitudes are calculated numerically with recursive methods [29,30]which keep the computational complexity under control, even for larger final-state multiplic-ities.A project is defined in a single user-defined input file, containing all the informationabout the desired center-of-mass energy, inclusive phase space cuts, and values of modelparameters like particles masses and widths. If the user wants to apply TMDPDFs that arenot included in TMDlib, this file must also include the paths to the files containing the hyper-rectangular grids. Finally, K A T IE does not generate a list of partonic sub-processes itself, andthe user must provide this list in the same input file.Event generation happens in two stages. During the fist stage, the phase space sampleris optimized for each sub-process separately. This stage is very cheap in terms of CPU timecompared to the second stage during which the actual event files are generated. This stagecan trivially be parallelized by running several instances of the executable with differentseeds for the random number generator. The complete set of transverse momentum dependent PDFs consistent with the matrix el-ements that we use can be obtained by applying Lipatov’s effective action approach com-bined with the Curci-Furmanski-Petronzio method, which allows to formally define newsplitting functions. The construction of a new set of evolution equations and the correspond-ing parton densities is still to be achieved. Only recently all the real contributions to theTMD splitting functions have been obtained [31]. At present, we obtain TMD parton densi-ties from collinear parton densities by the application of the KMRW procedure [22]. In thismethod the k t -dependent distributions are calculated from the DGLAP equation by takinginto account only the contribution corresponding to a single real emission. The virtual con-tributions between the scales k t and µ are resummed into a Sudakov factor, which describesthe probability that there are no emissions.The precise expressions for the TMD distributions read A i ( x, k t , µ ) = ∂∂k t (cid:2) xf i ( x, k t ) ∆ i ( k t , µ ) (cid:3) (1)with the Sudakov factors for quarks ∆ q ( k t , µ ) = exp (cid:32) − (cid:90) µ k t dκ t κ t α S ( κ t )2 π (cid:90) dζ P qq ( ζ )Θ(1 − z M − ζ ) (cid:33) (2)3nd for gluons ∆ g ( k t , µ ) = exp (cid:32) − (cid:90) µ k t dκ t κ t α S ( κ t )2 π (cid:90) dζ [ ζ P gg ( ζ )Θ(1 − z M − ζ )Θ( ζ − z M ) + n F P qg ( ζ ) ] (cid:33) . (3)Here, n F is the active number of quark–antiquark flavours into which the gluon may split,and we set n F = 5 . The infrared cutoff z M ≡ k t µ + k t arises because of the singular behaviourof the splitting functions P qq ( z ) and P gg ( z ) at z = 1 , which correspond to soft gluon emission. x f ( x , p ) − − − −
10 110 gluon, p = 500 GeV from 0.1 up to 500 GeV t MRW-der-v3-CT10, kCT10nlo T M D p l o tt e r . . gluon, p = 500 GeV x − − − x f ( x , p ) − − − −
10 110 up, p = 500 GeV from 0.1 up to 500 GeV t MRW-der-v3-CT10, kCT10nlo T M D p l o tt e r . . up, p = 500 GeV x − − − Figure 1: Comparison between the integrated TMD using the method of Ref. [22] and theunderlying collinear CT10nlo gluon PDFs [32] at a scale µ = 500 GeV for gluons (left) andu-quarks (right).The TMDs are defined only for k t > µ , where µ ∼ GeV is the minimum scale for thethe integrated (collinear) PDFs. In order to extend the TMD to the region k t < µ , we testedthree methods. One is to set the TMD proportional to k t , the second is to freeze the TMD at k t = µ and the third is taken from Ref. [22] and is used here: A i ( x, k t , µ ) = 1 µ xf i ( x, µ ) ∆ i ( µ , µ ) . (4)The TMDs used here (MRW-CT10nlo) are based on the CT10nlo collinear PDF set [32]including the appropriate running coupling α s . In fig. 1 we show a comparison of the originalCT10 parton density with the TMDs constructed here integrated over k t up to the scale µ using the TMDplotter tool [26, 33]. We observe reasonable agreement, except at large x ,4here the integration limits in the Sudakov form factor play a role. The large x region is,however, not relevant for the processes studied here.In fig. 2 we show the k t dependence of the TMD at a scale µ = 500 GeV for differentvalues of x . One can clearly see the treatment of the non-perturbative region of k t < GeV.The discontinuity at small k t comes from the matching procedure in eq.(4). [GeV] t k −
