Production of tau lepton pairs with high pT jets at the LHC and the TauSpinner reweighting algorithm
aa r X i v : . [ h e p - ph ] A p r IFJPAN-IV-2016-4
Production of t lepton pairs with high p T jets at the LHCand the TauSpinner reweighting algorithm
J. Kalinowski a , W. Kotlarski a , b , E. Richter-Wa¸s c and Z. Wa¸s da Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland b Institut für Kern- und Teilchenphysik, Technische Univesität Dresden, 01069 Dresden, Germany c Institute of Physics, Jagellonian University, Lojasiewicza 11, 30-348 Cracow, Poland d Institute of Nuclear Physics, PAN, Kraków, ul. Radzikowskiego 152, Poland
ABSTRACT
The purpose of the
TauSpinner algorithm is to provide a tool that allows to modify the physics model of the Monte Carlogenerated samples due to the changed assumptions of event production dynamics, but without the need of re-generating events.To each event
TauSpinner attributes the weights. In this way, for example, the spin effects of t -lepton production or decay aremodified, or the effect of the changes in the production mechanism are introduced according to a new physics model. Such anapproach is useful, because there is no need to repeat the detector response simulation with each variant of the physics modelconsidered. In addition, since only the event weights differ for the models, samples are correlated and statistical error of themodification is proportional to the reweighting only.We document the extension of the TauSpinner algorithm to (2 →
4) processes in which the matrix elements for the parton-parton scattering amplitudes into a t -lepton pair and two outgoing partons are used. The method is based on tree-level matrixelements with complete helicity information for the Standard Model processes, including the Higgs boson production. Forthis purpose automatically generated codes by MadGraph5 have been adapted. Consistency tests of the implemented matrixelements, reweighting algorithm and numerical results are presented.For the sensitive observable, namely the averaged t lepton polarisation, we perform a systematic comparison between(2 →
2) and (2 →
4) matrix elements used to calculate the spin weight in pp → tt j j events. We show, that for events with t -lepton pair close to the Z-boson peak, the t -lepton polarisation calculated using (2 →
4) matrix elements is very close tothe one calculated using (2 →
2) Born process only. For the m tt masses above the Z-boson peak, the effect from including(2 →
4) matrix elements is also marginal, however when taking into account only subprocesses qq , q ¯ q → tt j j , it can leadto a 10% difference on the predicted t -lepton polarisation. On the other hand, we have found that the appropriate choiceof electroweak scheme can have significant impact. We show that the modification of the electroweak or strong interactioninitialization (including change of the electroweak schemes or analytic form of scale dependence for a S ) can be performedwith the re-weighting technique as well.The new version of TauSpinner ver.2.0.0 presented here, allows also to introduce non-standard couplings for theHiggs boson and study their effects in the vector-boson-fusion processes by exploiting the spin correlations of t -lepton pairdecay products. The discussion of physics effects is however relegated to forthcoming publications. IFJPAN-IV-2016-4April 2016
Introduction
With the data collected so far by LHC experiments, there was not much interest to explore physics of t -lepton decays, withthe exception of exploiting t leptons in searches for rare or Standard-Model-forbidden decay channels, see eg. [1]. However, t -lepton signatures can provide a powerful tools in many areas, like studies of hard processes characteristics, measurements ofproperties of Higgs boson(s) [2, 3], or in searches for New Physics [4, 5, 6].The t leptons cannot be observed directly due to their short life-time. All decay products are observed, with the exceptionof n ’s. There are more than 20 different t decay channels, each of them leading to a somewhat distinct signature. Thismakes a preparation of observables involving t decays laborious. However, such efforts can be rewarding, because t -leptonspin polarization can be measured directly, contrary to the case of electron or muon signatures, giving better insight into thenature of its production mechanism, e.g. the properties of resonances decaying to t leptons. This is the main motivationfor developing TauSpinner , an algorithm to simplify the task of exploring the t physics potential, which could be used forevaluation/modification of event samples including t decays.In the first release, the program algorithms were focused on longitudinal spin effects only [7]. Already TauSpinnerver.1.1 handled these effects with the help of the appropriate spin weight attributed to each event. In this way, spin effectscould be introduced, or removed, from the sample. With time, variety of extensions were introduced. Since Ref. [8], a secondweight was introduced which allows to manipulate the production process by adding additional contributions or completelyreplacing the production process with an alternative one, including for example an exchange of a new intermediate particle.Ref. [9] brought a possibility of modifying transverse spin effects in the cascade t decays of intermediate Higgs boson. Later,Ref. [10] enabled the transverse spin effects for the case of t leptons produced in Drell-Yan processes to be studied as well.With time, technical options or important precision improvements were introduced too. In [9], an option to attributehelicity states to t -leptons was introduced. One should keep in mind, that because of quantum entanglement, the assignmentof a definite helicity state to intermediate t ’s is necessarily subject to an approximation. However, for spin weight calculation,the complete spin density matrix is taken into account and in general, approximation is not used. With later publication [10],one-loop electroweak (EW) corrections also became available for the Drell-Yan parton- process q ¯ q → Z / g ∗ → tt .Let us mention another technical option. Initially the program was expected to work for samples, where spin effects areeither taken into account in full, or are absent. One can however configure TauSpinner algorithm to work on generated sampleswhere only part of spin effects is taken into account (only some components of the density matrix used) and to correct them tofull spin effects.Until now, for calculations of spin weights,
TauSpinner algorithm was always using the Born-level (2 →
2) scatteringamplitudes convoluted with the corresponding parton distribution functions (PDFs). Kinematic configurations of the incom-ing/outgoing partons were reconstructed from the four-momenta of outgoing t leptons and incoming protons (using c.m.collision energy), and somewhat elaborated kinematical transformations were used for calculating an effective scattering angleof the assumed Born process.The validity and precision of this approximation became of a concern, especially for configurations with high momentumtransfers in the t-channel and for outgoing particles with high transverse momentum ( p T ) that accompany decay products ofthe electroweak bosons. In such cases, more elaborated description of the production process dynamics is needed. The aimof the present paper is to describe an improved version of TauSpinner 2.0.0 which now includes hard processes featuringtree-level parton matrix elements for production of a t -lepton pair and two jets. Numerical test, and some results of physicsinterest, will be also presented.The paper is organized as follows: In Section 2 we recall assumptions used for the Monte Carlo reweighting techniques,in particular for the modeling of kinematic distributions in the multi-dimensional phase-space. We then define the masterformula used by TauSpinner for modeling spin correlations of t -lepton decay products in events with different topologiesin proton-proton collisions. Section 3 documents details of the tree-level matrix elements used for the calculation of weightsin pp → tt j j events. The implemented functionality is based on automatically produced FORTRAN code from
MadGraph5 package [11] for processes of the Drell-Yan–type and of the Standard Model Higgs boson production in vector boson fusion(VBF) processes, which have been later manually modified and adapted. Numerical effects of different choices for electroweakand QCD interactions initialization are presented in the last two subsections. We classify parton level processes into groups,which are then used in the following Section 4 for technical tests. We explain details of the modification which we haveintroduced to the initialization of
MadGraph5 generated amplitudes and emphasize the necessity of using the effective sin q e f fW for the calculation of the coupling constants to correctly model the measured spin asymmetries in the Drell-Yan process. Thisis even more important for a correct generation of angular distributions of leptons in the decay frame of intermediate Z bosons.Then we discuss combinatorial and CP symmetries that allow us to reduce the number of parton subprocesses for whichdistinct codes of spin amplitudes are needed. (Appendix A is devoted to describe technical details of the introduced extensionof TauSpinner .) Numerical results shown in Section 5 are divided into three parts. The first one is devoted to the evaluationof systematic biases present if the (2 →
2) variant of
TauSpinner is used for spin effects (or for the matrix element weights) in pp → t t j j processes. Next, we present numerical consequences of the choice of the electroweak scheme, in particular: (i) inthe t t j j production, (ii) in the calculation of the spin correlation matrix used for the generation of t decays, for the observable1istributions. Section 6, closes the paper. Somewhat lengthy collection of tests are relegated to Appendices B and C.In the present paper we concentrate on physics oriented aspects of new implementations. All technical details and a descrip-tion of available options, resulting not only from the present work but also from the previous publications on TauSpinner , willbe collected in a forthcoming publication. The most important points for technical aspects of the program use are nonethelesspresented in Appendix A. Benchmark outputs from the programs are relegated to the project web page [12].
