Vector boson production at hadron colliders: a fully exclusive QCD calculation at NNLO

Stefano Catani, Leandro Cieri, Giancarlo Ferrera,

Daniel de Florian and Massimiliano Grazzini

INFN, Sezione di Firenze and Dipartimento di Fisica, Università di Firenze,

I-50019 Sesto Fiorentino, Florence, Italy

Departamento de Física, FCEYN, Universidad de Buenos Aires,

(1428) Pabellón 1 Ciudad Universitaria, Capital Federal, Argentina

Abstract

We consider QCD radiative corrections to the production of and bosons in hadron collisions. We present a fully exclusive calculation up to next-to-next-to-leading order (NNLO) in QCD perturbation theory. To perform this NNLO computation, we use a recently proposed version of the subtraction formalism. The calculation includes the – interference, finite-width effects, the leptonic decay of the vector bosons and the corresponding spin correlations. Our calculation is implemented in a parton level Monte Carlo program. The program allows the user to apply arbitrary kinematical cuts on the final-state leptons and the associated jet activity, and to compute the corresponding distributions in the form of bin histograms. We show selected numerical results at the Tevatron and the LHC.

March 2009

The production of and bosons in hadron collisions through the Drell–Yan (DY) mechanism [1] is extremely important for physics studies at hadron colliders. These processes have large production rates and offer clean experimental signatures, given the presence of at least one high- lepton in the final state. Studies of the production of bosons at the Tevatron lead to precise determinations of the mass and width [2]. The DY process is also expected to provide standard candles for detector calibration during the first stage of the LHC running.

Because of the above reasons, it is essential to have accurate theoretical predictions for the vector-boson production cross sections and the associated distributions. Theoretical predictions with high precisions demand detailed computations of radiative corrections. The QCD corrections to the total cross section [3] and to the rapidity distribution [4] of the vector boson are known up to the next-to-next-to-leading order (NNLO) in the strong coupling . The fully exclusive NNLO calculation, including the leptonic decay of the vector boson, has been completed more recently [5]. Full electroweak corrections at have been computed for both [6] and production [7].

In this Letter we present a new computation of the NNLO QCD corrections to vector boson production in hadron collisions. The calculation includes the – interference, finite-width effects, the leptonic decay of the vector bosons and the corresponding spin correlations. Our calculation parallels the one recently completed for Higgs boson production [8, 9], and it is performed by using the same method.

The evaluation of higher-order QCD corrections to hard-scattering processes is complicated by the presence of infrared (IR) singularities at intermediate stages of the calculation that prevents a straightforward implementation of numerical techniques. Despite this difficulty, general methods have been developed in the last two decades, which allow us to handle and cancel IR singularities [10, 11, 12] appearing in NLO QCD calculations. In the last few years, several research groups have been working on extensions of these methods to NNLO [13, 14, 15, 16, 17], and, recently, the NNLO calculation for was completed by two groups [18, 19]. Parallely, a new general method [20], based on sector decomposition [21], has been proposed and applied to the NNLO calculations of [22], Higgs [23] and vector [5] boson production in hadron collisions, and to some decay processes [24]. Our method [8] applies to the production of colourless high-mass systems in hadron collisions, and is based on an extension of the subtraction formalism [11, 12] to NNLO that we briefly recall below.

We consider the inclusive hard-scattering reaction

(1) |

where the collision of the two hadrons and produces the vector boson ( or ), with four-momentum and high invariant mass . At next-to-leading order (NLO), two kinds of corrections contribute: i) real corrections, where one parton recoils against ; ii) one-loop virtual corrections to the leading order (LO) subprocess. Both contributions are separately IR divergent, but the divergences cancel in the sum. At NNLO, three kinds of corrections must be considered: i) double real contributions, where two partons recoil against ; ii) real-virtual corrections, where one parton recoils against at one-loop order; iii) two-loop virtual corrections to the LO subprocess. The three contributions are still separately divergent, and the calculation has to be organized so as to explicitly achieve the cancellation of the IR divergences.

