Latest developments in the simulation of final states involving toppair and heavy bosons
Abstract
I give an overview of recent progress in the simulation of final states involving topquarks and vector bosons pair. First I’ll discuss the recently found solutions needed to simulate fully differential top pair production ( leptons) at NLO+PS accuracy, retaining offshellness and interference effects exactly. In the second part, I’ll review the MiNLO (Multiscale Improved NLO) method, and then show a recent application, namely the simultaneous NLO+PS description of and jet production.
Latest MC developments for simulating toppair and VV production
\FullConference38th International Conference on High Energy Physics
310 August 2016
Chicago, USA
In this article I will review a couple of recent results on the
inclusion of higherorder QCD corrections to the Monte Carlo
simulation of final states involving toppair and heavy electroweak
bosons. In section 1 I will focus on the recent progress
achieved in the matching of QCD NLO corrections with parton shower
simulations (NLO+PS) for the process , whereas
in section 2 I will discuss the NLO+PS merging of
and +1 jet production using the MiNLO
method.
1 toppair production
It is known that having an accurate simulation of the process is important for several reasons at the LHC, for instance to measure the top quark mass, or to have an unified treatment of and the socalled “singletop ” production.
At fixed order in QCD, hadronic production is very well known and much studied. Despite the fully differential cross section has been known for several years at NLO [1, 2, 3, 4, 5], with the exception of a first study appeared in [6], Monte Carlo NLO+PS event generators addressing all the issues related to the complete simulation of this final state started to be available only recently in the POWHEG approach [7, 8, 9]. In the rest of this section I’ll focus on these issues (and solutions thereof) within the POWHEG approach, although substantial work in this direction is also pursued within the MC@NLO matching scheme, and complete results for singletop tchannel production in the 4flavour scheme were published in ref. [10].
The problem with the simulation of production and the inclusion of finite width effects can be stated as follows: unless special care is taken, the intermediate topquark virtuality is not preserved among different parts of the computation, leading to the evaluation of matrix elements at phase space points which have different top virtualities. When this happens, three problems will in general occur:

at the level of computing NLO corrections with a subtraction method, the cancellation of collinear singularities associated to gluon emission off finalstate quarks can become delicate, eventually failing when approaching the narrowwidth limit.

when the hardest radiation is generated in POWHEG, the phasespace region associated to finalstate gluon emission off the quark is handled by a mapping that, in general, does not preserve the virtuality of the intermediate resonance. Unless , real and Born matrix elements ( and , respectively) will not be on the resonance peak at the same time, hence the ratio in the POWHEG Sudakov can become large when is on peak and is not, yielding a spurious “Sudakov suppression”.