Before we start the discussion of new implementations in the
TauSpinner and present numerical results, let us shortly recallthe basis of the approach being used. For the Monte Carlo techniques of calculating integrals or simulating series of events, tobe well established in the mathematical formalism, one has to define the phase-space and the function one is going to integrate.One can parametrize the integral in the following form G = Z n (cid:213) j = d ˆ x j g ( ˆ x , ˆ x , ..., ˆ x n ) = lim N → ¥ N N (cid:229) i = g ( ˆ x i , ˆ x i , ..., ˆ x in ) , (1)where on the right-hand side, the sum runs over n dimensional vectors ˆ x ij of random numbers (each ˆ x ij in the [ , ] range)which define the point in the hypercube of coordinates. The N denotes number of events used.The function g consists of several components: the phase-space Jacobian resulting from the use of ˆ x j coordinates for thephase-space parametrization; the matrix element squared calculated for a given process at prepared phase-space-point; andfinally the acceptance function which is zero outside the desired integration region. Uniformly distributed random numbers ˆ x j are used as Monte Carlo integration variables in formula (1). The average value of g , calculated over the event sample, givesthe value of integral G . In practical applications a lot of refinements are necessary to assure acceptable speed of calculationand numerical stability. From a single sample of events several observables can be obtained simultaneously, e.g. in the form ofdifferential distributions (histograms).For the convenience of calculating multi-dimensional observables one introduces rejection techniques. The event i (con-structed from random-number variables ˆ x ij ) is accepted if an additional randomly generated number is smaller than g ( ˆ x i , ˆ x i , ..., ˆ x in ) / g max ; otherwise the event is rejected. The result of the integral is then equal to g max × n accepted n generated . Statistical errorof this estimate can be calculated using standard textbook Monte Carlo methods. In such a method, one has to assure that forthe allowed ˆ x j range, the condition 0 ≤ g ≤ g max holds. The accepted events are distributed according to dG and can be usedas a starting sample for the next step of the generation of weighted (or weight 1) events.The principle goal of the TauSpinner program is to un-do, modify or supersede the discussed above rejection. Let usassume that the sample of events, for which the program will be used, are distributed accordingly, with all details , to the knownproduction mechanism described by the formula d s = (cid:229) i , j , k , l f i ( x ) f j ( x ) dx dx F f lux d W ( p , p ; p , p , p t + , p t − ) | M i , j , k , l ( p , p , p , p ) | . (2)In Eq. (2), the (cid:229) i , j , k , l extends over all possible configurations of incoming and outgoing partons for the processes of i ( p ) j ( p ) → k ( p ) l ( p ) t + t − . The p i stand for the 4-momenta of incoming/outgoing partons, x and x stand for energy fractions of thebeams carried by the incoming partons, parton distribution functions are denoted as f i ( x ) , f j ( x ) respectively for the firstand the second incoming proton. The parton-level flux factor is denoted as F f lux and the phase-space volume element as d W .Finally the parton-level matrix element M i , j , k , l completes the formula. Obviously parton distributions (PDFs) are dependenton parton flavour configurations. In Eq. (2) the t decay phase-space and the corresponding matrix elements are omitted. Eventhough it amounts to semi-factorization, exploited by TauSpinner algorithms, we omit for now also the discussion of t -spincorrelation matrix. They are not essential for the clarification of requirements needed for TauSpinner algorithms.For the calculation of
TauSpinner weights in the case of replacing one production mechanism A with another one B , onehas to take into account not only differences in the matrix elements and PDFs but also, potentially, in F f lux and d W . Thus therespective weight is calculated as follows wt A → Bprod = (cid:229) i , j , k , l f Bi ( x ) f Bj ( x ) | M Bi , j , k , l ( p , p , p , p ) | F flux d W ( p , p ; p . p , p t + , p t − ) (cid:229) i , j , k , l f Ai ( x ) f Aj ( x ) | M Ai , j , k , l ( p , p , p , p ) | F flux d W ( p , p ; p . p , p t + , p t − ) (3)Although the factors F f lux and d W may cancel between the numerator and denominator in the case when all incoming andoutgoing partons are considered to be massless, they still may differ due to symmetry factors which are different for identicalor distinct flavours of partons. In actual application to a sample of experimental events the assumption that events are distributed accordingly to Eq. (2), i.e. with head-on collision ofincoming partons, may not hold. As a result, the reweighing procedure of A to B according to Eq. (3) will not anymore be mathematically rigorous. Section4.3 is devoted to tests for this important issue. Physics and matrix elements of ( → ) processes. The physics processes of interest are the Standard Model processes in pp collision with two opposite-sign t leptons and 2jets (quarks or gluons) in the final state . Such processes are described at the tree level by (2 →
4) matrix elements, withintermediate states being single or double Z , W , g ∗ , H or fermion exchange in the s- or t-channel. Depending on the initialstate, tree-level matrix elements are of the order of a S a EW or a EW , involving sometimes triple WW Z couplings. More detailsare given in Table 1. We will limit our implementation to the tree-level only, but with the emphasis on controlling the spinconfigurations.
MadGraph generated code into TauSpinner
There are automated programs for generating codes of spin amplitudes calculation. In the development of
TauSpinner we haveused
MadGraph5 [13]. Let us recall some details of this step of the program development to explain the adopted procedure,which may be useful in future for introducing anomalous couplings or new physics models.The
FORTRAN code for calculating matrix elements squared ( ME ) is generated using MadGraph5 with the following com-mands:a) import model sm-ckm b) with default definition of "multiparticles" p = g u c d s u˜ c˜ d˜ s˜j = g u c d s u˜ c˜ d˜ s˜ c) for the Higgs signal processes generate p p > j j h, h > ta+ ta- d) for the Drell-Yan–type SM background processes generate p p > j j ta+ ta- / h QED=4 e) and print the output using output standalone "directory name".
Setting the parameter
QED=4 enforces generation of diagrams up to 4th order in the electroweak couplings. Other settings areinitialized as in the default version of the
MadGraph5 setup. The generated codes for the individual subprocesses are thengrouped together into subroutines, depending on the flavour of initial state partons, and named accordingly. For example,
SUBROUTINE UDX(P,I3,I4,H1,H2,KEY,ANS) corresponds to processes initiated by u ¯ d partons. X after the letter U,D,S and C means the antiquark, i.e. UXCX correspondsto processes initiated by ¯ u ¯ c , while GUX – processes initiated by g ¯ u . The input variables are: real matrix P(0:3,6) for four-momenta of incoming and outgoing particles, integers
I3,I4 for the Particle Data Group (PDG) identifiers for final partonflavours, integers
H1,H2 stand for outgoing t helicity states; integer KEY selects the requested matrix element for the SMbackground (
KEY=0 ), the SM Higgs boson ( KEY=1 ), ANS returns the calculated value of the matrix-element squared. Accordingto the value of
I3,I4,KEY the corresponding subroutine generated by
MadGraph5 is called . The TauSpinner user usuallywill not access to the
KEY variable.Before integrating these subroutines into the
TauSpinner program, a number of modifications have been done for thefollowing reasons:a) Since
MadGraph5 by default sums and averages over spins of incoming and outgoing particles, while we are interestedin t spin states, the generated codes have to be modified to keep track of the t polarization;b) Moreover, since the subroutines and internal functions generated by MadGraph5 have the same names for all subprocesses
SMATRIX(P,ANS) , the names had to be changed to be unique for each subprocess. To be more specific, for the Higgssignal subprocess u ¯ d → c ¯ d h , h → t + t − the generated subroutine name is changed to UDX_CDX_H(P,H1,H2,ANS) , whilefor the background u ¯ d → c ¯ d t + t − process, the generated subroutine name is changed to UDX_CDX_noH(P,H1,H2,ANS) . Here as jets we understand outgoing partons. The
KEY > Note that by convention, setting
I3 = 0 and
I4 = 0 returns the matrix element squared summed over all possible final state partons; as a default thisoption is not used, and the corresponding sum is performed explicitly in the code.
FORTRAN files with implemented subroutines for calculating the matrix element square,grouped by the flavour of incoming partons, are given in the second column. Examples of processes in each category are givenin the last column. Partially redundant codes for some of the processes are used for tests only, this is the case of amplitudesstored in files
UCX.f and
CUX.f , the amplitudes of these two files can be obtained from each other by CP symmetry.Category of Corresponding FORTRAN files ProcessesMatrix Elements(1) GG.f gg → (cid:229) f q f ¯ q f (2) GD.f, GU.f gq f ( ¯ q f ) → gq f ( ¯ q f ) (3) DD.f, UD.f, UU.f, q f q f ( ¯ q f ¯ q f ) → q f q f ( ¯ q f ¯ q f ) CC.f, CS.f,DC.f, DS.f, SS.f CD.f,CU.f, SD.f, SU.f, US.f(4) DDX.f, UDX.f, UUX.f q f ¯ q f ( ¯ q f ¯ q f ) → q f ¯ q f ( ¯ q f ¯ q f ) CCX.f, CSX.f, DCX.f, DSX.f, q f ¯ q f ( ¯ q f ¯ q f ) → gg SCX.f, SSX.f, UCX.f, USX.f,CDX.f, CUX.f, SDX.f, SUX.fFor other processes and internal functions similar convention is used, see Table 1. Note that for example for the processes with cs quarks in the initial state, exchange of W ′ s is allowed, the final states cannot include gluons and the only allowed final statesare: cs , cd , us , ud . After taking into account permutation of incoming and outgoing partons and CP symmetric states this givesin total 4 × × =
32 non-zero contributions to the sum of Eq. (2). This is the case both for Drell-Yan–type background andHiggs-boson production processes. For the remaining processes the codes listed in Table 1 are also used with the help of CP symmetry or re-ordering of partons.At the parton level each of the incoming or outgoing parton can be one of flavours: ¯ b ¯ c ¯ s ¯ u ¯ d g d u s c b , with Particle DataGroup (PDG) identifiers: -5, -4, -3, -2, -1, 21, 1, 2, 3, 4, 5 respectively. For processes with two incoming partons, two outgoing t leptons and two outgoing patrons that gives 11 possibilities, most of them with the zero contribution, and many available onefrom another by relations following from CP symmetries and/or permutations of incoming and/or outgoing partons. Groupedby the type of initial state partons, the subroutines listed in Table 1 are currently limited to the first two flavour families. Thematrix elements for processes involving b -quarks are not yet implemented .Also, for practical purposes, for a pair of final-state parton flavours k = l , the MadGraph5 generated codes have been ob-tained for a definite ordering ( k , l ) , but not for ( l , k ) , to reduce the number of generated configurations. When TauSpinner isinvoked, the flavour configuration of outgoing partons is unknown and it takes into account both possibilities: thus a compen-sating factor + d ij has to be introduced. This is because of the organization of the sum in Eq. (3). The number of contributing subprocesses is very large. For the case of the non-Higgs Drell-Yan–type background processes,in which the t -pair originates either from the vector boson decay (including also cascade decays) or from multi-peripheralvector-boson fusion processes, MadGraph5 generates 82 subprocesses with partons belonging to the first two generations ofquarks, or gluons. Subprocesses in which all partons are of the same flavour (like u ¯ u → u ¯ u t + t − ) receive contributions from64 Feynman diagrams, subprocesses with two pairs of flavours – either 43 diagrams (if one pair is of up-type and the otherdown-type, like u ¯ u → s ¯ s t + t − ) or 32 diagrams (if both pairs are either down- or up-type, like u ¯ u → c ¯ c t + t − ), subprocesseswith three or four different flavours – 11 diagrams (like us → ud t + t − ), and subprocesses