Logarithmic electroweak corrections to gauge-boson pair production at the LHC

TL;DR

This work provides a comprehensive treatment of the complete logarithmic electroweak $O(\alpha)$ corrections to gauge-boson pair production at the LHC within a double-pole framework. By combining a virtual correction analysis in the DPA with full real-photon emissions and a high-energy logarithmic expansion, the authors deliver a Monte Carlo tool capable of precision predictions for $pp\to 4f(+\gamma)$ in purely leptonic channels. They derive and implement detailed analytic forms for leading and subleading soft-collinear, collinear, and parameter-renormalization corrections, along with non-factorizable contributions, and rigorously address soft/collinear photon emission and initial-state mass-singularity absorption. The results demonstrate substantial negative EW corrections in high-energy regimes (roughly 5–30%), emphasizing their critical role in interpreting LHC data, constraining triple gauge couplings, and modeling backgrounds to new-physics searches. Collectively, this work enhances the accuracy of SM predictions for di-boson processes at the LHC, guiding both precision tests and new-physics explorations at high invariant masses and wide scattering angles.

Abstract

We have studied the effects of the complete logarithmic electroweak O(alpha) corrections on the production of vector-boson pairs WZ, ZZ, and WW at the LHC. These corrections are implemented into a Monte Carlo program for pp -> 4f (+γ) with final states involving four or two leptons using the double-pole approximation. We numerically investigate purely leptonic final states and find that electroweak corrections lower the predictions by 5-30% in the physically interesting region of large di-boson invariant mass and large angle of the produced vector bosons.

Figures (4)

• Figure 1: Structure of the process ${\rm p}$ p${\rm p}$ p$\to V_1 V_2+X \to 4f+X$
• Figure 2: The fundamental topologies for processes with six external fermions
• Figure 3: Dependence of the cross section for $\nu_e{\rm e^+}$ e^+$\mu^-\bar{\nu}_\mu$ production in the scenario (\ref{['eq:WWscenarioII']}) on the phase-space slicing cuts. Left: Dependence on $\delta_s$ for $\delta_c=10^{-4}$. Right: Dependence on $\delta_c$ for $\delta_s=10^{-4}$.
• Figure 4: Distributions for ${\rm W}$ W${\rm Z}$ Z$$ production: (a) Maximal transverse momentum of the charged leptons, (b) Missing transverse momentum. (c) Difference in rapidity between the reconstructed ${\rm Z}$ Z$$ boson and the charged lepton coming from the ${\rm W}$ W$$-boson decay. (d) Rapidity of the $\mu^-$. The contributions of the final states ${\rm e}$ e$^-\bar{$$\nu_{{\rm e}}\mu^-\mu^+$ and $\nu$ν$_{{\rm e}}{\rm e}$ e$^+\mu^-\mu^+$ are summed, and standard cuts as well as $P_{{\rm T}}({\rm Z}$ Z$)> 300\,{\rm GeV}$ are applied. The inset plots show the $\cal{O}\alpha)$O(α)$$ corrections relative to the Born results in per cent. We start discussing the scenario (\ref{['eq:WZscenarioI']}). In Fig. \ref{['fi:WZ_s1']} we have plotted the four distributions for the complete process ${\rm p}$ p${\rm p}$ p$\to {\rm e}$ e$^-\bar{$$\nu_{{\rm e}}\mu^-\mu^+, \nu$ν$_{{\rm e}}{\rm e}$ e$^+\mu^-\mu^+$, i.e. we sum over the two charge-conjugate final states. As a general feature the EW corrections are negative and lower the Born cross section by more than 10$\%$. For the individual distributions we observe the following. EW corrections reduce the distribution in $P_{{\rm T}}^{{\rm max}}(l)$ by the order of 10$\%$ at low to modest $P_{{\rm T}}^{{\rm max}}(l)$ values. This effect grows with increasing $P_{{\rm T}}^{{\rm max}}(l)$ as shown by the long tail where the contribution of EW corrections can amount to more than $30\%$. This is of course the result of enhanced EW logarithms at large energies, which are enforced by the large $P_{{\rm T}}^{{\rm max}}(l)$. The missing-transverse-momentum distribution shows the same qualitative behaviour. At low values the correction amounts to about $-12\%$, while at high $P_{{\rm T}}^{{\rm miss}}$ it increases up to $-40\%$. As stated in the literature, the large $P_{{\rm T}}$ region is an ideal place to look for new physics. As an example, the $P_{{\rm T}}({\rm Z}$ Z$)$ distribution has been found to be much more sensitive to new-physics effects than the ${\rm W}$ W${\rm Z}$ Z$$ invariant-mass distribution, which in principle should give a more direct access to the energy scale Baur:1995aj. This feature is shared by $P_{{\rm T}}^{{\rm max}}(l)$ and $P_{{\rm T}}^{{\rm miss}}$ we just discussed. As to angular distributions, EW corrections are maximal at low rapidity values in both cases, where once again effects due to new physics could be more enhanced. A low rapidity corresponds in fact to large scattering angles of the produced vector bosons in their rest frame. As shown in the lower left plot of Fig. \ref{['fi:WZ_s1']}, the distribution in the rapidity difference $\Delta y({\rm Z}$ Z$l)$ exhibits a characteristic dip, relic of an approximate radiation zero at high energy Baur:1994prl. Of course, new physics could have observable consequences on the shape of this variable Baur:1995aj. The general tendency is to fill in the dip, but in certain models the approximate zero may even become more pronounced. It is thus important to consider the impact of radiative corrections to this relevant signal. In the last decade, the effect of NLO QCD corrections has been extensively analysed Baur:1995ajDixon:1999di. It can completely spoil the significance of the dip, if one measures the inclusive ${\rm W}$ W${\rm Z}$ Z$+X$ production. By imposing a jet veto, the QCD corrections get drastically reduced to about 20$\%$ of the Born result, at the same time diminuishing the dependence of the NLO cross section on the factorization scale. As shown in the third plot of Fig. \ref{['fi:WZ_s1']}, EW corrections can be of the same order as QCD effects but with opposite sign. So, they slightly increase the dip. Distributions for ${\rm W}$ W${\rm Z}$ Z$$ production: (a) Maximal transverse momentum of the charged leptons. (b) Missing transverse momentum. (c) Difference in rapidity between the reconstructed ${\rm Z}$ Z$$ boson and the charged lepton coming from the ${\rm W}$ W$$-boson decay. (d) Rapidity of the $\mu^-$. The contributions of the final states ${\rm e}$ e$^-\nu_{\rm e}$ e$\mu^-\mu^+$ and ${\rm e}$ e$^+\bar{\nu}_{\rm e}$ e$\mu^-\mu^+$ are summed, and standard cuts as well as $M_{{\rm inv}}(ll^\prime\bar{l^\prime})> 500\,{\rm GeV}$ and $|\Delta y({\rm Z}$ Z$l)|< 3$ are applied. The last cut is omitted for the $\Delta y({\rm Z}$ Z$l)$ distribution in lowest order. The inset plots show the $\cal{O}\alpha)$O(α)$$ corrections relative to the Born results in per cent.Of course, radiative corrections do not only depend on the considered distribution but also on the selected cuts. Figure \ref{['fi:WZ_s2']} shows a second set of plots for the same set of distributions as above but in the scenario (\ref{['eq:WZscenarioII']}). The influence of the radiative corrections on the two momentum-like variables is analogous to the one observed in the previous case. The main difference between the two