Resonanceaware subtraction in the dipole method
Abstract
We present a technique for infrared subtraction in nexttoleading order QCD calculations that preserves the virtuality of resonant propagators. The approach is based on the pseudodipole subtraction method proposed by Catani and Seymour in the context of identified particle production. As the first applications, we compute the \(e^+e^ \rightarrow W^+W^b{\bar{b}}\) and \(pp \rightarrow W^+W^j_bj_b\) crosssection, which are both dominated by topquark pair production above the threshold. We compare the efficiency of our approach with a calculation performed using the standard dipole subtraction technique.
1 Introduction
The production and decay of heavy resonances like the top quark is of greatest interest in particlephysics phenomenology [1]. It presents a window into new physics, which is commonly believed to emerge in the form of new interactions at high energy. Precision measurements of Standard Model parameters at current collider energies may reveal parts of this structure if they can be made at high precision. The necessary theoretical predictions for topquark pair production have been computed at nexttoleading order (NLO) [2, 3, 4, 5] and nexttonextto leading order (NNLO) [6, 7, 8] QCD perturbation theory, and combined with NLO electroweak results [9]. In these calculations, top quarks are considered to be asymptotic final states, and finite width effects are neglected. When including top quark decays, a problem arises that is related to the very definition of the inclusive final state and can most easily be explained using Fig. 1. Diagram (a) represents the topquark pair production process in leading order perturbation theory. Diagram (b) can be obtained from diagram (a) by including the decay of one of top quarks. It may also be considered as a real radiative correction to the singletop quark production process represented by diagram (c). Quite obviously, diagram (b) is resonant when \(( p_W + p_b )^2  m_t^2\lessapprox \Gamma _t^2\). In this region of phase space the NLO calculation of \(pp\rightarrow Wt\) therefore overlaps with the calculation of \(pp\rightarrow t[{\bar{t}}\rightarrow Wb]\) and spoils the definition of a NLO cross section for \(pp\rightarrow Wt\). This problem has traditionally been addressed by techniques such as diagram removal or diagram subtraction [10]. However, both methods introduce theoretical uncertainties and violate gauge invariance. The natural approach is instead to not view singletop and toppair production as two separable channels and to consider only the fully decayed final state [11, 12, 13]. In addition to being theoretically robust, this technique matches the reality of performing an experimental measurement, where the existence of the top quark as an intermediate state is inferred from the decay products.
Higherorder radiative corrections to both the production and the decay of top quarks can also be simulated numerically in computer programs called event generators, which allow to map experimental signatures associated with topquark production to the parameters of the theory. It is the factorized approach of these simulations that presents a problem when the precision target of the experimental measurement lies below the resonance width, because the narrow width approximation can no longer be applied [14]. Again, the natural solution is to perform the computation for the complete final state and match the NLO fixedorder result to the parton shower [10, 15, 16, 17]. Since the process is an interplay of continuum contributions and resonant topquark production as in Fig. 1c, it could in principle be treated in the narrowwidth approximation if \(( p_W + p_b )^2  m_t^2\lessapprox \Gamma _t^2\). This mandates a special choice of kinematics mapping in the transition from Born to realemission final states in the matching procedure, which has been discussed in great detail [18, 19, 20, 21, 22] in the context of the FrixioneKunsztSigner subtraction method [23]. However, no attempt has so far been made to implement a solution based on Catani–Dittmaier–Seymour–Trocsanyi dipole subtraction [24, 25]. In this manuscript we therefore discuss a new technique that is based on the identified particle methods presented in [24] and apply the procedure to the computation of topquark pair production at a future linear collider [26, 27] and at the Large Hadron Collider (LHC), where measurements of singly and doublyresonant topquark pair production have just been reported [28].
The outline of this paper is as follows: Sect. 2 introduces the problem of resonances in NLO calculations, reviews the pseudodipole subtraction formalism as introduced in [24] and shows how it can be applied to resonanceaware subtraction. Section 3 presents first applications, and Sect. 4 gives an outlook.
