A technical element of the implementation considering photon-induced processes is treating the photon as either off-shell or on-shell particle in the process record. This is important with respect to renormalisation factors related to the photon at NLO EW as explained in Sec. <ref>. The off-shell photon's technical prescriptions of the OLP  which are essential for example for LHC physics are thus encoded in  for automated NLO EW corrections.

For processes containing light charged fermions in the final state infrared-safety which is dictated by the KLN theorem defined in Sec. <ref> demands phase-space cuts to be imposed on these fermions only if they are `dressed'. This means that photons appearing at NLO EW for this class of processes are not resolved but recombined with the charged fermions if they fulfil criteria such as

\begin{align}
\label{photonreco}
\Delta R_{f^{\pm}\gamma}\leq R_0
\end{align}

\begin{align}
\label{deltaR}
\Delta R_{ij}=\sqrt{(\Delta\phi_{ij})^2+(\Delta\eta_{ij})^2}
\end{align}

and $R_0$ is a user-defined real number[$R_0$ can be set in SINDARIN by the parameter  with $0.1$ as default value.]. $\Delta\phi_{ij}$ thereby denotes the difference of azimuthal angles and $\Delta\eta_{ij}$ of rapidities of particles $i$ and $j$. In case Eq. (<ref>) is fulfilled the sum of the momenta of the photon $\gamma$ and of the fermion $f^{\pm}$ is identified with the momentum of the dressed fermion. In any other case the non-recombined fermion is identical to the dressed one.

§ MIXED COUPLING EXPANSIONS

For the computation of NLO contributions $\delta\text{NLO}_{q_s,q_e}$ of generic coupling powers $q_s$ and $q_e$ the following technical procedures within the code have to be undertaken.

Starting from the LO coupling powers $p_s$ and $p_e$ specified in  as

\begin{align}
\label{couplingsSindarin}
\begin{split}
&\texttt{alphas\_power = <p\_s>}\\
&\texttt{alpha\_power = <p\_e>}
\end{split}
\end{align}

and the command of Eq. (<ref>) the calculation of cross sections and fixed-order event generation at NLO in $\alpha$ can be steered. Precisely, this means a phase-space integration over the integrand given formally by the expression of Eq. (<ref>). In this way, all contributions $\delta \text{NLO}_{p_s,p_e+1}$ can be computed. For the leading QCD coupling power contribution $\delta \text{NLO}_{n+1,m}$ (the left-most blob in the lower row in Fig. <ref>) the NLO QCD settings such as the correction type  (to be replaced with  in Eq. (<ref>)) as well as the coupling powers $p_s=n$ and $p_e=m$ for the LO contribution leading in $\alpha_s$ have to be used.

A systematic approach for the inclusion of subtraction terms to any correction type of the corresponding splittings contributing to $\delta \text{NLO}_{p_s,p_e+1}$ is realised in  by the following steps.

First of all, we note that the Born flavour structures  and those emerging from the  for the real  are completely independent from the coupling power. They purely depend on the process definition and the chosen model. For a fixed coupling order at NLO in $\alpha$ the raw real flavour structures  for this reason have to be filtered: All real flavour structures have to be rejected which do not exist for the chosen coupling order for which the NLO contributions are defined, i. e. $q_s=p_s$ and $q_e=p_e+1$.
This can be done via a coupling power counting algorithm (CPCA) presented in Sec. <ref> with which coupling powers can be declared as `forbidden' having a definite flavour structure at hand.
Thus all real flavour structures are eliminated from the set of  objects if the fixed NLO coupling powers $\{p_s,p_e+1\}$ corresponds to one coupling power combination which is forbidden from the flavour structure. Likewise, all Born flavour structures are vetoed if neither the coupling powers $\{p_s,p_e\}$ nor $\{p_s-1,p_e+1\}$ are allowed from the particle content of the corresponding objects  with the CPCA.

The resulting reduced set of  and  are used in a second step for the initialisation of FKS singular regions as described in Sec. <ref>.
A crucial ingredient for NLO contributions in mixed coupling expansions is the introduction of a dynamical correction type which varies between QCD and EW depending on the FKS region. A singular FKS region $\tilde{\alpha}$ is well-defined in the correction type due to a definite emitter $E_{\tilde{\alpha}}$ and FKS pair $(i,j)$ determining the splitting in Eq. (<ref>), and thus the SM vertex associated with it.
For the construction of singular regions in  as explained in Sec. <ref> this induces a new degree of freedom, i. e. considering possible splittings for both correction types. Hence, FKS regions are created from classifying external legs $(i,j)$ of real flavour structures into the sets $G^{\text{QCD}}$ and $G^{\text{QED}}$ defined in Eqs. (<ref>) to (<ref>) and constructing splittings from the associated correction types.
In the special case of FKS pairs $(q,\bar{q})$ (if $i$ and $j$ are final-states), $(q,{q})$ and $(\bar{q},\bar{q})$ (if $i$ is an initial-state), one is tempted to assume that this must imply two FKS regions associated with the two correction types.
However this is only the case if the underlying Born structure, i. e. tracing this FKS pair back to a photon or a gluon, is allowed for coupling powers $\{p_s,p_e\}$ or $\{p_s-1,p_e+1\}$ which can be found out via the CPCA. An example here would be the process $pp\to t\bar{t}j$: The real-emission process $q\bar{q}\to t\bar{t} q'\bar{q}'$ at $\mathcal{O}(\alpha_s^2\alpha^2)$ with the FKS pair $(q',\bar{q}')$ either is due to a $\gamma\to q'\bar{q}'$ splitting with the Born process $q\bar{q}\to t\bar{t} \gamma$ at $\mathcal{O}(\alpha_s^2\alpha)$ or a $g\to q'\bar{q}'$ splitting with the Born process $q\bar{q}\to t\bar{t} g$ at $\mathcal{O}(\alpha_s\alpha^2)$.

