Page 1

arXiv:1106.6019v1 [hep-ph] 29 Jun 2011

Preprint typeset in JHEP style - HYPER VERSION

CERN-PH-TH/2011-147

CP3-11-20

NSF-KITP-11-114

ZH-TH 13/11

W and Z/γ∗boson production in association with a

bottom-antibottom pair

Rikkert Frederix

Institut f¨ ur Theoretische Physik, Universit¨ at Z¨ urich, Winterthurerstrasse 190,

CH-8057 Z¨ urich, Switzerland

KITP, University of California Santa Barbara, CA 93106-4030, USA

Stefano Frixione∗

PH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland

ITPP, EPFL, CH-1015 Lausanne, Switzerland

Valentin Hirschi

ITPP, EPFL, CH-1015 Lausanne, Switzerland

Fabio Maltoni

Centre for Cosmology, Particle Physics and Phenomenology (CP3)

Universit´ e catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

Roberto Pittau

Departamento de F´ ısica Te´ orica y del Cosmos y CAFPE, Universidad de Granada

PH Department, TH Unit, CERN, CH-1211 Geneva 23, Switzerland

KITP, University of California Santa Barbara, CA 93106-4030, USA

Paolo Torrielli

ITPP, EPFL, CH-1015 Lausanne, Switzerland

Abstract: We present a study of ℓνb¯b and ℓ+ℓ−b¯b production at hadron colliders. Our re-

sults, accurate to the next-to-leading order in QCD, are based on automatic matrix-element

calculations performed by MadLoop and MadFKS, and are given at both the parton level,

and after the matching with the Herwig event generator, achieved with aMC@NLO. We

retain the complete dependence on the bottom-quark mass, and include exactly all spin

correlations of final-state leptons. We discuss the cases of several observables at the LHC

which highlight the importance of accurate simulations.

Keywords: LHC, NLO, Monte Carlo.

∗On leave of absence from INFN, Sezione di Genova, Italy.

Page 2

Contents

1.Introduction1

2. Results3

3.Conclusions and outlook14

4. Acknowledgments15

1. Introduction

The discovery and identification of new degrees of freedom and interactions at high-energy

colliders relies on the detailed understanding of Standard Model (SM) background pro-

cesses. Prominent among these is the production of electroweak bosons (W,Z) in associa-

tion with jets, of which one or more possibly contain bottom quarks. The prime example

is the observation of the top quark at the Fermilab Tevatron collider, produced either in

pairs [1, 2] or singly [3, 4], since both of these mechanisms typically lead to W plus b-jets

signatures. Other crucial examples involve the search for a SM Higgs in association with

vector bosons (WH/ZH), with the subsequent Higgs decay into a b¯b pair, a sought-for

discovery channel both at the Tevatron [5] and at the LHC [6, 7]. Finally, in models which

feature an extended Higgs sector, such as the MSSM or more generally a two-Higgs dou-

blet model, a typical Higgs discovery channel is through Hb¯b and Ab¯b final states, with

an H/A → τ+τ−decay. In this case, the SM process ℓ+ℓ−b¯b can provide an important

reference measurement.

Next-to-leading-order (NLO) QCD calculations for the production of a vector boson

in association with jets have by now quite a successful record. Accurate predictions for W

plus up to four light jets [8, 9, 10, 11, 12, 13] and for Z plus up to three light jets [14, 8, 9, 15]

have become available in the past few years. Associated production with heavy quarks,

and in particular with bottom quarks, has been studied using various approximations. The

Wb¯b and Zb¯b processes have been calculated at the NLO for the first time in refs. [16]

and [17] respectively, by setting the bottom-quark mass equal to zero. Such calculations

can be used only for observables that contain at least two b-jets. The same processes

have been considered again in refs. [18, 19, 20], where a non-zero bottom-quark mass has

been used; however, the matrix elements still involved on-shell vector bosons, thereby

neglecting spin correlations of the leptons emerging from W and Z decays. In the case of

W production, this limitation has been recently lifted in ref. [21], which presents the NLO

calculation for the leptonic process ℓνb¯b. Other NLO calculations for final states with one

b-jet, Wb and Zb [22, 23], and one b-jet plus a light jet, Wbj and Zbj [24, 25], are also

– 1 –

Page 3

available in the five-flavour scheme. All such calculations have played a role and/or have

been extensively compared to the data collected at the Tevatron [26, 27, 28, 29], and now

start to be considered in LHC analyses as well [30, 31].

