An essential part of any factorisation proof is the demonstration that the exchange of Glauber gluons cancels for the considered observable. We show this cancellation at all orders for double Drell-Yan production (the double parton scattering process in which a pair of electroweak gauge bosons is produced) both for the integrated cross section and for the cross section differential in the transverse boson momenta. In the process of constructing this proof, we also revisit and clarify some issues regarding the Glauber cancellation argument and its relation to the rest of the factorisation proof for the single Drell-Yan process.

Article funded by SCOAP3

R$ (e.g.\ when a collinear momentum in $R$ is hard in $R''$). This unwanted contribution is removed by the subtraction term $- T_{R''} C_R\ms\Gamma$ in the approximant $C_{R''} \Gamma$ for the larger region. As shown in~\cite{Collins-ml-2011zzd}, the subtractions also correctly treat the case where two regions intersect each other (collinear momenta in different directions have the soft region as a common smaller region and thus intersect). \item \textbf{Back to real momenta.}\label{enum:deform-back} After Grammer-Yennie approximations have been made, one can deform the soft momenta back to the real axis. This requires a suitable choice for the auxiliary vectors and the $i\varepsilon$ prescription in the Grammer-Yennie approximants for soft gluons. Because of the soft subtraction terms discussed in point~\ref{enum:soft-coll-subtr}, the same holds for the Grammer-Yennie approximants for collinear gluons. The $i\varepsilon$ prescriptions in~\eqref{GY-collin} and~\eqref{GY-soft}, together with the choice of auxiliary vectors specified in section~\ref{sec:GY-rearrange} are such that the poles in the Grammer-Yennie approximants do not obstruct the contour deformation for soft momenta. If one uses a different prescription, one must show that the contributions from poles crossed during the contour deformation are power suppressed or cancel in the final factorisation formula. Deforming contours back at this stage enables one to initially choose momentum routings for individual graphs as one finds suitable for establishing the cancellation of Glauber gluon contributions (cf.\ our comment in point~\ref{enum:kin}). In point~\ref{enum:ward}, the conditions on loop momentum routings are more stringent since we consider sums over different graphs, whose momentum routings must match. Having deformed soft gluon momenta back to the real axis, one can readily make the required changes of integration variables in the graphs. \item \textbf{Ward identities and Wilson lines.}\label{enum:ward} At this point Ward identities are applied to the contractions $\hat{\ell}^\nu H(\hat{\ell})_\nu$ obtained by the Grammer-Yennie approximation. This removes collinear gluons entering $H_1$ and $H_2$ (except for transversely polarised ones) and introduces Wilson lines along $v_A$ in $A$ and along $v_B$ in $B$, with one Wilson line attaching to each of the four physical partons. Likewise, Ward identities are applied to $\tilde{\ell}^{\nu}\bs A_{\nu}(\tilde{\ell})$ and $\tilde{\ell}^{\nu}\bs B_{\nu}(\tilde{\ell})$. If necessary, small light-cone components of loop momenta are shifted such that the corresponding arguments of collinear and soft factors are separate integration variables. This removes all soft gluons entering $A$ or $B$ and provides Wilson lines to $S$. The soft factor is then given as an expectation value of eight Wilson lines, one for each physical parton in $A$ and $B$. With the $i\varepsilon$ prescriptions and auxiliary vectors mentioned above, the Wilson lines in collinear and soft factors are past pointing, where ``past'' refers to the space-time variable over which the gluon potential is integrated. The result