selected kinematical regions is in the shape of the $\Delta y({\rm Z}$ Z$l)$ distribution. Here, the radiation-zero dip strongly increases. This is due to the fact that the requirement $M_{{\rm inv}}(ll^\prime\bar{l^\prime})> 500\,{\rm GeV}$ forces the reconstructed ${\rm Z}$ Z$$ boson and the charged lepton from the ${\rm W}$ W$$-boson decay to be produced at large separation angle. This effect translates into a depletion of events in the central region of low rapidity difference. Radiative corrections are more pronounced in this suppressed region. To measure the significance of the EW corrections, a naive but direct way is to compare their magnitude with the expected statistical error. In Table \ref{['ta:WZ_s1']} we have listed the relative deviation $\Delta$ and the statistical accuracy, estimated by taking as a luminosity $L=100\,{\rm fb}^{-1}$ for two experiments, in the scenario (\ref{['eq:WZscenarioI']}) for some values of the cut on the transverse momentum of the reconstructed ${\rm Z}$ Z$$ boson. To this purpose, we sum over all eight final states ${\rm e^-}$ e^-$\bar{$$\nu_{{\rm e}}\mu^-\mu^+$, $\nu$ν$_{{\rm e}}{\rm e^+}$ e^+$\mu^-\mu^+$, $\mu^-\bar{\nu}_\mu{\rm e^-}$ e^-${\rm e^+}$ e^+$$, $\nu_\mu\mu^+{\rm e^-}$ e^-${\rm e^+}$ e^+$$, $\mu^-\bar{\nu}_\mu\mu^-\mu^+$, $\nu_\mu\mu^+\mu^-\mu^+$, ${\rm e^-}$ e^-$\bar{$$\nu_{{\rm e}}{\rm e^-}$ e^-${\rm e^+}$ e^+$$, and $\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm e^-}$ e^-${\rm e^+}$ e^+$$. In Table \ref{['ta:WZ_s2']}, we give the same entries but for the scenario (\ref{['eq:WZscenarioII']}) and for different values of the cut on the invariant mass of the three charged leptons. $ p${\rm p}$ p${\rm p}$ l_ll^l^P_{{\rm T}}^{{\rm cut}}({\rm Z}{\rm Z})~[{\rm GeV}]\sigma_{{\rm Born}}~[{\rm fb}]\sigma_{{\rm AEWS}}~[{\rm fb}]\sigma_{{\rm virt}}^{{{\rm finite}}}~[{\rm fb}]~\sigma_{{\rm EW}} ~[{\rm fb}]~~\Delta~[\%]~~1/\sqrt{2L\sigma_{{\rm Born}}}~[\%]2501.6721.5631.5531.489-10.95.53000.8760.7940.7890.761-13.17.63500.4890.4310.4280.413-15.510.14000.2870.2460.2440.236-17.813.24500.1750.1460.1450.141-19.716.95000.1110.0900.0890.087-21.221.2$Cross section for ${\rm p}$ p${\rm p}$ p$\to l\nu_l l^\prime \bar{l^\prime}$ for various values of $P_{{\rm T}}^{{\rm cut}}({\rm Z}$ Z$)$. Here we have summed over all eight final states with $l,l'={\rm e}$ e$$ or $\mu$.$ p${\rm p}$ p${\rm p}$ l_ll^l^M_{{\rm inv}}^{{\rm cut}}(ll^\prime\bar{l^\prime})~[{\rm GeV}]\sigma_{{\rm Born}}~[{\rm fb}]\sigma_{{\rm AEWS}}~[{\rm fb}]\sigma_{{\rm virt}}^{{{\rm finite}}}~[{\rm fb}]~\sigma_{{\rm EW}} ~[{\rm fb}]~~\Delta~[\%]~~1/\sqrt{2L\sigma_{{\rm Born}}}~[\%]5001.7291.6891.6921.601-7.45.46000.8990.8580.8600.814-9.57.57000.5080.4740.4760.452-10.99.98000.3040.2780.2790.264-13.312.89000.1900.1700.1710.161-15.116.210000.1230.1080.1090.102-16.720.2$Cross section for ${\rm p}$ p${\rm p}$ p$\to l\nu_l l^\prime \bar{l^\prime}$ for $|\Delta y({\rm Z}$ Z$l)|< 3$ and various values of $M_{{\rm inv}}^{{\rm cut}}(ll^\prime\bar{l^\prime})$. Here we have summed over all eight final states with $l,l'={\rm e}$ e$$ or $\mu$.The integration errors in these and the following tables are at the level below 1%. The comparison of the expected statistical error with the EW corrections indicates that these are non-negligible and can be comparable with the experimental precision up to about $P_{{\rm T}}^{{\rm cut}}({\rm Z}$ Z$)=500\,{\rm GeV}$ or $M_{{\rm inv}}^{{\rm cut}}(ll^\prime\bar{l^\prime})=1\,{\rm TeV}$. In these regions the corrections range between $-7$ and $-22\%$, being slightly more enhanced in the first scenario. Of course, their significance depends on the available luminosity. This kind of accuracy is needed only in a high-luminosity run. Besides the lowest-order cross section $\sigma_{{\rm Born}}$ and the cross section $\sigma_{{\rm EW}}$ including the complete logarithmic EW corrections, we have also inserted two entries representing partial results in Tables \ref{['ta:WZ_s1']} and \ref{['ta:WZ_s2']} in order to give an idea of the individual contributions. The cross section including only the EW logarithms originating from above the EW scale, $M{\rm W}$ W$$M_${\rm W}$$$, is denoted by $\sigma_{{\rm AEWS}}$. This term neglects all IR- and mass-singular terms coming from the mass gap between the photon and the weak gauge bosons and is exactly the part computed in Ref. Accomando:2001fn for the same process. The column $\sigma_{{\rm virt}}^{{{\rm finite}}}$ contains instead the full finite virtual correction, i.e. the full logarithmic EW corrections with the IR- and mass-singular contribution (\ref{['eq:virt-sing']}) subtracted. The difference between $\sigma_{{\rm AEWS}}$ and $\sigma_{{\rm virt}}^{{{\rm finite}}}$ is numerically small, despite of the fact that it contains logarithmic contributions. The dominant contribution to this difference is in fact proportional to $\alpha/(2\pi)\ln(s/M^2)[\ln(s/M^2)-3]$ which is suppressed for energies between $500\,{\rm GeV}$ and $1\,{\rm TeV}$ owing to cancellations in the bracket. In this section we extend our analysis to the processes ${\rm p}$ p${\rm p}$ p$\to l\bar{l}l^\prime\bar{l^\prime}$ ($l,l'={\rm e}$ e$$ or $\mu$). This channel is proper for studying the impact of trilinear neutral gauge-boson vertices, ${\rm Z}$ Z${\rm Z}$ Z${\rm Z}$ Z$$ and ${\rm Z}$ Z${\rm Z}$ Z$\gamma$, on physical observables. While these couplings are absent in the SM Lagrangian, one-loop corrections induce small but not-vanishing values for them. Significantly larger couplings are predicted by non-standard models, where new physics appearing at energy scales much larger than those which can be directly probed at forthcoming experiments can be parametrized in terms of anomalous neutral self-interactions. At LEP2 and Tevatron, the ${\rm Z}$ Z${\rm Z}$ Z$\gamma$ vertex has been measured through ${\rm Z}$ Z$\gamma$ production. LEP2 has been able to produce also ${\rm Z}$ Z${\rm Z}$ Z$$ pairs but with poor statistics. At the LHC several thousands of such ${\rm Z}$ Z${\rm Z}$ Z$$ pairs will be produced, allowing for more stringent bounds on ${\rm Z}$ Z${\rm Z}$ Z${\rm Z}$ Z$$ and ${\rm Z}$ Z${\rm Z}$ Z$\gamma$ vertices. The envisioned increase in statistics, and the possibility to observe significant deviations due to new physics interactions have gathered a renewed interest in the literature NAC. ${\rm Z}$ Z${\rm Z}$ Z${\rm Z}$ Z$$ and ${\rm Z}$ Z${\rm Z}$ Z$\gamma$ couplings affect the production of longitudinal or transverse ${\rm Z}$ Z$$ bosons in a different way. Therefore, the helicity of the decay products coming from ${\rm Z}$ Z${\rm Z}$ Z$$ production constitutes a valuable information. Up to now, on one side the aforementioned studies have been performed in the production$\times$decay approach, neglecting all spin correlations and irreducible background contributions. On the other side, accurate calculations of QCD corrections have been carried out in Ref. Dixon:1999di. In this section, we illustrate the results of a complete calculation of four-fermion production mediated by ${\rm Z}$ Z${\rm Z}$ Z$$ production including logarithmic EW corrections. We focus, in particular, on the effect of the EW corrections on the distributions