TwoLoop Helicity Amplitudes for GluonGluon Scattering in QCD and Supersymmetric YangMills Theory
Abstract:
We present the twoloop helicity amplitudes for the scattering of two gluons into two gluons in QCD, which are relevant for nexttonexttoleading order corrections to jet production at hadron colliders. We give the results in the ‘t HooftVeltman and fourdimensional helicity variants of dimensional regularization. Summing our expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Glover, Oleari and TejedaYeomans. We also present the amplitudes for scattering in pure supersymmetric YangMills theory.
SLAC–PUB–9103
UCLA/02/TEP/1
January, 2002^{†}^{†}dedicated: Submitted to JHEP
1 Introduction
For at least the next decade, the energy frontier for acceleratorbased particle physics will be located at hadron colliders, the Tevatron at Fermilab and the Large Hadron Collider at CERN. At a given large momentum transfer, the most copious events at these colliders should be hadronic jets. To test the Standard Model at the shortest possible distances, therefore, the jet production cross section should be known with the highest possible precision. Existing calculations of jet production at nexttoleading order (NLO) in the strong coupling constant [1, 2, 3] agree well with the data over a broad range of transverse momentum. Still, the NLO predictions have an uncertainty from higher order corrections, traditionally estimated from dependence on the renormalization and factorization scales, which is of order 10% or more. For very large momentum transfer the predictions can be improved by resumming threshold logarithms [4]. There are also sizable uncertainties associated with the experimental input to the parton distribution functions [5], even though global fits to the data have recently been performed [6] within an approximate nexttonexttoleading order (NNLO) framework [7]. Nevertheless, an exact NNLO computation of jet production rates would be very welcome. Besides reducing the scale uncertainties for jet rates, the same numerical program should allow a better understanding of energy flows within jets, as a jet may consist of up to three partons at this order.
Several types of QCD amplitudes are required for a NNLO calculation of jet production at hadron colliders. Both the tree amplitudes for six external partons [8, 9] and the oneloop amplitudes for five external partons [10] have been known for some time now. Recently, in a tour de force series of calculations, Anastasiou, Glover, Oleari, and TejedaYeomans have provided the NNLO interferences of the twoloop amplitudes with the tree amplitudes, for all QCD fourparton processes, summed over all external helicities and colors [11, 12].
In this paper, we compute the amplitudes directly at two loops in the spinor helicity formalism [13], and expose their full dependence on external colors as well. The additional helicity and color information provided here is not necessary for the main phenomenological application, NNLO jet production in collisions of unpolarized hadrons. However, it still provides several benefits:

Jet production in collisions of polarized protons, as planned for the relativistic heavy ion collider (RHIC) at Brookhaven, may help to determine the poorlyknown polarized gluon distribution in the proton [14]. Theoretical predictions of the relevant observables require scattering amplitudes for polarized partons. Currently, predictions are available through NLO [15]; the helicity amplitudes presented here are a prerequisite for improving the predictions to NNLO accuracy.

Our results serve as a check of the results of ref. [12], and are useful for investigating the dependence of twoloop amplitudes on the variant of dimensional regularization used.
Here we also present the helicity amplitudes for scattering in pure supersymmetric gauge theory. Such amplitudes only differ from QCD with massless quarks in that the fermions are in the adjoint rather than the fundamental representation; yet they obey supersymmetry Ward identities [16] and are generally simpler than their QCD counterparts. They also provide useful auxiliary functions for describing the QCD results.
Several versions of dimensional regularization have been used for loop calculations in QCD, differing mainly in the number of gluon polarization states they assign in dimensions. The conventional dimensional regularization (CDR) scheme [20] assigns states to all gluons, whether internal or external, virtual or real. This scheme is traditionally employed in calculations of amplitude interferences, such as ref. [12]. In the helicity approach, the number of external, observed gluon states is necessarily 2 (helicity ), but there is some freedom in the number of virtual gluon polarizations. The ’t HooftVeltman (HV) scheme [21] contains virtual gluon states, while the fourdimensional helicity (FDH) scheme [22, 23] assigns . The FDH scheme is related to dimensional reduction () [24] but is more compatible with the helicity method, because it allows 2 transverse dimensions in which to define helicity. Of these variants, only the FDH scheme is fully compatible with supersymmetry Ward identities for helicity amplitudes, some of which have been verified through two loops [23]. Here we work primarily in the ’t HooftVeltman (HV) variant of dimensional regularization [21], but we also discuss the conversion to the CDR and FDH schemes.
