Quantifying quark mass effects at the LHC: A study of pp -> b anti-b b anti-b + X at next-to-leading order
The production of four bottom quarks is an important benchmark channel for Higgs analyses and searches for new physics at the LHC. We report on the calculation of the next-to-leading order QCD corrections to the process with the Helac-NLO automated framework, and present results for inclusive cross sections and differential distributions. We discuss the impact of the higher-order corrections and, in particular, the effect of the bottom quark mass. In addition, we provide an estimate of the theoretical uncertainty from the variation of the renormalisation and factorisation scales and the parton distribution functions. The results are obtained with a new subtraction formalism for real radiation at next-to-leading order, implemented in the Helac-Dipoles package.
]G. Bevilacqua, M. Czakon, M. Krämer, M. Kubocz and M. Worek \affiliationInstitute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, D-52056 Aachen, Germany \emailAddbevilacqua@physik.rwth-aachen.de \emailAddmczakon@physik.rwth-aachen.de \emailAddmkraemer@physik.rwth-aachen.de \emailAddkubocz@physik.rwth-aachen.de \emailAddworek@physik.rwth-aachen.de \dedicatedTTK-13-11 \keywordsNLO Computations, Standard Model, QCD Phenomenology, Hadronic Colliders, Monte Carlo Simulations
The production of four bottom quarks, , is an important background to various Higgs analyses and new physics searches at the LHC, including for example Higgs-boson pair production in two-Higgs doublet models at large , or so-called hidden valley scenarios where additional gauge bosons can decay into bottom quarks . Accurate theoretical predictions for the Standard Model production of multiple bottom quarks are thus mandatory to exploit the potential of the LHC for new physics searches. Furthermore, the calculation of the next-to-leading order (NLO) QCD corrections to provides a substantial technical challenge and requires the development of efficient techniques, with a high degree of automation. We have performed a NLO calculation of production at the LHC with the Helac-NLO system . In particular, we present results based on a new subtraction formalism [4, 5] for treating real radiation corrections, as implemented in the Helac-Dipoles package . Two calculation schemes have been employed, the so-called four-flavour scheme (4FS) with only gluons and light-flavour quarks in the proton, where massive bottom quarks are produced from gluon splitting at short distances, and the five-flavour-scheme (5FS)  with massless bottom quarks as partons in the proton. At all orders in perturbation theory, the four- and five-flavour schemes are identical, but the way of ordering the perturbative expansion is different, and at any finite order the results do not match. Comparing the predictions of the two schemes at NLO thus provides a way to assess the theoretical uncertainty from unknown higher-order corrections, and to study the effect of the bottom mass on the inclusive cross section and on differential distributions. First NLO results for in the 5FS have been presented in Ref. . We not only provide an independent calculation of this challenging process with a different set of methods and tools, but also a systematic study of the bottom quark mass effects by comparing the 5FS and 4FS results. We note that NLO results for the production of four top quarks in hadron collisions have been discussed in Ref. . In addition, NLO calculations for processes of similar complexity have recently been presented in the literature, including NLO QCD corrections to production in association with two jets , the production of a single gauge boson plus jets , double gauge boson production with two jets  and multi-jet production .
2 Theoretical Framework
The calculation of the process at NLO QCD comprises the parton processes and at tree-level and including one-loop corrections, as well as the tree-level parton processes , , and . In the four-flavour scheme , and the bottom quark is treated massive. The bottom mass effects are in general suppressed by powers of , where is the hard scale of the process, e.g. the transverse momentum of a bottom-jet. Potentially large logarithmic corrections could arise from nearly collinear splitting of initial-state gluons into bottom quarks, , where the bottom mass acts as a regulator of the collinear singularity. This class of -terms can be summed to all orders in perturbation theory by introducing bottom parton densities in the five-flavour scheme. The 5FS is based on the approximation that the bottom quarks from the gluon splitting are produced at small transverse momentum. However, in our calculation we require that all four bottom quarks can be experimentally detected, and we thus impose a lower cut on the bottom transverse momentum, . As a result, up to NLO accuracy the potentially large logarithms in the process are replaced by , with , and are thus much less significant numerically. Therefore, for the process at hand, the differences between the 4FS and 5FS calculations with massive and massless bottom quarks, respectively, should be moderate, but may not be completely negligible.
Our calculation is performed with the automated Helac-NLO framework , which includes Helac-1loop  for the evaluation of the numerators of the loop integrals and the rational terms, CutTools , which implements the OPP reduction method [16, 17, 18, 19] to compute one-loop amplitudes, and OneLoop  for the evaluation of the scalar integrals. The singularities for soft and collinear parton emission are treated using subtraction schemes as implemented in Helac-Dipoles , see the discussion below. The phase space integration is performed with the help of the Monte Carlo generators Helac-Phegas [21, 22, 23] and Kaleu , including Parni  for the importance sampling.