mostly discussed in the literature NACbaur. We consider the same kind of observables as in the previous section, with the only difference that we replace the distribution in the missing transverse momentum by the distribution in the maximal transverse momentum of the reconstructed ${\rm Z}$ Z$$ bosons. To be precise we plot distributions in: maximal transverse momentum of the four charged leptons,maximal transverse momentum of the two reconstructed ${\rm Z}$ Z$$ bosons,rapidity difference between the two reconstructed ${\rm Z}$ Z$$ bosons,rapidity of the $\mu^-$.The ${\rm Z}$ Z$$ bosons are reconstructed by imposing (\ref{['addcutsZ']}), and for identical particles in the final state we choose the possibility where the reconstructed ${\rm Z}$ bosons are closer to their mass shell. We have checked that the accuracy of the DPA is at the level of a few per cent for this case. We show results for the specific process ${\rm p}$ p${\rm p}$ p$\to{\rm e}$ e$^-{\rm e}$ e$^+\mu^-\mu^+$ and only for the scenario characterized by the requirement $M_{{\rm inv}}(l\bar{l}l^\prime\bar{l^\prime})> 500\,{\rm GeV},\qquad |\Delta y({\rm Z}{\rm Z}{\rm Z}{\rm Z})|< 3.$An analogous behaviour holds for the scenario with $P_{{\rm T}}(Z)>300\,{\rm GeV}$ for both reconstructed Z bosons. We have verified that for both these scenarios the conditions (\ref{['HEA']}) for the validity of the logarithmic high-energy approximation are fulfilled. Distributions for ${\rm Z}$ Z${\rm Z}$ Z$$ production: (a) Maximal transverse momentum of the charged leptons. (b) Maximal transverse momentum of the reconstructed ${\rm Z}$ Z$$ bosons. (c) Difference in rapidity between the two reconstructed ${\rm Z}$ Z$$ bosons. (d) Rapidity of the $\mu^-$. The final states is ${\rm e}$ e$^-{\rm e^+}$ e^+$\mu^-\mu^+$, and standard cuts as well as $M_{{\rm inv}}(l\bar{l}l^\prime\bar{l^\prime})> 500\,{\rm GeV}$ and $|\Delta y({\rm Z}$ Z${\rm Z}$ Z$)|< 3$ are applied. The last cut is omitted for the $\Delta y({\rm Z}$ Z${\rm Z}$ Z$)$ distribution in lowest order. The inset plots show the $\cal{O}\alpha)$O(α)$$ corrections relative to the Born results in per cent.As one can see in Fig. \ref{['fi:ZZ_s2']}, EW corrections modify the Born result in the same way as for ${\rm W}$ W${\rm Z}$ Z$$ production, but the effect is typically a factor of 1.5 larger. We note that ${\rm Z}$ Z${\rm Z}$ Z$$ production at tree level does not present any true or approximate radiation zero. The dip in the distribution of the rapidity difference of the two reconstructed ${\rm Z}$ Z$$ bosons results from the fact that the partonic process $q\bar{q}\rightarrow {\rm Z}$ Z${\rm Z}$ Z$$, dominated by the transversely polarized ${\rm Z}$ Z$$ bosons, is peaked forward and backward. Moreover, the dip is enhanced by the invariant mass cut in (\ref{['eq:ZZscenarioII']}). In Table \ref{['ta:ZZ_s2']} we compare the relative correction $\Delta$ to the Born cross section with the estimated experimental accuracy for some values of the cut on the partonic CM energy $M_{{\rm inv}}(l\bar{l}l^\prime\bar{l^\prime})$. To this purpose, we sum over all three final states ${\rm e}$ e$^-{\rm e}$ e$^+\mu^-\mu^+$, ${\rm e}$ e$^-{\rm e}$ e$^+{\rm e}$ e$^-{\rm e}$ e$^+$, and $\mu^-\mu^+\mu^-\mu^+$. The entries in Table \ref{['ta:ZZ_s2']} are defined as in the previous section. One can see that, compared to ${\rm W}$ W${\rm Z}$ Z$$ production, $\cal{O}\alpha)$O(α)$$ corrections manifest the same behaviour on the shown observables, but they are globally by a factor of ${\sim1.5}$ larger. At modest ${\rm Z}$ Z${\rm Z}$ Z$$ invariant masses, the effect of the EW corrections can amount to two standard deviations, while it becomes comparable to the experimental precision with increasing CM energy. Of course, final states coming from ${\rm Z}$ Z${\rm Z}$ Z$$ production involving only charged leptons will not be copiously generated at the LHC. A detailed study of their properties would be possible only during a high-luminosity run. Although it yields higher statistics, we have not investigated ZZ production leading to $l \bar{l} \nu_{l^\prime} \bar{\nu}_{l^\prime}$ final states because there the reconstruction of the Z bosons is more problematic. $ p${\rm p}$ p${\rm p}$ lll^l^M_{{\rm inv}}^{{\rm cut}}(l\bar{l}l^\prime\bar{l^\prime})~[{\rm GeV}]\sigma_{{\rm Born}}~[{\rm fb}]\sigma_{{\rm AEWS}}~[{\rm fb}]\sigma_{{\rm virt}}^{{{\rm finite}}}~[{\rm fb}]~\sigma_{{\rm EW}} ~[{\rm fb}]~~\Delta~[\%]~~1/\sqrt{2L\sigma_{{\rm Born}}}~[\%]5000.6920.6370.6330.588-15.08.56000.3560.3140.3120.291-18.311.97000.2030.1730.1720.160-21.015.78000.1230.1020.1010.094-23.820.19000.0780.0630.0620.058-26.125.310000.0510.0400.0400.037-28.131.2$Cross section for ${\rm p}$ p${\rm p}$ p$\to {\rm e}$ e$^-{\rm e}$ e$^+\mu^-\mu^+$, ${\rm e}$ e$^-{\rm e}$ e$^+{\rm e}$ e$^-{\rm e}$ e$^+$, and $\mu^-\mu^+\mu^-\mu^+$ for $|\Delta y({\rm Z}$ Z${\rm Z}$ Z$)|< 3$ and various values of $M_{{\rm inv}}^{{\rm cut}}(l\bar{l}l^\prime\bar{l^\prime})$Finally, we discuss the processes ${\rm p}$ p${\rm p}$ p$\to l\bar{\nu}_l\nu_{l^\prime}\bar{l^\prime}$ ($l,l'={\rm e}$ e$$ or $\mu$). This channel contains information on the charged gauge-boson vertices ${\rm W}$ W${\rm W}$ W${\rm Z}$ Z$$ and ${\rm W}$ W${\rm W}$ W$\gamma$. While LEP2 could establish the non-abelian nature of the SM by measuring these couplings, high-precision measurements are still missing. At the LHC, the precision will be sensitively improved, if the large background from ${\rm t}$ t$\bar{{\rm t}$ t$}$ production can be properly controlled. The ${\rm W}$ W${\rm W}$ W$$ channel has in fact the largest cross section among all vector-boson pair-production processes. Despite the presence of two neutrinos, which do not allow a clean and unambiguous reconstruction of the two ${\rm W}$ W$$ bosons, the sensitivity to anomalous couplings is not seriously reduced. One can in fact consider the distribution in the missing transverse momentum Baur:1995uv. For this channel, following the study of Ref. Dixon:1999di on the sensitivity to new-physics effects, we choose to discuss distributions in the following variables: maximal transverse momentum of the two charged leptons,missing transverse momentum,rapidity difference between the charged leptons,rapidity of the negatively charged lepton.Despite of the fact that we do not perform a reconstruction of the ${\rm W}$ bosons for these processes, the quality of the DPA is better than 10%. Since we apply the DPA only to the corrections and these are below 25%, at least where the cross section is appreciable, this introduces an error of only a few per cent. We consider the scenario $M_{{\rm inv}}(l\bar{l^\prime})> 500\,{\rm GeV},\qquad |\Delta y(l\bar{l^\prime})|< 3,$which fulfils the conditions (\ref{['HEA']}) for the validity of the logarithmic high-energy approximation, as we have verified. Possible ${\rm Z}$ Z${\rm Z}$ Z$$ intermediate states are heavily suppressed by the invariant-mass cut in (\ref{['eq:WWscenarioII']}). Therefore, we can safely neglect contributions of $l \bar{l} \nu_{l^\prime} \bar{\nu}_{l^\prime}$ final states with $l \neq l^\prime$. Distributions for ${\rm W}$ W${\rm W}$ W$$ production: (a) Maximal transverse momentum