We compute the three-loop master integrals required for the calculation of the triple-real contribution
to the N

Article funded by SCOAP3

0$. We can therefore introduce a cut-off $\mathcal{T}_0$ and divide the phase space for $V+X$ into two disjoint parts. We write schematically \begin{equation} \sigma_{pp \to V+X}^{\rm N^3LO} = \sigma_{pp \to V+X}^{\rm N^3LO}\left( \mathcal{T} \leq \mathcal{T}_0 \right) + \sigma_{pp \to V+X}^{\rm NNLO}\left( \mathcal{T} > \mathcal{T}_0 \right)\,. \label{eq:sigmasplit} \end{equation} Note the $\rm NNLO$ subscript in the second term on the right-hand side in~\cref{eq:sigmasplit}; the reason for its appearance is that by imposing the $\mathcal{T} > \mathcal{T}_0$ constraint, we exclude the situation where \emph{all} final-state partons become unresolved so that the calculation for $\mathcal{T} > \mathcal{T}_0$ reduces to the computation of the NNLO QCD corrections to $pp \to V+j$. Such calculations have already been performed for a variety of final states and we consider them to be known~\cite{Boughezal-ml-2015dra,Ridder-ml-2015dxa,Boughezal-ml-2015aha,Boughezal-ml-2015dva,Chen-ml-2016zka,Gehrmann-DeRidder-ml-2017mvr}. On the other hand, the first term on the right-hand side of~\cref{eq:sigmasplit} still receives contributions from those regions of phase space where the final-state radiation is fully unresolved. In general, the computation of these contributions can be as difficult as the full N${}^3$LO calculation itself. However, for 0-jettiness, this does not happen. Indeed, it was shown in ref.~\cite{Stewart-ml-2010tn} that the cross section for $pp \to V+X$ simplifies substantially in the limit $\mathcal{T} \to 0$ and can be written as a convolution of the hard cross section for $pp \to V$ with the so-called beam and soft functions~\cite{Berger-ml-2010xi,Gaunt-ml-2014xga,Gaunt-ml-2014cfa}. The cross section reads \begin{equation} \lim_{\mathcal{T}_0 \to 0} d\sigma_{pp \to V+X}^{\rm N^3LO}\left( \mathcal{T} \leq \mathcal{T}_0 \right) \sim \, B \otimes B \otimes S \otimes d\sigma_{pp \to V}^{\rm N^3LO}\,, \label{eq:fact0jet} \end{equation} where the two functions $B$ stand for the beam functions associated with each of the initial-state partons and $S$ represents the soft function. The general factorization formula for $N$-jettiness was originally derived in SCET~\cite{Bauer-ml-2000ew,Bauer-ml-2000yr,Bauer-ml-2001ct,Bauer-ml-2001yt,Bauer-ml-2002nz}. The factorization of soft and collinear radiation, made apparent in~\cref{eq:fact0jet}, is the key property of the 0-jettiness variable that simplifies the calculation of the differential cross section in the small-$\mathcal{T}$ limit. The cross-section formula~\cref{eq:fact0jet} implies that, in order to employ the $0$-jettiness slicing to compute the N$^3$LO corrections to $pp \to V+X$, the beam and soft functions must be known at the same perturbative order. While the soft function is a purely perturbative object and can, at least in principle, be computed order-by-order in perturbation theory, the beam-function computation requires a convolution of perturbative \emph{matching coefficients} $I_{ij}$ with the non-perturbative parton distribution functions (pdfs) $f_{j}$ \begin{equation} B_i = \sum_{\rm partons~j}\, I_{ij} \otimes f_j\,, \quad \mbox{where} \quad i,j=\{q,\bar{q},g\} \label{eq:matchI}. \end{equation} The computation of the N$^3$LO QCD corrections to the matching coefficient $I_{qq}$ is the main topic of this paper. At three loops, $I_{qq}$ receives contributions from three classes of partonic subprocesses: the emission of three collinear partons, which we will refer to as the triple-real contribution (RRR); the one-loop corrections to the emission of two collinear partons, or the double-real single-virtual contribution (RRV); and, finally, the two-loop virtual corrections to the emission of one collinear parton, or the single-real double-virtual contribution (RVV). In a previous paper~\cite{Melnikov-ml-2018jxb}, we presented the master integrals required for the calculation of the RRV contribution with two emitted gluons to the matching coefficient $I_{qq}$. In this paper, we focus on the master integrals required for the computation of the RRR contribution to the matching coefficient that originate from the process where the initial-state quark emits three collinear gluons before entering the hard scattering process. We note that the same master integrals can be used to compute the $N_f$-enhanced triple-real contribution to $I_{qq}$, caused by the emission of a gluon and a quark-antiquark pair collinear to the initial-state quark. The rest of the paper is organized as follows: in~\cref{sec:amp} we explain how to compute the RRR contribution to the matching coefficient $I_{qq}$ by considering collinear limits of scattering amplitudes and how reverse unitarity can be used to reduce this calculation to the computation of a large set of three-loop master integrals. We then show in~\cref{sec:masters} how these integrals can be computed using the method of differential equations. In~\cref{sec:checks}, we explain how the calculation was validated and we present our final results in~\cref{sec:results}. We conclude in~\cref{sec:concl}. The list of master integrals can be found in~\cref{app:listofmasters}. Some peculiar identities among master integrals are described in~\cref{app:extraids}. The results for the master integrals are provided in computer-readable format in the supplementary material attached to this paper. ]]>

