# Two-loop QCD corrections to massless identical quark
scattering^{1}^{1}1Work supported in part by the UK Particle Physics and
Astronomy Research Council and by the EU Fourth Framework Programme
‘Training and Mobility of Researchers’, Network ‘Quantum Chromodynamics
and the Deep Structure of Elementary Particles’,
contract FMRX-CT98-0194 (DG 12 - MIHT).
C.A. acknowledges
the financial support of the Greek government and
M.E.T. acknowledges financial support
from CONACyT and the CVCP. We thank
the British Council and German Academic Exchange Service for support
under ARC project 1050.

###### Abstract:

We present the two-loop virtual QCD corrections to the scattering of identical massless quarks, , in conventional dimensional regularisation and using the scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for the and the () scattering processes in terms of polylogarithms. The results presented here form a vital part of the next-to-next-to-leading order contribution to inclusive jet production in hadron colliders and will play a crucial role in improving the theoretical prediction for jet cross sections in hadron-hadron collisions.

^{†}

^{†}preprint: DTP/00/70,IPPP/00/08,MADPH-00-1200,hep-ph/0011094

## 1 Introduction

Jet production at large transverse energies is a direct test of parton-parton scattering processes in hadron-hadron collisions. At large jet energy scales, the point-like nature of the partons can be probed down to distance scales of about m by comparing data with QCD predictions. Within the experimental and theoretical uncertainties, data from the TEVATRON and CERN SS generally show good agreement with the state-of-the-art theoretical next-to-leading order estimates based on massless parton-parton scattering over a wide range of jet energies [1, 2]. It is anticipated that the forthcoming Run II starting at the TEVATRON in 2001 will yield a dramatic improvement in the quality of the data with increased statistics and improved detectors, leading to a significant reduction in both the statistical and systematic errors. Subsequently, the start of data taking at the LHC will lead to a much enlarged range of jet energies being probed.

It is a challenge to the physics community to improve the quality of the theoretical predictions to a level that matches the improved experimental accuracy. This may be achieved by including the next-to-next-to-leading order QCD corrections which both reduces the renormalisation scale dependence and improves the matching of the parton-level theoretical jet algorithm with the hadron-level experimental jet algorithm.

The full next-to-next-to-leading order prediction is a formidable task and requires a knowledge of the two-loop matrix elements as well as the contributions from the one-loop and tree-level processes. In the interesting large-transverse-energy region, , the quark masses may be safely neglected and we therefore focus on the scattering of massless partons. For processes involving up, down and strange quarks, which together with processes involving gluons form the bulk of the cross section, this is certainly a reliable approximation. The contribution involving charm and bottom quarks is only a small part of the total since the parton densities for finding charm and bottom quarks inside the proton are relatively suppressed. We note that the existing next-to-leading order programs [1, 2] used to compare directly with the experimental jet data [3, 4] are based on massless parton-parton scattering. Helicity amplitudes for the one-loop parton sub-processes have been computed in [5, 6, 7] while the amplitudes for the tree-level processes are available in [8, 9, 10, 11]. The parton-density functions are also needed to next-to-next-to-leading order accuracy. This requires knowledge of the three-loop splitting functions. At present, the even moments of the splitting functions are known for the flavour singlet and non-singlet structure functions and up to while the odd moments up to are known for [12, 13]. The numerically small non-singlet contribution is also known [14]. Van Neerven and Vogt have provided accurate parameterisations of the splitting functions in -space [15, 16] which are now starting to be implemented in the global analyses [17].

The calculation of the two-loop amplitudes for the scattering of light-like particles has proved more intractable due mainly to the difficulty of evaluating the planar and non-planar double box graphs. Recently, however, analytic expressions for these basic scalar integrals have been provided by Smirnov [18] and Tausk [19] as series in . Algorithms for computing the associated tensor integrals have also been provided [20] and [21], so that generic two-loop massless processes can in principle be expressed in terms of a basis set of known two-loop integrals. Bern, Dixon and Kosower [22] were the first to address such scattering processes and provided analytic expressions for the maximal-helicity-violating two-loop amplitude for . Subsequently, Bern, Dixon and Ghinculov [23] completed the two-loop calculation of physical scattering amplitudes for the QED processes and .

