Two-Loop Quark and Gluon Form Factors in Dimensional Regularisation

We compute the two-loop corrections to the massless quark form factor $\gamma^* \to q\bar q$ and gluon form factor $H\to gg$ to all orders in the dimensional regularisation parameter $\epsilon=(4-d)/2$. The two-loop contributions to the form factors are reduced to linear combinations of master integrals, which are computed in a closed form, expressed as $\Gamma$-functions and generalised hypergeometric functions of unit argument. Using the newly developed HypExp-package, these can be expanded to any desired order, yielding Laurent expansions in $\epsilon$. We provide expansions of the form factors to order $\epsilon^2$, as required for ultraviolet renormalisation and infrared factorisation of the three-loop form factors.


Introduction
The infrared pole structure of renormalised multi-loop amplitudes in dimensional regularisation with d = 4 − 2ǫ space-time dimensions can be predicted from an infrared factorisation formula, which was first conjectured in [1], where it was formulated up to two loops. A proof of the formula, together with an explicit formulation up to three loops was derived later in [2]. The simplest multi-loop amplitudes where the infrared factorisation formula can be applied are threepoint functions, involving two partons coupled to an external current: the quark form factor γ * → qq and the gluon form factor H → gg. The QCD corrections to these form factors can in particular be used to fix a priori unknown constants in the infrared factorisation formula, thus enabling an unambiguous prediction for multi-loop amplitudes involving more than two external partons.
In the infrared factorisation formula for a given form factor (or more generally for a given multi-leg amplitude) at a certain number of loops, infrared singularity operators act on the form factor evaluated with a lower number of loops. The infrared singularity operators contain explicit infrared poles 1/ǫ 2 and 1/ǫ. They do therefore project subleading terms in ǫ from the lower order form factors.
At present, two-loop corrections to the massless quark [3] and gluon [4] form factors are known to order ǫ 0 . Two-loop corrections to this order were also obtained for massive quarks [5]. The infrared structure of the massless form factors and infrared cancellations with real radiation contributions are described in detail in [6]. Very recently, results to order ǫ 2 were obtained for the quark form factor [7].
The calculation of these corrections proceeds through a reduction [8][9][10][11] of all two-loop Feynman integrals appearing in the form factors to a small set of master integrals. The reduction is exact in ǫ, such that the evaluation of the form factors is limited only by the order to which the master integrals can be computed. The massless two-loop form factors contain three two-loop master integrals, which can be computed either using various analytical methods [12] or numerically order-by-order in their Laurent expansion using the sector decomposition algorithm [13]. Up to now, exact expressions were known only for two of these master integrals, while the third (the so-called two-loop crossed triangle graph) was known only as a Laurent expansion up to finite terms [14].
In this letter, we derive an exact expression for the two-loop crossed triangle graph in terms of generalised hypergeometric functions of unit argument in Section 2. Using the HypExppackage [15] for the Laurent expansion of generalised hypergeometric functions, this can be expanded to any desired order in ǫ. Together with the exact expressions for the one-and twoloop quark and gluon form factors in Section 3, this allows the expansion of these form factors to higher orders in ǫ. For illustration, we list the one-loop form factors to order ǫ 4 and the two-loop form factors to order ǫ 2 in Section 4; these orders appear for example in the infrared factorisation of the corresponding three-loop from factors. Finally, Section 5 contains conclusions and an outlook.

Two-loop master integrals
The virtual two-loop vertex master integrals were first derived to order ǫ 0 in [14] in the context of the calculation of the two-loop quark form factor [3]. All but the crossed triangle graph A 6 can be expressed in terms of Γ-functions to all orders in ǫ.
Factoring out a common and introducing q 2 = (p 1 + p 2 ) 2 , they read No exact expression for A 6 was known up to now. Following the steps outlined in [14], we obtain While A 2,LO , A 3 and A 4 can be expanded using any standard computer algebra programme, the expansion of A 6 requires the expansion of generalised hypergeometric functions in their parameters. For this purpose, a dedicated package, HypExp [15], was developed recently. Using this, we obtain the eighth-order expansion: where we encountered a multiple zeta value in the last term.