We first note that, at LO, the transverse momentum of is exactly zero. As a consequence, as long as , the (N)NLO contributions are actually given by the (N)LO contributions to . Thus, we can write the cross section as

(2) |

This means that, when , the IR divergences in our NNLO calculation are those in : they can be treated by using available NLO methods to handle and cancel IR singularities (e.g., the general NLO methods in Refs. [10, 11, 12]). The only remaining singularities of NNLO type are associated to the limit . Following Ref. [8] we treat them by an additional subtraction. Our key point is that the singular behaviour of when is well known: it comes out in the resummation program [25] of logarithmically-enhanced contributions to transverse-momentum distributions. Therefore, the additional subtraction can be worked out by using a counterterm, , whose general structure [8] depends only on the flavour of the initial-state partons involved in the LO partonic subprocess ( annihilation in the case of vector-boson production, fusion in the case of Higgs boson production).

Our extension of Eq. (2) to include the contribution at is [8]:

(3) |

Comparing with the right-hand side of Eq. (2), we have subtracted the (N)LO counterterm and added a contribution at , which is needed to obtain the correct total cross section. The coefficient does not depend on and is obtained by the (N)NLO truncation of the hard-scattering perturbative function

(4) |

According to Eq. (3), the NLO calculation of requires the knowledge of and the LO calculation of . The general (process-independent) form of the coefficient is known: the precise relation between and the IR finite part of the one-loop correction to a generic LO subprocess is explicitly derived in Ref. [26]. At NNLO, the coefficient is also needed, together with the NLO calculation of . The calculation of the general structure of the coefficients is in progress. Meanwhile, by using the available analytical results at for the total cross section [3] and the transverse-momentum spectrum [27] of the vector boson, we have explicitly computed the coefficient of the DY process. Since the NLO corrections to are also available [28], using Eq. (3) we are able to complete our fully-exclusive NNLO calculation of vector-boson production.

We have encoded our NNLO computation in a parton level Monte Carlo program, in which we can implement arbitrary IR safe cuts on the final-state leptons and the associated jet activity.

In the following we present an illustrative selection of numerical results for and production at the Tevatron and the LHC. We consider quarks in the initial state. In the case of production, we use the (unitarity constrained) CKM matrix elements , , , , , from the PDG 2008 [29]. In the case of production, additional Feynman diagrams with fermionic triangles should be taken into account. Their contribution cancels out for each isospin multiplet when massless quarks are considered. The effect of a finite top-quark mass in the third generation has been considered and found extremely small [30], so it is neglected in our calculation. As for the electroweak couplings, we use the so called scheme, where the input parameters are , , . In particular we use the values GeV, GeV, GeV, GeV and GeV. We use the MSTW2008 [31] sets of parton distributions, with densities and evaluated at each corresponding order (i.e., we use -loop at NLO, with ). The renormalization and factorization scales are fixed to the value , where is the mass of the vector boson.

We start the presentation of our results by considering the inclusive production of pairs from the decay of an on-shell boson at the LHC. In Fig. 1 (left panel) we show the rapidity distribution of the pair at LO, NLO and NNLO, computed by using the MSTW2008 partons.

The corresponding cross sections^{†}^{†}†Throughout the paper, the errors
on the values of the cross sections and the error bars in the plots refer to an
estimate of the numerical errors in the Monte Carlo integration.
are nb,
nb and nb.
The total cross section is increased by about %
in going from NLO to NNLO. In Fig. 1 (right panel)
we also show
the results obtained
by using the
MRST2002 LO [32]
and MRST2004 [33]
sets of parton distribution functions. The corresponding cross sections are
nb, nb and
nb. In this case the total cross section
is decreased by about %
in going from NLO to NNLO.

We next consider the production of pairs from bosons at the Tevatron. For each event, we classify the lepton transverse momenta according to their minimum and maximum values, and . The leptons are required to have a minimum of 20 GeV and pseudorapidity . Their invariant mass is required to be in the range 70 GeV GeV. The accepted cross sections are pb, pb and pb. In Fig. 2 we plot the distributions in and at LO, NLO and NNLO. We note that at LO the and distributions are kinematically bounded by , where GeV is the maximum allowed invariant mass of the pairs. The NNLO corrections have a visible impact on the shape of the and distribution and make the distribution softer, and the distribution harder.

