hep-ph/1209.0673

JHEP12(2012)088

A next-to-next-to-leading order calculation

[0.5cm] of soft-virtual cross sections

Daniel de Florian^{*}^{*}* and
Javier Mazzitelli^{†}^{†}†

Departamento de Física, FCEyN, Universidad de Buenos Aires,

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

Abstract

We compute the next-to-next-to-leading order (NNLO) soft and virtual QCD corrections for the partonic cross section of colourless-final state processes in hadronic collisions. The results are valid to all orders in the dimensional regularization parameter . The dependence of the results on a particular process is given through finite contributions to the one and two-loop amplitudes, which have to be computed in a process-by-process basis. To evaluate the accuracy of the soft-virtual approximation we compare it with the full NNLO result for Drell-Yan and Higgs boson production via gluon fusion. We also provide a universal expression for the hard coefficient needed to perform threshold resummation up to next-to-next-to-leading logarithmic (NNLL) accuracy.

December 2012

## 1 Introduction

The development of accurate QCD calculations is a fundamental tool to properly test the Standard Model. Given the size of the perturbative corrections, leading order (LO) evaluations are insufficient, and higher perturbative orders must be taken into account. However, these calculations are highly non-trivial, and at present only a few processes have been computed analytically with full next-to-next-to-leading order (NNLO) precision. At hadronic colliders, only Drell-Yan [1, 2] and Higgs boson production [2, 3, 4] (within the effective vertex approach) have reached that stage of accuracy.

Higgs production is a particular example of an observable with a slow convergence for the perturbative expansion in the strong coupling constant . Next-to-leading (NLO) corrections [5, 6, 7] are as large as the Born result and the NNLO contribution still increases the cross section by about 25% at LHC energies. Analyses from scale variations [8, 9, 10, 11, 12] and soft-gluon expansion [13, 14, 15, 16, 17] indicate that the next orders (NLO and beyond) can contribute still at the level of .

An important step towards a complete NNLO calculation for both Drell-Yan and Higgs production has been the evaluation of the soft and virtual contributions [18, 19, 20], which provide the dominant terms for those processes. As a matter of fact, this is a general feature when a system of large invariant mass is produced in hadronic collisions. Since parton distributions grow very rapidly for small fractions of the hadron momentum , the partonic center-of-mass energy tends to be close to the invariant mass , and the remaining energy only allows for the emission of soft particles. For this reason the soft-virtual contributions are expected to be a very good approximation to the total cross section for a large number of processes.

In this paper we exploit the factorization properties of the QCD matrix elements to compute the soft and virtual contributions to the partonic cross sections at NNLO for a wide number of processes in hadronic collisions where a system of colourless particles is produced (as gauge bosons, Higgs, leptons, etc.). The computational approach presented here can be extended to higher orders in perturbation theory simplifying considerably the evaluation of the corrections.

We present a universal expression for the corresponding cross section at NNLO, valid to all orders in the dimensional regularization parameter . With this result, it is possible to evaluate the soft-virtual approximation for any process of the kind studied in this paper in an automatized way once the relevant one- and two-loop amplitudes become available and, therefore, provide a first estimate of the size of higher order corrections for a number of interesting observables at the LHC. Furthermore, since the results are valid to all orders in , they become necessary ingredients for ultraviolet and infrared factorization at NLO (and beyond) within the same approximation.

Another interesting use of the soft approximation is the relation to the soft-gluon threshold resummation approach. We profit from this calculation and obtain, for the first time, a universal expression for the hard coefficient needed to perform threshold resummation up to next-to-next-to-leading logarithmic (NNLL) accuracy.

The paper is organized as follows. In section 2 we present the notation for the QCD cross sections and show how phase space factorization occurs in the soft limit. In sections 3 and 4 we perform the calculation of the soft-virtual corrections at NLO and NNLO respectively. In section 5 we present the soft-virtual approximation in Mellin space. In section 6 we analyse the phenomenological results for Drell-Yan and Higgs boson production via gluon fusion to compare the soft-virtual approximation with the full result as a way to validate its dominance. In section 7 we profit from the previous result and present a universal expression for the hard coefficient required to perform threshold resummation up to NNLL accuracy. Finally, in section 8 we present our conclusions.