Twoloop scattering amplitudes in massless QCD possess strong infrared (soft and collinear) divergences. Using dimensional regularization with , the amplitudes generically contain poles in up to . However, these divergences have been organized by Catani [25] into a relatively simple form, which is completely predictable through at least order . We shall use Catani’s formulae and color space notation to organize the helicity amplitudes into singular terms (which do contain terms in their series expansion in ), plus finite remainders. We find that the general form of the divergences given in ref. [25] holds precisely in both the HV and FDH schemes; however, the numerical value of the coefficient , which appears at order , differs in the FDH scheme from its value [25] in the HV (or ) scheme.
The poles were not predicted a priori in ref. [25] for general processes at two loops. For the amplitude, ref. [12] computed the interference of the pole terms with the tree amplitude, summed over all colors and helicities. Here we extract the full color and helicity dependence of the pole terms. We find a term which is independent of color and helicity, and which agrees with that found by ref. [12] (when we use the HV scheme), plus a second term with nontrivial colordependence, which vanishes when the colorsummed interference is performed. A term with similar color structure has also been identified in contributions of oneloop factors for soft radiation to NNLO processes [26]. We shall also discuss how terms in the infrared decomposition of ref. [25] are modified, beginning at order , in other variants of dimensional regularization, such as the FDH scheme.
The paper is organized as follows. In section 2 we review the infrared and color structure of one and twoloop QCD amplitudes. In section 3 we describe the oneloop amplitudes in a form that is valid to all orders in [17, 27, 28, 29], and show how to expand them through . This accuracy is required because oneloop amplitudes enter the formulae for the singular parts of twoloop amplitudes multiplied by . Section 3.2 shows that apart from this requirement, only finite remainder terms in the oneloop amplitudes are needed, because of cancellations with other NNLO contributions. These remainder terms are then tabulated in section 3.3.
In section 4 we describe our method for computing the twoloop amplitudes. Section 4.1 summarizes how we evaluate loop integrals, especially those that arise only in the helicity method. Some consistency checks on the results are listed in section 4.2. Section 4.3 discusses the additional singular term appearing at order in the colordecomposed amplitude, which does not contribute to the colorsummed interference with the tree amplitude. The finite twoloop remainder functions in the HV scheme are then presented in section 4.4 and appendix A.
In section 5 we describe conversion of the HV results to different schemes, and the comparison with ref. [12], after our results are summed over all external colors and helicities. In section 6 and appendix B we give the twoloop amplitudes for pure supersymmetric YangMills theory, whose finite remainders also serve as auxiliary functions for describing the QCD results. In section 7 we present our conclusions.
2 Review of infrared and color structure
In this section we review the structure of the infrared singularities of dimensionally regularized one and twoloop QCD amplitudes, using Catani’s color space notation [25], as a prelude to presenting the finite remainders of the one and twoloop amplitudes.
The process considered in this paper is
(1) 
using an “alloutgoing” convention for the external momentum () and helicity () labeling. The Mandelstam variables are , , and .
We work with ultraviolet renormalized amplitudes, and employ the running coupling for QCD, . The relation between the bare coupling and renormalized coupling , through twoloop order, is [25]
(2) 
where is the renormalization scale, , and is Euler’s constant. The first two coefficients appearing in the beta function for QCD, or more generally gauge theory with flavors of massless fundamental representation quarks, are
(3) 
where , , and . (Note that ref. [25] uses the notation , .)
The perturbative expansion of the amplitude is
where is the loop contribution. Equation (2) is equivalent to the following renormalization prescriptions at one and two loops,
(5)  
(6) 
The infrared divergences of renormalized one and twoloop point amplitudes are given by [25],
(7)  
where the “ket” notation indicates that the loop amplitude is treated as a vector in color space. The actual amplitude is extracted via
(9) 
where the are color indices. The subscript indicates that a quantity depends on the choice of renormalization scheme. The divergences of are encoded in the color operator , while those of also involve the schemedependent operator .
In QCD, the operator is given by
(10) 
where if and are both incoming or outgoing partons, and otherwise. The color charge is a vector with respect to the generator label , and an matrix with respect to the color indices of the outgoing parton . For external gluons , so , and
(11) 
The operator is given by [25]
(12)  
where the coefficient in either the HV or CDR schemes is given by [25]
(13) 
Although no scheme dependence was assigned to this coefficient in ref. [25], we shall find in section 5 that it is scheme dependent. The function contains only single poles,
(14) 
but is not predicted a priori for general processes. The color and helicitysummed matrix element has previously been computed in the CDR scheme for [12] (and for some other multiparton processes [11, 30]). We shall extract the full color and helicity dependence of for in the HV scheme in section 4.3, and in the FDH scheme in section 5.