of the charged leptons. (b) Missing transverse momentum. (c) Difference in rapidity between the two charged leptons. (d) Rapidity of the negatively charged lepton. The final state is $\nu_e{\rm e^+}$ e^+$\mu^-\bar{\nu}_\mu$ with standard cuts as well as $M_{{\rm inv}}(l\bar{l^\prime})> 500\,{\rm GeV}$ and $|\Delta y(l\bar{l^\prime})|< 3$ applied. The last cut is omitted to the $\Delta y(l\bar{l^\prime})$ distribution in lowest order. The inset plots show the $\cal{O}\alpha)$O(α)$$ corrections relative to the Born results in per cent.In Fig. \ref{['fi:WW_s2']} we show the distributions for the final state $\nu_e{\rm e^+}$ e^+$\mu^-\bar{\nu}_\mu$ with our standard cuts applied. As in the previous two cases, $\cal{O}\alpha)$O(α)$$ corrections are enhanced at high energy and large scattering angles. This translates into larger radiative corrections in the tails of transverse momentum distributions and in the central region of rapidity distributions. Let us note that also in this case the partonic process at Born level does not vanish for any scattering angle, independently on the ${\rm W}$ W$$-boson polarization. The dip appearing in the distribution of the rapidity difference between the two charged leptons is this time exclusively due to the chosen set of cuts. In absence of any kinematical cuts, the ${\rm p}$ p${\rm p}$ p$\rightarrow{\rm W}$ W$^+{\rm W}$ W$^-$ process is dominated by the ${\rm u}$ u$$-quark contribution, and the rapidity-difference $\Delta y(l\bar{l^\prime})$ for the partonic process $\bar{{\rm u}$ u$}{\rm u}$ u$\rightarrow 4f$ is maximal and symmetric around zero. The requirement of having a large invariant mass of the two charged leptons, forces the two leptons to be produced at large separation angles. This fact depletes the number of events in the central region of $\Delta y(l\bar{l^\prime})$ and leaves events with larger rapidity difference. This gives rise to the shape of Fig. \ref{['fi:WW_s2']}. The general behaviour of EW corrections does not present novelties compared to the previous cases. The interesting feature of ${\rm W}$ W${\rm W}$ W$$ processes is the remarkable statistics of purely leptonic final states. As shown in Table \ref{['ta:WW_s2']}, where we sum over the four final states ${\rm e}$ e$^-\bar{\nu$ν$_{{\rm e}}}\nu_\mu\mu^+$, $\mu^-\bar{\nu}_\mu\nu$ν$_{{\rm e}}{\rm e}$ e$^+$, $\mu^-\bar{\nu}_\mu\nu_\mu\mu^+$, and ${\rm e}$ e$^-\bar{\nu$ν$_{{\rm e}}}\nu$ν$_{{\rm e}}{\rm e}$ e$^+$, the estimated experimental precision is around a few per cent at CM energies below 700$\,{\rm GeV}$. On the other hand, the deviation from the Born result given by the $\cal{O}\alpha)$O(α)$$ contributions ranges between $-14$ and $-18\%$ in the same energy domain. At larger invariant masses, the overall cross section decreases but radiative corrections are still of order 2--3 standard deviations. Thus, a reliable analysis of these final states requires the inclusion of EW corrections. Note also that, in contrast to previous processes, $\cal{O}\alpha)$O(α)$$ corrections can be relevant even in the low-luminosity run ($L=30 \,{\rm fb}^{-1}$). They are about twice the standard deviation for $M_{{\rm inv}}^{{\rm cut}}(l\bar{l^\prime})\le 700\,{\rm GeV}$, and become comparable with the experimental accuracy above that threshold. $ p${\rm p}$ p${\rm p}$ l_l_l^l^M_{{\rm inv}}^{{\rm cut}}(l\bar{l^\prime})~[{\rm GeV}]\sigma_{{\rm Born}}~[{\rm fb}]\sigma_{{\rm AEWS}}~[{\rm fb}]\sigma_{{\rm virt}}^{{{\rm finite}}}~[{\rm fb}]~\sigma_{{\rm EW}} ~[{\rm fb}]~~\Delta~[\%]~~1/\sqrt{2L\sigma_{{\rm Born}}}~[\%]5007.2356.5616.6826.235-13.82.66003.7233.2803.3503.131-15.93.77002.0591.7651.8081.688-18.14.98001.2011.0031.0310.959-20.26.59000.7310.5960.6130.570-22.08.310000.4600.3660.3780.352-23.410.4$Cross section for ${\rm p}$ p${\rm p}$ p$\to {\rm e}$ e$^-\bar{\nu$ν$_{{\rm e}}}\nu_\mu\mu^+,\mu^-\bar{\nu}_\mu\nu$ν$_{{\rm e}}{\rm e}$ e$^+, \mu^-\bar{\nu}_\mu\nu_\mu\mu^+,{\rm e}$ e$^-\bar{\nu$ν$_{{\rm e}}}\nu$ν$_{{\rm e}}{\rm e}$ e$^+$ for $|\Delta y(l\bar{l^\prime})|< 3$ and various values of $M_{{\rm inv}}^{{\rm cut}}(l\bar{l^\prime})$At the LHC, gauge-boson production processes will be used for precise measurements of the triple gauge-boson couplings. The relevant processes to investigate are ${\rm W}$ W${\rm Z}$ Z$$, ${\rm Z}$ Z${\rm Z}$ Z$$, and ${\rm W}$ W${\rm W}$ W$$ production, and the physically interesting region is the one of high di-boson invariant mass. We have examined these processes by means of a complete four-fermion calculation, i.e. by taking into account the decays of the gauge bosons, in the purely leptonic channels. The primary aim of our analysis was to investigate the influence of electroweak radiative corrections on the di-boson production processes at the LHC. The one-loop logarithmic corrections to the full four-fermion processes have been calculated in double-pole approximation. This includes corrections to the gauge-boson-pair-production processes, corrections to the gauge-boson decays, as well as non-factorizable corrections. In this study, we have included the full electromagnetic radiative corrections in the logarithmic approximation, which involve also the emission of real photons and therefore depend on the detector resolution. We have verified that the double-pole approximation and the high-energy approximation are applicable for the considered phase-space regions of large transverse momentum or large invariant mass of the gauge-boson pair. Thus, our approach is reliable in this region. The corrections have been implemented in a Monte Carlo program, so that arbitrary cuts and distributions can be studied. In order to illustrate the behaviour and the size of $\cal{O}\alpha)$O(α)$$ contributions, we have presented different cross sections and distributions. For ${\rm W}$ W${\rm Z}$ Z$$-, ${\rm Z}$ Z${\rm Z}$ Z$$-, and ${\rm W}$ W${\rm W}$ W$$-production processes, electroweak corrections turn out to be sizeable in the high-energy region of the hard process, in particular for large transverse momentum and small rapidity separation of the reconstructed vector bosons, which is the kinematical range of maximal sensitivity to new-physics phenomena. Electroweak radiative corrections lower the Born results for ${\rm W}$ W${\rm Z}$ Z$$, ${\rm Z}$ Z${\rm Z}$ Z$$, and ${\rm W}$ W${\rm W}$ W$$ production by 7--22%, 15--28%, and 14--24%, in the region of experimental sensitivity. Their size depends sensibly not only on the CM energy but also on the applied cuts and varies according to the selected observables and kinematical regions. Despite of the strong decrease of the cross section with increasing di-boson invariant mass, radiative effects are appreciable if compared with the expected experimental precision. This depends of course on the available luminosity. For ${\rm W}$ W${\rm Z}$ Z$$ and ${\rm Z}$ Z${\rm Z}$ Z$$ production, these effects are only relevant for a high-luminosity run of the LHC. Owing to their larger overall cross section, ${\rm W}$ W${\rm W}$ W$$-production processes can instead show a sensitivity to radiative effects even at a low-luminosity run.We thank M. Roth for his invaluable help concerning the Monte Carlo generator and S. Pozzorini for his contributions in the evaluation of the logarithmic corrections. This work was supported in part by the Swiss Bundesamt für Bildung und Wissenschaft, by the European Union under contract HPRN-CT-2000-00149, and by the Italian Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR) under contract Decreto MIUR 26-01-2001 N.13.In this appendix we generalize the results of Refs. Denner:2000bjDenner:1997ia for the virtual non-factorizable corrections to a general class of processes. We start by discussing non-factorizable corrections to the generic process $g_1(p_1) + g_2(p_2) \;\to\; \sum_{l=1}^N R_l(k_l) + \sum _{j=1}^{n_0} f_{0j}(q_{0j}) \;\to\; \sum _{l=1}^N \sum_{i=1}^{n_l} f_{li}(q_{li}) + \sum _{j=1}^{n_0} f_{0j}(q_{0j}).