Heavy quark radiation in NLO+PS POWHEG generators
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Abstract
In this paper we deal with radiation from heavy quarks in the context of nexttoleading order calculations matched to parton shower generators. A new algorithm for radiation from massive quarks is presented that has considerable advantages over the one previously employed. We implement the algorithm in the framework of the POWHEGBOX, and compare it with the previous one in the case of the hvq generator for bottom production in hadronic collisions, and in the case of the bb4l generator for top production and decay.
1 Introduction
The production and detection of bottom quarks play an important rôle in various contexts in LHC physics. Letting aside the very abundant direct production, that is exploited for flavour physics studies, bottom is used to identify top particles and to study their properties. Furthermore, it is the dominant decay mode of the Higgs boson, that can be used to study processes as the associate HV production [1, 2] and the large transverse momentum production [3]. In searches for physics beyond the Standard Model, bottom also appears often produced in association with newphysics objects.
Having a mass much larger than the typical hadronic scales, bottom quark production is calculable in perturbative QCD. In cases when the transverse momentum involved in the production is large compared to its mass, as, for example, in highenergy \(e^+e^\) annihilation, or in production at large transverse momentum in hadronic collisions, bottom can behave as a light parton, and give rise to a hadronic jet. Techniques for dealing with these regimes have been developed in the past [4], and have been applied to the LHC case [5]. They allow for the computation of the transverse momentum spectrum of promptly produced b quarks at nexttoleading order in QCD, including the resummation of large logarithms of the ratio of the transverse momentum over the bottom mass up to nexttoleadinglogarithmic accuracy. These large logarithms can arise both from initial state radiation, when, for instance, an incoming gluon splits into a \(b\bar{b}\) pair, with one of the b undergoing a largemomentumtransfer collision with a parton from the target, and from final state radiation. In the last case, an outgoing gluon can split into a \(b\bar{b}\) pair, or a directly produced b quark can emit a collinear gluon.
The large transverse momentum regime is treated consistently at the leading logarithmic level in parton shower generators. At the most basic level, heavy flavours are treated as light flavours, but with a shower cutoff scale of the order of the heavy quark mass. However, considerable work has been performed to better account for mass effects. In some generators, this is achieved by suitable modifications of the splitting kinematics and splitting kernels [6, 7]. The Sherpa dipole shower [8] makes use of the CataniSeymour dipoles for massive quarks [9]. The CataniSeymour formalism is also used in the DIRE shower [10]. In Ref. [11] a final state dipoleantenna shower for massive fermions is proposed, based upon the corresponding antenna subtraction formalism of Ref. [12].
In nexttoleading order (NLO) calculations matched to Shower generators (NLO+PS) for heavy flavour production [13, 14], one generally treats the heavy flavour as being very heavy. The heavy quark mass thus acts as a cutoff on collinear singularities, that are thus not resummed. This approach has in fact proven to be quite viable in heavy flavour production even at relatively large momentum transfer [5]. Consider, for example, heavy quark pair production in a POWHEG framework. By neglecting collinear singularities from heavy quarks, the only singular region that we have to consider has to do with initial state radiation involving only light partons. Since the POWHEG procedure guarantees that the matrix elements are given correctly for up to one hard radiation, gluon splitting, flavour excitation and radiation from the heavy flavour are included, so that the logarithmically enhanced terms are correctly reproduced at first order. Higher order leading logarithms, however, are not treated correctly. In particular, there are reasons to give an adequate treatment to final state radiation from a high transverse momentum bottom quark. In fact, this radiation process is intimately related to the physics of the bottom fragmentation function, and may have important effects in processes of considerable interest, like for example in top decay.
In the POWHEGBOX framework, a facility for the treatment of collinear radiation from a heavy particle was set up in Ref. [15], in the framework of electroweak corrections to W production. In that context, the purpose was to deal with mass effects in the final state radiation of photon from the lepton in W decays. The same framework is also appropriate for describing radiation from heavy quarks. In particular, it was adopted in Ref. [16], where the ttb_NLO_dec generator was introduced, and in Ref. [17] for the b_bbar_4l generator, for dealing with radiation from bottom quarks in top decays.