An explicit color basis for the amplitudes is given by
(15) 
where
(16) 
Here are generators in the fundamental representation, normalized according to the convention typically used in helicity amplitude calculations, . (The used in this color decomposition should not be confused with the appearing in , which are in the adjoint representation; nor should they be confused with the generators for the quark representation, which have the more “standard” normalization, , as mentioned above.)
We have also taken the opportunity in eq. (15) to remove some helicitydependent overall phases, which arise because we evaluate the amplitudes in the spinor helicity formalism [13],
(17) 
The spinor inner products [13, 9] are and , where are massless Weyl spinors of momentum , labeled with the sign of the helicity. They are antisymmetric, with norm , where . It follows that the are indeed phases. They will cancel out from (and therefore may be freely omitted from) all transition probabilities involving unpolarized gluons, or circularly polarized gluons.
A reflection identity implies that the coefficients of two color structures with reversed ordering are identical, so that
(20) 
Also, due to Bose symmetry, parity, and timereversal symmetry for the process (1), we only have to give results for the four helicity configurations
(21) 
The tree amplitudes are given in the basis (16) by
(22) 
A typical partonic cross section requires an amplitude interference, summed over all external colors. Such interferences are evaluated in the color basis (16) as
(23) 
where the symmetric matrix is [12, 31]
(24) 
with
(25) 
The unpolarized partonic cross section is obtained from the helicity sum
(26) 
after the usual averaging over initial spins and inclusion of flux factors. For example, the helicity sum for the treelevel cross section, constructed from eq. (22) in either the HV or FDH scheme, is
(27) 
3 Oneloop amplitudes
The oneloop amplitudes for were first evaluated through as an interference with the tree amplitude in the CDR scheme [32]. Later they were evaluated as helicity amplitudes in the HV and FDH schemes [22, 33].
Because contains terms of order , the term in the infrared decomposition (2) of the twoloop amplitudes requires the series expansions of the oneloop amplitudes through . In ref. [29], using results from refs. [17, 27, 28], the oneloop helicity amplitudes were presented in a representation valid to all orders in , in both the HV and FDH schemes. These results can easily be rewritten in terms of integral functions whose series expansions are known to the requisite order [34, 35]. In section 3.1 we present the allorder results in the color basis (16), with the normalizations implicit in eq. (2).
In section 3.2, we show that the only place that terms beyond in the oneloop amplitudes are required in an NNLO calculation is in the infrared decomposition (2) of the twoloop amplitudes.
Finally, in section 3.3 we list the finite remainders of the oneloop amplitudes in the HV scheme, after the renormalization (5) and subtraction of infrared divergences (7). The corresponding finite remainder in the oneloop/oneloop NNLO interference has already been computed in the CDR scheme, summed over all colors and helicities [31]. Our HV amplitude remainders lead to precisely the same result.
3.1 All orders in
Here we present the renormalized oneloop amplitudes in the color basis (16), with the normalizations implicit in eq. (2), in a form valid to all orders in .
The first coefficient in the color basis (16) for at one loop may be written in terms of “primitive” amplitudes for a gluon or quark in the loop, as [36]
(28) 
The remaining singletrace coefficients are obtained via crossing symmetry:
where appropriate analytic continuations are required to bring each function into the physical region. The double trace coefficients, to which only the gluon loops contribute, follow from a decoupling identity [36]:
(30) 
It is convenient to write the gluon and fermion loop contributions, and , in terms of a supersymmetric decomposition into scalar, chiral , and supersymmetric multiplets in the loop [37, 18]:
(31)  
(32) 
where for the HV scheme and for the FDH scheme.
For “maximally helicity violating” configurations, the supersymmetric components vanish by a supersymmetry Ward identity [16],
(33) 
The remaining independent components are [17, 27, 29]
(34)  
(35)  
(36) 
Here , and are the oneloop bubble, triangle and box scalar integrals, evaluated in dimensions. The bubble and box integrals are
(37) 
where
(38)  
with
(39) 
and we have kept the full dependence on in the integrals. In the channel (), expansions of the functions (37) are given by using the analytic continuation .