10 1 10 , p ) t x A ( x , k − − − − − −
10 1
MRW-CT10nlo, x = 0.01, p = 500 GeV gluonup T M D p l o tt e r . . MRW-CT10nlo, x = 0.01, p = 500 GeV [GeV] t k −
10 1 10 , p ) t x A ( x , k − − − − − −
10 1
MRW-CT10nlo, x = 0.1, p = 500 GeV gluonup T M D p l o tt e r . . MRW-CT10nlo, x = 0.1, p = 500 GeV
Figure 2: Transverse momentum distribution of the TMD at a scale µ = 500 GeV for gluonsand u-quarks at x = 0 . (left) and x = 0 . (right). 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 in α s , the scale in thecalculation of α s as well as the kinematic restrictions applied are identical in both the partonshower and the evolution of the parton densities.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 a transverse momentum of the initial statepartons during the backward evolution, the transverse momentum of the initial partons ofthe hard scattering process is fixed by the TMD and the parton shower does not change thekinematics. The transverse momenta during the cascade follow the behavior of the TMD.The hard scattering process is obtained directly using off-shell matrix element calculationsas described in section 2. The partonic configuration is stored in the form of an LHE (LesHouches Event) text file, but now including the transverse momenta of the incoming par-tons. This LHE files are input to the shower and hadronization interface of C ASCADE [15, 16](new version ) for the TMD shower where events in HEPMC [34] format are produced.5he backward evolution of the initial state parton shower follows very closely the de-scription in [7, 15–17]. The evolution scale µ is selected from the hard scattering process,with µ = ˆ p T or µ = Q t + ˆ s for an evolution in virtuality or angular ordering, with ˆ p T beingthe transverse momentum of the hard process, Q t being the vectorial sum of the initial statetransverse momenta and s being the invariant mass of the subprocess.Starting with the hard scale µ = µ i , the parton shower algorithm searches for the nextscale µ i − at which a resolvable branching occurs. This scale µ i − is selected from the Su-dakov form factor ∆ S making use of the TMD densities A a ( x (cid:48) , k (cid:48) t , µ (cid:48) ) which depend on thelongitudinal momentum fraction x (cid:48) = xz of parton a , its transverse momentum k (cid:48) t probed ata scale µ (cid:48) (see also [15]). The Sudakov form factor ∆ S for the backward evolution is given by(see fig. 3 left): ∆ S ( x, µ i , µ i − ) = exp (cid:34) − (cid:90) µ i µ i − dµ (cid:48) µ (cid:48) α s (˜ µ (cid:48) )2 π (cid:88) a (cid:90) dzP a → bc ( z ) x (cid:48) A a ( x (cid:48) , k (cid:48) t , µ (cid:48) ) x A b ( x, k t , µ (cid:48) ) (cid:35) (5)which describes the probability that parton b remains at x with transverse momentum k t when evolving from µ i to µ i − < µ . Please note, that the argument in α s is ˜ µ (cid:48) and dependson the ordering condition as discussed later. In the parton shower language, the selection of the next branching comes from solving theSudakov form factor eq.(5) for µ i − . However, to solve the integrals in eq.(5) numerically forevery branching would be too time consuming, instead the veto-algorithm [7, 35] is applied.The selection of µ i − and the branching splitting z i − follows the standard methods [7].The splitting function P ab as well as the argument ˜ µ in the calculation of α s is chosenexactly as used in the evolution of the parton density. In a parton shower one treats “resolv-able” branchings, defined via a cut in z < z M in the splitting function (see eq.