The Helac-Dipoles package has been based on the standard Catani-Seymour dipole subtraction formalism [26, 27]. We have now extended Helac-Dipoles by implementing a new subtraction scheme [4, 5] using the momentum mapping and the splitting functions derived in the context of an improved parton shower formulation by Nagy and Soper . Compared to standard dipole subtraction, the new scheme features a significantly smaller number of subtraction terms and facilitates the matching of NLO calculations with parton showers including quantum interference. The results presented here constitute the first application of the Nagy-Soper subtraction scheme for a scattering process with massive and massless fermions. A detailed description of the implementation of the new scheme, and a comparative study of the numerical efficiency and the speed will be presented elsewhere.
3 Numerical Results for the LHC
In this section we present cross-section predictions for the process at the LHC at the centre-of-mass energy of TeV. We discuss the impact of the NLO-QCD corrections, and study the dependence of the results on the bottom quark mass. We show results obtained with the standard dipole subtraction scheme for real radiation, and with the new subtraction formalism [4, 5] as implemented in Helac-Dipoles.
Let us first specify the input parameters and the acceptance cuts we impose. The top quark mass, which appears in the loop corrections, is set to GeV . We combine collinear final-state partons with pseudo-rapidity into jets according to the anti- algorithm  with separation . The bottom-jets have to pass the transverse momentum and rapidity cuts GeV and , respectively. The renormalisation and factorisation scales are set to the scalar sum of the bottom-jet transverse masses, , with and the transverse mass . For the five-flavour scheme calculation with massless bottom quarks the transverse mass equals the transverse momentum, . Note that the implementation of a dynamical scale requires a certain amount of care, as the subtraction terms for real radiation have to be evaluated with a different kinematical configuration specified by the momentum mapping of the subtraction scheme. Comparing our results as obtained with the Catani-Seymour subtraction and the Nagy-Soper scheme, which is based on a different momentum mapping, provides an important and highly non-trivial internal check of our calculation.
3.1 Massless bottom quarks within the five-flavour scheme
Results are presented for the NLO CT10  and MSTW2008  parton distribution functions (pdfs) with five active flavours and the corresponding two-loop . To study the impact of the higher-order corrections, we also show leading-order results obtained using the CT09MC1  and MSTW2008 LO pdf sets and one-loop running for .
We first discuss the impact of the bottom-quark induced processes, , on the hadronic cross section at leading-order. The difference between the initiated processes, , with and without bottom-quarks is at the level of . Moreover, at the central scale, , the hadronic cross section is completely dominated by gluon-fusion, with only about 1% contribution of all quark-antiquark annihilation processes. The bottom-induced contributions are thus negligible, and we decided to neglect bottom initial states in the computation of the cross section both at LO and NLO. Note that the suppression of the bottom-induced processes, which include for example potentially large forward scattering of bottom-quarks through -channel gluon exchange, depends crucially on the transverse momentum cuts we impose on the bottom-jets.
Let us first present our results for the inclusive cross section at the LHC ( = 14 TeV), including the transverse momentum and rapidity cuts specified at the beginning of the section. The NLO QCD corrections strongly reduce the scale dependence, as demonstrated in Figure 1. The central cross section predictions are collected in Table 1. Varying the renormalisation and factorisation scales simultaneously about the central scale by a factor of two, we find a residual scale uncertainty of approximately 30% at NLO, a reduction by about a factor of two compared to LO. The size of the -factor, , strongly depends on the pdf set, with for CT10 and for MSTW2008. We emphasise, however, that the -factor is an unphysical quantity and strongly sensitive to the choice of scale through the large LO scale dependence.
We observe a difference of about and between the CT10 and MSTW2008 pdf parametrisations at LO and NLO, respectively. The pdf uncertainty as estimated from the MSTW2008 error pdf sets  amounts to and at 68% C.L., and is significantly smaller than the scale uncertainty. A more systematic discussion of pdf and uncertainties will thus be referred to a forthcoming publication.
An important input for the experimental analyses and the interpretation of the experimental data are accurate predictions of differential distributions. Our calculation is set up as a parton-level Monte Carlo program and thus allows us to predict any infrared-safe observable at NLO. Figure 2 shows LO and NLO predictions for the invariant mass of the system (upper left panel), the total transverse energy (upper right panel), the transverse momentum of the hardest bottom jet (lower left panel) and the transverse momentum of the second hardest bottom jet (lower right panel). We also show the theoretical uncertainty through scale variation and the -factor as a function of the kinematic variable. It is evident from Figure 2 that the NLO corrections significantly reduce the theoretical uncertainty of the differential distributions, and that the size of the higher-order effects depends on the kinematics. For an accurate description of exclusive observables and differential distributions it is thus not sufficient to rescale a LO prediction with an inclusive -factor.
As discussed in section 2 we have performed the calculation with two different subtraction schemes, the standard Catani-Seymour (CS) dipole subtraction, and a new scheme based on the splitting functions and momentum mapping of an improved parton shower by Nagy and Soper (NS). The comparison between the two schemes for the inclusive 5FS cross section is presented in Table 2. For the Catani-Seymour scheme we show results without () and with () a restriction on the phase space of the subtraction as proposed in Ref. [34, 35]. As evident from Table 2, the cross sections obtained using the Catani-Seymour (CS) and Nagy-Soper (NS) subtraction schemes agree within the numerical uncertainty of the Monte Carlo integration. This result not only provides a validation of our implementation of the new subtraction scheme into Helac-Dipoles, but also provides a non-trivial internal cross check of the calculation.