In this paper we present a calculation of ℓνb¯b production that includes NLO QCD

corrections (analogous to that of ref. [21]), and the first calculation at the NLO of ℓ+ℓ−b¯b

production with massive bottom quarks; we retain the full spin correlations of the final-

state leptons1. Furthermore, we match both of these results to the Herwig event generator

by adopting the MC@NLO formalism [32]. Therefore, our results include all the relevant

features which are important in experimental analyses, and can be used in order to obtain

NLO predictions for a large class of observables, including those with zero, one and two

b-jets. All aspects of the calculations are fully automated and analogous to the calculation

recently appeared for Ht¯t/At¯t production [33]. One-loop amplitudes are evaluated with

MadLoop [34], whose core is the OPP integrand reduction method [35] as implemented

in CutTools [36]. Real contributions and the corresponding phase-space subtractions,

achieved by means of the FKS formalism [37], as well as their combination with the one-

loop and Born results and their subsequent integration, are performed by MadFKS [38].

The MC@NLO matching is also fully automated, and allows us to simulate for the first

time ℓνb¯b and ℓ+ℓ−b¯b production with NLO accuracy, including exactly spin correlations,

off-shell and interference effects, and hadron-level final states. All the computations are

integrated in a single software framework, which we have dubbed aMC@NLO in ref. [33].

We point out that the on-shell-W result of ref. [18] has recently been matched to showers

in ref. [39] in the framework of the POWHEG box [40].

The phenomenology of ℓνb¯b and ℓ+ℓ−b¯b final states is very rich and in fact transcends

the prosaic role of background for Higgs or top-quark physics. Thanks to the aMC@NLO

implementation, several QCD issues interesting on their own can now be addressed theoret-

ically and the results efficiently compared to experiments. In this work we limit ourselves

to providing a first evidence that reliable and flexible predictions for the ℓνb¯b and ℓ+ℓ−b¯b

processes need to include:

• NLO corrections;

• bottom quark mass effects;

• spin-correlation and off-shell effects;

• showering and hadronisation.

Detailed studies of these processes as backgrounds to specific signals, such as single-top

and Hb¯b or Ab¯b production respectively, are left to forthcoming investigations.

This paper is organised as follows. In the next section we present several distributions

relevant to ℓνb¯b and ℓ+ℓ−b¯b production at the LHC, and report the results for total rates

at both the Tevatron and the LHC. By working with a non-zero bottom mass, we are able

1In the rest of this paper, ℓνb¯b and ℓ+ℓ−b¯b production as predicted by our simulations may also be

denoted by Wb¯b and Zb¯b respectively.

– 2 –

Page 4

to obtain predictions for the cases in which one or two b’s are not observed, and can thus

have arbitrarily small transverse momenta. We also give one example of the comparisons,

at the level of hadronic final states, between the HW and HZ signals and their respective

irreducible backgrounds which we have computed in this paper. We draw our conclusions

in sect. 3.

2. Results

At the leading order (LO) in QCD ℓνb¯b and ℓ+ℓ−b¯b production at hadron colliders proceed

through different channels. Both final states can be obtained via a Drell-Yan-type mecha-

nism, i.e., q¯ q(′)annihilation in association with a gluon splitting in a b¯b pair, see fig. 1(a).

Zb¯b, however, can also be produced by gluon fusion, see fig. 1(b), a channel that at the

LO contributes a 30% of the total rate at the Tevatron, but turns out to be the dominant

one (80%) at the LHC, owing to the larger gluon luminosity there. As we shall see in the

following, the fact that Wb¯b and Zb¯b production are dominated by different channels at

the LHC leads to important differences in the kinematical properties of final states, and in

particular of b-jets.