of this procedure is shown in figure~\ref{fig:strategy:factorizedform}. The collinear factors $A$ and $B$ are a first version of double parton distributions, to be modified by subtractions as discussed below. They are given by operator matrix elements, which for two-quark distributions schematically read \begin{align} \label{dpd-matr-el} \langle\ms p |\, (\bar{q}_2^{} W^\dagger)_{k'}^{} \ms\Gamma_2^{}\ms (W\bs q_2^{})_{k}^{}\, (\bar{q}_1^{} W^\dagger)_{j'}^{} \ms\Gamma_1^{}\ms (W\bs q_1^{})_{j}^{} \,| p \ms\rangle \,, \end{align} where $k'$, $k$, $j'$ and $j$ are colour indices in the fundamental representation and $W$ is a Wilson line in the appropriate direction. $\Gamma_{1,2}$ are Dirac matrices, and the indices $1,2$ on the quark fields indicate that (after a Fourier transform) they create or annihilate a parton with momentum fraction $x_{1,2}$ in the proton. More detail is given in section 2.2 of~\cite{Diehl-ml-2011yj}. \begin{figure} \centering \includegraphics[width=0.5\textwidth]{diagrams/factorizeddoubledrellyan.eps} ]]>

0 \,, \end{align} whereas $a^+$ can have either sign, depending on the relative size of $\alpha_3^{} q_1^+$ and $\alpha_2^{} q_2^+$. By power counting, the region where $\ell^+ \sim \ell^- \sim \Lambda^2/Q$ could indeed give a leading contribution to $I$, and we have to show that this is not the case. According to the general rule explained in section~\ref{sec:scope-and-method} the integration over $\ell^-$ has a pinched pair of poles in that region, whereas the one over $\ell^+$ does not. To further analyse $I$ we carry out the integral over $\ell^+$ using Cauchy's theorem. For $\ell^- + a^- > 0$ both poles in $\ell^+$ are on the same side of the real axis and one gets zero. From the region $\ell^- + a^- < 0$ we obtain, picking up the residue at $\ell^+ = - i \eta$ for simplicity, \begin{align} \label{eq:ll1} I &= - 2\pi i\, \frac{3!\ms N}{D} \int [\df^4 \alpha] \int^{-a^-}_{-\infty} \frac{\df \ell^-}{\ell^- - i \eta'}\; \frac{1}{[2 a^+ \ell^- + A + i \varepsilon]^4} , \end{align} where we have dropped $i\eta$ in the last denominator, since it comes with the same sign as $i\varepsilon$ and thus does not change the position of the pole. We note that the evaluation of Feynman integrals in light-cone coordinates has to be done with some care, because in some cases the naive application of Cauchy's theorem can lead to wrong results. In the case at hand we have cross checked our results with the method described in appendix~\ref{app:feynmanint}. As explained there, we could not have used Cauchy's theorem for the $\ell^+$ integration in~\eqref{eq:ll3} if we had kept the full numerator of the graph given in~\eqref{double-box-full-num}, because terms with $\ell^+$ or $(\ell^+)^2$ would have cancelled the denominator $\ell^+ + i\eta$. Using the method just mentioned, we have checked that these terms give a power suppressed contribution and can hence be discarded, as we argued on the basis of power counting before~\eqref{eq:ll3}. Since $\ell^-$ is at least of size $Q$ in~\eqref{eq:ll1}, we have $2 a^+ \ell^- + A \sim Q^2$ or bigger and obtain a power suppressed result for $I$, as we needed to show. We observe that in~\eqref{eq:ll1} we have two poles \begin{align} \ell^-_1 &= i \eta' \,, & \ell^-_2 &= - A /(2 a^+) - \text{sgn}(a^+) \ms i \varepsilon \,, \end{align} which are pinched for $a^+ > 0$ in agreement with the general rule stated above. However, they are far away from the integration region and therefore do not matter. We can also explicitly calculate~\eqref{eq:ll1} by partial fractioning, which gives \begin{align} \label{eq:ll2} I = - 2\pi i\, \frac{3!