# The total top quark pair production cross-section at hadron colliders through

###### Abstract

We compute the next-to-next-to-leading order (NNLO) QCD correction to the total cross-section for the reaction . Together with the partonic channels we computed previously, the result derived in this letter completes the set of NNLO QCD corrections to the total top pair production cross-section at hadron colliders. Supplementing the fixed order results with soft-gluon resummation with next-to-next-to-leading logarithmic accuracy we estimate that the theoretical uncertainty of this observable due to unknown higher order corrections is about 3% at the LHC and 2.2% at the Tevatron. We observe a good agreement between the Standard Model predictions and the available experimental measurements. The very high theoretical precision of this observable allows a new level of scrutiny in parton distribution functions and new physics searches.

^{†}

^{†}preprint: CERN-PH-TH/2013-056, TTK-13-08

## I Introduction

Production of top quark pairs at hadron colliders is among the processes that are most challenging to theory. Bringing this process under good theoretical control therefore represents a significant step in our broader understanding of perturbative Quantum Chromodynamics (QCD) and its applications at hadron colliders.

The first step in this direction was made some 25 years ago, when the next-to-leading order (NLO) QCD corrections to production were computed in the groundbreaking works Nason:1987xz ; Beenakker:1988bq . The complexity of the NLO calculations required the application of purely numerical methods, and it took almost twenty years before the exact analytic result appeared Czakon:2008ii revealing the full complexity of the cross-section for massive fermion hadroproduction.

In the last few years we are witnessing a significant interest in computing next-to-next-to leading order (NNLO) corrections to hadron collider processes. Such a demand is dictated in part by the high-precision measurements available from the LHC and the Tevatron. The first hadron collider processes that were computed at NNLO, namely, Drell-Yan and vector boson Hamberg:1990np ; Anastasiou:2003yy ; Anastasiou:2003ds , Higgs Harlander:2002wh ; Anastasiou:2002yz ; Ravindran:2003um and diphoton Catani:2011qz production, all share the properties of (a) having massless QCD partons and (b) involving at leading order (LO) two partons meeting in a color singlet vertex. Tackling processes with higher complexity, among which production is a prominent example, proved to require new computational approaches.

About one year ago, the first step in this direction was made precisely in the context of production. Based on a new view Czakon:2010td about how to treat double-real radiation corrections, the first genuinely NNLO corrections to the total inclusive cross-section in were computed Baernreuther:2012ws . Later on, the partonic reactions involving at least one fermion in the initial state were also completed Czakon:2012zr ; Czakon:2012pz . In this work we report the calculation of the last missing NNLO correction to production, in the partonic reaction . With this calculation, the complete set of NNLO corrections to the total inclusive cross-section for top pair production at hadron colliders is now known. In this letter, for the first time, we quantify their phenomenological implications.

Before closing this section we would like to point out the very recent NNLO calculation of the process Boughezal:2013uia which was performed with methods similar to ours and, in particular, the subtraction scheme proposed by one of us Czakon:2010td . Moreover, a first partial result for dijet production at NNLO has just appeared Ridder:2013mf . We believe that this burst of precision applications at hadron colliders marks the outset of a new and lasting stage in precision physics at hadron colliders.

## Ii The production cross-section

In this letter we consider the total inclusive production cross-section

(1) |

The indices run over all possible initial state partons; ; is the c.m. energy of the hadron collider and , with , is the relative velocity of the final state top quarks with pole mass and partonic c.m. energy .

The function in Eq. (1) is the partonic flux

(2) |

expressed through the usual partonic luminosity

(3) |

As usual, are the renormalization and factorization scales. Setting , the NNLO partonic cross-section can be expanded through NNLO as

(4) |

In the above equation is the coupling renormalized with active flavors at scale and are functions only of . The procedure for restoring the dependence on is standard and has been detailed, for example, in Ref. Czakon:2012pz .

All partonic cross-sections are known exactly through NLO Nason:1987xz ; Beenakker:1988bq ; Czakon:2008ii . The NNLO corrections to the partonic reactions Baernreuther:2012ws ; Czakon:2012zr ; Czakon:2012pz . In the following we present the results for . were computed in Refs.

## Iii Parton level results for

Keeping the dependence on the number of light flavors explicit, the NNLO correction reads

(5) |

The functions read:

(6) | |||||

(7) | |||||

(8) | |||||

(9) | |||||

(10) | |||||

where and . The functions constitute the analytically known threshold approximation to Beneke:2009ye , including the exact Born term

(12) |

and with the constant (as defined in Ref. Beneke:2009ye ).

The functions are computed numerically, in 80 points on the interval . Details about the calculation are given in the next section.