We start by presenting results for the total cross sections at both the Tevatron,√s =

1.96 TeV, and the LHC,√s = 7 TeV; the ℓνb¯b results are the sums of the ℓ+νb¯b and ℓ−¯ νb¯b

ones (due to virtual-W+and W−production respectively). In our computations we have

set the lepton masses equal to zero, and is therefore not necessary to specify their flavour,

which we generically denote by ℓ (for the charged leptons) and ν (for the neutrinos); we

always quote results for one flavour. For the numerical analysis we have chosen:

µ2

F= µ2

R= m2

ℓℓ′ + p2

T(ℓℓ′) +m2

b+ p2

2

T(b)

+m2

b+ p2

2

T(¯b)

, (2.1)

with ℓℓ′= ℓν and ℓℓ′= ℓ+ℓ−in the case of Wb¯b and Zb¯b production respectively; the

value of the b-quark mass is that of the pole mass, mb= 4.5 GeV. We have used LO

and NLO MSTW2008 four-flavour parton distribution functions [41] for the corresponding

cross sections, and the SM-parameter settings can be found in table 1. Our runs are fully

inclusive and no cuts are applied at the generation level, except for mℓ+ℓ− > 30 GeV in

ParametervalueParametervalue

mZ

mW

mb

mt

α(LO,4)

s

α(NLO,4)

s

91.118

80.419

4.5

172.5

0.133551

0.114904

α−1

GF

CKMij

ΓZ

ΓW

132.50698

1.16639·10−5

δij

2.4414

2.0476(mZ)

(mZ)

Table 1: Settings of physical parameters used in this work, with dimensionful quantities given in

GeV.

– 3 –

Page 5

Figure 1: Representative diagrams contributing to ℓνb¯b and ℓ+ℓ−b¯b production at the leading

order. ℓνb¯b production can proceed only via a q¯ q′channel, diagram (a). For ℓ+ℓ−b¯b production the

q¯ q channel, diagram (a), is dominant at the Tevatron, while the gg channel, diagram (b), largely

dominates at the LHC.

Cross section (pb)

Tevatron√s =1.96 TeV LHC√s =7 TeV

LO NLOK factor LONLOK factor

ℓνb¯b 4.638.04 1.74 19.4 38.92.01

ℓ+ℓ−b¯b 0.8601.5091.759.6616.1 1.67

Table 2: Total cross sections for ℓνb¯b and ℓ+ℓ−b¯b production at the Tevatron (p¯ p collisions at

√s = 1.96 TeV) and the LHC (pp collisions at√s = 7 TeV), to LO and NLO accuracy. These

rates are relevant to one lepton flavour, and the results for ℓνb¯b production are the sums of those

for ℓ+νb¯b and ℓ−¯ νb¯b production. The integration uncertainty is always well below 1%.

the ℓ+ℓ−b¯b sample. The predicted production rates at the Tevatron and at the LHC are

given in table 2 where, for ease of reading, we also show the fully inclusive K factors. The

contribution of the gg → Zb¯b+X channels is clearly visible in these results: at the Tevatron

σ(ℓ+ℓ−b¯b)/σ(ℓνb¯b) is quite small (and of the same order of the ratio of the fully-inclusive

cross sections σ(Z)/σ(W)), whereas at the LHC ℓ+ℓ−b¯b and ℓνb¯b differ only by a factor of

two.

We now study the impact of NLO QCD corrections on differential distributions, at

both the parton level and after showering and hadronisation, and in doing so we limit

ourselves to the case of the LHC, where the kinematical differences between Wb¯b and Zb¯b

production are more evident. The parton shower in aMC@NLO has been performed with

fortran Herwig [42, 43, 44], version 6.5202.

We start by summarizing our results for b-jet rates. Jets are reconstructed at the parti-

2Automation of the matching to parton shower in the MC@NLO formalism to Herwig++ [45] and to

Pythia [46] (see refs. [47] and [48] respectively) is currently under way.