We have also compared the results obtained in the CS and NS subtraction schemes for various differential distributions. Some examples are collected in Figure 3. We observe full agreement between the predictions calculated with the two schemes.
3.2 Massive bottom quarks within the four-flavour scheme
Within the four-flavour scheme bottom quarks are treated massive and are not included in the parton distribution functions of the proton. We define the bottom quark mass in the on-shell scheme and use GeV, consistent with the choice made in the MSTW2008 four-flavour pdf. The central cross section prediction in LO and NLO for using the 4FS MSTW2008  pdf are shown in Table 3. Comparing with the 5FS results presented in Table 1, we observe that the bottom mass effects decrease the cross section prediction by at LO and at NLO. The residual scale dependence at NLO is approximately 30%, similar to the 5FS calculation.
The difference between the massless 5FS and the massive 4FS calculations has two origins. First, there are genuine bottom mass effects, the size of which depends sensitively on the transverse momentum cut. For GeV we find a 10% difference between the 5FS and 4FS from non-singular bottom-mass dependent terms. This difference decreases to about for GeV. Second, the two calculations involve different pdf sets and different corresponding . While a 4FS pdf has, in general, a larger gluon flux than a 5FS pdf, as there is no splitting, the corresponding four-flavour is smaller than for five active flavours. For the difference in is prevailing and results in a further reduction of the 4FS cross section prediction by about . This latter difference should be viewed as a scheme dependence rather than a bottom mass effect.
In Figure 4 we present the differential distribution in the transverse momentum of the hardest bottom jet, as calculated in the 5FS with massless bottom quarks and in the 4FS with GeV. We show the absolute prediction at LO and NLO, and the predictions normalised to the corresponding inclusive cross section. The latter plots reveal that the difference in the shape of the distributions in the 5FS and the 4FS is very small. We find similar results for other differential distributions.
Let us finally present the comparison of the massive bottom quark results as obtained with the Catani-Seymour and Nagy-Soper subtraction schemes, see Table 4. We observe full agreement between the two calculations within the numerical error of the Monte Carlo integration, and thereby validate our implementation of the NS subtraction scheme also for the case of massive fermions.
3.3 Comparison with results presented in the literature
A detailed comparison of our results with Ref.  has been performed. We find agreement for the virtual amplitude at one specific phase space point, but cannot reproduce the published results for the integrated hadronic LO and NLO cross sections with the setup as described in Ref. . When the CTEQ6.5 pdf set  is used rather than CTEQ6M  as specified in , and the factorization and renormalization scales are set to the common value
|94.88 0.14||94.74 0.20|
After comprehensive numerical checks and the communication with the authors of Ref.  it turned out that the NLO numbers published in  are based on a scale setting that mixes partons and jets and that is not consistent with what is specified in Ref. . Adopting , with defined in Eq. (1), and summing over the transverse momenta of all jets, we obtain the NLO numbers presented in Table 6. Note that at NLO the final state can consist of four or five jets, as determined by the jet algorithm. Our results are compared to the NLO number as published in Ref. , , and to a corrected number obtained by means of private communication from the authors of Ref. , . The corrected result agrees with our calculation.
|140.48 0.64||143.70 0.44||144.35 0.53||144.73 0.62|
Also note that in Ref.  an NLO pdf set has been used both for the LO and NLO result. Although this may be justified to study the impact of higher-order corrections to the partonic cross section, it can be misleading when establishing the genuine effect of higher-order corrections, the adequacy of the scale choice or the size/shape of integrated/differential -factors used in experimental analyses when comparing Monte Carlo simulations to the LHC data.
The production of four bottoms quarks, , provides an important background for new physics searches at the LHC. We have performed a calculation of the NLO QCD corrections to this process with the Helac-NLO framework, employing a new subtraction scheme for treating real radiation corrections at NLO implemented in Helac-Dipoles. Results have been presented for inclusive and differential cross-sections for at the LHC at the centre-of-mass energy of TeV. We find that the higher-order corrections significantly reduce the scale dependence, with a residual theoretical uncertainty of about 30% at NLO. The impact of the bottom quark mass is moderate for the cross section normalisation and negligible for the shape of distributions. The fully differential NLO cross section calculation for the process presented in this paper provides an important input for the experimental analyses and the interpretation of new physics searches at the LHC.
The calculations have been performed on the Grid Cluster of the Bergische Universität Wuppertal, financed by the Helmholtz-Alliance “Physics at the Terascale” and the BMBF. We acknowledge the correspondence with Nicolas Greiner on the results of Ref.  and discussions with Zoltan Nagy. This research was supported in part by the German Research Foundation (DFG) via the Sonderforschungsbereich/Transregio SFB/TR-9 “Computational Particle Physics”. The work of M.C. was supported by the DFG Heisenberg program. M.W. acknowledges support by the DFG under Grant No. WO 1900/1-1 (“Signals and Backgrounds Beyond Leading Order. Phenomenological studies for the LHC”).
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