with two quarks and two gluons –16 diagrams. As far as the dynamical structure of the amplitudes is concerned, there are all together seven different topologiesof Feynman diagrams, with representatives shown in Fig. 1. Which of them contribute to a given subprocess depends onflavours of incoming and outgoing partons. Irrespectively of their origin, in all processes the polarizations of t leptons arestrongly correlated due to the helicity-conserving couplings to the vector bosons. The spin correlations of the produced t pairdepend on the relative size of the subprocesses with vector and pseudo-vector couplings contributing to the given final stateconfiguration. For example, in the case of q ¯ q → t + t − q ¯ q , see Fig. 1, diagram (d) contributes with 100% polarised t ’s since The matrix elements with b quarks are set to zero in our default installation. However, the program has already been set up so that the user-provided codesfeaturing b -quark processes can be activated by a C++ pointer at any moment using the TauSpinner::set_vbfdistrModif() method, see Appendix A fordetails. a) (b) (c) (d)(e) (f) (g) Figure 1: Typical topologies of diagrams contributing to the Drell-Yan–type SM process in u ¯ d → t + t − u ¯ d : multi-pheripheral(a), double-t (b), t-cascade (c), s-cascade (d), double-s (e), mercedes (f) and fusion (g) type of diagrams. (cid:1) (cid:2) (cid:1)(cid:2) ττ (cid:3) (cid:3)(cid:4) (a) (cid:1)(cid:2) (cid:1) ττ (cid:2) (cid:3)(cid:3) (cid:4) (b) Figure 2: Topologies of diagrams contributing to the Higgs production process u ¯ d → H ( → t + t − ) u ¯ d : vector boson fusion (a),Higgs-strahlung (b). In general, depending on the flavour of incoming partons, mediating boson could be W or Z .they couple directly to W ± . In diagram (g), the polarisation of Z / g ∗ is different than in the Born-like production because Z / g ∗ decaying to t + t − originates from the WW Z / g ∗ vertex. This leads to a distinct polarisation of t leptons.For the Higgs signal processes the t pairs originate from the Higgs boson decay, as imposed at the generation level, andthe number of subprocesses is reduced to 67. Each subprocess receives contributions from at most two Feynman diagrams,since with massless quarks of the first two generations, the Higgs boson can originate either from the vector boson fusion orfrom Higgs-strahlung diagrams, as illustrated in Fig. 2. Depending on the flavour configuration of incoming partons, mediatingboson is W or Z , which leads to almost 10 GeV shift between resonance invariant mass of the outgoing pair of jets in case ofHiggs-strahlung process. The helicity-flipping scalar coupling to the Higgs boson results in the opposite spin correlation ascompared to the case of the Drell-Yan–process. The individual t polarization is absent.Concerning the analytic structure of the differential cross sections, it is determined by topologies of contributing diagramsto a particular subprocess. For example, s -channel propagators will result in a resonance enhancement, while the t -channelones may lead to collinear or soft singularities (in the limit m W / s ≪ m Z / s ≪
1) regulated either by the phase space cuts or bythe virtuality of the attached boson line. Understanding differences in analytic structures of subprocesses will turn importantwhen discussing tests of reweighing technique of
TauSpinner in Subsection 4.2.Technically speaking, the sums in Eq. (1) or (3) defining the production weights used in
TauSpinner consist of 9 (11 if b-quarks are allowed) elements, which are potentially distinct and require their own subroutines for the matrix elementcalculation. Since most of the elements are equal zero, or some matrix elements are related to others by permutation ofpartons and/or CP symmetries, special interfacing procedure is prepared to exploit those relations. It reduces significantly the5omputation time and size of the program code. Details are given in Appendix A. In early versions of
TauSpinner the electroweak interactions were embedded into an effective (2 →
2) Born process for q ¯ q → t + t − . Its analytic form is given by Eqs. (3)-(5) and Table 2 of Ref. [14]. The adopted scheme is fully compatible withthe one of Tauola universal interface [15]. It is using the lowest order ME for the q ¯ q → Z / g ∗ → tt process, howeverwith the effective value for the sin q e f fW and running Z-boson width. Such a choice corresponds to a partial resummation ofhigher order electroweak effects, exactly as it was adopted at the time of precision tests of the Standard Model at LEP [16],with the remaining loop weak corrections at the per mille level.However, since the effects of WW boxes can be numerically significant for t lepton pairs of large virtuality or large invariantmass, there is an option to include genuine weak loop effects into TauSpinner effective Born, already since its version 1.4.0of June 2014 as well. It can be done for
TauSpinner in a manner similar to
Tauola universal interface [15] because thisprocess is implemented in both codes in the same way.Table 2 compares numerical values of the input parameters for the (2 →
2) and (2 →
4) processes. Variants of initializationfor ( → ) processes are explained in Table 3. It is worth to point out that by using over-constrained set of parameters( EWSH=4 ): a QED ( M Z ) , M Z , sin q e f fW , M W , G F essential effects of the loop corrections are taken into account, providing theresults for t -lepton polarisation close to LEP measurements [16]. As the parameters are not independent, this can lead toproblems if the input values are not consistent, especially when applied to processes other than (2 → →
4) hard subprocessesentering the pp → tt j j . The code from MadGraph5 has its own initialisation module consistent with the so called G F scheme,which uses G F , a QED and m Z as input parameters, see Table 2 (and EWSH=1 scheme in Table 3). As such, it uses tree-level(equivalent to on-shell) definition of the weak mixing angle sin q W = − M W / M Z = . Z boson couplings to fermions; also the constant width of Z -boson is used. Since the t -lepton polarization isvery sensitive to the value of the mixing angle, and for both Tauola and
TauSpinner the t physics is important target, such aLO implementation in the G F scheme is not sufficiently realistic. This is even more serious issue for the angular distributionsof leptons themselves, making such a scheme phenomenologically inadequate to any observable that relies on directions ofleptons. Alternatively, one could adopt the scheme with G F , m Z and sin q e f fW as input parameters (and EWSH=2 scheme inTable 3), but then the predicted tree-level W -boson mass is away from the measured value which would result in distortedspectra of jets coming from W decays (and shift in the resonance structure of the matrix element). In some regions of the phasespace the distortion can reach 40%. One can also use scheme with G F , m Z and m W , as input parameters (and EWSH=3 schemein Table 3), but the on-shell definition sin q W = − M W / M Z = . q W .There are two options: either include EW loop corrections simultaneously with QCD corrections, or adopt an effective schemewhich would allow at tree-level to account correctly for the t -lepton polarization at the Z -boson peak and physical W -bosonmass. Since the former is beyond the scope of the present paper, we take the second option.To this end, we define an effective scheme with q e f fW , in which the effective weak mixing angle sin q e f fW = . G F , m Z , m W and sin q e f fW ( EWSH=4 scheme in Table 3). Although being in principle flavour dependent, the value sin q e f fW is flavour universal with an accuracyof order 0.1%. Effectively such a procedure amounts to the inclusion of some of higher order EW corrections to the Z t + t − vertex. This value is used in all vertices, also in the triple gauge-boson coupling since the
WW Z coupling is essential for thegauge cancellation and it must match the couplings in other Feynman diagrams, forming together the gauge invariant part of thewhole amplitude. In our case we are not aiming at a careful theoretical study of higher order corrections; instead we checkednumerically that the introduction of dominant loop corrections to Z t + t − vertex through the effective sin q e f fW does not lead tonumerically important consequences for the WW Z vertex. For example, the effect of the mismatch of
WW Z and
Z f ¯ f couplingsfor the case of qq → qq tt subprocess is small, see Fig. 7, in Section 4.4. Thus, we gain consistency with observables, suchas t -polarization or t -directions, which would otherwise be off by ∼
40% at the expense of breaking EW relations in higherorder of perturbation theory. Moreover, since
TauSpinner is used to reweigh events, as given in Eq. (3), the uncertainties ofour procedure should to a large degree cancel out.For the purpose of comparison of the predicted t -lepton polarisation at the Z-boson peak, we provide four initialisationoptions for the (2 →
4) matrix elements, the first three motivated by the schemes used in [18], and the fourth one correspondingto q e f fW . They are specified in Table 3. Scheme labeled EWSH = t polarisation, and taking into account configurations with two additional jets, as shown in Section 5. For technicaltesting purposes we also introduce a scheme like EWSH=4 but with modified
WW Z coupling by 5% which we label as
EWSH=5 . Although by itself the vertex correction is not gauge invariant, it has been shown for the case of e + e − → f ¯ f that near the Z -pole the box contribution,needed to cancel gauge dependence, is numerically negligible. See for example Ref. [17]. ( → ) in Tauola code and ( → ) in MadGraph5 code.Note that in
Tauola code a QED = a QED ( Q = ) is used as an input for calculation of the Z couplings as well. This leads, inprinciple, to an over-all missing factor of ( a QED ( Q ) a QED ( ) ) . It can be thus dropped off, as long as it cancels out in calculation ofweights, the ratios of differential cross-sections. The numerical values of CKM matrix are taken from Ref. [11].Type Tauola code Input/Calculated
MadGraph5 code Input/Calculated(default for 2 →
2) (SM default for 2 → m H —– 125.0 GeV Input G H —– 0.0057531 GeV Input m Z G Z m W —– 80.4190 Calculated G W —— 2.04760 GeV Input m t sin q W / a QED G F —– 1.16639 10 − GeV − InputTable 3: Implemented EW schemes, the recommended EW scheme is
EWSH=4 which gives the t lepton polarisation on theZ-boson mass peak, in agreement with the measurement at LEP1 [19], and physical W boson mass.Type EWSH=1 EWSH=2 EWSH=3 EWSH=4 input: G F , a QED , m Z input: G F , sin q W , m Z input: G F , m W , m Z input: G F , m W , m Z , sin q e f fW m Z m W sin q W / a QED G F − GeV − − GeV − − GeV − − GeV − .4 QCD scales and parton density functions The distribution version of
TauSpinner is interfaced with
LHAPDF v6 library [20]. User has the freedom of choosing renor-malization and factorization scales, within the constraint that µ F = µ R , otherwise minor re-coding is necessary. To this end wehave implemented four predefined choices for the scale µ as should be expected for our processes: scalePDFOpt=0
200 GeV scalePDFOpt=1 µ = √ ˆ s scalePDFOpt=2 µ = (cid:229) m T , m T = m + p ⊥ scalePDFOpt=3 µ = (cid:229) E ⊥ , E ⊥ = E p ⊥ / | ~ p | where sums are taken over final state particles of hard scattering process. For the a s ( µ ) we provide, as a default, a simplechoice of the µ dependence, following the leading logarithmic formula, a s ( µ ) = a s ( M Z ) + pa s ( M Z )( − N f / ) ln µ M Z (4)with the starting point a s ( M Z ) = . a s is used for the case of the fixed coupling constant, that is for scalePDFOpt=0 .The reweighting procedure of TauSpinner itself may be used to study numerically the effects of different scale choices, aswell as for the electroweak schemes, see the discussion later in Section 5 and Appendix A.2. → ) matrix elements. For the purpose of testing the consistency of implemented codes, generated with
MadGraph5 and modified as explained in Sect.3.1, we have chosen a fixed kinematic configuration at the parton level . For such kinematics we have calculated the matrixelement squared for all possible helicity configurations of all subprocesses using the codes implemented in TauSpinner andchecked against the numerical values obtained directly from
MadGraph5 . The agreement of at least 6 significant digits has beenconfirmed.