\ms N}{D} \int [\df^4 \alpha]\, &\biggl\{ \frac{1}{(A + i \varepsilon)^4} \ \log{\left| \frac{2 a^+ a^- - A}{2 a^+ a^-} \right|} \nonumber \\ &\quad + \frac{i \pi}{(A + i \varepsilon)^4}\, \bigl[ \Theta(2 a^+ a^- - A) + \Theta(a^-) - \Theta(a^+) - 1 \bigr] \nonumber \\ &\quad - \frac{1}{(A + i \varepsilon)^3 (A - 2 a^+ a^- + i \varepsilon)} - \frac{1}{2} \frac{1}{(A + i \varepsilon)^2 (A - 2 a^+ a^- + i \varepsilon)^2} \nonumber \\ &\quad - \frac{1}{3} \frac{1}{(A + i \varepsilon) (A - 2 a^+ a^- + i \varepsilon)^3} \biggr\} \,. \end{align} Since $a^+a^- \sim Q^2$, the potentially leading terms in~\eqref{eq:ll2} are those proportional to $1/A^4$. In the generic region of Feynman parameters, the logarithm gives a suppression by a factor $\Lambda^2/Q^2$ and the $\Theta$ function terms cancel exactly because $a^- > 0$. This confirms that $I$ is power suppressed. ]]>

0$ and $b^- > 0$: both poles are below the real axis, \item $a^+ > 0$ and $b^- < 0$: the integration contour is pinched in the Glauber region. \end{enumerate} If $\ell^+$ is at least of order $Q$ then $2 b^- \ell^+ + B \sim Q^2$ or bigger and we obtain a power suppressed integral. This holds in case 1, and also in case 2 if we deform the integration contour away from the real axis on the semi-circle going through the points $-a^+$, $i a^+$, $a^+$. It holds in case 3 if we deform the contour to the corresponding semi-circle in the lower half-plane, thus picking up the residue of the eikonal propagator. The additional term from the residue is proportional to $[A + i\varepsilon]^4\, [B + i \varepsilon]^3$, so that $I \sim 1/\Lambda^{10}$ as we expected. However, this contribution vanishes when one integrates over either $k_1^-$ or $\bar{k}_2^+$. To see this, we rewrite \begin{align} \label{combined-denoms} [A + i\varepsilon]^{-4}\, [B + i \varepsilon]^{-3} &= [2 a^+ k_1^- + 2 (\alpha_1^{} q_2^- - \alpha_1^{} p^- - \alpha_2^{} p^-)\ms \bar{k}_2^+ + \cdots + i\varepsilon]^{-4} \nonumber \\[0.1em] &\quad \times [2 (\beta_1^{} q_1^+ - \beta_1^{} \bar{p}^+ - \beta_2^{} \bar{p}^+)\ms k_1^- - 2 b^- \bar{k}_2^+ + \cdots + i \varepsilon]^{-3} \,, \end{align} where the ellipses denote terms independent of $k_1^-$ and $\bar{k}_2^+$. We see that for $a^+ > 0$ the poles of~\eqref{combined-denoms} in $k_1^-$ are on the same side of the real axis and thus give a zero integral (note that $a^+$ depends on $\bar{k}_2^+$ but not on $k_1^-$). The same result is obtained if one first integrates over $\bar{k}_2^+$ under the condition that $b^- < 0$. We have thus shown again that the Grammer-Yennie approximation works, but this required integration over several loop~\mbox{variables}. ]]>

0$, all poles lie on the same side of the real axis and one gets zero. For $\ell^- + a^- < 0$ we close the contour in the lower half plane. Approximating $(1- \delta^4) \approx 1$, we then get \begin{align} \label{eq:sl1} I_{\mathrm{SL}} &= - 2\pi i\, \frac{3!\ms N}{D} \int [\df^3 \alpha]\; \biggl\{ \int_{- \infty}^{-a^-} \frac{\df \ell^-}{\ell^- - i \eta''}\; \frac{1}{[2 \delta^2 (\ell^-)^2 + 2 \ell^- ( a^+ + \delta^2 a^-) + A + i \varepsilon]^4} \nonumber \\ & \qquad\quad - \int_{- \infty}^{-a^-} \frac{\df \ell^-}{\ell^- - i \eta''}\; \frac{1}{[2 \delta^{-2} (\ell^-)^2 + 2 \ell^- (a^+ + \delta^{-2} a^-) + A + i \varepsilon]^4} \biggr\} \end{align} with $\eta'' = \eta' - \delta^2 \eta$. Our previous argument for limiting ourselves to the generic region of Feynman parameters remains valid, so that we can use the properties in~\eqref{eq:powercounting}. With $a^- > 0$ the ambiguous sign of $\eta''$ does not matter since the pole $\ell^-_{\text{eik}} = i \eta''$ of the eikonal propagator is outside the integration region. Since $\ell^- \sim Q$ or larger in the integral~\eqref{eq:sl1}, the integrand is strongly power suppressed, as in the case of lightlike Wilson lines. It remains to discuss poles close to the integration path. With~\eqref{eq:powercounting} we find that that poles of the combined