We finally consider the production of a charged lepton plus missing through the decay of a boson () at the Tevatron. The charged lepton is selected to have GeV and and the missing of the event should be larger than 25 GeV. We define the transverse mass of the event as , where is the angle between the the of the lepton and the missing . The accepted cross sections are nb, nb and nb.

In Fig. 3 we show the distribution at LO, NLO and NNLO. We note that at LO the distribution has a kinematical boundary at GeV. This is due to the fact that at LO the is produced with zero transverse momentum: therefore, the requirement GeV sets GeV. Around the region where GeV there are perturbative instabilities in going from LO to NLO and to NNLO. The origin of these perturbative instabilities is well known [34]: since the LO spectrum is kinematically bounded by GeV, each higher-order perturbative contribution produces (integrable) logarithmic singularities in the vicinity of the boundary. We also note that, below the boundary, the NNLO corrections to the NLO result are large; for example, they are about % at GeV. This is not unexpected, since in this region of transverse masses, the result corresponds to the calculation at the first perturbative order and, therefore, our result is actually only a calculation at the NLO level of perturbative accuracy.

We have illustrated a calculation of the cross section for and boson production up to NNLO in QCD perturbation theory. An analogous computation was presented in Ref. [5]. Our calculation is performed with a completely independent method. In the quantitative studies that we have carried out, the two computations give results in numerical agreement. Our calculation is directly implemented in a parton level event generator. This feature makes it particularly suitable for practical applications to the computation of distributions in the form of bin histograms. A public version of our program will be available in the near future.

Acknowledgements. We thank Stefan Dittmaier for useful correspondence.