As further tests of the internal consistency of matrix element implementation in
TauSpinner we have used the reweightingprocedure by comparing a number of kinematic distributions obtained in two different ways: the first one obtained directly fromevents generated for a specified parton level process REF (a reference distribution REF), and the second one (GEN reweighted)obtained by reweighting with
TauSpinner events generated for a different process GEN. These tests have been performed in afew steps as follows. • Series of 10 million events each for a number of different processes in pp → tt j j (with specified flavours of final statejets, or for subprocesses with selected flavours of incoming partons) with MadGraph5_aMC@NLO [11] v2.3.3 at LO havebeen generated. Samples were generated for pp collisions at the c.m. energy of 13 TeV using CTEQ6L1 PDFs [21]linked through LHAPDF v6 interface. Renormalization and factorization scales were fixed to µ R = µ F = m Z . Only veryloose selection criteria at the generation level were applied: invariant mass of the tt pair was required to be in the range m tt = −
130 GeV, and jets to be separated by D R j j > . p jT > MadgraphCards.txt which is included for reference in
TAUOLA/TauSpinner/examples/example-VBF/benchfiles directory. • The testing program was reading generated events stored in the
LesHouches Event File format [22] filtering the onesof a given
ID1, ID2, ID3, ID4 configuration of flavour of incoming/outgoing partons corresponding to the processGEN. The weight wt ME allowing to transform this subset of events into the equivalent of reference REF one, wascalculated as wt ME = | ME ( ID , ID , ID ′ , ID ′ ) | | ME ( ID , ID , ID , ID ) | (5) This test is build into the
TauSpinner testing and can be activated with the hard-coded local variable of
TAUOLA/TauSpinner/src/VBF/vbfdistr.cxx by setting const bool DEBUG = 1; . Numerical results are collected on the project web page [12]. Several tests were repeated also for the full spectrum, i.e. starting from m tt >
10 GeV (ID1, ID2, ID3’, ID4’) . Note that for this test to be meaningful one has to select processes with thesame initial state partons, so that the dependence on the structure functions cancels out. A very good agreement betweenthe REF and GEN reweighted distributions was found for 10 different kinematic distributions for several configurationsof (ID1, ID2, ID3, ID4, ID3’, ID4’) . It has shown a very good numerical stability, which was not obvious fromthe beginning as events corresponding to the REF and GEN processes may have very different kinematic distributionsdue to their specific topologies and resonance structures of Feynman diagrams. • In the next step, the tests were repeated, but now reweighting the matrix elements convoluted with the structure functionsof the incoming patrons and summing over final states restricted to the selected sub-groups (named respectively C and D of parton level processes). In this case the weight is calculated as wt C → Dprod = (cid:229) Di , j , k , l f i ( x ) f j ( x ) | M i , j , k , l ( p , p , p , p ) | F flux d W ( p , p ; p . p , p t + , p t − ) (cid:229) Ci , j , k , l f i ( x ) f j ( x ) | M i , j , k , l ( p , p , p , p ) | F flux d W ( p , p ; p . p , p t + , p t − ) (6)where the notation as for Eq. (3) is used, except that now the (cid:229) C , D mean that summation is restricted to processesbelonging to sub-groups C , D , respectively. For testing the code implementation for the Drell-Yan process the groups,listed in the first column of Table 1, were reweighted, one to another.The reweighting tests performed between sub-groups of processes, and later, between groups of processes listed in Table 1,allowed to check relative normalization of amplitudes. Again, a good agreement has been found . For the tests, the followingkinematical distributions were used: − Pseudorapidity of an outgoing parton j . − Pseudorapidity gap of outgoing partons. − Rapidity of the tt and j j systems. − Transverse momentum of the tt and j j system. − Invariant mass of the tt and j j system. − Longitudinal momentum of the tt and tt j j . − Cosine of the azimuthal angle of t lepton in the tt rest frame.Let us discuss some of these results, shown in Fig. 3 and Fig. 4 (the complete set of distributions is shown in Appendices Band C). In each plot the distribution REF for the reference process is shown as a black histogram, while the red histogram showsthe distribution for a different process GEN. Both histograms are obtained directly from the MadGraph5 generated samples ofREF and GEN processes, respectively. Now the histogram GEN is reweighted using
TauSpinner and the resulting reweightedhistogram is represented by the red points with error bars. For the test to be successful the red points should follow the blackhistogram; the ratio of the REF and GEN reweighted distributions is shown in the bottom panel of each figure.Let us note, that in our tests, we reweight events of substantially different dynamical structures over the multi-dimensionalphase-space. This may be not evident from the histograms shown in figures, which can be both for the REF and GENreweighted distributions rather regular and similar. Nevertheless, several bins of GEN reweighted distributions with smallerrors can be found to lie below the REF distribution, whereas a few above with large errors. This second category of bins ispopulated by a few events, which originate from the flat distribution of the GEN process, receiving high weight due to someresonance/collinear configuration of the REF process. This is a technical difficulty for the testing, but is not an issue of theactual use of
TauSpinner when all subprocesses are used together. To confirm that the observed deviations are not significantstatistically we have reproduced plots from Fig. 3 and Fig. 4 for four independent series of events. We observed that bins withlarge error or sequences of few bins with large deviations were randomly distributed between these series strongly indicatingthat observed deviations are of statistical origin. As primarily we are not interested in use of implemented code to reweightbetween the groups of parton level processes, for checking general correctness of its implementation it was sufficient to usefour statistically independent samples only. In practical applications, contributions from all processes will be merged togetherand weights will become less dispersed.Similar tests have been performed for the Higgs boson production. Fig. 5 shows the comparison of generated and reweigheddistributions for the jet pseudorapidity and for the pseudorapidity gap between jets in the case of qq and q ¯ q processes. As theresonant structure in the m j j distribution coming from Z → q ¯ q and W → q ¯ q is different in REF and GEN processes, results ofsome bins feature unexpectedly large statistical fluctuations.Finally, let us stress that simple, but nonetheless, necessary check have been done as well: from the inspection of thecontrol outputs we confirmed that the dominant contributions to cross sections are distinct for Drell-Yan and Higgs productionprocesses, and that the slopes of energy spectra of t -decay products are of a proper sign. That confirms that our installation isfree of possible trivial errors in spin implementation. Technical point is worth mentioning: we had to randomize order of final state partons in events generated by
MadGraph5 , as such order in not imposed inthe matrix elements implemented in
TauSpinner . j h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (2): g q (g qx) jj t t fi GEN (1): g g GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 jj t t fi REF (3): q q (qx qx) jj t t fi GEN (1): g g GEN re−weighted -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (4): q qx jj t t fi GEN (3): q q (qx qx) GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (1): g g jj t t fi GEN (4): q qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (2): g q (g qx) jj t t fi GEN (4): q qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o Figure 3: Shown are distributions of the pseudorapidity gap between outgoing partons for the GEN sub-process (thin red line)and after its reweighting to the reference one (GEN reweighted, red points). Reference distribution REF is shown with a blackline. GEN and REF sub-processes are grouped as listed in Table 1. The qx on plots, denote antiquark i.e. ¯ q . More plots forother distributions are given in Appendix B. 10 (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (2): g q (g qx) jj t t fi GEN (1): g g GEN reweighed R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (1): g g GEN re−weighted R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (4): q qx jj t t fi GEN (3): q q (qx qx) GEN reweighed R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (1): g g jj t t fi GEN (4): q qx GEN reweighed R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (2): g q (g qx) jj t t fi GEN (4): q qx GEN reweighed R a t i o Figure 4: Shown are distributions of transverse momenta of t pairs, p tt T with labeling as in Fig. 3.11 h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o t h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o Figure 5: Shown are the distributions of the jet pseudorapidity (left plots) and the pseudorapidity gap between outgoing partons(right plots) with labeling as in Fig. 3 but for the processes of Higgs boson production. More plots for other distributions aregiven in Appendix C. 12 spin -wt
ISRspin wt-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 E v en t s -1 Drell-Yan processes j j t t fi p p ) prod / wt ISRprod (wt log-3 -2 -1 0 1 2 3 E v en t s -1 Drell-Yan processes j j t t fi p p Figure 6: Impact on the matrix element calculation of parton shower smearing, as explained in the text. On the left, thedifference of spin weights calculated with and without ISR parton shower kinematic smearing is shown. On the right, the ratioof matrix element weights calculated for the two cases is shown. Sample of 10000 events was used.