further problems can arise during the partonshowering stage: from the second emission onward, the shower should be instructed to preserve the mass of the resonances. This could be done easily if there was an unique mechanism to “assign” the radiation to a given resonance. For processes where interference(s) is(are) present, no obvious choice is possible.
An intermediate solution to the previous issues was presented in ref. [7], where a fully consistent NLO+PS simulation for production was obtained in the narrowwidth limit, and offshellenss and interference effects were implemented in an approximate way. I refer to the original paper, or to the review [11], for more details. Here it suffices to say that, by using the narrowwith approximation to compute NLO corrections, production and decay can be clearly separated (no interference arises), thereby allowing a nonambiguous “resonance assignment” for finalstate radiation, as well as the use of an improved (“resonance aware”) subtraction method, where radiation in the decay is generated by first boosting momenta in the resonance restframe. In this way, and are always evaluated with the same virtuality for the intermediate resonance, so that the subtraction can be safely performed, and no spurious Sudakov suppression can arise.
More recently, a general solution to include offshellness and
interference effects in the
POWHEG approach was
proposed in ref. [8], and later applied to the
process in ref. [9], where
matrix elements were obtained using OpenLoops [12]. The main new concept introduced
in [8] is that one separates all contributions to the
cross section into terms with definite resonance structure,
i.e. each term should only have peaks associated to a given
resonance structure (“resonance history”). For
production, for instance, one has two types of resonance histories:
one where at least one channel toppropagator appears (this
includes both doubly and singleresonant contributions) and another
associated to nonresonant production. By means of projectors
built by combining BreitWigner like functions ,
a partition of the unit can be constructed, so that a given Born (and
virtual) partonic subprocess can be separated into contributions
that are, individually, dominated by one and only one resonance history
(labeled by ):
(1) 
Since each Born matrix element is separated according to resonance histories, one needs to setup a similar mechanism for real matrix elements, such that, eventually, each projected real matrix element can be associated to an unique resonance history, with a counterpart in the corresponding list of Born’s ones. As usual, real matrix elements also need be separated according to their collinear singularities: to this end, one requires that a collinear region is admitted only if the two collinear partons both arise either from the same resonance, or from the hard interaction. This separation is achieved schematically as
(2) 
In eq. (2), denotes a given resonance history assignment for , when the collinear region is approached, and the sums in the denominator run only on the possible resonance histories present in , and on the compatible singular regions associated to a given : hence a given becomes dominant only if the collinear partons of region have the smallest and the corresponding resonance history is the closest to its mass shell.
The above prescriptions allowed to build a POWHEG generator able to
simulate processes with intermediate resonances, keeping all
finitewidth effects and interferences. In fact, having separated each
contribution as explained above, for the singular regions associated
to a radiation in a resonance decay, it becomes now possible to safely
use the “resonanceaware” subtraction method developed
in [7], thereby avoiding the mismatches mentioned
at the beginning of this section. Similarly, because an index
is naturally associated to the hardest radiation generated
by POWHEG, it’s possible to unambiguously assign the radiation to a
given resonance, preventing the parton shower to distort the mass of
the resonances.
The left panel of Fig. 1
shows the differences in shape of the reconstructed top peak obtained using three tools, namely the new generator of ref. [9] (), the resonanceimproved generator based on an approximate treatment of offshell effects [7] (decay), and the original generator [13] based on onshell NLO matrix elements for production (). As expected, the “” and “decay” generators are fairly consistent, especially close to the resonance peak, whereas the “” generator shows larger deviations. The spectrum of the hardest jet is instead shown in the right panel of fig. 1, without imposing any particular cuts. The shape difference at small can be attributed to the fact that the “Wt” contribution is missing (approximate) in the “” (“decay”) generator, whereas it’s fully taken into account in the “” one.
2 boson pair production
The study of vector boson pairproduction is central for the LHC Physics program. Not only is production measured to access anomalous gauge couplings, but it’s also an important background for several searches, notably for those where the decay is present. For these and other similar reasons, it is important to have flexible and fully realistic theoretical predictions that allow to simultaneously model, with high accuracy, the production of , inclusively as well as in presence of jets. The methods aiming at this task are usually referred to as “NLO+PS merging”. NLO+PS merging for +jet(s) was achieved using the MEPS@NLO [14, 15] and FxFx [16, 17] methods. In this section, I’ll review the MiNLO formalism and show how it was used to merge at NLO the processes and jet [18].
The MiNLO (Multiscale Improved NLO) procedure [19] was originally introduced as a prescription to apriori choose the renormalization () and factorization () scales in multileg NLO computations: since these computations can probe kinematic regimes involving several different scales, the choice of and is indeed ambiguous, and the MiNLO method addresses this issue by consistently including CKKWlike corrections [20, 21] into a standard NLO computation. In practice this is achieved by associating a “mostprobable” branching history to each kinematic configuration, through which it becomes possible to evaluate the couplings at the branching scales, as well as to include (MiNLO) Sudakov form factors (FF). This prescription regularizes the NLO computation also in the regions where jets become unresolved, hence the MiNLO procedure can be used within the POWHEG formalism to regulate the function for processes involving jets at LO.
In a single equation, for a induced process as production, the MiNLOimproved POWHEG function reads:
(3) 
where is the colorsinglet system ( in this case), is its transverse momentum, is set to , and is the MiNLO Sudakov FF associated to the jet present at LO. Convolutions with PDFs are understood, is the leadingorder matrix element for the process jet (stripped off of the strong coupling), and (the expansion of ) is removed to avoid double counting. We also notice that is a function of , i.e. the phase space to produce the system and an extra parton, which can be arbitrarily soft and/or collinear.
In ref. [22] it was also realized that, if is a
color singlet, upon integration over the full phase space for the
leading jet, one can formally recover NLO+PS accuracy for the process
by properly applying MiNLO to NLO+PS simulations for
processes of the type jet.
The coefficient is processdependent, and formally also a function of , because part of it stems from the 1loop correction to the process. For Higgs, DrellYan, and production, these 1loop corrections can be expressed as form factors: becomes just a number as its dependence upon disappears, and the analogous of eq. (3) can be easily implemented [22, 24]. For diboson production, the situation is more delicate. First, extracting for the case is more subtle, as the virtual corrections to the process don’t factorize on the Born squared amplitude, hence . As a consequence, a mismatch between different phase spaces becomes apparent, because in eq. (3) depends upon , whereas needs to be computed as a function of . In ref. [18] these two issues were handled as follows:

to compute we started from the relatively simple expression used for the DrellYan case, and replaced its processdependent part with the corresponding term for production: .

in order to evaluate , we defined on an eventbyevent basis a projection of the jet state onto a one, using the FKS mapping relevant for initialstate radiation as implemented in the POWHEG BOX [25]. For real emission events, a similar mapping was used. In all cases, in the limit, the effect of these projections on the final state kinematics smoothly vanishes, making sure that the precise numerical determination of is affected only beyond the required accuracy.
In ref. [18] we have built a POWHEG generator
for the jet process, and upgraded it with MiNLO,
according to the aforementioned procedure. We worked in the 4flavour
scheme, including exactly the vector bosons’ decay products, as well
as finitewidth effects and singleresonant contributions. Treelevel
matrix elements were obtained with an interface to
MadGraph 4 [26, 27], whereas one loop
corrections were computed with GoSam 2.0 [28, 24].
The left panel of Fig. 2
shows the transverse momentum spectrum of the system as obtained with the WWJMiNLO generator against the one obtained with the original POWHEG generator for [29]. The importance of NLO corrections is manifest in the high tail, whereas the differences at small can be attributed to the differences among the POWHEG and the MiNLO Sudakovs. The right panel shows instead a comparison for the leading jet spectrum between the partonlevel computation jet at NLO, and the MiNLO result. This observable is formally described with the same accuracy (NLO) by both predictions, as shown in the plot. The effect of resumming collinear logarithms at small is reflected in the difference between MiNLO and the pure NLO, where no resummation is included. At high , the small differences are due to the fact that different central values for the and scales are used, namely for MiNLO (as by prescription) and at NLO.
Footnotes
 Unless otherwise stated, throughout this document, we indicate with “” the leptonneutrino finalstate pair arising from a bosons, i.e. bosons are treated as unstable, have a finite width and they decay leptonically.
 For simplicity we suppressed the labels and used in [8], which represent the “bare” structure, i.e. the flavour of external partons of Born and real matrix elements. Moreover in [8] the symbols and represent a “full” structure, since they label a given resonance history related to a given set of external partons, i.e. they also contain the explicit information on the external partons, which we are suppressing in this document.
 I want to mention that another technical but crucial issue was addressed in [8], related to the computation of softcollinear contributions to be added to to the virtual terms. In [7], these terms were computed independently for production and for each radiating resonance decay, and in different frames. This posed no problem, because in the narrowwidth limit no interferences are present. When interferences are present, this is clearly no longer possible, and a substantial generalization of the subtraction scheme adopted in POWHEG was worked out in [8], leading to the development of a new framework, dubbed POWHEGBOXRES.
 The idea has been generalized recently in ref. [23].
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