$Two incoming particles $g_1$ and $g_2$ with momenta $p_1$ and $p_2$, masses $m'_1$ and $m'_2$, and charges $Q'_1$ and $Q'_2$ scatter into $N$ resonances $R_l$ with momenta $k_l$, masses $M_l$, decay widths $\Gamma_l$, and charges $Q_l$ and $n_0$ stable particles $f_{0j}$ with momenta $q_{0j}$, masses $m_{0j}$, and charges $Q_{0j}$. Each resonance $R_l$ then decays into $n_l$ stable massless particles with momenta $q_{li}$, masses $m_{li}$, and charges $Q_{li}$. Whereas the charges $Q'_k$ are incoming, all charges $Q_l$ and $Q_{li}$ are assumed to be outgoing. The masses of the external particles, which are typically light fermions, are neglected, except where this would lead to mass singularities. The complex masses squared of the resonances are denoted by $\overline{M}_l^2 = M_l^2-{{\rm i}} M_l \Gamma_l,$and we introduce the off-shellness variables $K_l = k_l^2-M_l^2.$We want to give the non-factorizable corrections to the process (\ref{['process']}) in leading-pole approximation (LPA). The LPA takes into account only the leading terms in an expansion around the poles originating from the propagators of the resonances. For two resonances, the LPA is just the double-pole approximation used in the main part of this paper. In LPA, the lowest-order matrix element for process (\ref{['process']}) factorizes into the matrix element for the production of the $N$ on-shell resonances, ${\cal{M}}^{g_1g_2\to R_1\ldots R_N f_{01}\ldots f_{0n_0}}_{\rm Born}(p_1,p_2,k_l,q_{0j})$, the propagators of these resonances, and the matrix elements for the decays of these on-shell resonances, ${\cal{M}}^{R_l\to f_{l1}\ldots f_{ln_l}}_{\rm Born}(k_l,q_{li})$: ${\cal M}_{{\rm Born}} = \sum_{{\rm pol}} {\cal{M}}^{g_1g_2\to R_1\ldots R_{N} f_{01}\ldots f_{0n_0}}_{\rm Born} \prod_{l=1}^{N}\frac{{\cal{M}}^{R_l\to f_{l1}\ldots f_{ln_l}}_{\rm Born}}{K_l}.$The sum runs over the physical polarizations of the resonances.The non-factorizable EW corrections result exclusively from the exchange of photons that connect the production and decay subprocesses or two decay subprocesses Denner:1997iaAeppli:1993rsBeenakker:1997bp. Only photons with energies of the order of the decay widths or smaller are relevant so that an extended soft-photon approximation, which takes into account the dependence of the resonant propagators on the photon momenta, can be used. Consequently, the non-factorizable corrections ${{\rm d}}\sigma_{{\rm nf}}$ to the fully differential lowest-order cross section ${{\rm d}}\sigma_{\rm Born}$ resulting from the matrix element (\ref{['mborn']}) take the form of a correction factor to the lowest-order cross section: ${{\rm d}} \sigma_{{{\rm nf}},{{\rm LPA}}}^{\rm virt} = \delta^{{\rm virt}}_{{{\rm nf}},{{\rm LPA}}}\, {{\rm d}}\sigma_{{\rm Born},{{\rm LPA}}} .$The non-factorizable corrections get contributions from virtual photons exchanged between a resonance and its decay products (${\rm }$ mf$$), between the production process and a resonance (${\rm }$ im$$), between the production process and the decay products of a resonance (${\rm }$ if$$), between two resonances (${\rm '}$ mm'$$), between a resonance and the decay products of another resonance (${\rm '}$ mf'$$), between decay products of different resonances (${\rm '}$ ff'$$), and virtual photons attached to one resonance (${\rm }$ mm$$). Examples can be found in Figures 1 and 2 of Ref. Denner:1997ia or in Figure 2 of Ref. Denner:2000bj. Upon splitting the contributions that result from photons coupled to the charged resonances according to $Q_l=\sum_i Q_{li}$ into contributions associated with definite final-state fermions and using $Q'_1+Q'_2= \sum_{l=0}^{N} \sum_{i=1}^{n_l} Q_{li}$ to rewrite the terms originating from the (${\rm }$ mf$$) and (${\rm }$ mm$$) contributions, the complete correction factor to the lowest-order cross section can be written as \delta^{{\rm virt}}_{{{\rm nf}},{{\rm LPA}}}=-\sum_{l=1}^{N-1} \,\sum_{m=l+1}^{N} \sum_{i=1}^{n_l}\sum_{j=1}^{n_m} \, Q_{li} Q_{mj} \, \frac{\alpha}{\pi} \, \mathop{{\rm Re}}\nolimits\{\Delta_1(k_{l},q_{li};k_{m},q_{mj})\}{}- \sum_{k=1}^2 \sum_{l=1}^{N} \, \sum_{i=1}^{n_l} \, Q'_{k} Q_{li} \, \frac{\alpha}{\pi} \, \mathop{{\rm Re}}\nolimits\{\Delta_2(p_k;k_{l},q_{li})\}{}+ \sum_{j=1}^{n_0} \sum_{l=1}^{N} \, \sum_{i=1}^{n_l} \, Q_{0j} Q_{li} \, \frac{\alpha}{\pi} \, \mathop{{\rm Re}}\nolimits\{\Delta_2(-q_{0j};k_{l},q_{li})\} .The quantity $\Delta_1$ gets contributions from $\Delta^{\rm virt}_{{\rm '}$ mm'$}$, $\Delta^{\rm virt}_{{\rm '}$ mf'$}$, $\Delta^{\rm virt}_{{\rm '}$ ff'$}$, $\Delta^{\rm virt}_{{\rm }$ mf$}$, and $\Delta^{\rm virt}_{{\rm }$ mm$}$, \Delta_1(k_{l},q_{li};k_{m},q_{mj})=(\Delta^{\rm virt}_{{\rm '}{\rm mm}'}+\Delta^{\rm virt}_{{\rm '}{\rm mf}'}+\Delta^{\rm virt}_{{\rm '}{\rm ff}'})(k_{l},q_{li};k_{m},q_{mj}){}- (\Delta^{\rm virt}_{{\rm }{\rm mf}}+\Delta^{\rm virt}_{{\rm }{\rm mm}})(k_{l},q_{li})- (\Delta^{\rm virt}_{{\rm }{\rm mf}}+\Delta^{\rm virt}_{{\rm }{\rm mm}})(k_{m},q_{mj}),and $\Delta_2$ gets contributions from $\Delta^{\rm virt}_{{\rm }$ im$}$, $\Delta^{\rm virt}_{{\rm }$ if$}$, $\Delta^{\rm virt}_{{\rm }$ mf$}$, and $\Delta^{\rm virt}_{{\rm }$ mm$}$, \Delta_2(p_k;k_{l},q_{li})=(\Delta^{\rm virt}_{{\rm }{\rm im}}+\Delta^{\rm virt}_{{\rm }{\rm if}})(p_k;k_{l},q_{li})+ (\Delta^{\rm virt}_{{\rm }{\rm mf}}+\Delta^{\rm virt}_{{\rm }{\rm mm}})(k_{l},q_{li}).The contributions of the different types of diagrams are given by \Delta^{\rm virt}_{{\rm '}{\rm ff}'}\sim{} -2(q_{li}q_{mj}) K_lK_{m} E_0(-q_{mj},-k_{m},k_l,q_{li},\lambda,m_{mj},\overline{M}_{m},\overline{M}_l,m_{li}),\Delta^{\rm virt}_{{\rm '}{\rm mf}'}\sim{} - 2(k_lq_{mj})K_{m} D_0(-q_{mj},-k_{m},k_l,0,m_{mj},\overline{M}_{m},\overline{M}_l){} - 2(k_{m}q_{li})K_l D_0(-q_{li},-k_l,k_{m},0,m_{li},\overline{M}_l,\overline{M}_{m}),\Delta^{\rm virt}_{{\rm if}}\sim{} -2(p_kq_{li})K_l D_0(p_k,k_l,q_{li},\lambda,m'_{k},\overline{M}_l,m_{li}),\Delta^{\rm virt}_{{\rm '}{\rm mm}'}\sim{} -2(k_lk_{m})\biggl\{ C_0(k_l,-k_{m},0,\overline{M}_l,\overline{M}_{m}) - \Bigl[C_0(k_l,-k_{m},\lambda,M_l,M_{m})\Bigr]_{k_{l,m}^2=M_{l,m}^2} \biggr\}, \space\Delta^{\rm virt}_{{\rm im}}\sim{} -2(p_kk_l)\biggl\{C_0(p_k,k_l,0,m'_k,\overline{M}_l) -\Bigl[C_0(p_k,k_l,\lambda,m'_k,M_l)\Bigr]_{k_l^2=M_l^2} \biggr\}, \space\Delta^{\rm virt}_{{\rm mf}}\sim{} -2(k_lq_{li})\biggl\{ C_0(k_l,q_{li},0,\overline{M}_l,m_{li}) -\Bigl[C_0(k_l,q_{li},\lambda,M_l,m_{li})\Bigr]_{k_l^2=M_l^2}\biggr\} ,\Delta^{\rm virt}_{{\rm