After technical tests at the hard process level (convoluted with structure functions), we turn to check the algorithm on eventswhere the incoming parton momenta can not be assumed to be along the beam direction due to the presence of the partonshower in the initial state (ISR). For that purpose, we have taken events generated with
MadGraph5 and added ISR with thedefault version of
Pythia 8.2 (as described in Ref. [23]). The two statistically correlated samples were constructed and usedby
TauSpinner for calculation of spin weights ( wt spin ) and production weights ( wt prod ). Fig. 6 shows the number of eventsas a function of differences for the spin weights calculated for each event from configurations with and without ISR partonshower. Similarly, shown is the ratio of wt prod weights calculated for configurations with and without ISR parton shower. Onecan see from Fig. 6 (left plot), that the spin weights for the cases with and without ISR are strongly correlated. Majority ofevents reside in central bins of the distribution and the difference in weights is smaller than the bin width. Also the matrixelement weights for the two cases are strongly correlated, see Fig. 6 (right plot). Majority of events reside in central bins. Wecan conclude that, similarly as in the past [7] for the (2 →
2) process, the algorithm which is applied to kinematics of the hardprocess particles effectively removes impact of the initial state transverse momentum and leads to results which are stable withrespect to the presence of extra showering. This test is of more physical nature, since in such a case Eq. (2) does not hold forthe distribution of reweighted events and, as a consequence, reweighting with Eq. (3) is featuring an approximation, which wehave validated with this test. Note that adding ISR means only that the system of partons and t leptons outgoing from the hardprocess underwent (as a whole) a boost and rotation before calculating matrix elements and PDF’s. This justifies the evaluationof x , x , fraction of proton energies carried by the incoming partons in collinear approximation. For (2 →
2) process, resummation of higher order effects into effective couplings is well established. In (2 →
4) case, care isnecessary, one may destroy gauge cancellations where matching of Z emissions from quark lines with the ones of the t -channelW must be preserved. In Fig. 7 we demonstrate results, where effective sin q W is used in otherwise G F scheme, one can see thatvarying arbitrarily of WW Z coupling by ± .
05 bring marginal effects only, even for the q q , ¯ q ¯ q processes, chosen to maximizethe relative effect of WW Z coupling mismatch. The effect is negligible for the shown, most sensitive kinematical distributionstudied. The estimate of the average polarisation remains unchanged. This is an expected result as for our amplitudes conditionsin q W = − M W / M Z is in principle not needed for gauge cancelation. Once we have completed our technical tests, and gained confidence in the functioning of the 2 → TauSpinner algorithms, let us turn to presentation of numerical results. In spite of a limited scope of the present version, like lack of theloop-induced gluon coupling to the Higgs boson, or subprocesses with b-quarks,
TauSpinner can already be used as a toolto obtain numerical results of interest for phenomenology. Note that b-quarks as final jets can be tagged, and should thus betreated separately, while the contribution from the b-quark PDFs is rather small. Possible applications of
TauSpinner arepresented below. 13 j h D -10 -8 -6 -4 -2 0 2 4 6 8 10 jj t t fi REF (3): q q (qx qx) jj t t fi GEN (3): q q (qs qx) GEN reweighed with GC_53 * 0.95 -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 jj t t fi REF (3): q q (qx qx) jj t t fi GEN (3): q q (qs qx) GEN reweighed with GC_53 * 1.05 -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (4): q qx jj t t fi GEN (4): q qx GEN reweighed with GC_53 * 0.95 -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (4): q qx jj t t fi GEN (4): q qx GEN reweighed with GC_53 * 1.05 -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o Figure 7: Distribution of D j j , reweighted to the one corresponding to WW Z coupling (internal
MadGraph5 notation GC _53)multiplied by factor 0.95 (right) and 1.05 (left), shown for q q , ¯ q ¯ q (top) and q ¯ q (bottom) Drell-Yan processes.14able 4: Comparison of the t -lepton polarisation in tt j j events, calculated using TauSpinner weight wt spin of ( → ) and(2 →
4) processes and G F EW schemes with sin q W = 0.22222. Required invariant mass of the t pair of m Z ±
10 GeV andlow threshold on outgoing partons transverse momenta, p T > MadGraph5 , selected accordingly to flavours of incoming partons.
TauSpinner algorithm is not usingthis information and the average of all possible configurations is used. In case of the last collumn, we restrict the average to theones actually used for the selected subset of events.Process Fraction Polarisation Polarisation Polarisationof events ( → ) ( → ) ( → ) Average Average Process specificAll processes -0.2142 ± ± ± g g → t t j j ± ± ± g q , g ¯ q → t t j j ± ± ± q q , ¯ q ¯ q → t t j j ± ± ± q ¯ q → t t j j ± ± ± t -lepton polarisation in tt j j events, calculated using TauSpinner weight wt spin of (2 →
2) and(2 →
4) processes and G F EW schemes with sin q W = 0.22222. Required invariant mass of the t pair of 100 −
130 GeV andlow threshold on outgoing partons transverse momenta, p T = ( → ) ( → ) ( → ) Average Average Process specificAll processes -0.4837 ± ± ± g g → t t j j ± ± ± g q , g ¯ q → t t j j ± ± ± q q , ¯ q ¯ q → t t j j ± ± ± q ¯ q → t t j j ± ± ± t lepton polarisation For calculation of weights earlier versions of
TauSpinner used the elementary ( → ) parton level q ¯ q ( gg ) → Z / g / ( H ) → t − t + amplitudes factorized out from the complex event processes. This approach can now be verified with the explicitly implemented ( → ) matrix elements when two hard jets are present in the calculation of the amplitudes. The physics of interest is themeasurement of the Standard Model Higgs boson properties in decays to the t leptons and its separation from the Drell-Yanbackground of t -pair production.We start by confirming the overall consistency of the calculations, comparing results from ( → ) and ( → ) calculationson inclusive tt j j events, with t -pair around the Z-boson mass peak, but with very loose requirements on the accompanying jets, p jetT > ( → ) and ( → ) forfour categories of hard processes and for cuts selecting events at the Z peak or above. For the ( → ) implementation shownis also the difference when estimating polarisation using an average of all hard processes, or for only specific category.To verify that not only the calculation of spin averaged amplitudes, but the contributions from specific helicity configu-rations are properly matched between (2 →
2) and (2 → E p / E t spectra in the t ± → p ± n decays.This variable is sensitive to the polarisation of the tt system and longitudinal spin correlations. To introduce spin effects tothe sample, otherwise featuring non-polarized t decays, we have used weights calculated by TauSpinner . The spin weightdistribution, the visible mass of t ’s decay products combined and the energy fraction carried by the p ± in t → pn decays arecompared for two different EW schemes in Fig. 8.To emphasize possible differences between using the ( → ) or ( → ) matrix elements for calculating spin weightsfor tt j j events, we have applied simplified kinematic selection inspired by the analysis of [2], called in the following VBF-like selection: transverse momenta of outgoing jets above 50 GeV; pseudorapidity gap between jets, | Dh j j | > .
0; Transversemomenta of outgoing t leptons of 35 GeV and 30 GeV, respectively and pseudorapidity | h t | < .
5. It is also required that theinvariant mass of the t -lepton pairs and jj pair is above the Z -boson peak. Results for the average polarization, are shown inTable 6. 15 WT weight 0 1 2 3 4 5 6 7 8 9 10 N o r m a li s ed t o un i t y j j t t fi p p 2), EWSH=4 fi (2 4), EWSH=1 fi (2 4), EWSH=4 fi (2 (GeV) tt visible m0 20 40 60 80 100 120 140 N o r m a li s ed t o un i t y j j t t fi p p 2), EWSH=4 fi (2 4), EWSH=1 fi (2 4), EWSH=4 fi (2 + t /E + p E0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N o r m a li s ed t o un i t y j j t t fi p p 2), EWSH=4 fi (2 4), EWSH=1 fi (2 4), EWSH=4 fi (2 Figure 8: Distribution of the spin weight (top), of the invariant mass of visible decay products of t -pairs (bottom-left) and theenergy fraction of the decaying t lepton carried by p ± in t → p ± n (bottom-right), weighted with (2 →
2) and (2 →
4) matrixelements and for different EW schemes. 16able 6: Polarisation of the t -lepton in tt j j events, calculated using TauSpinner weight wt spin and (2 →
2) and (2 → EWSH=1 scheme with sin q W = 0.22222. For this comparison the initialisation of ( → ) process was alsoadopted to EWSH=1 scheme. Required is the invariant mass of the t -pair and j j -pair above 120 GeV and VBF-like selection(see text). Process Fraction Polarisation Polarisation Polarisationof events ( → ) ( → ) ( → ) Average Average Process specificAll processes -0.5026 ± ± ± g g → t t j j ± ± ± g q , g ¯ q → t t j j ± ± ± q q , ¯ q ¯ q → t t j j ± ± ± q ¯ q → t t j j ± ± ± t -lepton in tt j j events, calculated using TauSpinner weight wt spin of ( → ) and ( → ) processes and different EW schemes. Required is the invariant mass of the t pair of m Z ±
10 GeV and low threshold on gluontransverse momenta of p T > ( → ) ( → ) sin q W = 0.222246 EWSH=1 -0.2140 ± ± q W = 0.231470 EWSH=2 -0.1488 ± ± q W = 0.222246 EWSH=3 -0.2140 ± ± q W = 0.231470 EWSH=4 -0.1488 ± ± qq , ¯ q ¯ q → tt j j processes to about 25% of the total cross section. Thehighest discrepancy found between the predicted t lepton polarisation with (2 →
2) and (2 →
4) matrix element is at the levelof 4% in absolute value, being relative 10% of the polarisation. Using the average (2 →
4) matrix element i.e. assuming thatthe initial state is not known, reduces the discrepancy by factor 2. For the polarisation averaged over all production processesthe difference between (2 →
2) and (2 →
4) matrix element is at the level of 1.0 - 1.5% in absolute value, which is only 2%relative effect.The above results indicate strongly that
TauSpinner in the (2 →
2) mode is sufficient for the evaluation of spin effectsobservable in t decays. The (2 →
4) mode is useful mainly for validations or systematic studies.Please note, that results were obtained with the G F -on-shell scheme EWSH=1 , thus are different from physically expectedvalues. Let us continue now with the discussion of typical initializations used in calculations of matrix elements for (2 → In initialization of programs like
MadGraph5 , the tree level formula for weak mixing angle, sin q W = − M W / M Z = . EWSH=1 or EWSH=3 schemes described previously. This theoretically motivated choice is quite distantfrom the sin q e f fW = . Z -boson to fermions and which is usedin the EWSH=2 scheme. The LO approximation used in
MadGraph5 initialization and in our tests so far, can not be used for theprogram default initialization. We are constrained by the measured values of the M W , M Z and sin q e f fW and that is why the EWSH=4 scheme is chosen as a default.One must keep in mind, that the t -lepton polarisation in Z-boson decays is very sensitive to the scheme used for theelectroweak sector. Table 7 gives numbers for the average polarisation in the case of G F and effective EW schemes. Resultsfor using ( → ) and ( → ) matrix elements coincide within statistical error, for event sample with rather loose kinematicalcuts. On the contrary, results of calculations strictly following the G F scheme are off by
50 % with respect to experimentallymeasured value, -0.1415 ± Summary and outlook
In this paper, new developments of the
TauSpinner program for calculation of spin and matrix-element weights for the previ-ously generated events have been presented. The extension of the program enables the calculation of spin and matrix-elementweights with the help of (2 →
4) amplitudes convoluted with parton distribution functions. Required only are kinematicalconfigurations of the outgoing t leptons, their decay products and two accompanying jets.The comparisons of results of the new version of TauSpinner , where matrix elements feature additional jets, and theprevious one where the Born-level (2 →
