The double-soft integral for an arbitrary angle between hard radiators

The European Physical Journal C August 2018, 78:687 | Cite as The double-soft integral for an arbitrary angle between hard radiators AuthorsAuthors and affiliations Fabrizio CaolaMaximilian DeltoHjalte FrellesvigKirill Melnikov Open Access Regular Article - Theoretical Physics First Online: 29 August 2018 Received: 14 August 2018 Accepted: 22 August 2018 39 Downloads Abstract We consider the double-soft limit of a generic QCD process involving massless partons and integrate analytically the double-soft eikonal functions over the phase-space of soft partons (gluons or quarks) allowing for an arbitrary relative angle between the three-momenta of two hard massless radiators. This result provides one of the missing ingredients for a fully analytic formulation of the nested soft-collinear subtraction scheme described in Caola et al. (Eur Phys J C 77(4):248, 2017). 1 Introduction A precise description of hard processes offers an exciting opportunity to discover or constrain physics beyond the Standard Model at the LHC using indirect methods. Such a description is based on the collinear factorization framework that emphasizes the importance of understanding partonic cross sections in higher orders of perturbative QCD. Currently, it is possible to compute most processes of phenomenological interest at a fully-differential level at leading and next-to-leading orders in perturbative QCD, and \(2 \rightarrow 1\) and \( 2 \rightarrow 2 \) processes at next-to-next-to-leading order (NNLO). An important recent development in the field of precision collider physics is the start of “mass production” of NNLO QCD results for major \(2 \rightarrow 2\) LHC processes such as \(pp \rightarrow t \bar{t}\) [1, 2], \(pp \rightarrow 2j\) [3, 4], \(pp \rightarrow V+j\) [5, 6, 7, 8, 9], \(pp \rightarrow H+j\) [5, 10, 11, 12], the t-channel single top production [13, 14], Higgs production in weak boson fusion [15, 16] etc. General purpose public numerical codes also became available recently [17, 18]. This progress happened because a number of computational schemes, both of the slicing type and the subtraction type [5, 15, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], have matured enough to be used in complex realistic calculations. Nevertheless, in spite of these successes, it is fair to say that none of the suggested schemes are fully optimal. This is unfortunate as it can limit our ability to make precise predictions for higher multiplicity processes at the LHC in the future. Hence, further developments of subtraction methods are welcome. Motivated by these considerations, two of us in collaboration with R. Röntsch have recently proposed [19] a modification of the subtraction scheme described in Refs. [29, 30, 31]. A key element in our proposal is the double-soft limit defined as follows. We consider the double-real emission contribution to NNLO QCD corrections to the production of an arbitrary final state X in hadron collisions. Specifically, we are interested in the \(X + f\) final state, where f are either two gluons or a quark–antiquark pair. We assign four-momenta \(k_{4,5}\) to the two additional partons, and consider the double soft configuration \(k_{4,5} \rightarrow 0\), with no particular hierarchy between \(k_4\) and \(k_5\). It is well known that soft emissions factorize. Indeed, in the soft approximation parton emission does not change the kinematics of the final state X and does not affect infra-red safe observables. Moreover, the matrix element squared of the process \( ij \rightarrow X+f\) factorizes into a color-correlated emissionless matrix element squared for the process \(ij \rightarrow X\) and a universal eikonal function that depends on momenta of hard radiators that are present in either the initial or the final state, and the momenta \(k_4\) and \(k_5\) of the soft partons. An important ingredient of any NNLO subtraction scheme is the integral of the double-soft eikonal function over the phase-space of the two extra partons f, subject to kinematic constraints. In the framework of Ref. [19], the following constraints on energies of the soft partons are imposed $$\begin{aligned} k_4^0< E_\mathrm{max},\quad k_5^0 < k_4^0. \end{aligned}$$ (1) In Ref. [19], double-soft integrals with constraints as in Eq. (1) were computed numerically for the case when hard emittors are back-to-back. Although this is adequate for the color-singlet production processes considered in [19], in more complicated cases the numerical approach becomes cumbersome, since it requires a non-trivial continuation of phase-space integrals beyond four space-time dimensions (see [29, 30, 31, 38] for details). Moreover, beyond the back-to-back limit, the double-soft integrals become functions of an angle between the three-momenta of the hard radiators. Since these angles change from event to event, the required numerical computations become quite expensive. In what follows, we show how to overcome these issues and present an analytic computation of the double-soft integrals required for the description of NNLO real emission contributions to an arbitrary process. The rest of this paper is organized as follows. In Sect. 2 we introduce our notation, present relevant formulas for the double-soft limit and define the integrals that need to be computed. In Sect. 3 we discuss how to use differential equations to find the phase-space integrals. In Sect. 4 we explain how to fix the boundary conditions needed to fully reconstruct the required integrals from the differential equations. In Sect. 5 we present our final results for the integrals of the double-soft eikonal functions. We conclude in Sect. 6. 2 The double-soft current and its integration In this section, we consider the double-soft limit of a generic scattering process. It is well known that soft emissions factorize. We now recall basic features of this factorization, following closely Ref. [39]. Interested readers should consult Ref. [39] for further details. In QCD, soft emissions involve non-trivial color correlations. It is then convenient to introduce a color basis \(|c_1,\ldots ,c_n\rangle \), and write a generic scattering amplitude as $$\begin{aligned} {\mathcal {M}}^{c_1,\ldots ,c_n}(p_1,\ldots ,p_n) = \left\langle c_1,\ldots ,c_n|{\mathcal {M}}(p_1,\ldots ,p_n)\right\rangle ,\nonumber \\ \end{aligned}$$ (2) where \(c_i\) are the color indices. It is also useful to associate a color charge \(\mathbf{T}_i\) with the emission of soft gluons off a parton i. Its action is defined as $$\begin{aligned}&\left\langle c_1,\ldots ,c_i,\ldots ,c_m,a|\mathbf{T}_{i}|b_1,\ldots ,b_i,\ldots b_m\right\rangle \nonumber \\&\quad =\delta _{c_1 b_1}\ldots T^{a}_{c_i b_i}\ldots \delta _{c_m b_m}, \end{aligned}$$ (3) where a is the gluon color index (\(a=1,\ldots ,N_c^2-1)\) and \(T^a_{c_i b_i} = i f_{a c_i b_i}\) if parton i is a gluon, \(T^a_{c_i b_i} = t^a_{c_i b_i}\) if i is a quark, (...truncated)