2 Pseudodipole subtraction
Here \(\int _{m} \mathrm {d}\sigma ^\mathrm {I} = \int _{m+1} \mathrm {d}\sigma ^{\mathrm {S}}\) is the subtraction term, which is analytically integrated over the phasespace of the additional parton in the real correction and \(\int _m\) indicates that the phasespace integral corresponds to m finalstate partons.^{1} In the remainder of this paper we will focus on CS dipole subtraction [24]. In processes with intermediate resonances, this technique exhibits an undesired feature, which can most easily be explained using a concrete example, say \(e^+e^ \rightarrow W^+W^b{\bar{b}}\). If the centerofmass energy is greater than the top pair threshold \(\sqrt{s} > rsim 2m_t\), this process is dominated by onshell \(t{\bar{t}}\)production and decay. One possible real emission correction to this process is depicted on the lefthand side of Fig. 2. The subtraction term associated to the \({\bar{b}}g\) collinear sector is constructed from the Borndiagram on the righthand side of Fig. 2, and its kinematics is obtained by mapping the onshell finalstate momenta of the real correction to Born kinematics using the algorithm in [24]. In the canonical method, the momenta of the emitter (the \({\bar{b}}\)quark) and the spectator (the bquark) are adjusted, while all other momenta remain the same. This procedure generates a recoil that is indicated by the dashed line in Fig. 2. The recoil leads to the subtraction term being evaluated at different virtualities of the intermediate topquarks than the realemission diagrams whose divergences it counteracts. As the topquark propagator scales like \((p_t^2  m_t^2 + im_t\Gamma _t)^{1}\) and \(\Gamma _t \ll m_t\), the change in virtuality may cause numerically large deviations between the realemission corrections and the corresponding subtraction terms. Though the cancellation of infrared divergences still takes place, the associated large weight fluctuations may significantly affect the convergence of the MonteCarlo integration. The problem becomes manifest when interfacing the fixedorder NLO calculation to a parton shower. The difference in matrixelement weights arising from resonant propagators being shifted off resonance by means of adding radiation and mapping momenta from Born to realemission kinematics bears no relation with the logarithms to be resummed by the parton shower, yet its numerical impact may be similar. This motivates the usage of an improved kinematics mapping by means of pseudodipoles.
2.1 Catani–Seymour pseudodipole formalism
The concept of pseudodipoles was introduced in [24] to cope with the situation where a subset of the finalstate partons lead to the production of identified hadrons. In such a scenario, both emitter and spectator of a dipole may be “identified” in the sense that they fragment into identified hadrons. Because the directions of the identified hadrons are measurable, neither emitter nor spectator parton in the dipole can be allowed to absorb the recoil when mapping the momenta of the realemission final state to a Born configuration. Instead the kinematics is balanced by adjusting the momenta of all nonidentified finalstate particles (not just partons). This idea is reminiscent of standard dipoles with initialstate emitter and initialstate spectator. In fact, pseudodipoles may be thought of as a generalization of these configurations.
2.1.1 Finalstate singularities: differential form
2.1.2 Initialstate singularities: differential form
2.2 Application to resonanceaware subtraction
 1.
If the emitter is the decay product of a resonance and the spectator is not a decay product of the same resonance, the dipole is replaced by a pseudodipole where the emitter and all particles except for the emission and the remaining decay products of the resonance are identified.
 2.
If the emitter is not a decay product of a resonance but the spectator is, the dipole is replaced by a pseudodipole where the emitter and all particles, which are not decay products of the resonance to which the spectator belongs are identified.
 3.
If emitter and spectator are decay products of the same resonance, the standard CSsubtraction formalism is used.
 1.
If the emitter is in the inital state and the spectator is the decay product of a resonance, the dipole is replaced by a pseudodipole where no particle is identified.
In view of our application example \(pp \rightarrow W^+W^j_bj_b\) in Sect. 3, we like to comment on the situation in which one has to deal with multiple (sub)processes. To ensure that pseudodipoles are used only if potentially resonant diagrams occur, we replace standard CS with pseudodipoles only if both real and underlying Bornconfiguration comprise at least one b and one \({\bar{b}}\)quark in the final state. In this concrete example this means that subprocesses, which do not feature a resonance, like \(bb \rightarrow W^+W^ bb\), are treated with standard CSdipoles.

We do not identify particles throughout the calculation, but identify different particles depending on the subtraction term.

We integrate over the momenta of the identified partons by means of adding partonic fragmentation functions.
2.2.1 Finalstate singularities: integrated form
For simplicity, we first consider a configuration with no initialstate partons and m finalstate (anti)quarks at Born level. In the following, the integration over nonQCD particles shall be understood whenever we write \(\int _m \mathrm {d}\phi _m\).