2) matrix element is used, offer the possibility to evaluate systematic errors due to theneglect of transverse momentum of jets in calculating spin weights. We have found that for observables sensitive to spin, thebias was not exceeding 0.01 for sufficiently inclusive observables with tagged jets.Numerical tests and technical details on how the new option of the program can be used were discussed. Special emphasiswas put on spin effects sensitive to variants for SM electroweak schemes used in the generation of samples and available ininitialization of
TauSpinner . The effect of using different electroweak schemes can be as big as 50% of the spin effect andcan be even larger for angular distribution of outgoing t leptons. For the configurations of final states with a pair of jets closeto the W mass the effect can be also high, up to 40%. For applications of TauSpinner , we recommend the effective schemeleading to results on t polarisation, m Z and m W close to measurements.The phenomena of t decay and production are separated by the t lifetime. This simplifying feature is used in organizingthe programs. As a consequence for the generated Monte Carlo sample, different variant of the electroweak initialization maybe used for generation of t lepton momenta and later, for implementation of spin effects in the t ’s decays. Such a flexibility ofthe code may be a desirable feature: TauSpinner weight calculation can be also adjusted to situation when the matrix elementweight and spin weight have to be calculated with distinct initializations.Numerical results in the paper were obtained with the help of weights. Not only spin weight, but also the production weighthas been used to effectively replace the matrix element of the generation. This feature, introduced and explained in Ref. [8], wastargeting an implementation of anomalous contributions. However, its use can easily be adopted for studies of the electroweaksector initialization. This helps to get results quicker thanks to correlated sample method. It provides technical advantage forthe future, namely the possibility for the use of externally provided matrix elements or initialisation of EW schemes.In tests discussed in this paper we have used
MadGraph5 generated events for the pp → t + t − j j process. The incomingpartons were distributed according to PDFs, but in most cases neither the transverse momentum of incoming state nor additionalinitial state jets were allowed. We will return to this point in the future, with greater attention. We may also be able to extend,with the help of the program developed for the present paper, the work on factorization of the effective Born of (2 → p T activity of our parton parton → t + t − j j hardprocesses from the pp collision. The special case of processes with a single hard jet in the final state and its correspondingmatrix elements will be also useful for such tests and we plan to return to such topic in the near future.The program is now ready for studies with matrix elements featuring extensions of SM amplitudes. Systematic errorshave been discussed. Let us stress, that results of such studies depend on the definition of observable and need to be repeatedwhenever new observables or selection cuts are introduced. In the evaluation of impact of new physics or variation of SMparameters on experimentally accessible distribution it is necessary to compare results of calculations which differ by suchchanges. The Monte Carlo simulations are used, whenever detector acceptance and other effects are to be taken into account. Acknowledgments
We thank Tomasz Przedzi´nski for work on the early versions of
TauSpinner (2 →
4) implementation and help with docu-mentation of technical aspects.This project was supported in part from funds of Polish National Science Centre under decisions UMO-2014/15/ST2/00049and by PLGrid Infrastructure of the Academic Computer Centre CYFRONET AGH in Krakow, Poland, where majority ofnumerical calculations were performed. JK, ERW and ZW were supported in part by the Research Executive Agency (REA) ofthe European Union under the Grant Agreement PITNGA2012316704 (HiggsTools). WK was supported in part by the GermanDFG grant STO 876/4-1.
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Comments on the code organization and how to use it.
In this Section, we collect information on how to use the (2 →
4) option of the
TauSpinner program. We will concentrateon aspects, which are important to demonstrate the general scheme and organization of the new functionality of the code. Weassume that the reader is already familiar with previous versions of
TauSpinner or Refs. [9, 10].
A.1 Technical implementation
The general strategy of the reweighting technique of
TauSpinner for the case of configurations with tt j j final states does notdiffer much from the previous one in which only four-momenta of outgoing t leptons and their decay products have been used.Nevertheless a few extensions with respect to Refs. [9, 10] have been introduced as explained below.1. For calculation of t polarimetric vectors from their decay products and for the definition of boost routines from t -lepton’srest frames to the laboratory frame the same algorithms as explained in Refs. [9, 10] are used.2. Before evaluation of production matrix elements and numerical values of PDF functions, one has to reconstruct the fourmomenta of the incoming partons. For that purpose the following assumptions are made:(a) For calculation of the hard process virtuality ¯ Q and p z , the four-momenta of t leptons and jets are summed to afour-momentum vector ¯ Q µ .(b) The ¯ Q µ determined from experimental data, or from events generated by another Monte Carlo program, may havesizable transverse momentum which has to be taken into account when the directions ˆ e jz of the two incomingpartons j = , Q µ . The time-like components of boosted versors are dropped and the remaining space-like partis normalized to unity. Note that ˆ e , z obtained in this way do not need to remain back-to-back.(c) Four-momenta of t leptons and of accompanying jets/partons are forced to be on mass-shell to eliminate all possibleeffects of the rounding errors. This is necessary, to assure the numerical stability of spin amplitude calculations.(d) The four-momenta for the incoming partons are constructed using the direction of the versors ˆ e , z and enforcingfour-momentum conservation.This exhausts list of steps and changes to the components for production and t decay matrix elements of the TauSpinner in (2 →
4) mode with respect to (2 →
2) one. Only in steps (a) and (b) there are differences with respect to the original(2 →
2) case.3. The new source code for the matrix elements library and interfaces is stored in
TAUOLA/TauSpinner/src/VBF
4. An exemplary code example-VBF.cxx showing how to use
TauSpinner with (2 →
4) matrix elements can be foundin directory
TAUOLA/TauSpinner/examples/example-VBF . The extract of this code is given in Subsection A.4. In thesame directory the code read_particles_for_VBF.cxx, read_particles_for_VBF.h for reading the events fromthe file in the
HepMC format, as well as a file events-VBF.dat with a sample of 100 events, are also stored. Some furthertechnical details can be found in the
README file of that directory.5. At the initialization step, basic information on the input sample like the center of mass energy for the pp collisions orthe set of parton density functions, PDF’s, should be configured. This part of the configuration has not changed sinceprevious version of TauSpinneR . See Section A.2 and A.46. The example-VBF.cxx provides also a prototype for implementation of the user code to replace the default ( → ) matrix element of TauSpinner .7. The spin weight WT (denoted in this paper as wt spin ) is calculated using (2 →
4) matrix element by invoking method
WT = calculateWeightFromParticlesVBF(p3,p4,X,tau1,tau2,tau1_daughters,tau2_daughters);
Note that only final state four-vectors of t ’s, their decay products and outgoing jets, are passed to calculate the weight.8. The method getME2VBF(p3,p4,X,tau1,tau2, W,KEY) returns a double-precision table W[2][2] , which contains partonic-level cross sections for t + t − helicity states ( , − ) , ( − , ) , ( − , − ) , ( , ) , respectively. They are obtained by sum-ming matrix element squared over all parton flavour configurations and convoluted with the corresponding PDFs. Directuse of this method is optional. It is invoked internally by TauSpinner though. We take weighted average of the two ˆ e , z directions, see in the code of vbfdistr.cxx method getME2VBF() definition of P[6][4] .
20. Several scenarios (models) of the hard process for calculating corresponding spin weight (note that at the same time theweight for the production matrix elements is calculated), are possible. At the initialisation step the choice is made andthe technical internal parameter
KEY is set. We give below some details: • KEY=1 for the Standard Model Higgs process, matrix elements explained in Section 3.1 is used. • KEY=0 for non-Higgs Drell-Yan–like processes, matrix elements explained in Section 3.1 is used. • KEY > nonSM=true option is used , and the matrix elementcalculations are modified with the routines provided by the user. Provisions with KEY=3 have been prepared forthe non-standard Higgs-like production process and
KEY=2 for the Drell-Yan–like. The required choice is madeimplicitly, at the initialisation step when setting the pointer to the user provided function vbfdistrModif andselecting initialization variable nonSM2=1 which will set internal global variable nonSM=true . The result of defaultcalculations will be passed to vbfdistrModif function to be overwritten with user-driven modifications to thematrix elements, without the need of re-coding and recompiling standard
TauSpinner library. • The internal parameter
KEY in general does not need explanation. However, as it is passed latter to the methodsfor calculating a s or matrix elements, which may be replaced from user main program by re-setting the pointer,documentation was necessary.10. The method double getTauSpin() returns helicities attributed to t leptons on the statistical basis. It is the same methodas already implemented for the (2 →
2) case.11. The value which is returned by the method double getWtNonSM() depends on the configuration of two flags: nonSM and relWTnonSM . • For relWTnonSM=true : in case of nonSM=true the method returns the weight obtained from Eq. (3), for nonSM=false the value of 1 is returned. • For relWTnonSM=false : in case of nonSM=true the method returns the numerator of Eq. (3), for nonSM=false the value of Eq. (3) denominator is returned.The above discussed weight features matrix elements squared and summed over spin degrees of freedom. It has sim-ilar functionality as already implemented for the ( → ) case. It is supposed to supplement the spin weight WT of TauSpinner . In general, spin weight differs for the SM and nonSM calculation. The ratio of these two has to be used formodifying decay product kinematic distributions. Finally let us point out, that also the helicity of t ’s will be attributed atthe nonSM step of the calculation, corresponding to the chosen nonSM model.Let us bring some further points on the details of the use of the example. • This example has been prepared to read events in
HepMC format. An additional tool lhe-to-hepmc.exe convertin
LHE event to
HepMC has been provided as well. • To read events form the data file, the method read_particles_for_VBF stored in file
TAUOLA/TauSpinner/examples/example-VBF/read_particles_for_VBF.cxx is used. It is invoked as follows: int status =read_particles_for_VBF(input_file,p1,p2,X,p3,p4,tau1,tau2,tau1_daughters,tau2_daughters); which reads consecutive event and retrieves the following information: four-momenta of incoming and outgoing partons,denoted as p1,p2 , and p3,p4 , respectively; four-momenta of outgoing t leptons, tau1, tau2 and tau1_daughters,tau2_daughters which stand for lists of decay products (their four-momenta and PDG-id’s ) and, ifavailable, also the four-momentum of an intermediate resonance X and its PDG-id . It returns status=1 if no event toread was found (this is specific to the method chosen for reading the events and is used in the user program only).Let us stress, that the above interface to read event record is not a part of
TauSpinner library. It is used in the demon-stration program and is adopted to the particular conditions. It is expected to be replaced by the user with the customizedone. Implementation of such a method must match conventions for the format and information on different particles inthe stored event. For example, for formats (eg.