– 4 –

Page 6

Figure 2: Fractions of events (in percent) that contain: zero b-jets, exactly one b-jet, and exactly

two b-jets. The rightmost bin displays the fraction of b-jets which are bb-jets. The two insets show

the ratio of the aMC@NLO results over the corresponding NLO (solid), aMC@LO (dashed), and

LO (symbols) ones, separately for Wb¯b (upper inset) and Zb¯b (lower inset) production.

cle level. In the case of MC simulations, this means giving all final-state stable hadrons3in

input to the jet algorithm. We adopt the anti-kTjet clustering algorithm [49] with R = 0.5,

and require each jet to have pT(j) > 20 GeV and |η(j)| < 2.5. A b-jet is then defined as a

jet that contains at least one b-hadron; a bb-jet is a jet that contains at least two b-hadrons

(hence, a bb-jet is also a b-jet). This implies that we make no distinction between the b

quark and antiquark contents of a jet. We point out that at least another definition of

b-jets exists [50] which has a better behaviour in the mb→ 0 limit, in the sense that it

gives (IR-safe) results consistent with the naive picture of “quark” and “gluon” jets. In

practice, this is relevant only in the pT ≫ mblimit. Since this region is not our primary

interest in this paper, we stick to the usual definition; however, it should be obvious that

any jet definition can be used in our framework.

In fig. 2 we present b-jet rates, as the fractions of events that contain zero, exactly

one, or exactly two b-jet(s). In the case of MC-based simulations, there are also events

with more than two b-jets and more than one bb-jet, but they give a relative contribution

to the total rate equal to about 0.4% (for Wb¯b) and 0.6% (for Zb¯b), and are therefore not

reported here. The rightmost bin of fig. 2 shows the fraction of b-jets which are bb-jets.

There is an inset for each of the two histograms shown in the upper part of fig. 2. Each

of the insets presents three curves, obtained by computing the ratio of the aMC@NLO

results over the NLO (solid), aMC@LO4(dashed), and LO (symbols) corresponding ones.

3In order to simplify the Herwig analyses, weakly-decaying B hadrons are set stable.

4We call aMC@LO the analogue of aMC@NLO, in which the short-distance cross sections are computed

at the LO rather than at the NLO. Its results are therefore equivalent to those one would obtain by using,

e.g., MadGraph/MadEvent [51] interfaced to showers.

– 5 –

Page 7

Figure 3: Invariant mass distribution of final-state lepton pairs, as predicted by aMC@NLO.

The b-jet fractions are fairly similar for Wb¯b and Zb¯b production, and the effects of the

NLO corrections are consistent with the fully-inclusive K factors. On the other hand, the

bb-jet contribution to the b-jet rate is seen to be more than three times larger for ℓνb¯b

than for ℓ+ℓ−b¯b final states. This fact is again related to the different mechanisms for the

production of a b¯b pair in the two processes considered here. At variance with the case of

ℓνb¯b production, in a ℓ+ℓ−b¯b final state the two b’s may come from the separate branchings

of two initial-state gluons, and thus the probability of them ending in the same jet is much

smaller than in the case of a g → b¯b final-state branching. We conclude this discussion

by pointing out that the zero- and one-b-jet rates can only be obtained with a non-zero

b-quark mass, since the one or two “untagged” b’s must be integrated down to pT= 0, and

hence mb?= 0 is required in order to screen initial-state collinear divergences. This fact

is a severe test condition for the computer programs used in the computations, because it

may induce numerical instabilities. We stress that we did not impose low-pT cuts on any

of the final-state particles in MadLoop, MadFKS, and in the generation of hard events

in aMC@NLO; our results are therefore completely unbiased, which is what gives us the

possibility of computing quantities such as those reported in table 2 and in fig. 2.

We now turn to studying differential distributions, and start by considering those

defined in terms of final-state leptons. Observables sensitive to the hadronic activity of

the events, be either relevant to b-jets or to B-hadrons, will follow later. In the case of

MC-based simulations, several leptons can appear in the final state. We use the MC-truth

information to select the two which emerge from the hard process, and we shall simply

refer to them as “the leptons” henceforth. A more realistic analysis may select leptons on

the basis of their hardness, measured e.g. with their pT’s; in practice, for the processes we

are considering here (and thanks to the fact that the b-hadrons have been set stable) the

two approaches are equivalent.