(3)) to avoidthe singular behavior of the terms − 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 ∆ S .The longitudinal momentum fraction x i − = x i z i − is calculated by generating z i − accord-ing to the splitting function. With z i − and µ i − all variables needed for a collinear partonshower are obtained.The calculation of the transverse momentum k t is sketched in fig. 3 right. The transversemomentum q t i can be obtained by giving a physical interpretation to the evolution scale µ i (see fig. 3 right), and q t i can be calculated in case of angular ordering ( µ is associated with theangle of the emission) in terms of the angle Θ of the emitted parton wrt the beam directions q t,c = (1 − z ) E b sin Θ : q t,i = (1 − z ) µ i . (6)Once the transverse momentum of the emitted parton q t is known, the transverse mo- In equation eq.(5) ordering in µ is assumed, if angular ordering, as in CCFM [11–14], is applied then the ratioof parton densities would change to x (cid:48) A a ( x (cid:48) ,k (cid:48) t ,µ (cid:48) /z ) x A b ( x,k t ,µ (cid:48) ) as discussed in [15]. ti , µ i q ti − , µ i − k ti − , z i − k ti − , z i − k ti , z i q ti − , µ i − acz = x a /x b x b p + , k t,b x a p + , k t,a q t,c → µb Figure 3: Left: Schematic view of a parton branching process. Right: Branching process b → a + c .mentum of the propagating parton can be calculated from k t i − = k t i + q t i − (7)with a uniformly distributed azimuthal angle φ is assumed for the vector components of k and q .The whole procedure is iterated until one reaches a scale µ i − < q with q being a cut-offparameter, which can be chosen to be the starting evolution scale of the TMD. However, itturns out that during the backward evolution the transverse momentum k t can reach largevalues, even for small scales µ i − , because of the random φ distribution. On average thetransverse momentum decreases, and it is of advantage to continue the parton shower evo-lution to a scale q ∼ Λ qcd ∼ . GeV, to allow enough emissions to share the transversemomenta generated. p t dijets in pp at the LHC We show predictions obtained with off-shell matrix elements of → QCD processes usingthe TMDs obtained in sec. 3. The results of the parton level calculation are fed via LHEfiles to the shower and hadronization interface of C
ASCADE [15, 16] (new version ) forthe TMD shower where events in HEPMC format are stored for further processing as viaRivet [36].First we show parton level results of azimuthal de-correlations of high p t dijet productionat the LHC at √ s = 7 TeV [37]. In fig. 4 we compare predictions obtained from our calculation(without parton shower) with the one from P
OWHEG dijet (without parton shower). Onecan observe reasonable agreement between both parton level calculations at high ∆ φ . TheP OWHEG prediction shows a sharp drop at ∆ φ = 2 π/ , which is the kinematic limit for a3 parton configuration. The prediction using TMDs shows a smooth distribution to smallervalues of ∆ φ which is typical for a configuration where more partons are radiated in theinitial state. The distribution of our prediction depends entirely on the shape of the TMD.7hus, with a precise determination of the TMD, we expect the ∆ φ distribution to be welldescribed, without any tuning and without any adjustment of additional parameters. DataTMD noPS µ = Q t + ˆ s Powheg jet NLO noPS . . . . . . − − − Di-jet azimuthal decorrelation, 110 < p leading T <
140 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] DataTMD noPS µ = Q t + ˆ s Powheg jet NLO noPS . . . . . . − − − Di-jet azimuthal decorrelation, 140 < p leading T <
200 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] Figure 4: ∆ φ distribution for high p t dijet production [37]. The solid (blue) histogram showsthe prediction using off-shell → matrix elements with TMD parton densities, the dashed(red) line is a 3-parton configuration obtained with P OWHEG . Both predictions are withoutparton shower and hadronization.