HepMC [25] or lhe [22]) distinct conventions are used in Monte Carlogenerators: i.e. for relations among particles and intermediate states (resonances) which may be explicitly written intoevent record or omitted. The same is true for production/decay vertices, status codes, etc. With a variety of conventionsused, it can be highly error-prone and lead to necessity of non trivial implementations in the code, like can be seenin read_particles_for_VBF method provided in the distribution tar-ball. See eg. [15, 26] for discussion of similardifficulties in other projects. In fact, for
TauSpinner algorithms this is less an issue, as the information on intermediateand incoming states is not used, even though it can be very useful for testing purposes. See later Appendix A.2, the first and the second bullets.
TauSpinner algorithms for ( → ) processes which must be read in, is limited to: four-momenta of outgoing jets and of t ± including all their decay products and PDG identifiers of t ± and their decay products.The four momenta of incoming partons and the intermediate resonance are not needed but can be used for tests. Theimminent next step is to exploit also, if available, information on the four-momentum of the intermediate Z / g / H state(or any other non-standard resonance) which decays to t ± lepton pair as well, as they can be used to tackle the effect ofthe QED bremsstrahlung in its decay, similar as it was done for the TauSpinner algorithms in ( → ) case. A.2 Initialisation methods • Matrix elements:
The
TauSpinner library includes codes for calculation of matrix elements squared for all (first two families) parton levelcross sections of ( → ) processes. Use of this library functions can be over-loaded, with the user’s own matrix elementsimplementation by providing respective function vbfdistrModif(...) . Its usage is activated at initialization withcommand TauSpinner::set_vbfdistrModif(vbfdistrModif) , which sets the pointer to vbfdistrModif(...) . Askeleton function, vbfdistrModif(...) , for user provided calculation of matrix elements squared is included in /TAUOLA/TauSpinner/examples/example-VBF/example-VBF.cxx .The invocation inside
TauSpinner library of the function vbfdistrModif(int I1,int I2,int I3,int I4,int H1,int H2,double P[6][4],int KEY,double result) includes among its arguments the result of the default
TauSpinner
Standard Model ( → ) calculation.The vbfdistrModif(...) of the demonstration program returns directly the result .The following arguments are passed to this function, and this must be obeyed in its declaration: – The first four arguments
I1,I2,I3,I4 denote PDG identifiers [27] of incoming and outgoing partons (for gluon
ID=21 ) . – The following two
H1,H2= ± denote helicities of outgoing t + and t − . – Matrix
P[6][4] encapsulates four-momenta of all incoming/outgoing partons and t ± leptons. They are for mass-less partons and for massive t leptons. Energy momentum conservation is required at the double precision level.The partons are not expected to be in the phase-space regions close to the collinear/soft boundaries. – The parameter
KEY=0 is reserved for the SM default processes of Drell-Yan–type (all Feynman diagrams included,but the ones with H → t + t − ), while KEY=1 for SM processes with the Higgs production and its decay to t -leptonpair. In these two cases vbfdistrModif is not activated. For KEY=2,3 the SM calculation (again respectivelyfor Drell-Yan and Higgs processes) is performed first and the result is passed into vbfdistrModif() where itcan be just modified, before being used for final weight calculations. The user may choose to modify the valueof the default calculations for all or only for subset of processes involved. This is why complete information ofthe initial and final state configurations is exposed. If a completely new calculation is to be performed using theabove method, then it is advised to use options
KEY=4,5 , so that the Standard Model calculation will be avoided(to save CPU) and result=0 will be passed to vbfdistrModif() . The
KEY=4,5 is reserved for optional use of vbfdistrModif() . • Electroweak schemes:
In Table 3 options of initialization for the EW schemes implemented in
TauSpinner are explained. The particular choicecan be made with vbfinit_(&ref,&variant) as follows: int EWSH_ref=4; // EW scheme to be used for the default 2 -> 4 calculation.int EWSH_variant =5; // EW scheme to be used for non-standard 2 -> 4 calculation.vbfinit_(&ref,&variant);
The
EWSH_ref will set initialization as used for the default calculation, and
EWSH_variant for the reweighting withmodified amplitudes. The choices 1, 2, 3, 4 correspond to
EWSH=1, EWSH=2, EWSH=3, EWSH=4 respectively. Thedefault
EWSH=4 , as explained in the main text, leads to correct t lepton polarisations and angular distributions. As itcauses at tree-level inconsistencies in the calculation of the WW Z coupling, we provide an additional option,
EWSH=5 ,for which parameter setting as for
EWSH=4 is used, but with the
WW Z coupling modified by 5%. It can be used fortesting sensitivity of the analysed distributions to the missed higher order corrections to the
WW Z coupling. For morediscussion, see Section 3.3. This enables possibility to obtain weights for different setting of the electroweak initialization, but calculated otherwise with the same matrix elements. PDFs and a s : Any PDF set from
LHAPDF5 library [28] can be used for calculating spin weight. The choice can be configured by setting, string name="cteq6ll.LHpdf";LHAPDF::initPDFSetByName(name);
The choice of renormalization and factorization scales (imposed is case of µ F = µ R ) can be set with the help of thefollowing command: int QCDdefault=1; // QCD scheme to be used for default 2 ->4 calculation.int QCDvariant=1; // QCD scheme to be used in optional matrix element reweighting (nonSM2=1).setPDFOpt(QCDdefault,QCDvariant); The choice can be different for the default (SM) calculation and the variant one ( nonSM=true ), see Appendix A.1, point9. The Q evolution and starting value of a s used in PDF’s is internally defined by the LHAPDF5 library. For the matrixelement calculations we do not impose consistent definition of a s but it can be enforced by the user, see next point. As adefault, we fix starting point at a s ( M Z ) = . Q with a simple formula of Eq. (4). • User own a s in matrix element calculation :User can supersede the simple, leading logarithmic function provided by us for a s ( Q ) used in the matrix element calcu-lation (Eq. (4)) with his preferred one, and pass it to the program. The function calculating a s has to have the followingarguments: alphasModif(double Q2,int scalePDFOpt, int KEY) In alphasModif one can also use directly a method LHAPDF::alphasPDF(sqrt(Q2)) of LHAPDF5 library [28], the sameassuring consistency between value of a s in the matrix element and the structure functions. Such function can be usedby executing set_alphasModif(alphasModif); . An example of such setup has been provided in example-VBF.cxx program. A.3 Random number initialization
In most of the calculations the
TauSpinner algorithms are not using random numbers. However, there are two exceptions. Inboth cases random generators from
TAUOLA are used, see Appendix C.12 of Ref. [15]. • The helicity states attribution uses
Tauola::RandomDouble . It should be replaced by the user, with the help of
Tauola::setRandomGenerator(double (*gen)()) method and then properly initialized, with distinct seed for eachparallel run. In our example program the actual command is
Tauola::setRandomGenerator( randomik ); • If the read_particles_for_VBF.cxx code is required to generate t decays, then a second random generator, coded in FORTRAN has to be also initialized with distinct seed for each individual parallel run:
Tauola::setSeed(int ijklin, int ntotin, int ntot2n) . A.4 Main program – an example
The following files are prepared for the user prototype program in the
TAUOLA/TauSpinner/examples/example-VBF direc-tory • The user example program example-VBF.cxx . • The prototype method read_particles_for_VBF.cxx to read in events stored in
HepMC format is prepared specificallyfor
MadGraph5 generated events. • The separate program lhef-to-hepmc.cxx for translating
MadGraph5 events from lhe [22] to
HepMC [25] format. • The
README file which contains auxiliary information.Only the program example-VBF.cxx is generic, and does not depend on the specific environment for event generation. This iswhy we provide an extract from this code below. For the
TauSpinner library to work, the t decay products must be present inthe event. In case they are absent, like e.g. in events generated with t ’s as final states in MadGraph5 , we prepared settings for23heir decays in
Tauola library using the mode of not-polarised t decays and Tauola universal interface . Such additionalprocessing is implemented in read_particles_for_VBF.cxx code.In our demonstration program for
TauSpinner spin correlations between t leptons are then introduced, using ( → ) matrix elements and calculating respective spin weight. The purpose of the example is to demonstrate the default initialisationof the TauSpinner program and a flow of the main event loop.