In fig. 3 the invariant mass of the final-state lepton pairs is shown. The effect of the γ∗

contribution to ℓ+ℓ−b¯b production is clearly visible at small invariant masses. In this plot,

with limit ourselves to presenting only the aMC@NLO results, since the other simulations

– 6 –

Page 8

Figure 4: Transverse momentum of the charged leptons in ℓνb¯b (left panel) and ℓ+ℓ−b¯b (right

panel) production, shown separately for positive and negative charges. The upper and middle

insets follow the same patterns as those in fig. 2. The lower inset (magenta solid histogram) is

the ratio of the aMC@NLO results relevant to positively-charged leptons over those relevant to

negatively-charged ones.

give results which are essentially identical to the present ones.

In figs. 4 and 5 the transverse momenta and the pseudorapidities of the charged leptons

are shown separately according to their electric charges. In the two upper insets we have

used the same patterns and conventions as in fig. 2 – these will be used throughout this

paper. In the case of Wb¯b production, the effects of the NLO corrections are especially

pronounced at large pT’s, where they are the signal of new partonic subprocesses opening

up at this order, and in particular of those which include an initial-state gluon, such as

qg. Results after matching with showers consistently show a similar behaviour. The same

large enhancement is not present in the case of Z production, which receives gluon-initiated

contributions already at the LO; again, this trend is seen also after matching with showers.

The lowest insets (solid magenta curves) show the ratios of the aMC@NLO results relevant

to positively-charged leptons over those relevant to negatively-charged ones. In the case of

Wb¯b production the behaviour is similar to what has been described recently in ref. [52],

while for Zb¯b this distribution is flat, as expected.

A complementary aspect of the different parton luminosities that contribute to Wb¯b

and Zb¯b production can be appreciated by looking for example at the transverse momentum

distributions of the ℓν and ℓ+ℓ−pairs (i.e. of the virtual W and Z bosons respectively),

shown in the left panel of fig. 6. In the case of ℓνb¯b final states, the aMC@LO and LO

results are very close to each other, which is not the case for ℓ+ℓ−b¯b production. This is

due to the fact that the gg-initiated channel in the latter case is responsible for much more

QCD radiation in MC-based simulations than the q¯ q channel (the latter being identical to

the mechanism that induces Wb¯b production). This larger amount of radiation hardens the

virtual-Z pTspectrum predicted by aMC@LO, making it more similar to the aMC@NLO

result than in the case of Wb¯b production. The NLO pTspectra of the virtual vector bosons

– 7 –

Page 9

Figure 5: As in fig. 4, for the pseudorapidity of the charged leptons.

are closer to the aMC@NLO results, because at that order one does get contributions from

gluon-initiated channels to both ℓνb¯b and ℓ+ℓ−b¯b final states, and thus the relative changes

obtained when matching with parton showers have a milder impact. This is an example

of the “stabilizing” pattern that one observes when higher-order perturbative results are

taken into account. On the other hand, the rapidity distributions of the lepton pairs do

not change significantly under QCD radiation, as is shown in the right panel of fig. 6.

We remark that we do not find any significant enhancement in the large-pT tails of

vector bosons when going from the NLO to the aMC@NLO predictions, at variance with

the POWHEG Wb¯b result of ref. [39] which necessitates an ad-hoc perturbative tuning for

this reason. It should further be stressed that the almost perfect coincidence between the

aMC@LO and LO results for pT(ℓν) may not occur by simply changing the tuning of the

shower parameters in Herwig: indeed, we have verified that the spectrum predicted by

Pythia 6 is slightly different w.r.t. the aMC@LO or LO ones. It is also interesting to

notice that the features discussed here and that affect the low- and intermediate-pTregions

of lepton pairs are not visible in the case of the individual lepton pTspectra, fig. 4. We have

verified that kinematical correlations are such that, for a fixed small or intermediate value

of pT(ℓ±), one integrates over a pT(ℓ+ℓ−) range that causes the local differences between

the aMC@LO and LO results for the latter transverse momentum to be averaged out.

The left panel of fig. 7 presents the cosθ∗distribution, computed by separating the

positively- and negatively-charged lepton contributions. We remind the reader that such

an observable is defined as the cosine of the angle between the chosen charged lepton, and

the direction of flight of the parent vector boson, in the rest frame of the latter. Clearly

visible are the strong angular correlations and charge asymmetry in the ℓνb¯b case. For

ℓ+ℓ−b¯b production such correlations and asymmetries, while present, are much milder, and

likely not observable in a real experiment. For both processes, the aMC@NLO results are

basically identical to those of aMC@LO, NLO, and LO, and thus we refrain from showing

the latter here.