Feynman propagators are approximately at \begin{align} \ell^-_1 &\approx - A /(2 a^+) - \text{sgn}(a^+) \ms i \varepsilon \,, & \ell^-_2 &\approx - \delta^{-2} a^+ + \text{sgn}(a^+) \ms i \varepsilon \intertext{for the first term in \protect\eqref{eq:sl1} and at} \ell^-_3 &\approx - \delta^2 A /(2 a^-) - i \varepsilon \,, & \ell^-_4 &\approx - a^- - \delta^2 a^+ + i \varepsilon \end{align} for the second term. With $A/(2 a^+) \sim \Lambda^2/Q$ and $\delta^2 A/(2 a^-)\sim \Lambda^4/Q^3$, the poles at $\ell^-_1$ and $\ell^-_3$ are close to zero and thus far away from the integration region. For $a^+ > 0$ the pole at $\ell^-_2$ (with a huge real part of order $Q^3/\Lambda^2$) is infinitesimally close to the integration path but can easily be avoided by contour deformation. The pole at $\ell^-_4$ is located at a short distance of order $\Lambda^2/Q$ from the endpoint $\ell^- = a^-$ of the integration and therefore deserves closer examination. If $a^+ < 0$ then the pole is to the right of the endpoint, so that the closest point on the integration path is the endpoint itself. There we have \begin{align} \label{denom-at-endpoint} 2 \delta^{-2} (\ell^-)^2 + 2 \ell^- (a^+ + \delta^{-2} a^-) + A &= - 2 a^+ a^- + A & \mbox{for $\ell^- = a^-$} \,, \end{align} which is of order $Q^2$ and provides a lower limit for the size of the left hand side on the entire path (i.e.\ an upper limit for the size of the integrand, where this expression is raised to the fourth power in the denominator). This is sufficient to suppress the second integral in~\eqref{eq:sl1}. If $a^+ > 0$ then the pole is to the left of the integration endpoint. It can however be avoided by contour deformation, and on a semi-circle around $\ell^-_4$ going to the endpoint (see figure~\ref{fig:poles}a) the upper limit for the denominator based on~\eqref{denom-at-endpoint} remains valid. Thus the contribution to the integral is power suppressed, as we anticipated. \begin{figure} \centering \subfigure[]{\includegraphics[height=0.25\textwidth]{diagrams/poles_sl_apgz.eps}} \phantom{xx} \subfigure[]{\includegraphics[height=0.25\textwidth]{diagrams/poles_tl_apgz.eps}} ]]>

0$.]]>

0$ and take the residue at $\ell^+ = \ell^+_{\text{eik},2}$. Approximating $(1- \delta^4) \approx 1$ we then get \begin{align} \label{eq:tl1} I_{\mathrm{TL}} &= - 2\pi i\, \frac{3!\ms N}{D} \int [\df^3 \alpha]\; \biggl\{ \int_{- \infty}^{-a^-} \frac{\df \ell^-}{\ell^- - i \eta''}\; \frac{1}{[- 2 \delta^2 (\ell^-)^2 + 2 \ell^- ( a^+ - \delta^2 a^-) + A + i \varepsilon]^4} \nonumber \\ & \qquad\quad + \int_{-a^-}^{\infty} \frac{\df \ell^-}{\ell^- - i \eta''}\; \frac{1}{[- 2 \delta^{-2} (\ell^-)^2 + 2 \ell^- (a^+ - \delta^{-2} a^-) + A + i \varepsilon]^4} \biggr\}\,, \end{align} where now $\eta'' = \eta' + \delta^2 \eta$ is always positive. In addition to the eikonal pole at \begin{align} \ell^-_{\text{eik}} = i \eta'' \end{align} we now have poles at \begin{align} \ell^-_1 &= - A /(2 a^+) - \text{sgn}(a^+)\ms i \varepsilon \,, & \ell^-_2 &= \delta^{-2} a^+ + \text{sgn}(a^+)\ms i \varepsilon \intertext{for the first term in \protect\eqref{eq:tl1} and at} \ell^-_3 &= \delta^2 A /(2 a^-) + i \varepsilon \,, & \ell^-_4 &= - a^- + \delta^2 a^+ - i \varepsilon \end{align} for the second term. For the first integral in~\eqref{eq:tl1}, the situation is essentially identical to that for the first integral in~\eqref{eq:sl1} (except that $\ell_2^-$ now has a negative real part for $a^+ < 0$ rather than $a^+>0$), and we can use the same strategy to avoid the poles. In the second integral, we have the poles at $\ell^-_{\text{eik}}$ and $\ell^-_3$ near the origin. These can be avoided by contour deformation: a semi-circle around the origin with radius of order $\Lambda^2/Q$, as in figure~\ref{fig:poles}b, is sufficient to ensure that the integrand remains power suppressed. Finally, there is the pole at $\ell_4^-$ near the endpoint of integration. For this pole we can use the same logic as in the spacelike case to determine that it does not result in a leading contribution. In summary then, there is no leading contribution to $I_{\text{TL}}$, and the factorised expression for the double box is valid to leading power with timelike Wilson lines as well. ]]>