The purpose of the present paper is twofold: we present a new algorithm for radiation from a heavy quark, that has proven superior to the old one; furthermore we perform a thorough investigation of the behaviour of this component of the POWHEG generator, also by comparing the two methods, both in the framework of bottom quarks generated in top decay, and in inclusive bottom quark pair production. In the last case, such a study was never carried out.
The paper is organized as follows: in Sect. 2 we describe the new algorithm, in Sect. 3 we illustrate our phenomenological studies, and in Sect. 4 we give our conclusions.
2 Description of the new algorithm
2.1 The POWHEG mapping for the massive emitter case
In what follows, we will construct a onetoone map from a real configuration with radiation variables \((\xi ,y,\phi )\) into a Born one. This leads to a factorisation of the real phase space in term of Born and radiation variables.
2.2 Inverse map
In Fig. 4 we show the partition o the kinematic region represented in the more familiar Dalitz plane. Notice that in the massless limit the physical region in the Dalitz plot develops an acute angle in the lower right, corner corresponding to the gluon being anticollinear with the b quark. Thus, the problematic region \(\xi >\xi (0)\) is not a singular one.
2.3 Full kinematic reconstruction of the real emission
So far, we have got the length of the trivectors \(\mathbf {k}_n\) and \(\mathbf {k}_{n+1}\). It is a standard kinematical problem to determine their directions in such a way that their sum \(\mathbf {k}\) is parallel to \(\mathbf {\overline{k}}_n\). We do not enter in further details about it.

when approaching the curve \(\xi =\xi (0)\) for \(y>0\), behaving as \(\xi (0)\xi \)

when approaching the curve \(\xi =\xi (y)\), as \(\sqrt{\xi (y)\xi }\).
2.4 Generation of radiation
We model the upper bound function on the asymptotic singular behavior of the real matrix element near the soft and collinear singularities. We recall that the jacobian of the mapping has a divergent behaviour near the curve \(\xi =\xi (y)\). The upper bound function should have a behaviour not weaker than the Jacobian near the singular regions, and furthermore, it should be simple enough to allow us to perform an analytical integration in the constrained radiation phase space given by the cut \(K_T^2>t\).
It is convenient to perform a change of integration variables from \(\xi ,\,y\) to \(\xi ,\,K_T^2\). Indeed, it turns out that \(K_T^2\) is a monotonic decreasing function of y at fixed \(\xi \), i.e. \({\partial K_T^2}/{\partial y} < 0\).
2.4.1 Upper bound function
2.4.2 Integral of the upper bound function
2.4.3 Generation of radiation kinematics
 1.
We set the initial scale \(t_0=K^2_{\mathrm{max}}\).
 2.We generate a uniform random numberand get t from Eq. (67). If t is below \(t_{\mathrm{min}}\), no radiation is generated, and the event is emitted as is.$$\begin{aligned} 0<r<\exp [2\pi N I(t_0)], \end{aligned}$$
 3.We pick a new uniform random number \(0<r'<1\) and we generate a value for \(\xi \) asThis is consistent with the distribution of \(\xi \) at fixed \(K_T^2\) according to Eq. (66).$$\begin{aligned} \xi = \xi _\text {m}(t)\exp (y_0r'). \end{aligned}$$(68)
 4.
If \(\xi >\xi _\text {max}\), we set \(t_0=t\), and go back to the step 2.
 5.If the veto condition is passed, given t and \(\xi \), we solve numerically for y the implicit equationIf a solution does not exist, we set \(t=t_0\) and go back to step 2.$$\begin{aligned} K_T^2(\xi ,y) = t. \end{aligned}$$(69)
 6.
Now that \(\xi \) and y are available, we generate a random \(\phi \), and compute the ratio \(R=[R/B J(\xi ,y)]/U(\xi ,y)]\), with U given in terms of \(U'\) in Eq. (56), and generate a new random number \(0<r'''<1\). If \(r'''>R\) we set \(t_0=t\) and go back to the step 2. Otherwise, the event is accepted.