We remark that the introduction of the collinear counterterms in Eq. (2.19) is actually unnecessary, since they give a vanishing contribution to the crosssection. This is due to \(P_{qq}(z)\) being a pure plusdistribution and the testfunction with which it is convoluted not being zdependent after we substituted \(p_a = z {\tilde{p}}_{ai}\). This is also true for differential crosssections, since any partonic observable can be expressed without reference to z. The vanishing effect of collinear counterterms may also be understood from another point of view: As we do not actually restrict the momenta of the “identified” partons, but integrate over them eventually, we have already collected all singularities necessary to cancel those present in the virtual corrections. Hence, no collinear mass factorization counterterms are required.
2.2.2 Initialstate singularities: integrated form
3 Application to \(W^+W^b{\bar{b}}\) production
We have tested the above described resonanceaware subtraction by means of pseudodipoles in two reactions: \(e^+e^ \rightarrow W^+W^b{\bar{b}}\) and \(pp \rightarrow W^+W^j_bj_b\). In the following we are going to compare results obtained with standard CS dipoles to those obtained with pseudodipoles for fixed NLO QCD predictions. In Sect. 3.1, we first examine the cancellation of divergences between the realemission matrix elements and the different dipoles using ensembles of trajectories in phasespace, which approach the collinear and soft limits in a controlled way. Following this, we compare physical crosssections calculated with the different subtraction techniques while paying special attention to the rate of convergence in the MonteCarlo integration.
3.1 Singular limits
NLO cross sections for \(e^+e^ \rightarrow W^+W^b{\bar{b}}\) at \(\mu _R = m_t\) and varying centerofmass energy, computed using standard CS subtraction terms (CS) or pseudodipoles (ID). The subtracted realemission contributions (RS) were calculated using \(10^7\) phasespace points. The Born, virtual corrections and integrated subtraction terms (BVI) were calculated using \(3\cdot 10^6\) phasespacepoints
\(\sigma \mathrm {[fb]}\)  \(\sqrt{s}=3m_W\)  \(\sqrt{s}=2m_t\)  \(\sqrt{s}=4m_t\)  

CS  ID  CS  ID  CS  ID  
RS  \(\,0.00772\,(6)\)  \(\,0.00140\,(5)\)  \(\,0.52\,(3)\)  \(\,2.85\,(1)\)  \(\,9.5\,(4)\)  \(\,5.3\,(1)\) 
BVI  \(0.16143\,(13)\)  \(0.15506\,(13)\)  \(148.07\,(9)\)  \(150.55\,(9)\)  \(230.0\,(2)\)  \(226.0\,(2)\) 
\(\sum \)  \(0.15371\,(14)\)  \(0.15366\,(14)\)  \(147.55\,(9)\)  \(147.70\,(9)\)  \(220.5\,(4)\)  \(220.7\,(2)\) 
3.1.1 \(e^+e^ \rightarrow W^+W^b{\bar{b}}\)
Figure 7 shows the values of the realemission matrixelement, \(R=\left {\mathcal {M}}_R\right ^2\) and the sum of the associated dipoles (\(S=\sum {\mathcal {D}}\)), as well as their ratio for the soft limit (left) and the bg–collinear limit (right) limit. We have averaged – for all subplots – over all entries within a bin. In doing so, the ratio plots enable us to assess the pointwise convergence of S / R best. It can be seen that the cancellation of divergences occurs in both the standard CS subtraction method and in the pseudodipole approach, but that the pseudodipoles converge faster towards the realemission matrix element, which can be seen in the lower panels of the figures.
3.1.2 \(pp \rightarrow W^+W^j_bj_b\)
Figures 8 and 9 display the average values of the realemission matrix element and the corresponding dipole subtraction terms, as well as their ratio – averaged within each bin – for processes with an additional gluon (Fig. 8) and an additional quark (Fig. 9) in the final state. The former develop both soft and collinear singularities, shown in the left and right panels of Fig. 8, respectively, while the latter only feature collinear singularities. Combining all tests, we validate all four initialstate splitting functions in Eq. (2.15).
3.2 Physical cross sections
In this section we present first results validating the pseudodipole subtraction method for resonanceaware processes at the level of observable cross sections and distributions. Results are crosschecked using two different implementations of our new algorithm within the public event generation framework Sherpa [31, 32], one using the matrixelement generator AMEGIC++ [29, 33], and one using the new interface between Sherpa and OpenLoops [34]. In this interface, the colorcorrelated Born matrixelements are imported from OpenLoops libraries [35], while the splitting function is calculated in Sherpa and the integration is performed using the techniques implemented in AMEGIC++ [33].
