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arXiv:1105.3624v1 [hep-ph] 18 May 2011

FREIBURG-PHENO-2011-008, Nikhef2011-014, MPP-2011-55, DESY 11-079

NLO QCD corrections to the production of two bottom-antibottom pairs at the LHC

Nicolas Greiner,1Alberto Guffanti,2Thomas Reiter,3,4and J¨ urgen Reuter5,2

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana IL, 61801, USA

2Physikalisches Institut, Albert-Ludwigs-Universit¨ at, 79104 Freiburg, Germany

3Nikhef, 1098 XG Amsterdam, The Netherlands

4Max-Planck-Institut f¨ ur Physik, 80805 M¨ unchen, Germany

5DESY, 22607 Hamburg, Germany

(Dated: May 19, 2011)

We report the results of a computation of the full next-to-leading order QCD corrections to the

production of two b¯b pairs at the LHC. This calculation at the parton level provides predictions

for well separated b-jets. The results show that the next-to-leading order corrections lead to an

enhancement of the cross-section for the central scale choice by roughly 50% with respect to the

leading order result. The theoretical uncertainty estimated by variation of the renormalization and

factorization scales is strongly reduced by the inclusion of next-to-leading order corrections.

PACS numbers: 12.38.Bx, 13.85.Hd, 14.65.Fy

INTRODUCTION

The search for the Higgs boson, and more in general

the study of the Electroweak Symmetry breaking mech-

anism, is a major goal of the experiments at the LHC

collider at CERN. In various extensions of the Standard

Model the signature of two light Higgs bosons decaying in

two pairs of b-quarks, hh → b¯bb¯b, is a viable channel for

the Higgs Boson discovery. Examples of these models are

the Minimally Supersymmetric Standard Model (MSSM)

for large values of tanβ and moderate mA[1–4], hidden

valley scenarios where the decay of hadrons of an addi-

tional gauge group can produce additional b-jets [5, 6]

and two Higgs doublet models. The possibility of mea-

suring the Higgs self-coupling through H → hh → b¯bb¯b

has been investigated in [7]. This and other related stud-

ies, however show that such a measurement would be

extremely difficult, primarily due to the large Standard

Model background. The precise knowledge of the b¯bb¯b

final state within the Standard Model is therefore an im-

portant factor for the success of these measurements.

Because of its importance this process has been added

to the Les Houches wish list of relevant next-to-leading

order calculations [5].

In an earlier publication [8], we presented the next-to-

leading order (NLO) QCD corrections to the production

of b¯bb¯b via quark-antiquark annihilation. In the present

Letter we complete the existing work including the gluon

initiated contributions and present the results for the full

NLO QCD corrections to pp → b¯bb¯b at the LHC.

We show that the inclusion of the NLO corrections

reduces the unphysical scale dependence of the leading

order (LO) prediction greatly, improving the precision

of this prediction and allowing a better estimation of the

Standard Model background to possible New Physics sig-

nals in this channel.

METHOD

A complete NLO QCD description requires the calcu-

lation of the 2 → 4 subprocesses q¯ q → b¯bb¯b and gg → b¯bb¯b

at the tree and the one-loop level as well as the 2 → 5 par-

ticle processes q¯ q → b¯bb¯bg, gg → b¯bb¯bg and

at tree level.

We sum over four massless quark flavours q

{u,d,s,c} in the initial state. Neglecting the contribu-

tion from initial state b-quarks is justified by the small-

ness of the b parton distribution function (PDF) with

respect to the other quark PDFs. Moreover the fact that

the gluon-gluon channel is the dominant contribution at

LHC energies further reduces the relative importance of

the quark channels. We treat the b-quarks as massless,

which is a very good approximation for LHC kinematics

also due to the cuts imposed in order for the final state

b-quarks to be detected and separated in phase space.

Effects of the heavy top quark are neglected altogether

in the final result after having shown that they are nu-

merically not important.

The LO and the real radiation matrix elements are gen-

erated using MadGraph [9]. For the subtraction of the

infrared singularities we use Catani-Seymour dipoles [10],

supplemented with a slicing parameter α as proposed

in [11, 12], implemented in the MadDipole package [13,

14].

As described in our earlier work [8], we compute the

one loop corrections to scattering matrix elements us-

ing an approach based on Feynman diagrams. The code

for the numerical evaluation of the virtual corrections is

generated using the automated one-loop matrix element

generator golem-2.0 [15–17] which employs QGraf [18],

Form [19], the Form library Spinney [20] and the code

generator Haggies [21] at intermediate levels of the dia-

gram and code generation. The reduction and evaluation

of the loop integrals is performed using the Samurai [22]

and OneLoop [23] packages respectively.