In an earlier paper [24], we derived expressions for the two-loop contribution to unlike quark scattering, , as well as the crossed and time reversed processes. The infrared pole structure agreed with that predicted by Catani [25] and we provided explicit formulae for the finite parts in the -, - and -channels in terms of logarithms and polylogarithms. Matrix elements for the other parton-parton scattering processes remain to be evaluated. In this paper we extend the work of [24] to describe the case of identical quark scattering. We use the renormalisation scheme and conventional dimensional regularisation where all external particles are treated in dimensions to provide dimensionally regularised and renormalised analytic expressions at the two-loop level for the scattering process

together with the time-reversed and crossed processes

As in the unlike quark case, we present analytic expressions for the infrared pole structure, as well as explicit formulae for the finite remainder decomposed according to powers of the number of colours and the number of light-quark flavours . For the contributions most subleading in , there is an overlap with the two-loop contribution to Bhabha scattering described in [23] and the analytic expressions presented here provide a useful check of some of their results.

Our paper is organised as follows. We first establish our notation in Section 2. The results are collected in Section 3 where we provide analytic expressions for the interference of the two-loop and tree-level amplitudes as series expansions in . In Section 3.1 we adopt the notation of Catani [25] to isolate the infrared singularity structure of the two-loop amplitudes in the scheme. We give explicit formulae for the pole structure obtained by direct evaluation of the Feynman diagrams and show that it agrees with the pole structure expected on general grounds. The finite remainder of the two-loop graphs is the main result of our paper and expressions appropriate for the and () scattering processes are are given in Section 3.2. Finally Section 4 contains a brief summary of our results.

## 2 Notation

For calculational convenience, we treat all particles as incoming so that

(1) |

where the light-like momentum assignments are in parentheses and satisfy

As stated above, we work in conventional dimensional regularisation treating all external states in dimensions. We renormalise in the scheme where the bare coupling is related to the running coupling at renormalisation scale via

(2) |

In this expression

(3) |

is the typical phase-space volume factor in dimensions, and are the first two coefficients of the QCD beta function for (massless) quark flavours

(4) |

For an gauge theory ( is the number of colours)

(5) |

The renormalised four point amplitude in the scheme is thus

(6) | |||||

where the represents the colour-space vector describing the -loop amplitude for the -channel graphs, and the -channel contribution is obtained by exchanging the roles of particles 2 and 4:

(7) |

The dependence on both renormalisation scale and renormalisation scheme is implicit.

We denote the squared amplitude summed over spins and colours by

(8) | |||||

where the Mandelstam variables are given by

(9) |

The squared matrix elements for the process are obtained by exchanging and

(10) |

The function is related to the squared matrix elements for unlike quark scattering

(11) | |||||

(12) |

while represents the interference between -channel and -channel graphs that is only present for identical quark scattering.

The function can be expanded perturbatively to yield

where

(14) | |||||

(15) | |||||

(16) |

Expressions for are given in Ref. [26] using dimensional regularisation to isolate the infrared and ultraviolet singularities. Analytical formulae for the two-loop contribution to , , are given in Ref. [24].

Similarly, the expansion of can be written

where, in terms of the amplitudes, we have

(18) | |||||

(19) | |||||

As before, expressions for which are valid in conventional dimensional regularisation are given in Ref. [26]. Here, in order to complete the calculation of the two-loop contribution to quark-quark scattering, we concentrate on the next-to-next-to-leading order contribution and in particular the interference of the two-loop and tree graphs.

As in Ref. [24], we use QGRAF [27] to produce the two-loop Feynman diagrams to construct either or . We then project by or respectively and perform the summation over colours and spins. Finally, the trace over the Dirac matrices is carried out in dimensions using conventional dimensional regularisation. It is then straightforward to identify the scalar and tensor integrals present and replace them with combinations of the basis set of master integrals using the tensor reduction of two-loop integrals described in [20, 21, 29], based on integration-by-parts [30] and Lorentz invariance [31] identities. The final result is a combination of master integrals in . The basis set we choose comprises

(21) | |||||

(22) | |||||

(23) | |||||

(24) | |||||

(25) | |||||

(26) | |||||

(27) | |||||

(28) |

and^{2}^{2}2Reference [20] describes the procedure for reducing the tensor integrals down to a basis
involving the planar box integral

(29) |

where represents the planar box integral with one irreducible numerator associated with the left loop. The expansion in for all the non-trivial master integrals can be found in [18, 19, 20, 21, 28, 29, 33, 34, 35].