Following the approach of Refs. Baernreuther:2012ws ; Czakon:2012zr ; Czakon:2012pz , the functions are derived as fits to the difference . The functions together with the discrete values for (including the numerical errors) are shown in fig. 1.

As can be seen from fig. 1 the functions vanish smoothly at threshold , which implies that our calculation agrees with the exactly known threshold behavior Beneke:2009ye . This is a strong check of our result.

To assess the size of the newly derived NNLO correction, in fig. 2 we compare: (a) the exact NNLO result, (b) the approximate NNLO result with exact Born term and (c) the approximate NNLO result with Born term restricted to its leading power of . Each of these three partonic cross-sections is multiplied by the partonic flux Eq. (2) for LHC 8 TeV. We observe that the power corrections derived in the present work are very large. In fact their contribution to the integrated cross-section is virtually as large as the one due to pure soft gluon corrections.

The partonic cross-section’s leading power behavior in the high-energy limit reads Nason:1987xz ; Catani:1990xk ; Collins:1991ty ; Catani:1990eg ; Catani:1993ww ; Catani:1994sq

(13) |

The constant is known exactly Ball:2001pq . To improve the accuracy of the partonic result (5) in the high-energy limit, we have imposed on it the logarithmic behavior implied by Eq. (13). Numerical prediction for the constant term was given in Ref. Moch:2012mk . Our fits return the value which falls within the range estimated in Ref. Moch:2012mk .

The parton level results derived in this section can be used to derive an estimate for the so-far unknown constant appearing in the threshold approximation Beneke:2009ye . Expanding Eq. 5 around the limit we obtain

(14) |

As explained in Ref. Hagiwara:2008df , the estimate (14) for has to be used with caution and a sizable uncertainty should be assumed. We have no good way of estimating the error on the extracted constant and to be reasonably conservative in the following we take this error to be .

The constant is related Cacciari:2011hy to the hard matching coefficients needed for NNLL soft gluon resummation matched to NNLO. However, since our calculation deals with the color averaged cross-section, we cannot extract both constants . We proceed as follows.

Close to threshold, the color singlet and color octet contributions to have independent constant terms , with the constant in Eq. (14) being their color average. We parameterize the second, unknown, combination of by their ratio , which has the advantage of being normalization independent. For any guessed value of , together with Eq. (14), we can extract values for the hard matching constants . As a guide for a reasonable value of we take the one-loop result (see Beneke:2009ye ; Hagiwara:2008df ): .

In the following we vary in the range ; for each value of we then vary the color averaged constant by additional . We observe that as a result of this rather conservative variation, the NNLO+NNLL theoretical prediction for LHC 8 TeV changes by 0.4% (in central value) and by 0.2% (in scale dependence). Given the negligible phenomenological impact of these variations, we choose as our default values:

(15) |

derived from Eq. (14) and the mid-range value .

## Iv Calculation of through

The calculation of the corrections to is performed in complete analogy to the calculations of the remaining partonic reactions Baernreuther:2012ws ; Czakon:2012zr ; Czakon:2012pz . The two-loop virtual corrections are computed in gg-two-loop , utilizing the analytical form for the poles Ferroglia:2009ii . We have computed the one-loop squared amplitude; it has previously been computed in Anastasiou:2008vd . The real-virtual corrections are derived by integrating the one-loop amplitude with a counter-term that regulates it in all singular limits Bern:1999ry . The finite part of the one-loop amplitude is computed with a code used in the calculation of at NLO Dittmaier:2007wz . The double real corrections are computed in Czakon:2010td . Factorization of initial state collinear singularities as well as scale dependence is computed in a standard way; see Refs. Czakon:2012zr ; Czakon:2012pz .

## V Phenomenological applications

In table 1 we present our most precise predictions for the Tevatron and LHC at 7, 8 and 14 TeV. All numbers are computed for GeV and MSTW2008nnlo68cl pdf set Martin:2009iq with the program Top++ (v2.0) Czakon:2011xx . Scale uncertainty is determined through independent restricted variation of and .

Collider | [pb] | scales [pb] | pdf [pb] |
---|---|---|---|

Tevatron | |||

LHC 7 TeV | |||

LHC 8 TeV | |||

LHC 14 TeV |

Our best predictions are at NNLO and include soft gluon resummation at NNLL Beneke:2009rj ; Cacciari:2011hy .

In this letter we take as a default value for the constant introduced in Ref. Bonciani:1998vc . The reason for switching to a new default value for (compared to in Cacciari:2011hy ; Baernreuther:2012ws ; Czakon:2012zr ; Czakon:2012pz ) is that this constant is consistently defined only through NLO. Nonetheless it contributes at NNLO too, and a consistent NNLO treatment would require the analysis of Ref. Bonciani:1998vc to be extended to NNLO, which is now possible with the help of the results derived in this letter as well as Ref. Baernreuther:2012ws . Given the numerical effect is small (a shift at LHC 8 TeV and a shift at the Tevatron), in this work we take .