– 8 –

Page 10

Figure 6: Transverse momentum (left panel) and rapidity (right panel) of the ℓν and ℓ+ℓ−pairs

(i.e. of the virtual W and Z bosons respectively) in ℓνb¯b and ℓ+ℓ−b¯b production. The insets follow

the same patterns as those in fig. 2.

In the right panel of fig. 7, where we consider only leptons with positive electric charge

to be definite, we plot the ratio of the lepton transverse momentum over the same quantity,

obtained by imposing a phase-space (i.e., flat) decay of the parent vector boson; hence,

this ratio is a measure of the impact of spin correlations on the inclusive-lepton pT. We

see that differences between correlated and uncorrelated decays can be as large as 20%,

and vary across the kinematical range considered. This confirms that the inclusion of spin-

correlation effects is necessary when an accurate description of the production process is

required. We stress again that our computations feature spin correlations exactly at the

matrix-element level, including one-loop ones. It is interesting to observe that, while in the

case of Zb¯b production all four calculations give similar results (see the lower inset), this

happens in Wb¯b production only for pT(ℓ+) ? 50 GeV (see the upper inset). At pTvalues

larger than this, aMC@NLO and NLO predict ratios that differ from the corresponding

aMC@LO and LO ones. Once again, this is a manifestation of the significant impact of

gluon-initiated, NLO partonic processes on Wb¯b cross sections.

In figs. 8 and 9 the transverse momenta and the pseudorapidities of the two hardest

b-jets are shown. Differences in normalisation are consistent with what we expect on the

basis of inclusive K factors; differences in shapes are typically small, but visible. We point

out that for an event to contribute to the hardest-b-jet observables shown here it is sufficient

that one b-jet be present in the event; the other b quark emerging from the hard process

can have arbitrarily small momentum.

In the left panel of fig. 10, the ∆R separation between the two hardest b-hadrons

(for the MC-based simulations) or between the b and¯b quarks (for the NLO and LO

computations) is shown. Differences between the Wb¯b and Zb¯b processes are manifest. In

the former case the two b’s originate from a final-state gluon splitting, and they will thus

tend to be quite close in pseudorapidity. On the other hand, the two b’s in Zb¯b production

can arise from the uncorrelated branchings of the initial-state gluons in the gg channel, and

– 9 –

Page 11

Figure 7: Left panel: cosθ∗distribution of final-state charged leptons for different charges. All

histograms have been obtained with aMC@NLO. See the text for the observable definition. Right

panel: ratio of the results for the pT of the positively-charged lepton over the same quantity com-

puted by neglecting production spin correlations. The insets follow the same patterns as those in

fig. 2.

Figure 8: Transverse momentum of the hardest (left panel) and second-hardest b-jet (right panel)

in Wb¯b and Zb¯b production. The insets follow the same patterns as those in fig. 2.

in this way they will naturally acquire a large separation in pseudorapidity, which is directly

related with large-∆R values. However, a b¯b pair arising from a final-state gluon branching

can be easily separated in pseudorapidity by QCD radiation. This is the reason why the

parton-level LO result in the case of Wb¯b production is so different from the other three

predictions (as shown by the symbols in the upper inset). Both parton-level NLO (through

radiation present at the matrix-element level) and aMC@LO (through radiation due to

parton showers) results are in fact much closer to aMC@NLO than parton-level LO result

is. This does not happen in the case of Zb¯b production, since as discussed before the b and

¯b quarks can be well-separated in pseudorapidity already at the LO. It should be stressed

– 10 –

Page 12

Figure 9: As in fig. 8, for the pseudorapidity of the hardest and the second-hardest b-jet.