## 3 Results

In this section, we give explicit formulae for the -expansion of the two-loop contribution to the next-to-next-to-leading order term . To distinguish between the genuine two-loop contribution and the squared one-loop part, we decompose as

(30) |

The one-loop-square contribution is vital in determining but is relatively straightforward to obtain. For the remainder of this paper we concentrate on the technically more complicated two-loop contribution .

As in Ref. [24], we divide the two-loop contributions into two classes: those that multiply poles in the dimensional regularisation parameter and those that are finite as

(31) |

contains infrared singularities that will be analytically canceled by the infrared singularities occurring in radiative processes of the same order (ultraviolet divergences are removed by renormalisation).

### 3.1 Infrared Pole Structure

Catani has made predictions for the singular infrared behaviour of two-loop amplitudes. Following the procedure advocated in [25], we find that the pole structure in the scheme can be written as

(32) | |||||

where the constant is

(33) |

In Eq. (32), the symmetrisation under and exchange represents the additional effect of the -channel tree graph interfering with the -channel two-loop graphs.

The colour algebra is straightforward and we find that the - symmetric contributions proportional to

(34) |

are given by

(36) | |||||

and

(37) | |||||

where the constant is

(38) |

Here is the Riemann Zeta function, , and

(39) |

The square bracket in Eq. (37) is a guess simply motivated by summing over the antennae present in the quark-quark scattering process and on dimensional grounds. Different choices affect only the finite remainder.

The bracket of between the -channel tree graph and the finite part of the -channel one-loop graphs is not symmetric under the exchange of and and is given by

The functions and appearing in Eq. (3.1) are finite and are given by

and

with

(43) | |||||

(44) | |||||

These expressions are valid in all kinematic regions. However, to evaluate the pole structure in a particular region, they must be expanded as a series in . We note that in Eq. (32) these functions are multiplied by poles in and must therefore be expanded through to . In the physical region , , has no imaginary part and is given by [23]

(45) | |||||

where , and and the polylogarithms are defined by

(46) | |||||

(47) |

Analytic continuation to other kinematic regions is obtained using the inversion formulae for the arguments of the polylogarithms (see for example [29]) when

(48) |

Finally, the one-loop bubble integral in dimensions is given by

(49) |

The leading infrared singularity is and it is a very stringent check on the reliability of our calculation that the pole structure obtained by computing the Feynman diagrams agrees with that anticipated by Catani through to . We therefore construct the finite remainder by subtracting Eq. (32) from the full result.

### 3.2 Finite contributions

In this subsection, we give explicit expressions for the finite two-loop contribution to , , which is given by

(50) |

The identical-quark processes probed in high-energy hadron-hadron collisions are the mixed - and -channel process

controlled by (as well as the distinct quark matrix elements and as indicated in Eq. (10)), and the mixed - and -channel processes

which are determined by the . We need to be able to evaluate the finite parts for each of these processes. Of course, the analytic expressions for different channels are related by crossing symmetry. However, the master crossed boxes have cuts in all three channels yielding complex parts in all physical regions. The analytic continuation is therefore rather involved and prone to error. We therefore choose to give expressions describing and which are directly valid in the physical region and , and are given in terms of logarithms and polylogarithms that have no imaginary parts.

Using the standard polylogarithm identities [36] we retain the polylogarithms with arguments , and , where

(51) |

For convenience, we also introduce the following logarithms

(52) |

where is the renormalisation scale. The common choice corresponds to setting .

For each channel, we choose to present our results by grouping terms according to the power of the number of colours and the number of light quarks , so that in channel

(53) |

Here () to denote the mixed - and -channel (- and -channel) processes respectively.

#### 3.2.1 The process

We first give expressions for the mixed -channel and -channel annihilation process, . We find that

(54) | |||||