## 2 QCD cross sections

We consider the following general process in hadronic collisions:

(1) |

where denotes any colourless final state (i.e. without quarks or gluons), and stands for any inclusive final hadronic state. The center-of-mass energy is , and is the invariant mass of the system which can involve a combination of gauge bosons, Higgs, isolated photons, leptons, etc. The inclusive cross section can be written as

(2) |

where , and are the factorization and renormalization scales respectively, and is the Born level partonic cross section. The parton densities of the colliding hadrons are denoted by and the subscripts label the type of massless partons (, with different flavours of light quarks).

According to Eq. (2), the cross section for the partonic subprocess at the center-of-mass energy is

(3) |

where the term arises from the flux factor and leads to an overall factor, being the partonic equivalent of . The hard coefficient function has a perturbative expansion in terms of powers of the QCD renormalized coupling :

(4) |

In the following, the dependence of on the renormalization scale is understood. We always use the scheme for the renormalization of the strong coupling.

At leading-order the partonic subprocess is , and since the final state is colourless we can only have or (and ). The LO contribution is then

(5) |

At higher orders, other parton subprocesses can contribute to the total cross section.

In the soft-virtual approximation, however, we are only interested in the same parton subprocess present at LO, and we compute only those contributions in that give rise to the distributions and in the coefficient function, where we have defined

(6) |

The indicates the usual plus-prescription,

(7) |

These two types of contributions are the most singular terms when , and then dominate the cross section in the soft limit.

The aim of this paper is to calculate the NNLO soft-virtual corrections for any process of the type of Eq. (1). The parton subprocesses that contribute up to second order in are

with or , depending on the process.

### 2.1 Phase-space factorization

To compute the real corrections of the processes we are interested in, we have to perform the phase-space integration of the corresponding matrix elements in the limit in which the emitted QCD-partons become soft. In this limit, the phase-space can be written in a factorized form.

Let us consider that the final state has non-QCD particles with momenta , and soft QCD massless partons with momenta . The -dimensional phase-space is then

(9) |

We introduce the momentum [21], with . Multiplying the above equation by the identity

(10) |

we arrive at

(11) | |||||

In the soft limit we have

(12) |

where the symbol indicates that the equality is valid when the emitted QCD partons are soft. Within this approximation we obtain in the first line of Eq. (11) the corresponding leading-order phase-space , which contains the dependence on the internal variables of the system . The second line in Eq. (11) is the phase-space of a process with one particle of invariant mass in the final state plus soft partons, . Then Eq. (11) can be rewritten in the following way:

(13) |

arriving to a factorized expression for the phase-space in the soft limit.

## 3 Nlo

At NLO we have to consider the one-loop corrections to the partonic subprocess , and also the real gluon emission subprocess, . We begin by computing the latter.

Let be the LO matrix element, and the correction corresponding to the real gluon emission subprocess at tree-level. In the limit where the momentum of the gluon becomes soft, can be written in the following factorized way [22]:

(14) |

where and are the momenta of the incoming QCD-partons, and the dependence of the matrix elements on other non-QCD particles momenta is understood. The symbol stands for the bare coupling constant, and is the dimensional-regularization scale. Renormalization is achieved by the replacement

(15) |

where is the first coefficient of the QCD beta function and is the typical phase-space volume factor in dimensions:

(16) |

being the Euler number.

The scalar eikonal function contains all the dependence of on the soft gluon momentum, and takes the form [22]

(17) |

while the constant depends on the nature of the radiating parton, being

(18) |

where is the number of colours.