i} \kappa_{ij}^- - \rho_i^- + \rho^-_{i-1}
\bigr)$ at the vertex $i$, although this $\delta$ function now includes
the fictitious momenta $\rho_i^-$ and $\rho_{i-1}^-$. Using the
$\delta$ functions to perform the $\rho_i^-$ integrals, we generate the
``light-cone energy denominators'' in~\eqref{eq:LCdenomdef} from the
denominator factors in~\eqref{eq:thetamta}:
\begin{align} \label{eq:denomv1}
\frac{i}{\rho_i^- + i\epsilon} &=
\frac{i}{\sum\limits_{j\le i} p_j^-
- \sum\limits_{j\le i,\, i

\xi}}\ell_j^- - \sum\limits_{\substack{\text{lines } L \\[0.1em] L \in \xi}} \kappa_L^- + i \epsilon} \\* \nonumber & \quad \times 2\pi \delta \biggl( p^- - k^- - \sum\limits_{\substack{\text{vertices } j \\ j > F_A}} \ell_j^- - \sum\limits_{\substack{\text{lines } L \\[0.1em] L \in F_A}} \kappa_L^- \biggr) \\* \nonumber & \quad \times \prod_{\substack{\text{states } \xi \\ F_A < \xi < H'}} \dfrac{1}{p^- -k^- - \sum\limits_{\substack{\text{vertices } j \\ j > \xi}}\ell_j^- - \sum\limits_{\substack{\text{lines } L \\[0.1em] L \in \xi}} \kappa_L^- - i \epsilon} \,, \\ I'_{T}(\tilde{\ell}_j) &= \prod_{\substack{\text{states } \xi \\ H' < \xi}} \dfrac{1}{p^- - \sum\limits_{\substack{\text{vertices } j \\ j > \xi}}\ell_j^- - \sum\limits_{\substack{\text{lines } L \\[0.1em] L \in \xi}} \kappa_L^- - i \epsilon} \,, \end{align}}where $p$ is the proton momentum and (in contrast to section~\ref{sec:LCPTderivation}) we simply number the on-shell minus momenta of the lines $\kappa_L$ sequentially. The initial state factors $I_T^{}$ and $I'_T$ only have poles in the lower half plane for the momenta $\ell^-_j$ entering prior to $H$ or $H'$. However, the $\ell^-_j$ poles of the factor $F_T$ are pinched and seem to prevent us from deforming the $\ell^-_j$ integration out of the Glauber region. To proceed, we label the $N$ states $\xi_f$ in $F_T$ by an index $f = 1,\ldots,N$ running from left to right, and abbreviate the sum of on-shell minus momenta in state $f$ as \begin{equation} D_f = \sum\limits_{\substack{\text{lines } L \\[0.1em] L \in \xi_f}} \kappa_L^- \,. \end{equation} The sum over all final state cuts $F_A$ for $F_T$ gives \begin{align} \label{eq:FTcutsum} \sum_{F_A} F_T(k,\tilde{\ell}_j) &= \sum_{c=1}^{N} \, \Biggl[ \; \prod_{f=1}^{c-1} \dfrac{1}{p^- - k^- - \sum\limits_{j>f} \ell_j^- -D_f + i \epsilon} \\ \nonumber & \quad\qquad \times 2\pi \delta \biggl( p^- - k^- - \sum\limits_{j>c} \ell_j^- - D_c \biggr) \prod_{f=c+1}^{N} \dfrac{1}{p^- -k^- - \sum\limits_{j>f} \ell_j^- -D_f - i \epsilon} \, \Biggr] \\ \nonumber & = i \, \Bigg[ \, \prod_{f=1}^{N} \dfrac{1}{p^- -k^- - \sum\limits_{j>f} \ell_j^- -D_f + i \epsilon} - \prod_{f=1}^N \dfrac{1}{p^- - k^- - \sum\limits_{j>f} \ell_j^- -D_f - i \epsilon} \, \Biggr] \,, \end{align} which is essentially the Cutkosky identity~\cite{Cutkosky-ml-1960sp, Veltman-ml-1994wz} in the light-cone formalism and can readily be obtained by using \begin{align} 2\pi\delta(x) = i\, \biggl[ \dfrac{1}{x+i\epsilon} - \dfrac{1}{x-i\epsilon} \biggr] \,. \end{align} It is now easy to perform perform the integral of~\eqref{eq:FTcutsum} over the external momentum $k^-$ (note that no other factor in~\eqref{eq:CApart} depends on this variable): \begin{align} \label{eq:one-or-zero} \sum_{F_A} \int \dfrac{\df k^-}{2\pi}\, F_T(k,\tilde{\ell}_j) = & \begin{cases} \, 1 \quad \text{if } N =1 \\ \, 0 \quad \text{otherwise} \end{cases} . \end{align} For $N>1$ each product on the r.h.s.