Figure 10: Left panel: ∆R separation between the two hardest b-hadrons (aMC@NLO and

aMC@LO) or the b and¯b quarks (NLO and LO) in the event. Right panel: invariant mass of the

b-jets, inclusive over all b-jets in the event. The insets follow the same patterns as those in fig. 2.

that the b-hadrons that contribute to the ∆R separation shown in fig. 10 are not subject

to any lower cuts in pT. Thus, one expects that the effects of extra radiation be diminished

when imposing a pT cut or, which is equivalent, by studying the same distribution in the

case of b-jets. We have verified that this is indeed the case, i.e. that when a minimum-pTcut

is imposed on the two b-hadrons the pattern of NLO QCD corrections in Wb¯b production is

more similar to that observed in Zb¯b production. This is another example of the possibility

of testing detailed properties of QCD radiation by considering low-pTevents. It should be

clear that from the theoretical viewpoint such studies can be sensibly performed only by

retaining the full b-mass dependence.

The right panel of fig. 10 shows the mass of the b-jets in the events. The observable

is inclusive over all b-jets, which implies that a given event may enter more than once

– 11 –

Page 13

Figure 11: Transverse momentum fraction carried by b-jets. See the text for details.

in the plot.

mjb≈ 12 GeV. In the case of Zb¯b production this effect is almost invisible, a consequence

of the fact that the fraction of events where a b-jet is actually a bb-jet is much smaller than

for Wb¯b production, see fig. 2. The distribution discussed here measures the activity inside

a jet, and one cannot expect fixed-order parton-level results, where a jet consists of one

or two particles, to be particularly sensible in this case. In fact, we see that fixed-order

results are very different from MC-based ones. On the other hand, the differences in shape

when going from aMC@LO to aMC@NLO are small, in particular for Zb¯b production, as

expected for observables which are insensitive to emissions at large relative pT’s. We also

point out that the knee at mjb≈ 12 GeV would appear as a feature of gluon jets if the b-jet

definition of ref. [50] were used. This stresses again the fact that, at small and moderate

pT’s, the usual definition gives more intuitive results. On the other hand, at large jet pT’s

the onset of the bb-jet contribution to mjbis largely smeared out.

In fig. 11 we show the ratio of the total transverse momentum PT[jb] of b-jets, over the

total transverse hadronic momentum PT5. In the context of parton-level computations, by

“hadrons” we simply understand QCD partons. At the parton-level LO, the configurations

with one bb-jet or with two b-jets (each of which contains one b quark) give contribution

at PT[jb]/PT= 1. Configurations with one b-jet that contains only one b quark contribute

to 0.5 < PT[jb]/PT < 1 if the other b quark has pT < 20 GeV (i.e., it is softer than a jet

is required to be), while values PT[jb]/PT < 0.5 can be obtained when the other b quark

has pT> 20 GeV and |η(b)| > 2.5 (i.e., it is outside the b-jet tagging region in pseudora-

pidity). What was said above implies that PT[jb]/PT = 1 is an infrared-sensitive region,

which gives rise to Sudakov logarithms at higher order; this explains the behaviour of the

parton-level NLO results there. Furthermore, the LO contributions to the PT[jb]/PT< 0.5

region decrease when increasing the maximum-pseudorapidity cut on jets. This is only

marginally the case at the NLO (because of the presence of a hard light parton in the real-

emission contributions), which explains the longer tail of the latter results w.r.t. the LO

Striking is the onset of the bb-jet contribution in the Wb¯b result around

5We stress that PT is defined without including the underlying event and pile-up contributions.

– 12 –

Page 14

Figure 12: Invariant mass of the pair of the two leading b-jets. WH(→ ℓνb¯b), ZH(→ ℓ+ℓ−b¯b),

ℓνb¯b, and ℓ+ℓ−b¯b results are shown, with the former two rescaled by a factor of ten.

ones. The arguments above obviously do not apply to the context of an event generator;

this is confirmed by the similarity of the aMC@NLO and aMC@LO results. Firstly, at

PT[jb]/PT= 1 Sudakov logarithms are properly resummed. Secondly, the extra radiation

generated by parton showers implies that quite a few hadrons will lie outside b-jets, hence

shifting further the PT[jb]/PTresults to the left of those relevant to parton-level NLO com-

putations. This shift is also present when passing from the aMC@LO to the aMC@NLO

predictions in Wb¯b production, while in the case of Zb¯b production these two results are

very similar (up to an overall rescaling by the inclusive K factor). We are finding here the

same pattern already discussed for a few observables in this paper. Namely, the opening of

gluon-initiated partonic channels at the NLO in Wb¯b production implies a richer hadronic

activity w.r.t. the corresponding LO case, which is only marginal in the case of Zbb produc-

tion owing to the dominance of the gg channel already at the LO there. Hence, the relative

enhancement of the hadronic activity outside the b-jets when going from aMC@LO to

aMC@NLO is stronger for Wb¯b production than is for Zb¯b production.