Combining Eq. (14) with the phase-space factorization of Eq. (13) we arrive at the following expression for the NLO tree-level real gluon emission cross section :

(19) |

The phase-space integration in Eq. (19) can be performed in a closed form. After some simple algebra we arrive at

(20) |

Using polar coordinates in dimensions we can write

(21) |

with

(22) |

where and for the others. Parametrizing the momenta in the center-of-mass of the incoming partons, and setting , it can be shown that

(23) |

where is the angle between the soft gluon and the -th axis. Using the variable , the remaining integral can be carried out as a particular case of

(24) |

Finally, the tree-level real gluon emission cross section in the soft limit has the following expression:

(25) |

This formula is valid to all orders in for any reaction of the kind of Eq. (1), and its only dependence on a particular process is in the Born-level cross section . The factor only depends on the nature of the incoming partons. The expansion of the factor leads to the appearance of and terms, according to the following relation:

(26) |

We now have to evaluate the one-loop correction term. Even though this cannot be done in a process independent way, the infrared-singular behaviour of QCD amplitudes at one-loop (and two-loop) order is well known [23, 24, 25, 26]. For the processes we are interested in, the renormalized one-loop order amplitude can be written in terms of the Born-level amplitude in the following way^{†}^{†}†For the sake of simplicity we omit the explicit dependence of the matrix elements on the partons momenta. [26]:

(27) |

where

(28) |

and the contribution is finite when ^{‡}^{‡}‡We explicitly keep higher order terms in originated in the one-loop amplitude as they contribute to the final result to the same order in the dimensional regularization parameter. Those in are implicitly included in the definition of .. The coefficient depends on the initial-state partons, being

(29) |

For the NLO calculation we only need the term of the squared matrix element, that is . Performing the formal phase-space integration of this term we arrive at the following expression for the one-loop virtual contribution to the cross section:

(30) |

where is a one-loop finite contribution to the cross section defined by

(31) |

To obtain the NLO soft-virtual contribution to the coefficient function we still have to add to Eqs. (25) and (30) the counterterms coming from mass factorization. Keeping terms up to second order in powers of , relevant for the NNLO calculation in the next section, we have

where for simplicity we have set . The symbol “” stands for the usual convolution. The Altarelli-Parisi splitting functions in the soft limit () take the form [27]

(33) | |||||

(34) |

where we have defined

(35) |

Combining the results of Eqs. (25) and (30) with the counterterms coming from Eq. (3) we arrive at a closed expression valid to all orders in for the NLO soft-virtual coefficient function . For simplicity, we only write the first three terms of its expansion in powers of ^{§}^{§}§Contributions up to are needed to build the renormalization and factorization counterterms for a calculation to NLO accuracy.:

where we have set . The dependence of this expression on a particular process is contained only in the one-loop coefficients defined by

(37) |

## 4 Nnlo

At second order in the perturbative expansion we have to consider the four parton subprocesses of Eq. (2). The one-loop correction to the subprocess can be obtained in a very similar way to the tree-level one. Let be the one-loop gluon emission matrix element. The analogous to Eq. (14) is given by the soft limit of one-loop amplitudes as [28, 29, 30, 31, 32]

(38) |

The phase-space integrals we have to perform are

(39) |

and their calculation can be achieved with the tools discussed in section 3. The result is

where can be written as in Eq. (30), using the delta function to perform the integral.

We continue by computing the double real emission subprocesses, that is and . For the NNLO squared amplitudes the following infrared factorization formulae hold [33, 34]:

where all the dependence on the momenta and of the soft particles is embodied in the functions and , which take the form [33]

(45) |

To perform the phase-space integration of both contributions we use the parametrization of introduced in [18] for the calculation of the second order corrections to the Drell-Yan process. Introducing the variables and we can write

(46) | |||||

The last line of the above equation is most easily computed in the center-of-mass of and . In this frame, orientated so that is in the direction of the axis and lies in the plane defined by the and axes, the momenta can be parametrized as follows [18]:

(47) |

where we have defined and (see Eq. (22)), and

(48) |

where is the Källen function, . The dots in and represent unspecified components of momentum, which are trivially integrated with the present parametrization. Using momentum conservation, we obtain for the following expression:

(49) |

Considering the above parametrization and introducing the variables , and :

(50) |