\alpha_1 \beta_2$ and $\alpha_3 \beta_1 < \alpha_1 \beta_3$; we will call it $X_9^{(a)}$. After applying the transformations in~\cref{eq:FvA}, we find \begin{align} X_9^{(a)} &= \left(\prod_{i=1}^{3}\int\limits_0^1 d \alpha_i d\beta_i \, (\alpha_i \beta_i)^{-\eps}\right) \frac{\delta(\alpha_{123}-1)\delta(\beta_{123}-1)} {\alpha_{12} \, \beta_{13}\,\alpha_1\,\alpha_2 \,\beta_1\,\beta_3} \theta(\alpha_2 \beta_1 - \alpha_1 \beta_2) \label{eq:X9av1} \\&\qquad \times\theta(\alpha_1 \beta_3 - \alpha_3 \beta_1)\, {}_2F_1\left(1,1+\eps;1-\eps;\frac{\alpha_1 \beta_2}{\alpha_2 \beta_1}\right) {}_2F_1\left(1,1+\eps;1-\eps;\frac{\alpha_3 \beta_1}{\alpha_1 \beta_3}\right) \,. \nonumber \end{align} Upon changing variables $\beta_2 \to r = \alpha_1 \beta_2/(\alpha_2 \beta_1)$ and $\beta_3 \to \mu = \alpha_3 \beta_1/(\alpha_1 \beta_3)$ and integrating over $\beta_1$ to remove the delta function, we obtain \begin{align} \begin{aligned} X_9^{(a)} &= \int \limits_0^1 \frac{dr d\mu \, r^{-\eps}\mu^{-2\eps}}{(1-r)^{1+2\eps}(1-\mu)^{1+2\eps}} {}_2F_1\left(-2\eps,-\eps;1-\eps;r\right) {}_2F_1\left(-2\eps,-\eps;1-\eps;\mu\right) \\&\quad \times\left(\prod_{i=1}^{3}\int \limits_0^1 d \alpha_i \, \alpha_i^{-\eps}\right) \frac{\delta(\alpha_{123}-1)}{\alpha_{12}\,\alpha_1\,\alpha_2} (\alpha_3+\alpha_1\mu+\alpha_2 r \mu)^{3\eps}\, \,. \label{eq:X9av2} \end{aligned} \end{align} In~\cref{eq:X9av2} we have re-written the hypergeometric functions to make them regular in the $r \to 1$ and $\mu \to 1$ limits. We proceed by integrating out $\alpha_2$ and change the integration variables $\alpha_1 \to \xi = \alpha_1/(1-\alpha_3)$ and $\alpha_3 \to f = \alpha_3/( \mu (1-\alpha_3))$. We obtain \begin{align} \begin{aligned} X_9^{(a)} &= \int \limits_0^1 \frac{dr d\mu \, r^{-\eps}\mu^{-\eps}}{(1-r)^{1+2\eps}(1-\mu)^{1+2\eps}} {}_2F_1\left(-2\eps,-\eps;1-\eps;r\right) {}_2F_1\left(-2\eps,-\eps;1-\eps;\mu\right) \\&\quad \times\int \limits_0^1 \frac{d\xi \, (1-\xi)^{-2\eps}}{\xi^{1+2\eps}} \int \limits_0^\infty \frac{df \, f^{-2\eps} (1+\mu f)^{3\eps}}{f+\xi} (f + r + \xi - \xi r)^{3\eps}\,. \label{eq:X9av3} \end{aligned} \end{align} The integral in~\cref{eq:X9av3} is singular; the overlapping logarithmic singularities appear at $r = 1, \mu = 1, \xi = 0$ and $f\in\{0,\infty\}$. These singularities are disentangled by performing suitable (iterated) subtractions, after which the resulting integrals are carried out using the program \texttt{HyperInt}~\cite{Panzer-ml-2014caa}. The other independent contributions are obtained in a similar fashion. Upon adding all the contributions, we obtain the result for $X_9$, \begin{align} X_9 &= \frac{3}{4 \epsilon ^4} -\frac{5 \pi ^2}{4 \epsilon ^2} -\frac{42 \zeta _3}{\epsilon } -\frac{13 \pi ^4}{10} +\left(43 \pi ^2 \zeta _3-720 \zeta _5\right) \epsilon +\left(429 \zeta _3^2-\frac{129 \pi ^6}{140}\right) \epsilon ^2 + \mathcal{O}(\eps^3)\,. \label{eq:B9_X9} \end{align} The boundary constant $\widetilde{C}_9$ is easily obtained from this result. \boldmath ]]>