3 Phenomenology
3.1 Comparison in the bb4l case
We found that the efficiency and the generation rate of the new implementation are comparable with those of the POWHEG default.
3.2 b production in hadronic collisions
We will now compare the results obtained with the default hvq generator, that we will label nol (for “no light”, meaning that the heavy quark is treated as very heavy), that treats as singular regions only the radiation from massless partons (i.e. initial state radiation); hvq with the inclusion of the radiation from the heavy quark as a singular region will be labeled asl (for “as light”, meaning that the heavy quark is treated as a light parton). Furthermore, the default treatment of the heavy quark radiation region will be denoted as def, while the new implementation presented here will be called alt. In Figs. 8, 9 and 10 we show a comparison of def and alt. We can immediately see that we do not find important differences between the two methods, consistently with what was found in the bb4l case. The settings are similar to the bb4l case: we make the B hadrons stable, and define the b (\({{\bar{b}}}\)) jets as the jets containing the hardest b (\({{\bar{b}}}\)) flavoured hadron, with the jets defined as in the bb4l case. However, we do not exclude the case when both hardest bflavoured hadrons are in the same jet. We perform the calculation for the LHC at 8 TeV, using NNPDF30_nlo_as_0118 pdf set [23]. As one can see, the two implementations are in excellent agreement. Observe the jump at 10 GeV in the \({j_\mathrm{\mathchoice{\displaystyle }{\scriptstyle }{\scriptscriptstyle }{\scriptscriptstyle } B}} \) mass. It is due to the case in which the b and \({{\bar{b}}}\) flavoured hadrons are both in the jet cone. From Fig. 10 we also see that for jet masses above 10 GeV the gluon splitting configuration dominates.
We found that the new implementation has a generation efficiency, which is estimated from the numbers of vetoes in FSR generation, three times greater than the default one. This leads to a generation rate of 1316 events per minute, against the 298 events per minute of the POWHEG default, which corresponds to a gain more than a factor of 4.
3.2.1 Problematic regions
In case the transverse momentum of the gluon is small, the scale assignments and the Sudakov form factor describe the process appropriately. It can happen however, that the real emission kinematics is near the gluon splitting, flavour excitation or quark radiation regimes. In these cases the gluon transverse momentum is not small. Furthermore, the numerator R in the integrand may be enhanced with respect to the denominator, thus yielding a damping of the real cross section that is not justified. Also the scale choices are not appropriate. For example, in the case of production of a high transverse momentum heavy quark pair according to the gluon splitting mechanism, the appropriate scale should correspond to two powers of \(\alpha _{\mathchoice{\displaystyle }{\scriptstyle }{\scriptscriptstyle }{\scriptscriptstyle } \mathrm S}\) evaluated at the gluon transverse momentum, and one power of \(\alpha _{\mathchoice{\displaystyle }{\scriptstyle }{\scriptscriptstyle }{\scriptscriptstyle } \mathrm S}\) evaluated at the scale of the order of the invariant mass of the heavy quark pair.
The result of this procedure is shown in the right panels of Figs. 11, 12 and 13. We notice a remarkable improvement in the agreement, although some important differences do remain. This is not unexpected, since in the two cases radiation from the heavy quark is treated in a very different way. It is interesting to notice that the B and the \({j_\mathrm{\mathchoice{\displaystyle }{\scriptstyle }{\scriptscriptstyle }{\scriptscriptstyle } B}} \) spectra computed with the nol without remnants (which is the default in the standard hvq generator), is in fair agreement with the alt one when the enhanced regions are separated using the remnants. Since the default hvq program gives a description of the transverse momentum distribution of B hadrons that is in fair agreement with the FONLL calculation, we infer that also the alt prediction will display a similar agreement, provided the gluon splitting and flavour excitation region are treated separately as remnants.