mm}}\sim{} -2M_l^2\Biggl\{ \frac{B_0(k_l^2,0,\overline{M}_l)-B_0(\overline{M}_l^2,0,\overline{M}_l)}{k_l^2-\overline{M}_l^2} -B'_0(M_l^2,\lambda,M_l) \Biggr\}.Note that these definitions deviate partially in sign and form from those used in Refs. Denner:2000bjDenner:1997ia. The symbol "$\sim$" in (\ref{['eq:Delta1']}) indicates that the limits $k_l^2\to M_l^2$ and $\Gamma_l\to 0$ are implicitly understood whenever possible. The definition of the scalar integrals $B_0$, $C_0$, $D_0$, $E_0$ and of their arguments can be found in Refs. Denner:ktDenner:1997ia. The explicit expressions of these functions have been given in Refs. Denner:2000bjDenner:1997ia for equal masses of the resonances. The generalized expressions for arbitrary masses are listed in Appendix \ref{['app:scalints']}.Using the explicit expression for the loop integrals given in Appendix \ref{['app:scalintsgen']}, the terms in the correction factor can be simplified. The results given in the following are only valid for $(k_l-q_i)^2=0$, which holds if the resonances decay into two massless particles. The sum $\Delta^{\rm virt}_{{\rm '}$ mf'$}+\Delta^{\rm virt}_{{\rm '}$ ff'$}$ can be simplified by inserting the decompositions of the 5-point function (\ref{['vE5redpole']}). In LPA this leads to \lefteqn{(\Delta^{\rm virt}_{{\rm '}{\rm mf}'}+\Delta^{\rm virt}_{{\rm '}{\rm ff}'})(k_m,q_j;k_l,q_i) \sim \frac{K_lK_ms_{ij}\det(Y_0)}{\det(Y)}D_0(0)}\quad{} +\frac{K_l\det(Y_3)}{\det(Y)} \left\{[K_l\tilde{s}_{mi}+K_mM_l^2]D_0(1) + K_ms_{ij}D_0(3)\right\}{} +\frac{K_m\det(Y_2)}{\det(Y)} \left\{[K_m\tilde{s}_{lj}+K_lM_m^2]D_0(4) + K_ls_{ij}D_0(2)\right\} ,where $s_{ij}$ and $\tilde{s}_{lj}$ are defined in (\ref{['eq:shorthands']}) and $D_0(l)$ in (\ref{['D0_ints']}). Note that $\Delta^{\rm virt}_{{\rm '}$ mf'$}$ is exactly cancelled by the contributions of the last two terms in (\ref{['vE5redpole']}). Inserting the expressions for the scalar integrals into the different contributions, we find using the first relation in (\ref{['reldetYipole']}) \lefteqn{(\Delta^{\rm virt}_{{\rm '}{\rm mf}'}+\Delta^{\rm virt}_{{\rm '}{\rm ff}'})(k_m,q_j;k_l,q_i) -\Delta^{\rm virt}_{{\rm }{\rm mf}}(k_m,q_j)-\Delta^{\rm virt}_{{\rm }{\rm mf}}(k_l,q_i) }\qquad\sim{} \frac{K_lK_ms_{ij}\det(Y_0)}{\det(Y)}D_0(0) + \frac{K_l \det(Y_3)}{\det( Y)} F(k_m,q_j;k_l,q_i){} + \frac{K_m \det(Y_2)}{\det( Y)} F(k_l,q_i;k_m,q_j) +\ln\biggl(\frac{\lambda^2}{M_lM_m}\biggr) \ln\biggl(-\frac{s_{ij}}{M_lM_m}-{{\rm i}}\epsilon\biggr)with $D_0(0)$ given in (\ref{['D00']}) and \lefteqn{F(k_m,q_j;k_l,q_i)= [K_l\tilde{s}_{mi}+K_mM_l^2]D_0(1) + K_ms_{ij}D_0(3) }\qquad{}+\ln\biggl(\frac{\lambda^2}{M_lM_m}\biggr) \ln\biggl(-\frac{s_{ij}}{M_lM_m}-{{\rm i}}\epsilon\biggr){}-M_l^2\biggl\{ C_0(k_l,q_i,0,\overline{M}_l,m_i) - \Bigl[C_0(k_l,q_i,\lambda,M_l,m_i)\Bigr]_{k_l^2=M_l^2} \biggr\}{}-M_m^2\biggl\{ C_0(k_m,q_j,0,\overline{M}_m,m_j) - \Bigl[C_0(k_m,q_j,\lambda,M_m,m_j)\Bigr]_{k_m^2=M_m^2} \biggr\}=\sum\limits_{\tau=\pm 1}\biggl[ \mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_lM_m}{K_mM_l},r_{lm}^\tau\biggr)- \mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\tilde{s}_{mi}}+{{\rm i}}\epsilon,r_{lm}^\tau\biggr)\biggr]{} -2\mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_lM_m}{K_mM_l},-\frac{\tilde{s}_{mi}}{M_lM_m}-{{\rm i}}\epsilon\biggr) -\mathop{{\rm Li}_2}\nolimits\biggl(1-\frac{\tilde{s}_{mi}}{s_{ij}}\biggr) - \ln^2\biggl(-\frac{\tilde{s}_{mi}}{M_lM_m}-{{\rm i}} \epsilon\biggr){} +\ln\biggl(-\frac{s_{ij}}{M_lM_m}-{{\rm i}} \epsilon\biggr) \biggl[ \ln\biggl(-\frac{K_m}{M_lM_m}\biggr) + \ln\biggl(-\frac{K_m}{M_m^2}\biggr)\biggr] .The dilogarithms $\mathop{{\rm Li}_2}\nolimits$, $\mathop{{\cal L}i_2}\nolimits$ are defined in (\ref{['cLi']}) and (\ref{['Li']}). The quantity $\Delta_1$ is then obtained from (\ref{['eq:simp']}) and \Delta^{\rm virt}_{{\rm }{\rm mm}}(k_l,q_i)=-2\ln\biggl(\frac{\lambda M_l}{-K_l}\biggr)-2,\Delta^{\rm virt}_{{\rm '}{\rm mm}'}(k_m,q_j;k_l,q_i)=-\bar{s}_{lm}\biggl\{ C_0(k_l,-k_m,0,\overline{M}_l,\overline{M}_m) - \Bigl[C_0(k_l,-k_m,\lambda,M_l,M_m)\Bigr]_{k_l^2=M_l^2} \biggr\}\spaceas \Delta_1(k_m,q_{j};k_l,q_{i})\sim\frac{K_lK_ms_{ij}\det(Y_0)}{\det(Y)}D_0(0) + \frac{K_l \det(Y_3)}{\det( Y)} F(k_m,q_j;k_l,q_i){} + \frac{K_m \det(Y_2)}{\det( Y)} F(k_l,q_i;k_m,q_j) +\ln\biggl(\frac{\lambda^2}{M_lM_m}\biggr) \ln\biggl(-\frac{s_{ij}}{M_lM_m}-{{\rm i}}\epsilon\biggr){}-\bar{s}_{lm}\biggl\{ C_0(k_l,-k_m,0,\overline{M}_l,\overline{M}_m) - \Bigl[C_0(k_l,-k_m,\lambda,M_l,M_m)\Bigr]_{k_l^2=M_l^2} \biggr\}{}+2\ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) +2\ln\biggl(\frac{\lambda M_m}{-K_m}\biggr)+4 .When inserting the explicit expressions for the integrals, the quantity $\Delta_2$ simplifies to \Delta_2(p_k;k_l,q_i)=(\Delta^{\rm virt}_{{\rm }{\rm im}}+\Delta^{\rm virt}_{{\rm }{\rm if}})(p_k;k_l,q_{i})+ (\Delta^{\rm virt}_{{\rm }{\rm mf}}+\Delta^{\rm virt}_{{\rm }{\rm mm}})(k_l,q_{i})=2\ln\biggl(\frac{\lambda M_l}{-K_l}\biggr)\biggl[\ln\biggl(\frac{\tilde{t}_{kl}}{t_{ki}}\biggr)-1\biggr] -2 -\mathop{{\rm Li}_2}\nolimits\left(1-\frac{\tilde{t}_{kl}}{t_{ki}}\right)with $t_{ki}$ and $\tilde{t}_{kl}$ defined in (\ref{['eq:shorthands']}).Using the expressions for the scalar integrals in the high-energy limit given in Appendix \ref{['se:scalint_he']}, we find \lefteqn{\Delta_1(k_m,q_j;k_l,q_i)\sim \frac{1}{2}(s_{ij}\bar{s}_{lm}-\tilde{s}_{lj}\tilde{s}_{mi}) D_0(q_j-k_m,q_j+k_l,q_i+q_j,0,M_m,M_l,0)}\quad{}+ \ln\biggl(\frac{K_mM_l}{K_lM_m}\biggr) \ln\biggl(\frac{\tilde{s}_{mi}}{\tilde{s}_{lj}}\biggr) +\biggl[2+\ln\biggl(\frac{s_{ij}}{\bar{s}_{lm}}\biggr)\biggl] \biggl[\ln\biggl(\frac{\lambda M_m}{-K_m}\biggr) + \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr)\biggr]with $D_0$ defined in (\ref{['D00he']}) and \Delta_2(p_k;k_l,q_i)=2\ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) \biggl[\ln\biggl(\frac{\tilde{t}_{kl}}{t_{ki}}\biggr)-1\biggr].Note that we always assume that the resonances decay into a pair of massless particles.In Sect. \ref{['se:correctionfactors']} we have given the virtual non-factorizable corrections in terms of scalar one-loop integrals. In this appendix we list the explicit expressions for these integrals. We have the on-shell conditions for the external particles $p_k^2=(m'_k)^2,\qquad q_i^2=m_i^2,$and all expression are given for $k_l^2\to M_l^2$ and $\Gamma_l\to 0$, i.e. we neglect $k_l^2-M_l^2$ and $\Gamma_l$ everywhere where this does not give rise to singularities. Moreover, we assume $(k_l-q_i)^2=0,$which holds if the resonances decay into a pair of massless particles. We introduce the shorthand notations K_l=k_l^2 - \overline{M}_l^2t_{ki}=-2(p_kq_i) = (p_k-q_i)^2, \qquad \tilde{t}_{kl} = -2(p_kk_l) \sim (p_k-k_l)^2 - M_l^2 , \qquads_{ij}=2(q_iq_j) = (q_i+q_j)^2,\qquad \tilde{s}_{lj} = 2(k_lq_j) \sim (k_l+q_j)^2-M_l^2,\bar{s}_{lm}=2(k_lk_m) \sim (k_l+k_m)^2-M_l^2-M_m^2,w_{lm}=\sqrt{\lambda[(k_l+k_m)^2,M_l^2,M_m^2]},r_{lm}=\frac{1}{2M_lM_m}(-\bar{s}_{lm}+w_{lm}) \left(1-\frac{{{\rm i}}\epsilon}{w_{lm}}\right),\kappa_{lmij}=\sqrt{\lambda[s_{ij}(\bar{s}_{lm}+s_{ij}-\tilde{s}_{lj}-\tilde{s}_{mi}), (\tilde{s}_{lj}-s_{ij})(\tilde{s}_{mi}-s_{ji}),M_l^2M_m^2]}=\sqrt{\lambda[4(q_iq_j)((k_l-q_i)(k_m-q_j)), 4((k_l-q_i)q_j)(q_i(k_m-q_j)),M_l^2M_m^2]}\,,where ${{\rm i}}\epsilon$ is an infinitesimal imaginary part, and use the definitions $\lambda(x,y,z) = x^2+y^2+z^2-2xy-2xz-2yz$and \mathop{{\cal L}i_2}\nolimits(x,y)=\mathop{{\rm Li}_2}\nolimits(1-xy)+[\,\ln(xy)-\ln(x)-\ln(y)]\ln(1-xy),|\mathop{{\rm arc}}\nolimits(x)|,|\mathop{{\rm arc}}\nolimits(y)|<\piwith the usual dilogarithm $\mathop{{\rm Li}_2}\nolimits(z) = -\int_0^z\,\frac{{{\rm d}} t}{t}\,\ln(1-t), \qquad |\mathop{{\rm arc}}\nolimits(1-z)|<\pi.