## References

- [1] S. D. Drell and T. M. Yan, Phys. Rev. Lett. 25 (1970) 316 [Erratum-ibid. 25 (1970) 902].
- [2] See e.g. The Tevatron Electroweak Working Group for the CDF and D0 Collaborations, Combination of CDF and D0 results on the boson mass and width, preprint FERMILAB-TM-2415, arXiv:0808.0147 [hep-ex].
- [3] R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B 359 (1991) 343 [Erratum-ibid. B 644 (2002) 403]; R. V. Harlander and W. B. Kilgore, Phys. Rev. Lett. 88 (2002) 201801.
- [4] C. Anastasiou, L. J. Dixon, K. Melnikov and F. Petriello, Phys. Rev. D 69 (2004) 094008.
- [5] K. Melnikov and F. Petriello, Phys. Rev. Lett. 96 (2006) 231803, Phys. Rev. D 74 (2006) 114017.
- [6] S. Dittmaier and M. Kramer, Phys. Rev. D 65 (2002) 073007; U. Baur and D. Wackeroth, Phys. Rev. D 70 (2004) 073015; V. A. Zykunov, Phys. Atom. Nucl. 69 (2006) 1522 [Yad. Fiz. 69 (2006) 1557]; A. Arbuzov et al., Eur. Phys. J. C 46 (2006) 407 [Erratum-ibid. C 50 (2007) 505]; C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini, JHEP 0612 (2006) 016.
- [7] U. Baur, O. Brein, W. Hollik, C. Schappacher and D. Wackeroth, Phys. Rev. D 65 (2002) 033007; V. A. Zykunov, Phys. Rev. D 75 (2007) 073019; C. M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini, JHEP 0710 (2007) 109; A. Arbuzov et al., Eur. Phys. J. C 54 (2008) 451.
- [8] S. Catani and M. Grazzini, Phys. Rev. Lett. 98 (2007) 222002.
- [9] M. Grazzini, JHEP 0802 (2008) 043.
- [10] W. T. Giele and E. W. N. Glover, Phys. Rev. D 46 (1992) 1980; W. T. Giele, E. W. N. Glover and D. A. Kosower, Nucl. Phys. B 403 (1993) 633.
- [11] S. Frixione, Z. Kunszt and A. Signer, Nucl. Phys. B 467 (1996) 399; S. Frixione, Nucl. Phys. B 507 (1997) 295.
- [12] S. Catani and M. H. Seymour, Nucl. Phys. B 485 (1997) 291 [Erratum-ibid. B 510 (1997) 503].
- [13] D. A. Kosower, Phys. Rev. D 57 (1998) 5410, Phys. Rev. D 67 (2003) 116003, Phys. Rev. D 71 (2005) 045016.
- [14] S. Weinzierl, JHEP 0303 (2003) 062, JHEP 0307 (2003) 052, Phys. Rev. D 74 (2006) 014020.
- [15] A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, Nucl. Phys. B 691 (2004) 195, Phys. Lett. B 612 (2005) 36, Phys. Lett. B 612 (2005) 49, JHEP 0509 (2005) 056; A. Daleo, T. Gehrmann and D. Maitre, JHEP 0704 (2007) 016.
- [16] S. Frixione and M. Grazzini, JHEP 0506 (2005) 010.
- [17] G. Somogyi, Z. Trocsanyi and V. Del Duca, JHEP 0506 (2005) 024, JHEP 0701 (2007) 070; G. Somogyi and Z. Trocsanyi, JHEP 0701 (2007) 052, JHEP 0808 (2008) 042; U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi and Z. Trocsanyi, JHEP 0809 (2008) 107.
- [18] A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover and G. Heinrich, Phys. Rev. Lett. 99 (2007) 132002, JHEP 0711 (2007) 058, JHEP 0712 (2007) 094, Phys. Rev. Lett. 100 (2008) 172001;
- [19] S. Weinzierl, Phys. Rev. Lett. 101 (2008) 162001.
- [20] C. Anastasiou, K. Melnikov and F. Petriello, Phys. Rev. D 69 (2004) 076010.
- [21] T. Binoth and G. Heinrich, Nucl. Phys. B 585 (2000) 741, Nucl. Phys. B 693 (2004) 134; K. Hepp, Commun. Math. Phys. 2 (1966) 301.
- [22] C. Anastasiou, K. Melnikov and F. Petriello, Phys. Rev. Lett. 93 (2004) 032002.
- [23] C. Anastasiou, K. Melnikov and F. Petriello, Phys. Rev. Lett. 93 (2004) 262002, Nucl. Phys. B 724 (2005) 197; C. Anastasiou, G. Dissertori and F. Stockli, JHEP 0709 (2007) 018.
- [24] C. Anastasiou, K. Melnikov and F. Petriello, JHEP 0709 (2007) 014; K. Melnikov, report UH-511-1123-08 [arXiv:0803.0951].
- [25] See the list of references in Sect. 5 of S. Catani et al., hep-ph/0005025, in Proceeding of the CERN Workshop on Standard Model Physics (and more) at the LHC, eds. G. Altarelli and M.L. Mangano (CERN 2000-04, Geneva, 2000), p. 1.
- [26] D. de Florian and M. Grazzini, Phys. Rev. Lett. 85 (2000) 4678, Nucl. Phys. B 616 (2001) 247.
- [27] R. K. Ellis, G. Martinelli and R. Petronzio, Nucl. Phys. B 211 (1983) 106; P. B. Arnold and M. H. Reno, Nucl. Phys. B 319 (1989) 37 [Erratum-ibid. B 330 (1990) 284]; R. J. Gonsalves, J. Pawlowski and C. F. Wai, Phys. Rev. D 40 (1989) 2245.
- [28] W. T. Giele, E. W. N. Glover and D. A. Kosower, Nucl. Phys. B 403 (1993) 633; J. Campbell, R.K. Ellis, MCFM - Monte Carlo for FeMtobarn processes, http://mcfm.fnal.gov.
- [29] C. Amsler et al. [Particle Data Group], Phys. Lett. B 667 (2008) 1.
- [30] D. A. Dicus and S. S. D. Willenbrock, Phys. Rev. D 34 (1986) 148.
- [31] A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt, report IPPP/08/190 [arXiv:0901.0002].
- [32] A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J. C 28 (2003) 455.
- [33] A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Phys. Lett. B 604 (2004) 61.
- [34] S. Catani and B. R. Webber, JHEP 9710 (1997) 005.