(−)

q g → b¯bb¯b

(−)

q

∈

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2

The integration over phase space is carried out using

MadEvent [24] and it has been split up in independent

parts in order to optimize the computational time re-

quired.

The first contribution consists of the real emission ma-

trix element supplemented with the subtraction terms.

The integration of this contribution over the correspond-

ing 13-dimensional phase space is one of the main compu-

tational bottlenecks of such a calculation. This integra-

tion has been performed using up to 2 · 109phase space

points. For the q¯ q and gg subprocess the evaluation of

a single phase space point requires the evaluation of 30

subtraction terms for each partonic channel and an addi-

tional 10 subtraction terms are needed for the qg channel.

This means that a substantial fraction of CPU time is

spent calculating the dipole contributions. In such a sit-

uation the use of a value smaller than one for the slicing

parameter α, as proposed in [11, 12], speeds up substan-

tially the computation by avoiding that each subtraction

term is evaluated for each phase-space point. Besides the

reduction of the computational time per point this set-

ting has a second advantage. If not close to a singularity

the integrand is given just by the real emission matrix

element. Close to a singularity, where also subtraction

terms are calculated, these subtraction terms per defi-

nition have the same kinematical structure as the real

emission matrix element which is not necessarily true for

an arbitrary point in phase space. So for each point the

integrand is either exactly the real emission matrix el-

ement or something with the same structure but with

one singularity subtracted. But as this is an integrand

where our integration routine is optimized for, choosing

a value for α smaller than one leads to an improvement

of the convergence of the integral. In our calculation we

set α = 0.01.

The second contribution to the integration combines

the tree-level contribution and the integrated subtrac-

tion terms. The virtual matrix element is integrated over

phase space by reweighting a sample unweighted Born

level events, as described in [15]. This leads to a con-

siderable reduction of the required CPU-time since less

phase space points have to be evaluated. For the results

shown below, event samples consisting of 104− 105un-

weighted events have been used. The LO event samples

used for the reweighting have been generated with Mad-

Event [24] and WHIZARD [25, 26].

In order to establish the correctness of the results ob-

tained we have performed a number of non-trivial tests.

The dipole contributions for single phase space points

and at the phase space integration level have been com-

pared with the HELAC code [27, 28] and agreement has

been established up to double precision accuracy for sin-

gle phase space points and within integration errors for

the integrated results. The phase space integration of the

dipole contributions is validated by checking the indepen-

dence of the result of the slicing parameter α. Also the

cancellation of the single and double poles between the

virtual amplitude and the integrated subtraction terms

has been verified. Finally, the virtual matrix element

computation for a single phase space point has been com-

pared to the result published in [29]. In order to perform

this comparison the contribution from top quark loops

has been added, even though it is neglected in the re-

sults presented in the following section. Our result is in

agreement with the result of [29], providing a very strong

test of our virtual contributions computations.

RESULTS

In the following we consider the process pp → b¯bb¯b +

X at the LHC at a center of mass energy of

14TeV.The final state jets are defined by apply-

ing the kT-algorithm as explained in [30] with a ra-

dius in R-space of 0.8.More precisely, the jet algo-

rithm requires exactly four b-jets in the final state for

the event to be accepted.

within a rapidity range of |η(bj)| < 2.5 and to have

a transverse momentum pT(bj) > 30GeV.

pose a separation cut between the jets of ∆R(bi,bj) =

?(φi− φj)2+ (ηi− ηj)2> 0.8. All results have been

tions [31] with two-loop running of αs both for the LO

and the NLO cross-section evaluations and αs(MZ) =

0.118.

The unphysical renormalization and factorization

scales are usually chosen to be in the vicinity of the typ-

ical scale of the process. For processes where heavy par-

ticles such as top-quarks or W/Z-bosons are involved the

masses of these particles provide a natural choice. In

our case, dealing with massless particles only there is no

such scale, the only scale involved in this process is the

pT-cut imposed to define the b-jets. In this respect the

process considered here is similar to the production of

four light jets. In [12] it has been shown that the aver-

age transverse momentum pTof a jet is a good choice for

the production of three jets in hadron-hadron collisions.

On the basis of the scale choice we made earlier for the

quark initiated case [8], we define the central scale to be

√s =

All jets are required to lie

We im-

obtained using the CTEQ6M parton distribution func-

µ0=1

4

??

i

p2

T,i,(1)

which turns out to be of the same order of magnitude as

the average pT of the jets.