3.2.1 \(e^+e^ \rightarrow W^+W^ b{\bar{b}}\)
Again we investigate first the reaction \(e^+e^ \rightarrow W^+W^b{\bar{b}}\). We vary the centerofmass energy of the collider to obtain predictions below, at and above the topquark pair production threshold, and we do not include the effects of initialstate radiation. We require two hard jets at \(y=(5\text { GeV}/E_\mathrm{cms})^2\) defined according to the Durham jet algorithm [30]. The running of the strong coupling is evaluated at two loops, and the reference value is set to \(\alpha _s(M_Z^2)=0.118\), where \(M_Z=91.1876~\mathrm{GeV}\).
Table 1 shows the total cross sections as well as the individual contributions from the subtracted realemission terms (RS) as well as Born, virtual corrections and integrated subtraction terms (BVI). As expected, the RS and BVI contributions differ between the standard CS subtraction method and the pseudodipole approach, but their sum agrees within the statistical accuracy of the MonteCarlo integration. The cross section is significantly enhanced at and above the production threshold for a topquark pair. For those two centerofmass energies, we expect the pseudodipoles to give a more physical interpretation of the subtraction term and thus a reduced variance during the integration. This is confirmed in Fig. 10, which shows the evolution of the MonteCarlo error during the integration. In the case of pseudodipole subtraction at or above threshold, the uncertainty is indeed substantially lower than for standard CSdipoles. Below threshold the performance of pseudodipole subtraction is similar to the standard technique.
Our validation is completed by a comparison of a few selected differential cross sections in the two subtraction schemes. Figure 11 displays the invariant mass of the (anti)top quark reconstructed at the level of the \(W^+W^b{\bar{b}}\) final state from the Wboson and a bjet with a matching signed flavor tag. The deviation plot shows excellent statistical compatibility between the two simulations. It also displays clearly that the pseudodipole subtraction technique generates smaller statistical uncertainties than the standard CS subtraction method.
3.2.2 \(pp \rightarrow W^+W^j_bj_b\)
NLO cross section for \(pp \rightarrow W^+W^j_bj_b\) at \(\mu _R = \mu _F = m_t\) and \(\sqrt{s}=13\) TeV, computed using standard CS subtraction terms (CS) or pseudodipoles (ID). The subtracted realemission contributions (RS) were calculated using \(10^8\) phasespace points. The Born, virtual corrections and integrated subtraction terms (BVI) were calculated using \(8.5\cdot 10^6\) phasespacepoints
\(\sigma \mathrm { [pb]}\)  \(\sqrt{s}=13\) TeV  

CS  ID  
RS  \(\,62.03\,(59)\)  \(\,93.21\,(13)\) 
BVI  \(360.02\,(39)\)  \(391.53\,(39)\) 
\(\sum \)  \(297.99\,(71)\)  \(298.32\,(41)\) 
4 Conclusions
We have presented a technique that allows to preserve the virtuality of intermediate propagators in the computation of subtracted realemission corrections to processes involving resonances. We have validated this approach in a simple fixedorder calculation and outlined how it can be generalized to more complicated processes. Due to the close correspondence with standard Catani–Seymour dipole subtraction, a matching to parton showers can be carried out in the MC@NLO or POWHEG methods in the future, thus paving the way for a precision measurement of processes involving for example singletop and topquark pair production.
Footnotes
 1.
Our analysis focuses on NLO QCD corrections, though the method can be extended to include NLO electroweak corrections.
 2.
In particular, it would not be possible in this method to define a resonanceaware subtraction for the full sixjet (fourjet) final state in fully hadronic (semileptonic) \(t{\bar{t}}\) production at hadron colliders.
 3.
In order to be applicable to initialstate spectators, we have to rephrase the second rule to: If the emitter is not the decay product of a resonance and the spectator is an initialstate parton, the dipole is replaced by a pseudodipole where the emitter and all particles, which are not decay products of any resonance are identified.
Notes
Acknowledgements
This work was supported by the German Research Foundation (DFG) under grant No. SI 2009/11 and by the U.S. Department of Energy under contract DEAC0276SF00515. It used resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE–AC02–07CH11359.
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