Quark and gluon form factors at two loops
The tree-level quark and gluon form factors are obtained by normalising the corresponding tree-level vertex functions to unity: The unrenormalised one-loop and two-loop form factors are calculated from the relevant Feynman diagrams. Using integration-by-parts [8] and Lorentz invariance [10] identities (which can be solved symbolically for massless two-loop vertex integrals, see the appendix of [16]), these can be reduced [9][10][11] to the master integrals listed in Section 2.
The unrenormalised one-loop quark and gluon form factors read: where N = 3 is the number of colours and g is the bare QCD coupling parameter. The unrenormalised two-loop quark and gluon form factors for N F massless quark flavours are: 192d 10 − 6947d 9 The renormalised form factors are obtained by introducing the renormalised QCD coupling constant and the renormalised effective coupling of H to the gluon field strength [4], and subsequent expansion in powers of the renormalised coupling.

Expansion of two-loop form factors
The renormalised form factors are expanded in the renormalised coupling constant. In the MS scheme, the bare coupling α 0 = g 2 /(4π) is related to the renormalised coupling α s ≡ α s (µ 2 ), evaluated at the renormalisation scale µ 2 by where S ǫ = (4π) ǫ e −ǫγ with the Euler constant γ = 0.5772 . . . and µ 2 0 is the mass parameter introduced in dimensional regularisation to maintain a dimensionless coupling in the bare QCD Lagrangian density. For simplicity, we set µ 2 = q 2 . If the squared momentum transfer q 2 is space-like (q 2 < 0), the form factors are real, while they acquire imaginary parts for time-like q 2 . These imaginary parts (and corresponding real parts) arise from the ǫ-expansion of The renormalised form factors can then be written as Expanding the first and second order coefficients of the form factors to ǫ 4 and ǫ 2 respectively, we obtain:

Conclusions and outlook
In this letter, we computed the two-loop quark and gluon form factors to all orders in the dimensional regularisation parameter ǫ. The principal ingredient to this calculation is the twoloop crossed triangle graph A 6 , for which we computed an exact expression in terms of generalised hypergeometric functions of unit argument, which can be expanded to any desired order in ǫ using the HypExp-package.
A potential application of the form factors derived here is the extraction of the complete set of infrared pole terms of the genuine three-loop quark form factor from the recently derived three-loop splitting and coefficient functions in deep inelastic scattering [17]. In turn, these allow to fix the yet unknown hard radiation constants in the infrared factorisation formula at three loops. Parts of these constants were derived previously from N = 4 supersymmetry relations [18,19].
The two-loop vertex master integrals feature as subtopologies in the reduction of the threeloop form factor contributions, appearing if one of the three loops is disconnected from the others by pinching the connecting propagators. In this case, their terms to ǫ 2 are required.
The calculation presented here illustrates the applicability of the HypExp-package in the calculation of multi-loop corrections in quantum field theory. Functions similar to those which were expanded here appear also in multi-particle phase space integrals in massless [20] and massive decay processes [21]. Since all these integrals correspond to particular cuts of multiloop two-point functions, one might expect that three-loop and four-loop two-point functions could also be expanded using HypExp to high orders in ǫ, as required for multi-loop calculations of fully inclusive observables [22].
Note added: While finalising this letter, an independent paper addressing very similar issues appeared. In hep-ph/0507039 [7], Moch, Vermaseren and Vogt compute the two-loop quark form factor to order ǫ 2 and apply it in the extraction of the pole parts of the three-loop quark form factor from deep inelastic coefficient functions. In this paper, the hard radiation constants for infrared factorisation at three-loops and related resummation coefficients are extracted for processes involving quarks only. Expanding our unrenormalised quark form factor (10) to order ǫ 2 , we confirm the result (B.1) of [7].