As can be concluded from table 1 the precision of the theoretical prediction at full NNLO+NNLL is very high. At the Tevatron, the scale uncertainty is as low as 2.2% and just slightly larger, about 3%, at the LHC. The inclusion of the NNLO correction to the -initiated reaction increases the Tevatron prediction of Ref. Baernreuther:2012ws by about 1.4%, which agrees well with what was anticipated in that reference.

Collider | [pb] | scales [pb] | pdf [pb] |
---|---|---|---|

Tevatron | |||

LHC 7 TeV | |||

LHC 8 TeV | |||

LHC 14 TeV |

To assess the numerical impact from soft gluon resummation, in table 2 we present results analogous to the ones in table 1 but without soft gluon resummation, i.e. at pure NNLO. Comparing the results in the two tables we conclude that the effect of the resummation is a increase in central values and decrease in scale dependence for, respectively, (Tevatron, LHC7, LHC8, LHC14).

Next we compare our predictions with the most precise experimental data available from the Tevatron and LHC.

The comparison with the latest Tevatron combination tev-sigma_exp is shown in fig. 3. The measured value pb is given, without conversion, at the best top mass measurement Aaltonen:2012ra GeV. From this comparison we conclude that theory and experiment are in good agreement at this very high level of precision.

In fig. 4 we show the theoretical prediction for the total cross-section at the LHC as a function of the c.m. energy. We compare with the most precise available data from ATLAS at 7 TeV Atlas7TeV , CMS at 7 Chatrchyan:2012bra and 8 TeV CMS8TeV as well as the ATLAS and CMS combination at 7 TeV Atlas_and_CMS-7TeV . We observe a good agreement between theory and data. Where conversion is provided Chatrchyan:2012bra , the measurements have been converted to GeV.

Finally, we make available simplified fits for the top mass dependence of the NNLO+NNLL cross-section, including its scale and pdf uncertainties:

The coefficient can be found in table 3.

GeV | [pb] | |||
---|---|---|---|---|

Central | 7.1642 | 1.46191 | 0.945791 | |

Scales | 7.27388 | 1.46574 | 0.957037 | |

Tevatron | Scales | 6.96423 | 1.4528 | 0.921248 |

PDFs | 7.33358 | 1.4439 | 0.930127 | |

PDFs | 7.04268 | 1.4702 | 0.936027 | |

Central | 172.025 | 1.24243 | 0.890776 | |

Scales | 176.474 | 1.24799 | 0.903768 | |

LHC 7 TeV | Scales | 166.193 | 1.22516 | 0.858273 |

PDFs | 176.732 | 1.22501 | 0.861216 | |

PDFs | 167.227 | 1.2586 | 0.918304 | |

Central | 245.794 | 1.1125 | 0.70778 | |

Scales | 252.034 | 1.11826 | 0.719951 | |

LHC 8 TeV | Scales | 237.375 | 1.09562 | 0.677798 |

PDFs | 251.968 | 1.09584 | 0.682769 | |

PDFs | 239.441 | 1.12779 | 0.731019 |

## Vi Conclusions and Outlook

In this work we compute the NNLO corrections to . With this last missing reaction included, the total inclusive top pair production cross-section at hadron colliders is now known exactly through NNLO in QCD. We also derive estimates for the two-loop hard matching coefficients which allows NNLL soft-gluon resummation matched consistently to NNLO. All results are implemented in the program Top++ (v2.0) Czakon:2011xx .

The theoretical precision achieved in this observable is very high. To illustrate this, we compare to a NLO level calculation. At LHC 8 TeV we observe a decrease in scale dependence by a factor of when compared to, respectively, (NLO, NLO+LL, NLO+NLL). The corresponding numbers for the Tevatron are (3.9, 4.1, 2.0).

The predicted cross-section agrees well with all available measurements. We are confident that its very high precision will enable a new generation of precision collider applications to, among others, parton distributions and searches for new physics.

###### Acknowledgements.

We thank S. Dittmaier for kindly providing us with his code for the evaluation of the one-loop virtual corrections in Dittmaier:2007wz . The work of M.C. and P.F. was supported by the DFG Sonderforschungsbereich/Transregio 9 Computergestützte Theoretische Teilchenphysik . M.C. was also supported by the Heisenberg and by the Gottfried Wilhelm Leibniz programmes of the Deutsche Forschungsgemeinschaft. The work of A.M. is supported by ERC grant 291377 “LHCtheory: Theoretical predictions and analyses of LHC physics: advancing the precision frontier”.## References

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