The alt (or equivalently the def generator), with the remnant separation discussed above, seems to be at this point the generator that may give the best description of b production data at hadron collider. We should not forget, however, that some flexibility still remains in the treatment of the remnant (in this work we have made a definite scale choice for the remnants in order to have a clearer comparison with the nol generator). We also notice from Figs. 11 and 12 that after the remnants are introduced, the Bhadron and bjet \(p_T\) spectra become softer. This seems to be in contrast with the discussion at the beginning of Sect. 3.2.1. On the other hand, this result may be due to the particular scale choice that we have performed for the real graphs, and that POWHEG applies automatically also to the remnants. This scale turns out to be higher than the typical scale involved in the region discussed at the beginning of Sect. 3.2.1. A better approach would be to introduce the possibility of alternative scale choices in the remnants, including the possibility of performing a different scale choice depending upon which enhanced region one is considering. We refrain here from carrying out such study, since we believe that comparison with data on single inclusive bhadron and bjet production (see Refs. [27, 28, 29] and references therein) and on correlations of \(b\bar{b}\) pairs [30, 31], would be needed in order to make progress in this direction, and this is beyond the scope of the present work. Such study would however be very valuable, and not only for the purpose of testing QCD in bottom production. The production of top pairs is one of the most important background process at the LHC, including also its future high luminosity and eventually high energy developments. An accurate simulation of the \(t\bar{t}\) background would be very valuable for the LHC experimental collaborations, and it is quite clear that a model of b production yielding a good description of the data from low to high transverse momenta should be also well suited to describe top production in all the needed phase space. Furthermore, these studies could lead to an improved description of top production at large transverse momentum, that, as reported currently by CMS and ATLAS (see [32, 33] and references therein) seems to be not well described by theoretical models.
4 Conclusions
In this paper we have presented a method for implementing radiation from a heavy quark in the POWHEG framework. This method is considerably simpler and more transparent than the one presented in Ref. [15], and it has a much better numerical performance. The present method overcomes a problem related to the fact that the most natural map from an underlying Born configuration and a set of FKSlike radiation variables for radiation from a massive quark does not have a unique inverse in the whole kinematic region. The POWHEG inverse map has thus two solutions in some region of phase space, and in the present work it is shown how, by giving up the physical connection of the y variable to the gluon emission angle in a very limited region of phase space one can pick one of the two solutions in such a way that we still have a single valued inverse mapping. We have examined the output of the new method in the framework of the generators of Refs. [17] and [14]. We found that the new method yields results that are very consistent with the previous one, that is at this moment the POWHEG default. This is reassuring, since it shows that details of the implementation do not impact in a visible way the physics result, and also it shows that the previous implementation, in spite of being quite contrived, is in fact correct.
In this work we have also examined for the first time the impact of the inclusion of the singular region associated with radiation from the heavy quark in the case of the hvq generator. We have shown that, unless one separates the enhanced gluon splitting and flavour excitation contribution from the real contribution that are dealt with by the POWHEG radiation formula, and treats them as remnants, one finds results that are in considerable disagreement with the traditional hvq implementation. On the other hand, it seems that performing this separation is the appropriate thing to do for a consistent modeling of the process. We notice that such modeling would be of great interest also for its potential application to top pair production, that is a very important background to many LHC physics studies.
Footnotes
 1.
The system we are considering can be either the full final state, or the system of decay products of a resonance, according to the origin of the heavy quark.
 2.
\(y_\text {phy}\) must not be confused with the y variable of the mapping. More specifically, in the region \(\xi (0) \le \xi \le \xi _\text {max},\, y>0\) we have \(y_\text {phy}=y\), while in all the remaining region \(y_\text {phy}=y\).
 3.
In fact, rather than proving analytically that \(K_T^2\) is a monotonic decreasing function of y at fixed \(\xi \), we demonstrated it numerically by checking it a large number of times for random values of the input parameters.
Notes
Acknowledgements
We thank Matteo Cacciari for useful exchanges. The work of L.B. and F.T. has been supported in part by the Italian Ministry of Education and Research MIUR, under Project No. 2015P5SBHT.
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