Extract from an example for main user program, example-VBF.cxx file. //-----------------------------------------------------------------//replacement of default (not best quality) random number generator// .5 New option for the ( → ) case To synchronize the old code with the equivalent method implemented now for (2 →
4) process
TauSpinner::set_vbfdistrModif(vbfdistrModif) which enables introduction of the user-defined function for matrixelements which are sensitive to flavours of incoming partons, we provide such an option for the (2 →
2) variant of
TauSpinner as well. Just from now on the first argument of user-defined function nonSM_adopt denotes the incoming parton flavour
PDGid and is respectively treated when calculating matrix element for (2 →
2) process.The necessary changes were introduced, and from now on the first argument ID passed by TauSpinner library to the user-defined function nonSM_adopt activated by the pointer: set_nonSM_born( nonSM_adopt ) of /TAUOLA/TauSpinner/examples/tau-reweight-test.cxx denotes the incoming parton flavour PDGid . In constrast, in earlier version of
TauSpinner library [8], it was possible toinvoke user-defined function nonSM_adopt activated by the pointer in set_nonSM_born( nonSM_adopt )
The first argumentof this method was passing to the user function the information if incoming parton was up- or down-type quark only, withoutspecifyingits family affiliation.
B Tests of reweighting the differential cross-sections for Drell-Yan–like processes
In this Appendix we show in Figs. 9 and 10 a complete set of kinematic distributions validating implementation of (2 → pp → tt j j events into four groups, depending on the initial partons, see Table 1for definition of parton level processes. We use the implemented (2 →
4) matrix elements to calculate per event a weight, wt C → Dprod = d s D / d s C , see Eq. (6), defined as a ratio of the cross-sections for events of groups C and D . The expression is similarto Eq. (3) except that the sum is over subprocesses which belong to the chosen groups C or W . We apply wt C → Dprod to events fromthe group C and compare both the absolute normalisations and shapes of the re-weighted distributions with the distributions ofevents from the group D .These tests were done on the large statistics samples and have been repeated between each groups of processes and withingroups between subgroups. The achieved agreement between the reference and re-weighted distributions validates the correct-ness of the implemented matrix elements. C Tests of reweighting differential cross-sections for Higgs boson production
Similar tests, as discussed in Appendix B, have been repeated for the pp → H ( → tt ) j j processes. Results are shown in Figs.11 - 14. Very good agreement between the reference and re-weighted distributions is observed, both for shapes and relativenormalisations. D Optimalization of interface to Standard Model matrix element calculation
The steering function for calculating (2 →
4) matrix elements squared
REAL*8 FUNCTION VBFDISTR(ID1,ID2,ID3,ID4,HH1,HH2,PP,KEYIN) is coded in
FORTRAN and stored in the
VBF_distr.f file. Before invoking calculation of particular matrix element squared ofthe Standard Model, it performs several steps of filtering to speed up the CPU needed for numerical calculations by settingmatrix element squared to zero without calculation for the cases when configuration of partonic PDG identifiers for incomingand outgoing partons imply that it is the case. The following conditions are consecutively checked (strictly in the given order ).Each condition must be passed to go to the next one, and finally to invoke the matrix element calculation. In the case of non-standard calculations, these checks are not performed, because the function
VBFDISTR is not invoked. h -5 -4 -3 -2 -1 0 1 2 3 4 5 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o t t Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jjT p0 50 100 150 200 250 300 350 400 450 500 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o jj Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj m0 100 200 300 400 500 600 700 800 900 1000 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o Figure 9: Shown generated g q → tt j j (thin red line) after reweighing to q q ( qx qx ) → tt j j (red points). Reference q q ( qx qx ) → tt j j distribution shown with black line. 26 t m0 20 40 60 80 100 120 140 jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o (GeV) jj t t m0 100 200 300 400 500 600 700 800 900 1000 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o (GeV) t t z p-1000 -800 -600 -400 -200 0 200 400 600 800 1000 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -1000 -800 -600 -400 -200 0 200 400 600 800 1000 R a t i o (GeV) jj t t z p-10000-8000-6000-4000-2000 0 2000 4000 6000 800010000 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000 R a t i o j h -5 -4 -3 -2 -1 0 1 2 3 4 5 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · jj t t fi REF (3): q q (qx qx) jj t t fi GEN (2): g q (g qx) GEN reweighed R a t i o Figure 10: Shown generated g q → tt j j (thin red line) after reweighting to q q ( qx qx ) → tt j j (red points). Reference q q ( qx qx ) → tt j j distribution shown with black line. 27 h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o t t Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jjT p0 50 100 150 200 250 300 350 400 450 500 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed R a t i o jj Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj m0 100 200 300 400 500 600 700 800 900 1000 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed R a t i o Figure 11: Shown example of tests distributions for generated process q q , qx qx → H ( → tt ) j j (thin red line) after reweightingto q qx → H ( → tt ) j j process (red points). Reference q qx → H ( → tt ) j j distribution shown with black line.28 t m0 20 40 60 80 100 120 140 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed R a t i o (GeV) jj t t m0 100 200 300 400 500 600 700 800 900 1000 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed R a t i o (GeV) t t z p-1000 -800 -600 -400 -200 0 200 400 600 800 1000 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -1000 -800 -600 -400 -200 0 200 400 600 800 1000 R a t i o (GeV) jj t t z p-10000-8000-6000-4000-2000 0 2000 4000 6000 800010000 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000 R a t i o j h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · ) jj t t fi H( fi REF (4): q qx, qx q ) jj t t fi H( fi GEN (3): q q, qx qx GEN reweighed R a t i o Figure 12: Shown example of tests distributions for generated process q q , qx qx → H ( → tt ) j j (thin red line) after reweightingto q qx → H ( → tt ) j j (red points). Reference q qx → H ( → tt ) j j distribution shown with black line.29 h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o jj h D -10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o t t Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jjT p0 50 100 150 200 250 300 350 400 450 500 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed R a t i o jj Y-10 -8 -6 -4 -2 0 2 4 6 8 10 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -10 -8 -6 -4 -2 0 2 4 6 8 10 R a t i o jj m0 100 200 300 400 500 600 700 800 900 1000 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed R a t i o Figure 13: Shown example of tests distributions for generated process q qx → H ( → tt ) j j (thin red line) after reweighting to q q , qx qx → H ( → tt ) j j process (red points). Reference q q , qx qx → H ( → tt ) j j distribution shown with black line.30 t m0 20 40 60 80 100 120 140 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed R a t i o (GeV) jj t t m0 100 200 300 400 500 600 700 800 900 1000 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed R a t i o (GeV) t t z p-1000 -800 -600 -400 -200 0 200 400 600 800 1000 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -1000 -800 -600 -400 -200 0 200 400 600 800 1000 R a t i o (GeV) jj t t z p-10000-8000-6000-4000-2000 0 2000 4000 6000 800010000 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000 R a t i o j h -5 -4 -3 -2 -1 0 1 2 3 4 5 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed -5 -4 -3 -2 -1 0 1 2 3 4 5 R a t i o (GeV) t t T p0 20 40 60 80 100 120 140 · ) jj t t fi H( fi REF (3): q q, qx qx ) jj t t fi H( fi GEN (4): q qx, qx q GEN reweighed R a t i o Figure 14: Shown example of tests distributions for generated process q qx → H ( → tt ) j j (thin red line) after reweighting to q q , qx qx → H ( → tt ) j j (red points). Reference q q , qx qx → H ( → tt ) j j distribution shown with black line.31 heck if Matrix Element can be set to zero
1. Two incoming (or two outgoing) parton identifiers correspond to gluons and the sum of the other two identifiers is zero,otherwise
ID1 · ID2 · ID3 · ID4 must be positive, to pass to the next step.2. mod(ID1 + ID2 + ID3 + ID4, 2) = 0 ,3. If both
ID1,ID2 are negative or both
ID3,ID4 are negative, and at least one of the other two ID ’s is positive, then theresult is zero.4. Charge conservation imposes that for processes without gluons the following condition must be fulfilled: mod(ID1,2) · sign(ID1)+ mod(ID2,2) · sign(ID2)=mod(ID3,2) · sign(ID3) +mod(ID4,2) · sign(ID4) .5. Number of gluons in the process must be zero or two.6. If there are two gluons, then for the process to give a non zero contribution it is required that ID1 + ID2 = ID3 + ID4 or ID1 + ID2 =0 or ID3 + ID4 =0 .For some configurations it is enough to change the order of partons (arguments of
VBFDISTR routine) or to apply CP symmetry, to avoid duplicating routines for matrix element calculations. It is achieved by first copying kinematic variables intothe local ones of VBFDISTR routine and then performing the following permutations/modifications of the parton positions andmomenta:
Reorder arguments and apply CP symmetry for convenient choice of
ID1, ID2
1. For incoming quark-quark pair, we interchange the order, if necessary, to assure |ID1| ≥ |ID2| .2. If all ID ’s which do not correspond to gluons are negative, we change their signs. At the same time we interchangepositions of t + with t − and flip signs of helicities. Finally we change signs of all 3-momenta to complete the CP transformation.3. For incoming quark-antiquark pair where at least one is non-first family, we require that |ID2| ≤ |ID1| . For the firstfamily quarks we require that ID1=-1 or ID1=2 or |ID2|=|ID1| . To achieve that goal, if condition is not fulfilled, wechange the signs of all ID ’s. At the same time we interchange positions of t + with t − their helicities signs changed aswell. Finally, we change signs of all 3-momenta to complete the CP transformation.4. We enforce (by reordering) that the first parton is not an antiquark, nor a gluon in the case of gluon-fermion initial state.5. If both ID1, ID2 are non gluon and positive, we enforce that
ID1 ≥ ID2
That completes transformations triggered by the configuration of identifiers of incoming partons. Note that if the thirdfamily is to be taken into account, also the sign of the CP symmetry breaking phase will have to be changed to complete the CP transformation. Reorder arguments for convenient choice of
ID3, ID4
1. The
ID3 can not be negative and
ID4 can not be alone the gluon.2. If both
ID3 and
ID4 are non-gluon and positive, then
ID3 can be even and
ID4 odd but not the other way.3. If both
ID3 and
ID4 are odd non-gluon and also
ID3 · ID4 >0 , then
ID4 must be larger/equal
ID3 .4. If
ID3, ID4 are simultaneously even and also
ID3 · ID4 >0, then
ID4 must be larger/equal