$The various combinations of scalar integrals read: case ${\rm }$ mm$$\frac{B_0(k_l^2,0,\overline{M}_l)-B_0(\overline{M}_l^2,0,\overline{M}_l)}{k_l^2-\overline{M}_l^2} -B'_0(M_l^2,\lambda,M_l) \sim \frac{1}{M_l^2}\biggl\{ \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr)+1 \biggr\}, \qquadcase ${\rm }$ mf$$\lefteqn{C_0(k_l,q_i,0,\overline{M}_l,m_i) -\Bigl[C_0(k_l,q_i,\lambda,M_l,m_i)\Bigr]_{k_l^2=M_l^2}}\qquad\space\sim-\frac{1}{M_l^2} \biggl\{ \ln\biggl(\frac{m_i^2}{M_l^2}\biggr) \ln\biggl(\frac{-K_l}{\lambda M_l}\biggr) +\ln^2\biggl(\frac{m_i}{M_l}\biggr) + \frac{\pi^2}{6} \biggr\}, \qquadcase ${\rm }$ im$$\lefteqn{C_0(p_k,k_l,0,m'_k,\overline{M}_l) -\Bigl[C_0(p_k,k_l,\lambda,m'_k,M_l)\Bigr]_{k_l^2=M_l^2}}\qquad\space\sim\frac{1}{\tilde{t}_{kl}} \biggl\{ \ln\biggl(\frac{m'_kM_l}{-\tilde{t}_{kl}}+{{\rm i}}\epsilon\biggr) \biggl[ \ln\biggl(\frac{K_l}{\tilde{t}_{kl}}\biggr) +\ln\biggl(\frac{-K_l}{\lambda^2}\biggr) +\ln\biggl(\frac{m'_k}{M_l}\biggr) \biggr] + \frac{\pi^2}{6} \biggr\}, \qquadcase mm'${\rm mm}'$\lefteqn{ C_0(k_l,-k_m,0,\overline{M}_l,\overline{M}_m) - \Bigl[C_0(k_l,-k_m,\lambda,M_l,M_m)\Bigr]_{k_l^2=M_l^2,{k_m}^2=M_m^2} } \qquad\space\sim\frac{1}{w_{lm}}\biggl\{ \mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_lM_m}{K_mM_l},\frac{1}{r_{lm}}\biggr) -\mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_lM_m}{K_mM_l},r_{lm}\biggr) +\mathop{{\cal L}i_2}\nolimits(r_{lm},r_{lm}){} +\ln^2(r_{lm}) +2\ln(r_{lm})\ln\biggl(\frac{-K_m}{M_m\lambda}\biggr) \biggr\},case if${\rm if}$\lefteqn{ D_0(p_k,k_l,q_i,\lambda,m'_k,\overline{M}_l,m_i) \sim -\frac{1}{t_{ki} K_l} \biggl\{ 2\ln\biggl(\frac{-t_{ki}}{m'_k m_i}-{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) } \qquad\space{} +\ln^2\biggl(\frac{-\tilde{t}_{kl}}{m'_k M_l}-{{\rm i}}\epsilon\biggr) +\ln^2\biggl(\frac{m_i}{M_l}\biggr) +\frac{\pi^2}{3}+\mathop{{\rm Li}_2}\nolimits\biggl(1-\frac{\tilde{t}_{kl}}{t_{ki}}\biggr) \biggr\}, \qquadcase mf'${\rm mf}'$\lefteqn{D_0(1) = D_0(-q_i,-k_l,k_m,0,m_i,\overline{M}_l,\overline{M}_m) = D_0(-k_m,k_l,q_i,0,\overline{M}_m,\overline{M}_l,m_i) } \qquad\space\sim\frac{1}{K_l \tilde{s}_{mi}+K_m M_l^2} \biggl\{ \sum_{\tau=\pm 1} \biggl[ \mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_l M_m}{K_m M_l},r_{lm}^\tau\biggr) -\mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\tilde{s}_{mi}}+{{\rm i}}\epsilon,r_{lm}^\tau\biggr) \biggr]{} -2\mathop{{\cal L}i_2}\nolimits\biggl(\frac{K_l M_m}{K_m M_l},-\frac{\tilde{s}_{mi}}{M_lM_m}-{{\rm i}}\epsilon\biggr){} -\ln\biggl(\frac{m_i^2}{M_l^2}\biggr) \biggl[ \ln\biggl(\frac{K_l M_m}{K_m M_l}\biggr) +\ln\biggl(-\frac{\tilde{s}_{mi}}{M_lM_m}-{{\rm i}}\epsilon\biggr) \biggr] \biggr\},case ff'${\rm ff}'$\lefteqn{ D_0(0) = D_0(-k_m+q_j,k_l+q_j,q_i+q_j,0,M_m,M_l,0) } \qquad\space\sim\frac{1}{\kappa_{lmij}} \sum_{\sigma=1,2} (-1)^\sigma \biggl\{ \mathop{{\cal L}i_2}\nolimits\biggl(-\frac{\tilde{s}_{lj}}{M_lM_m}-{{\rm i}}\epsilon,-x_\sigma\biggr) +\mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\tilde{s}_{mi}}+{{\rm i}}\epsilon,-x_\sigma\biggr)\qquad {} -\mathop{{\cal L}i_2}\nolimits\biggl(r_{lm},-x_\sigma\biggr) -\mathop{{\cal L}i_2}\nolimits\biggl(r_{lm}^{-1},-x_\sigma\biggr) -\ln\biggl(\frac{\tilde{s}_{mi}}{s_{ij}}\biggr)\ln(-x_\sigma) \biggr\},\hbox{with} \qquad x_1 = \frac{(\tilde{s}_{mi}-s_{ji})z}{M_lM_m} -\frac{s_{ij}}{\kappa_{lmij}}{{\rm i}}\epsilon , \quad x_2 =\frac{M_lM_m}{(\tilde{s}_{lj}-s_{ij})z} +\frac{s_{ij}}{\kappa_{lmij}}{{\rm i}}\epsilon, \quad\qquad\qquad z = \frac{M_l^2M_m^2+\tilde{s}_{lj}\tilde{s}_{mi}-\bar{s}_{lm}s_{ij}+\kappa_{lmij}}{2(\tilde{s}_{lj}-s_{ij})(\tilde{s}_{mi}-s_{ji})},\lefteqn{ D_0(2) = D_0(-q_j,k_l,q_i,\lambda,m_j,\overline{M}_l,m_i) }\qquad\sim-\frac{1}{K_ls_{ij}} \biggl\{ 2\ln\biggl(-\frac{s_{ij}}{m_i m_j}-{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) +\ln^2\biggl(-\frac{\tilde{s}_{lj}}{m_jM_l}-{{\rm i}}\epsilon\biggr){} +\ln^2\biggl(\frac{m_i}{M_l}\biggr) +\frac{\pi^2}{3} +\mathop{{\rm Li}_2}\nolimits\biggl(1-\frac{\tilde{s}_{lj}}{s_{ij}}\biggr) \biggr\}.The 5-point function $E_0 = E_0(-q_j,-k_m,k_l,q_i,\lambda,m_j,\overline{M}_m,\overline{M}_l,m_i)$can be reduced to the five 4-point functions D_0(0)=D_0(-k_m+q_j,k_l+q_j,q_i+q_j,m_j,M_m,M_l,m_i),D_0(1)=D_0(-k_m,k_l,q_i,0,\overline{M}_m,\overline{M}_l,m_i),D_0(2)=D_0(-q_j,k_l,q_i,\lambda,m_j,\overline{M}_l,m_i),D_0(3)=D_0(-q_j,-k_m,q_i,\lambda,m_j,\overline{M}_m,m_i) ,D_0(4)=D_0(-q_j,-k_m,k_l,0,m_j,\overline{M}_m,\overline{M}_l)according to $E_0 = -\sum_{l=0}^4 \frac{\det(Y_l)}{\det Y} D_0(l).$The symmetric matrix $Y$ reads (using $\tilde{s}_{li}=M_l^2$) Y=\left({}00-K_m-K_l0{}*0M_m^2\;(-K_l-\tilde{s}_{lj})\;-s_{ij}{}**2M_m^2\; (-\bar{s}_{lm}-K_l-K_m)\;(-K_m -\tilde{s}_{mi}){}***2M_l^2M_l^2{}****0\right),and $Y_i$ is obtained from $Y$ by replacing all entries in the $i$th column with 1. Neglecting terms that do not contribute to the correction factor in LPA, the corresponding determinants are given by \det(Y)\sim2s_{ij} \Bigl[ K_lK_m(s_{ij}\bar{s}_{lm}-\tilde{s}_{lj}\tilde{s}_{mi}-M_l^2M_m^2)\qquad {} + K_l^2M_m^2(s_{ij}-\tilde{s}_{mi}) + K_m^2M_l^2(s_{ij}-\tilde{s}_{lj})\Bigr],\det(Y_0)\sim\lambda(s_{ij}\bar{s}_{lm},\tilde{s}_{lj}\tilde{s}_{mi},M_l^2M_m^2) +4s_{ij}M_l^2M_m^2(\tilde{s}_{lj}+\tilde{s}_{mi}-s_{ij}) ,{} \det(Y_1)\simK_l\left[M_l^2M_m^2(\tilde{s}_{mi}-2s_{ij})+ \tilde{s}_{mi}(s_{ij}\bar{s}_{lm}-\tilde{s}_{lj}\tilde{s}_{mi}) \right]{} - K_m M_l^2(s_{ij}\bar{s}_{lm}+\tilde{s}_{lj}\tilde{s}_{mi} -2s_{ij}\tilde{s}_{mi}-M_l^2M_m^2),\det(Y_2)\sims_{ij}\left[ K_l(\tilde{s}_{lj}\tilde{s}_{mi}-s_{ij}\bar{s}_{lm} +M_l^2M_m^2) +2K_mM_l^2(\tilde{s}_{lj}-s_{ij}) \right],\det(Y_3)=\det(Y_2)|_{i\leftrightarrow j, l\leftrightarrow m},\det(Y_4)=\det(Y_1)|_{i\leftrightarrow j, l\leftrightarrow m}.The specific determinants (\ref{['detYvpole']}) appearing in the reduction for the IR-singular $E_0$ (\ref{['E0']}) with massless external lines obey the relations 0=\det(Y) + K_m\det(Y_2) + K_l\det(Y_3),0=-\tilde{s}_{mi}\det(Y) + K_ms_{ij}\det(Y_1) - \left[ K_l\tilde{s}_{mi}+K_mM_l^2 \right]\det(Y_3),0=-\tilde{s}_{lj}\det(Y) + K_ls_{ij}\det(Y_4) - \left[ K_m\tilde{s}_{lj}+K_lM_m^2 \right]\det(Y_2).These relations allow us to eliminate $\det(Y_1)$ and $\det(Y_4)$ from (\ref{['E0redf']}), resulting in: \lefteqn{E_0(-q_j,-k_m,k_l,q_i,\lambda,m_j,M_m,M_l,m_i))= -\frac{\det(Y_0)}{\det(Y)} D_0(0)}\quad{}- \frac{\det(Y_3)}{\det(Y)K_ms_{ij}} \Bigl\{[K_l\tilde{s}_{mi}+K_mM_l^2]D_0(1)+K_ms_{ij}D_0(3)\Bigr\}{}- \frac{\det(Y_2)}{\det(Y)K_ls_{ij}} \Bigl\{[K_m\tilde{s}_{lj}+K_lM_m^2]D_0(4)+K_ls_{ij}D_0(2)\Bigr\}-\frac{\tilde{s}_{lj}}{K_ls_{ij}} D_0(4) -\frac{\tilde{s}_{mi}}{K_ms_{ij}} D_0(1).In this section we list the integrals with the additional approximation that all invariants are large compared with the masses, i.e. $s_{ij},\tilde{s}_{lj},\bar{s}_{lm},t_{ki},\tilde{t}_{kl} \gg M_l^2.