In fig. 4 we have shown the advantage in using TMD parton densities compared to a fixedorder collinear calculation: due to the resummation of multiple parton emissions in the TMDparton density, the phase space for multi-jet production is covered, as seen in the tail tosmall ∆ φ . Of course, the experimental measurement is different from a purely 2-parton finalstate, even using TMDs, since the jet clustering is based on multiple partons (hadrons). Infig. 5 we show a comparison of the prediction using TMDs with and without initial stateTMD parton showering and including final state parton shower and hadronization (takenfrom P YTHIA [38]), with a final state parton shower scale of µ fps = 2ˆ p t being the averagetransverse momentum of the outgoing matrix element partons. While even without partonshower a tail towards small ∆ φ is observed, the simulation of the parton shower, both initialTMD and final state parton shower contributes to the shape of the distribution and brings itclose to the measurement.In fig. 6 we show predictions for the azimuthal de-correlation ∆ φ for high p t dijets for dif-ferent regions of p leadingt using TMD parton densities with off-shell matrix elements, partonshower and hadronization in comparison with measurements at √ s = 7 TeV in pp collisionsat the LHC [37]. We show predictions for two different factorization scales: µ = Q t + ˆ s ,where Q t is the vectorial sum of the initial state transverse momenta and √ ˆ s is the invariantmass of the partonic subsystem and µ = ˆ p t . The first scale choice is motivated by angularordering (see Ref. [39]), the second one is the conventional scale choice. The scale choicemotivated from angular ordering describes the measurements significantly better than the8 ataTMD IFPSTMD noPSTMD FPS . . . . . . − − − Di-jet azimuthal decorrelation, 110 < p leading T <
140 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] DataTMD IFPSTMD noPSTMD FPS . . . . . . − − − − Di-jet azimuthal decorrelation, 140 < p leading T <
200 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] Figure 5: ∆ φ distribution of high p t dijet events for different regions of p leadingt : withoutparton shower (noPS, dashed red line), with final state parton shower (FPS, dashed-dottedbrown line), with initial TMD shower and final state parton shower (IFPS, blue solid line).The factorization scale µ = Q t + ˆ s was chosen.conventional one.It is important to note, that there are no free parameters left: once the TMD parton densityis determined, the initial state parton shower follows exactly the TMD parton distribution.The TMD parton distribution is the essential ingredient in the present calculation, and aprecise determination of the TMD parton distribution over a wide range in x , k t and scale µ is an important topic. First steps towards a precision determination of the TMD densitiesfrom HERA measurements have been performed in Ref. [20, 21]. A new calculation using off-shell matrix elements with TMD parton densities supplementedwith a newly developed initial state TMD parton shower has been presented. The calcu-lation is based on the K A T IE package for an automated calculation of the partonic processin high-energy factorization, making use of TMD parton densities implemented in TMDlib.The partonic events are stored in an LHE file, similar to the conventional LHE files, but nowcontaining the transverse momenta of the initial partons. The LHE files are read in by theC ASCADE package for the full TMD parton shower where events in HEPMC format are pro-duced for further processing, like with Rivet.We have determined a full set of TMD parton densities using the KMRW approach, whichinclude all flavours and are valid over a wide range in x , k t , and µ . These TMD partondensities are available in TMDlib.We have developed an initial state TMD parton shower, including all flavors and follow-ing the TMD distribution, without the need for adjusting further parameters.As an example of application we have calculated the azimuthal de-correlation of high p t dijets as measured at the LHC and found very good agreement with the measurement.9 ataTMD PS µ = Q t + ˆ s TMD PS µ = ˆ p t . . . . . . − − − Di-jet azimuthal decorrelation, 110 < p leading T <
140 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] DataTMD PS µ = Q t + ˆ s TMD PS µ = ˆ p t . . . . . . − − − − Di-jet azimuthal decorrelation, 140 < p leading T <
200 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] DataTMD PS µ = Q t + ˆ s TMD PS µ = ˆ p t . . . . . . − − − − Di-jet azimuthal decorrelation, 200 < p leading T <
300 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] DataTMD PS µ = Q t + ˆ s TMD PS µ = ˆ p t . . . . . . − − − Di-jet azimuthal decorrelation, p leading T >
300 GeV ∆ φ [rad] σ d σ d ∆ φ [ r a d − ] Figure 6: ∆ φ distribution as measured by [37] for different regions of p leadingt . The dataare compared with predictions using off-shell → matrix elements with TMD partondensities, an initial state TMD parton shower, conventional final state parton shower andhadronization. Shown are predictions for two different choices of the factorization scale, asdiscussed in the text.It is remarkable, that using TMDs with off-shell matrix element calculations covers alreadya larger phase space than is accessible in collinear higher order calculations. Including ini-tial state TMD parton showers together with conventional final state parton showers givesa remarkably good description of the measurements, which opens the floor for a rich phe-nomenology at the LHC making use of the advantages of automatic off-shell matrix elementcalculations with a fully TMD consistent parton shower. Acknowledgements
HJU thanks the Foundation for Polish Science (FNP) for support with an Alexander vonHumboldt Polish Honorary Research Scholarship, allowing extensive stays in Cracow wherethe present work was completed. MB, KK, MS acknowledge the support of NCN grant DEC-2013/10/E/ST2/00656. AvH was supported by grant of National Science Center, Poland,No. 2015/17/B/ST2/01838. MS is also partially supported by the Israeli Science Foundation10hrough grant 1635/16, by the BSF grants 2012124 and 2014707, by the COST Action CA15213THOR and by a Kreitman fellowship from the Ben Gurion University of the Negev.
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