Finally, as a simple application to Higgs searches of the calculations presented in this

paper, we show in fig. 12 the invariant mass of the two leading b-jets in WH(→ ℓνb¯b),

ZH(→ ℓ+ℓ−b¯b), ℓνb¯b, and ℓ+ℓ−b¯b events. The former two processes (the “signal”) have

been simulated with MC@NLO [32]6, with a Higgs mass mH = 120 GeV. The tail at

m[jb,1,jb,2] > mHis due to the fact that the jet momenta are typically larger than those of

the b-hadrons they contain, owing to the contributions of other final-state hadrons emerging

from initial-state showers. This is compensated by the fact that the b-hadron momenta

are only a fraction of those of their parent b quarks, the complementary fraction being

6In the process of validating aMC@NLO, we had checked that it gave results identical to MC@NLO for

all the processes implemented in the latter. Hence, we could have equally well employed aMC@NLO to

simulate the signal here.

– 13 –

Page 15

lost to radiation which may end up outside the jets. These two effects smear the Higgs

peak. Furthermore, in some events the b quarks entering the two hardest b-jets do not arise

from the Higgs decay, but from a g → b¯b branching in the shower phase. Although rare

indeed, these events may result in invariant masses much larger than the Higgs pole mass.

The comparison given here is just an example of an analysis in which both the signal and

its irreducible backgrounds can be computed at the same precision with (a)MC@NLO,

improving upon both fixed-order and LO-based Monte Carlo descriptions.

3. Conclusions and outlook

In this work we have presented results for the ℓνb¯b and ℓ+ℓ−b¯b production processes,

accurate to the NLO in QCD and that include the matching to parton showers according

to the MC@NLO formalism. Our approach is fully general, completely automated, and

opens the way to performing comparisons with experimental data from the Tevatron and

the LHC at the highest theoretical accuracy attainable nowadays.

By studying a limited but representative set of observables, we have shown that several

are the elements to be kept into account in order to achieve reliable and flexible predictions

for this class of processes: spin correlations of the final state leptons emerging from the

decays of the vector bosons, heavy-quark mass effects, and a realistic description of the

final states, obtained thanks to the interface with a shower and hadronisation program. As

we have seen, NLO QCD corrections have a highly non-trivial impact, since they lead not

only to large enhancements of total rates, but also to significant changes in the shapes of

distributions. In this respect, the opening at the NLO of new partonic channels, and in

particular of those involving gluons, plays a fundamental role. In general and apart from

well-understood cases in which pure perturbative results are not meaningful, one observes

that at the NLO level fixed-order and MC-based results are closer to each other than the

corresponding LO ones. This is in keeping with naive expectations based on perturbation

theory, and it is significant in that it shows that the very large corrections affecting the

processes considered here do not pose problems when the matching with parton shower

Monte Carlos is carried out according to the MC@NLO method.

Thanks to the aMC@NLO implementation, several QCD issues interesting on their

own can now be addressed. One example over all is the study of NLO corrections, mass

effects and radiation pattern in final-state gluon splitting, for which Wb¯b production offers

a particularly clean environment. Gluon splitting in the initial state and the role of the b

PDF (and therefore of different schemes for the predictions of total and differential cross

sections) can be assessed by considering Zb¯b production. The outcome of such a study

can then be applied to the Hb¯b case. In particular, available predictions in the five-flavour

scheme at the NNLO for the fully-inclusive production of a Z in association with bottom

quarks [53], and for Z+1 b-jet at the NLO [23], can now compared with our four-flavour-

scheme results. In addition, QCD radiation effects on high-pTb¯b pairs, which can be merged

into one jet, are also of interest in boosted-Higgs searches [54]. Finally, spin correlation

effects may also be investigated to gather more insight on the production mechanisms

in QCD, and possibly to distinguish them from other competing hard reactions, such as

– 14 –