$We keep only the logarithmic terms and omit also the constant terms. The various combinations of scalar integrals read: case ${\rm }$ mm$$\frac{B_0(k_l^2,0,\overline{M}_l)-B_0(\overline{M}_l^2,0,\overline{M}_l)}{k_l^2-\overline{M}_l^2} -B'_0(M_l^2,\lambda,M_l) \sim \frac{1}{M_l^2}\ln\biggl(\frac{\lambda M_l}{-K_l}\biggr), \qquadcase ${\rm }$ mf$$\lefteqn{C_0(k_l,q_i,0,\overline{M}_l,m_i) -\Bigl[C_0(k_l,q_i,\lambda,M_l,m_i)\Bigr]_{k_l^2=M_l^2}}\qquad\space\sim-\frac{1}{M_l^2} \ln\biggl(\frac{m_i}{M_l}\biggr)\biggl[ 2\ln\biggl(\frac{-K_l}{\lambda M_l}\biggr) +\ln\biggl(\frac{m_i}{M_l}\biggr) \biggr], \qquadcase ${\rm }$ im$$\lefteqn{C_0(p_k,k_l,0,m'_k,\overline{M}_l) -\Bigl[C_0(p_k,k_l,\lambda,m'_k,M_l)\Bigr]_{k_l^2=M_l^2}}\qquad\space\sim\frac{1}{\tilde{t}_{kl}} \ln\biggl(\frac{m'_kM_l}{-\tilde{t}_{kl}}+{{\rm i}}\epsilon\biggr) \biggl[\ln\biggl(\frac{K_l}{\tilde{t}_{kl}}\biggr) +\ln\biggl(\frac{-K_l}{\lambda^2}\biggr) +\ln\biggl(\frac{m'_k}{M_l}\biggr) \biggr], \qquadcase mm'${\rm mm}'$\lefteqn{ C_0(k_l,-k_m,0,\overline{M}_l,\overline{M}_m) - \Bigl[C_0(k_l,-k_m,\lambda,M_l,M_m)\Bigr]_{k_l^2=M_l^2,{k_m}^2=M_m^2} } \qquad\space\sim\frac{1}{\bar{s}_{lm}}\biggl\{ \ln\biggl(\frac{M_lM_m}{-\bar{s}_{lm}}+{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{-\bar{s}_{lm}}{\lambda^2}-{{\rm i}}\epsilon\biggr) +\ln\biggl(\frac{\bar{s}_{lm}}{K_l}\biggr) \ln\biggl(\frac{\bar{s}_{lm}}{K_m}\biggr) -\frac{1}{2}\ln^2\biggl(\frac{M_l^2}{-K_l}\biggr){} -\frac{1}{2}\ln^2\biggl(\frac{M_m^2}{-K_m}\biggr) +\frac{1}{4}\ln^2\biggl(\frac{-\bar{s}_{lm}}{M_l^2}-{{\rm i}}\epsilon\biggr) +\frac{1}{4}\ln^2\biggl(\frac{-\bar{s}_{lm}}{M_m^2}-{{\rm i}}\epsilon\biggr) \biggr\},case if${\rm if}$\lefteqn{ D_0(p_k,k_l,q_i,\lambda,m'_k,\overline{M}_l,m_i)} \qquad\space\sim-\frac{1}{t_{ki} K_l} \biggl\{ 2\ln\biggl(\frac{-t_{ki}}{m'_k m_i}-{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) +\ln^2\biggl(\frac{-t_{ki}}{m'_k m_i}-{{\rm i}}\epsilon\biggr) +\ln^2\biggl(\frac{m_i}{M_l}\biggr) \biggr\},\spacecase mf'${\rm mf}'$\lefteqn{D_0(1) = D_0(-q_i,-k_l,k_m,0,m_i,\overline{M}_l,\overline{M}_m) } \qquad\sim\frac{1}{K_l \tilde{s}_{mi}} \biggl\{ \ln^2\biggl(\frac{-\tilde{s}_{mi}}{M_l^2}-{{\rm i}}\epsilon\biggr) -\frac{1}{2}\ln^2\biggl(\frac{-\bar{s}_{lm}}{M_l^2}-{{\rm i}}\epsilon\biggr) -\ln\biggl(\frac{-\tilde{s}_{mi}}{M_l^2}-{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{m_i^2}{M_l^2}\biggr){} +\frac{1}{2}\ln^2\biggl(\frac{K_l}{K_m}\biggr) +\ln\biggl(\frac{K_l}{K_m}\biggr) \biggl[\ln\biggl(\frac{-\tilde{s}_{mi}}{m_i^2}-{{\rm i}}\epsilon\biggr) +\ln\biggl(\frac{\tilde{s}_{mi}+{{\rm i}}\epsilon}{\bar{s}_{lm}+{{\rm i}}\epsilon}\biggr) \biggr] \biggr\}, \qquadcase ff'${\rm ff}'$\lefteqn{ D_0(0) = D_0(-k_m+q_j,k_l+q_j,q_i+q_j,0,M_m,M_l,0) } \qquad\sim-\frac{1}{\tilde{s}_{lj}\tilde{s}_{mi}-\bar{s}_{lm}s_{ij}} \biggl\{ \mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\tilde{s}_{mi}}+{{\rm i}}\epsilon,-x_1\biggr) +\mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\tilde{s}_{lj}}+{{\rm i}}\epsilon,-\frac{1}{x_2}\biggr)\qquad {} -\mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\bar{s}_{lm}}+{{\rm i}}\epsilon,-x_1\biggr) -\mathop{{\cal L}i_2}\nolimits\biggl(-\frac{M_lM_m}{\bar{s}_{lm}}+{{\rm i}}\epsilon,-\frac{1}{x_2}\biggr)\qquad {} -\biggl[\ln\biggl(\frac{\tilde{s}_{mi}+{{\rm i}}\epsilon}{s_{ij}+{{\rm i}}\epsilon}\biggr) +\ln\biggl(\frac{\tilde{s}_{lj}+{{\rm i}}\epsilon}{\bar{s}_{lm}+{{\rm i}}\epsilon}\biggr) \biggr]\Bigl(\ln(-x_1)-\ln(-x_2)\Bigr) \biggr\}\hbox{with} \qquad x_1 = \frac{(\tilde{s}_{mi}-s_{ij})z}{M_lM_m}(1-{{\rm i}}\epsilon s_{ij}(\tilde{s}_{lj}-s_{ij})), \quad\space x_2 = \frac{M_lM_m}{(\tilde{s}_{lj}-s_{ij})z}(1+{{\rm i}}\epsilon s_{ij}(\tilde{s}_{mi}-s_{ij})), \quad\qquad\qquad z = \frac{\tilde{s}_{lj}\tilde{s}_{mi}-\bar{s}_{lm}s_{ij}}{(\tilde{s}_{lj}-s_{ij})(\tilde{s}_{mi}-s_{ij})},\lefteqn{ D_0(2) = D_0(-q_j,k_l,q_i,\lambda,m_j,\overline{M}_l,m_i) }\qquad\sim-\frac{1}{K_ls_{ij}} \biggl\{ 2\ln\biggl(-\frac{s_{ij}}{m_i m_j}-{{\rm i}}\epsilon\biggr) \ln\biggl(\frac{\lambda M_l}{-K_l}\biggr) +\ln^2\biggl(-\frac{\tilde{s}_{lj}}{m_jM_l}-{{\rm i}}\epsilon\biggr) +\ln^2\biggl(\frac{m_i}{M_l}\biggr) \biggr\}.\spaceWe note that $D_0(0)$ does not involve large logarithms and could be replaced by zero in the logarithmic approximation.In this appendix we list the explicit values for the couplings $I^{V_a}_{\phi_i\phi_{i'}}$ introduced in Ref. Denner:2001jv required for our calculation. Note that a bar over a field indicates the charge-conjugated field. For quarks the couplings $I^{V}_{q_{i,\sigma} q_{j,\sigma}}$ depend on the helicity $\sigma$ and involve the quark-mixing matrix $V_{ij}$: I^{A}_{q_- q_-}=I^{A}_{q_+ q_+} = -Q_q,I^{Z}_{q_- q_-}=\frac{I^3_q-s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}^2 Q_q}{s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}} c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}}},\qquad I^{Z}_{q_+ q_+} = -\frac{s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}} Q_q}{c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}}},I^{W^+}_{u_{i,-} d_{j,-}}=\frac{1}{\sqrt{2}s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}}V_{ij} ,\qquad I^{W^-}_{d_{j,-} u_{i,-}} = \frac{1}{\sqrt{2}s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}}V^*_{ij} .All other quark couplings vanish, and the couplings for antiquarks are obtained by $I^{V}_{\bar{q}_{i,\sigma} \bar{q}_{j,\sigma}} = -I^{V}_{q_{j,-\sigma} q_{i,-\sigma}}.$For gauge bosons the couplings $I^{V_1}_{\bar{V}_2V_3}$ are totally antisymmetric in the field indices $V_1,V_2,V_3$ and read I^{A}_{W^-W^-}=I^{W^+}_{W^+A} = I^{W^-}_{AW^+} = -I^{A}_{W^+W^+} = -I^{W^-}_{W^-A} = -I^{W^+}_{AW^-} =1,I^{Z}_{W^-W^-}=I^{W^+}_{W^+Z} = I^{W^-}_{ZW^+} = -I^{Z}_{W^+W^+} = -I^{W^-}_{W^-Z} = -I^{W^+}_{ZW^-} = -\frac{c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}}}{s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}}.For Higgs and would-be Goldstone bosons, the couplings $I^{V}_{\bar{S}_1S_2}$ are antisymmetric in the scalar fields $S_1$ and $S_2$ and read I^{Z}_{\chi H}=-I^{Z}_{H\chi} = \frac{{{\rm i}}}{2c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}} s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}},I^{A}_{\phi^-\phi^-}=-I^{A}_{\phi^+\phi^+} = 1, \qquad I^{Z}_{\phi^-\phi^-} = -I^{Z}_{\phi^+\phi^+} = -\frac{c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}}^2-s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}^2}{2c{{\rm W}{\rm W}}c_{{\rm W}{\rm W}} s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}},I^{W^\pm}_{\phi^\pm H}=-I^{W^\pm}_{H\phi^\mp} = \pm\frac{1}{2s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}}, \qquad I^{W^\pm}_{\phi^\pm \chi} = -I^{W^\pm}_{\chi\phi^\mp} = \frac{{{\rm i}}}{2s{{\rm W}{\rm W}}s_{{\rm W}{\rm W}}}.The couplings appearing in the lowest-order matrix elements for four-fermion production can be read off from Appendix A of Ref. Denner:kt and are related to the quantities $I^{V}_{ff'}$ via $C^{\sigma\sigma'}_{V\bar{f} f'} = \delta_{\sigma,-\sigma'} I^{V}_{f_{\sigma'}f'_{\sigma'}}.$