\ of~\eqref{eq:FTcutsum} decreases faster than $1/k^-$ at infinity, so that one obtains zero by the theorem of residues, whereas the result for $N=1$ simply corresponds to the original $\delta$ function on the l.h.s. Once we sum over all cuts, $F_T$ thus gives either $0$ or $1$. The pinched poles in $\ell_j^-$ are then removed, and we can deform $\ell_j^-$ into the upper half plane as we set out to show. Notice that if one does not consider the inclusive Drell-Yan cross section (differential or not in the transverse momentum of the vector boson), but instead some observable that is not constant for all the cuts of $F_T$, then this argument fails and Glauber gluon exchange may not cancel~\cite{Gaunt-ml-2014ska, Zeng-ml-2015iba}. As already stated at the beginning of section~\ref{sec:allorder}, the preceding derivation works not only for the original terms in the region decomposition of a graph, but also for the subtraction terms. In the graph of figure~\ref{fig:subt-terms}b one neglects $\tvec{\ell}_j$ and the masses of certain lines inside $A'$, which leads to different values of the on-shell momenta $\kappa_L^-$ in the above expressions, but does not affect the arguments otherwise. ]]>

H_1} \ell_j - D_f + i\epsilon}
\nonumber \\
& + \frac{1}{p^- + k^- - K ^-\!/\,2 + \sum_{j

0$. In the second line of~\eqref{int-final}, the subtraction term leads to a suppression for $\ell^- \sim |\tvec{\ell}|$ or larger, whereas for $\ell^- \ll |\tvec{\ell}|$ the terms with $\pm \delta^2$ in the eikonal propagators are negligible. Thus the sign of the $\delta^2 \ell^-$ term is not important here either. The preceding arguments also work if the gluon couples to a Wilson line left of the final state cut and then crosses the cut; in that case the gluon propagator in~\eqref{int-start},~\eqref{int-coll-region} and the starting expression of~\eqref{int-final} is to be replaced by $2\pi \delta(\ell^2)\, \Theta(\ell^-)$. This completes our proof of~\eqref{diff-AT-AS}. Note that this result cannot be obtained by power counting alone. For $|\tvec{\ell}| \sim \Lambda$, the region with $\ell^- \sim \Lambda^2/Q$ and $\ell^+ \sim \Lambda^4/Q^3$ gives a leading-power contribution to~\eqref{int-start} since $\ell^-$ is too small for the Grammer-Yennie approximation in the subtraction term to work, and the smallness of the eikonal propagator compensates the smallness of the integration volume of $\ell^+$ in that region, where one cannot neglect $\pm \delta^2 \ell^-$. The position of poles in $\ell^+$ was essential to pick up only the pole of the gluon propagator using Cauchy's theorem, and then the mass-shell condition for $\ell$ excluded the above dangerous region. Now let us consider the soft and collinear-to-$A$ or $B$ approximants to the full set of graphs in which one gluon extends between the left and right moving collinear sectors. We collect together graph approximants in which a gluon attaches in all different places to the upper and lower parts of the graph, with these attachments being either to the left or to the right of the final-state cut in each case. This allows us to use Ward identities to convert contractions of $\ell^\mu$ with $A_\mu(\ell)$ or with $B_\mu(\ell)$ to collinear factors $A$ and $B$ without an external gluon. We write $A^{i}$ and $B^{i}$ with $i=T,S$ for the unsubtracted collinear factor with one gluon coupling to a time- or spacelike Wilson line, and $S^{ij}$ for a graph contributing to the one-loop expression of the soft factor with time- or spacelike Wilson lines (with the gluon coupling to $v_L$ and $v_R$ on a definite side of the final state cut). The difference between the factorised expressions with timelike and spacelike auxiliary Grammer-Yennie vectors (or Wilson lines), after the sum over