In Figure 1 we plot the variation of the total cross sec-

tion for the production of two bottom-antibottom pairs

at the LHC, with the cuts described previously, when the

renormalization scale µrand the factorization µFare var-

ied together, with x defined as the ratio to the the central

scale, µr= µF= x · µ0.

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3

x

0.2 0.30.4 0.51234

[pb]

σ

0

50

100

150

200

250

300

350

LO

NLO

Graph

FIG. 1. Total cross section as a function of the scale µ =

µr = µF = x·µ0. Renormalization and factorization scale are

varied in the same direction.

If we set the renormalization and factorization scale

to the value µ0as in Eq. (1) we find for the total cross

section with the cuts described above

σNLO

pp→b¯bb¯b= 140.48 ± 0.64 pb .

This means that for our preferred choice of scales we

find that the inclusion of the NLO contribution leads to

an increase of nearly 50% of the total cross section with

respect to the LO result of σLO= 94.88 ± 0.14 pb.

However one observes that, as expected, the dependence

of the result on the unphysical scales is strongly reduced

in the NLO result with respect to the leading order one.

(2)

) [GeV]

2

,b

1

m(b

50 100 150 200250 300 350400 450

/dm [pb/GeV]

σ

d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

LO

NLO

FIG. 2. Invariant mass distribution of the two b-jets with

the highest pT. The black shaded area denotes the tree level

contribution, the red area denotes the NLO cross-section. The

error bands for both histograms are determined by a scale

variation between µ0/2 and 2µ0.

In Figure 2 we plot the invariant mass distribution of

the two b-jets with the highest transverse momentum.

The error bands are obtained by a variation of the scales

between µ0/2 and 2µ0. With respect to the LO result

one observes a shift of the distribution to lower energies

due to the inclusion of the radiative corrections. The pT

[GeV]

T,1

p

50 100150200 250

[pb/GeV]

T,1

/dp

σ

d

0

0.5

1

1.5

2

2.5

LO

NLO

FIG. 3. pT distribution of the hardest jet. The error bands

are defined as in Figure 2.

distribution of the jet with the highest pTis shown in Fig-

ure 3. Here, the radiative corrections enhance the distri-

bution at higher momentum. Both Figure 2 and Figure 3

show a significant reduction of the error induced by scale

uncertainties on differential distributions. Moreover, the

distortion in the shapes of differential distributions when

going from LO to NLO suggests that the application of

a global K-factor is not sufficient in order to accurately

describe the higher-order effects.

A complete phenomenological study of the production

of two bottom-antibottom pairs at the LHC, including

the study of PDF uncertainties and the effects of varying

the cuts on the final state b-jets is beyond the scope of

the present Letter and will be the subject of an upcoming

publication.

CONCLUSIONS

We have calculated the next-to-leading order QCD

corrections to the production of two bottom-antibottom

quark pairs at the LHC. This calculation has been imple-

mented in a highly automated framework for the compu-

tation of NLO QCD corrections (the golem-2.0 frame-

work) which is based on a Feynman diagrammatic ap-

proach for the evaluation of virtual corrections imple-

mented in the Samurai and OneLoop packages, inter-

faced to the Madgraph/Madevent and MadDipole pro-

grams for the evaluation of the leading-order and dipole

subtraction contributions and the phase space integra-

tion.

The inclusion of the NLO corrections leads to a sig-

nificant reduction of the uncertainties due to unphysical

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4

scale dependence of the LO result, while enhancing the

cross section by 50% for our central scale choice. Fur-

thermore, we have shown that the radiative corrections

lead to changes in the overall shape of the distributions,

which cannot be accounted for in a reliable way by a

simple rescaling of the leading order predictions.

This reduced theoretical uncertainty improves the

prospects for the use of the b¯bb¯b channel in searches of

Higgs bosons in various extensions of the the standard

model like SUSY, two Higgs doublet models or hidden

valley models.

Acknowledgments: The authors would like to thank

Gudrun Heinrich for useful discussion and for her support

in cross-checking parts of the amplitude. N.G. wants to

thank Francesco Tramontano for helpful advice. N.G and

A.G would like to thank Nikhef for kind hospitality. The

work of T.R. was supported by the Dutch Foundation for

Fundamental Research on Matter (FOM), project FORM

07PR2556 and by the Alexander von Humboldt Founda-

tion, in the framework of the Sofja Kovaleskaja Award

Project ”Advanced Mathematical Methods for Particle

Physics”, endowed by the German Federal Ministry of

Education and Research.N.G was supported by the

U. S. Department of Energy under contract No. DE-

FG02-91ER40677.

This paper is dedicated to the memory of our friend

Thomas Binoth who initiated this project and has always

been a driving force in our collaboration.

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