gluon attachments as specified, is \begin{align} \label{diff-T-S1} \Delta &= (A^T - A\ms S^{TT}) B + A (B^T - S^{TT} B) + A\ms S^{TT} B \nonumber \\ & \quad - (A^S - A\ms S^{SS}) B - A (B^S - S^{SS} B) - A\ms S^{SS} B \,. \end{align} We know that the factorised expression with spacelike Wilson lines is a good approximation to the full expression, so if this difference is negligible, the expression with timelike Wilson lines is good, too. The result~\eqref{diff-AT-AS} for subtracted collinear factors reads $A^T - A\ms S^{TS} \approx A^S - A\ms S^{SS}$ in this notation, and its analogue for left collinear factor reads $B^T - S^{ST} B \approx B^S - S^{SS} B$. Using this in~\eqref{diff-T-S1}, we obtain \begin{align} \label{diff-T-S} \Delta &\approx - A (S_{TT} + S_{SS} - S_{TS} - S_{ST}) B \,. \end{align} It is easy to evaluate the graphs for the one-loop soft factor explicitly (see e.g.\ section~3.3.1 of~\cite{Diehl-ml-2011yj}). We find that, up to power corrections, the result for each graph is a factor independent of $v_L$ and $v_R$ times $\log(- v_L^-/v_L^+ + \sigma_L^{}\ms i \varepsilon) + \log(- v_R^+/v_R^- + \sigma_R^{}\ms i \varepsilon)$, where $\sigma_{L,R}^{} = -1$ ($+1$) if the corresponding eikonal line is left (right) of the final state cut. Plugging this result into~\eqref{diff-T-S}, we get zero. This result can also be shown in a simple way as follows. Up to global factor the combination of soft factors can be written either as \begin{align} \label{TS-softfacts} \int \df\ell^+\ms \df\ell^- & \biggl[ \frac{1}{\ell^+ + \delta^2 \ell^- - i\varepsilon} - \frac{1}{\ell^+ - \delta^2 \ell^- - i\varepsilon} \biggr] \frac{i}{2\ell^+ \ell^- - \tvec{\ell}{}^2 + i\epsilon} \nonumber \\ \times & \biggl[ \frac{1}{\ell^- + \delta'^2 \ell^+ + i\varepsilon} - \frac{1}{\ell^- - \delta'^2 \ell^+ + i\varepsilon} \biggr] \end{align} or as its analogy with the gluon propagator replaced by $2\pi \delta(\ell^2)\, \Theta(\ell^-)$. The differences of eikonal propagators yield a power suppression unless $\ell^+ \lsim \delta^2 \ell^-$ (for the first square bracket) and $\ell^+ \gsim \ell^- /\delta'^{2}$ (for the second square bracket). However, both conditions cannot be satisfied at the same time, so one always has a suppression in~\eqref{TS-softfacts}. Let us now move to the double Drell-Yan process. In each collinear factor we now have twice as many Wilson lines to which the single gluon may attach. Demonstrating that~\eqref{diff-AT-AS} still holds at the one-gluon level is a straightforward copy of the single Drell-Yan argument, provided that one routes $\ell$ through the relevant Wilson line, down into $A$ through the gluon and back up through the parton associated with the Wilson line. In the formulae analogous to~\eqref{diff-T-S1} and~\eqref{diff-T-S}, both $A^{i}$ and $B^{j}$ are now summed over the two possible gluon attachments to Wilson lines. Likewise, the one-loop expression for $S^{ij}$ in~\eqref{diff-T-S1} and~\eqref{diff-T-S} now contains a sum over the four possibilities how the gluon can attach to the different left- and right-moving Wilson lines (always on a definite side of the final-state cut). For each of these possibilities, the longitudinal structure of the loop integral is exactly the same as for single Drell-Yan production. The argument of~\eqref{TS-softfacts} works on a diagram-by-diagram basis, and of course also for the relevant sums over graphs. Thus our overall argument carries over to the double Drell-Yan process. The simple structure in~\eqref{diff-T-S} is only obtained with a single exchanged gluon. We can therefore not decide from the above whether timelike Wilson lines are generally suitable for factorisation. However, the proof in the present section shows that a counter-example would require at least two exchanged gluons. ]]>