Simultaneously, assigning a correction type to an FKS region the underlying Born flavour structure is attributed to it. The dynamical treatment of the correction type is visualised by assignments  and  to each region  in the FKS table shown for the exemplary process $pp\to t\bar{t}$ with $p=\{b,\bar{b},g,\gamma\}$ in App. <ref>.
If no underlying Born structure from the reduced set of  can be found to be attributed to either QCD or QED splittings of legs of a real flavour structure , a pseudo-FKS region is created. This means that for this real process no singularity from collinear or soft splittings exist, but the tree-level $n+1$ process for this flavour structure is still contributing at NLO which is explained in detail in Sec. <ref>.
Concerning the technical bookkeeping, for the dynamical correction type the string  is assigned to these pseudo-FKS regions. Due to this labeling the subtraction terms are omitted for these regions.

According to the labels  or  assigned to the other FKS regions, the corresponding group factors entering the real subtraction terms are chosen. This labelling is reused for the subtraction terms in integrated form due to the linking of the underlying Born flavour structures to the dynamical correction types in each FKS region.
If a Born flavour structure is linked to both correction types, e. g. for the process $b\bar{b}\to t\bar{t}$ once at $\mathcal{O}(\alpha_s^2)$ with correction type  and once at $\mathcal{O}(\alpha_s \alpha)$ with correction type , the integrated subtraction terms have to be evaluated for both of these cases. Note that for this special case the opposite applies for the virtual finite loop terms entering NLO contributions at coupling powers $\{p_s,p_e+1\}$, for which QCD and EW corrections overlap in the loop-tree interfering amplitudes. Due to this, virtual loop contributions from QCD and EW corrections can not be evaluated separately but only by a single OLP request for loop-tree interfering matrix elements. This request per default contains the Born flavour structure, the coupling powers $\{p_s,p_e\}$ and the correction type  as OLP input parameters.

In the context of steering  via the BLHA interface the input of coupling powers is required not only for the loop-tree interfering matrix elements but for squared matrix elements for each element of the NLO calculation. Subtraction terms for NLO contributions of order $\{p_s,p_e+1\}$ can be constructed with respect to either QCD or QED splittings and thus need squared factorised Born tree amplitudes at the corresponding coupling order, i. e. either at $\{p_s,p_e\}$ or $\{p_s-1,p_e+1\}$. Applying the CPCA to the set of  the allowed coupling powers from the particle content of the flavour structures are checked for compatibility with either $\{p_s,p_e\}$ or $\{p_s-1,p_e+1\}$. Note that for processes containing two or more $q\bar{q}$ pairs the coupling powers are not uniquely defined due to interfering diagrams with two possibilities for intermediate states, gluons and EW bosons.
Tracking these Born-level process structures and requesting the two different tree-level squared amplitudes is treated as a special case in  throughout all parts of the NLO calculation.[The code implementation for the extraction of squared tree-level matrix elements of coupling powers $\{p_s-1,p_e+1\}$ for this special case is based on sums over off-diagonal entries of the colour-correlated squared matrix elements for this coupling order.]

For Born-level processes at the subleading QCD coupling powers $\{p_s-1,p_e+1\}$, colour correlated squared matrix elements must be requested from  which enter the subtraction terms corresponding to QCD soft splittings. These need the correction type input `'. Due to this, for each  BLHA request associated with Born flavour structures – except for those entering the  file – the correction type, either  or , as well as the coupling powers have to be set.
An example for squared amplitudes requests in the  file for $pp\to t\bar{t}$ with $p=\{b,\bar{b},g,\gamma\}$ is shown in App. <ref>. The  file contains only tree-level squared matrix elements requests for Born processes at the coupling powers $\{p_s,p_e\}$ which are selected by means of the CPCA. The squared matrix elements accessed by this file enter the $\text{LO}_{p_s,p_e}$ contribution of the complete NLO calculation.

§.§ Coupling power counting algorithm

An algorithm determining coupling powers from the particle content of the flavour structures is useful as it flexibly provides the information of contributing subprocesses throughout an NLO calculation with mixed QCD-EW corrections. The algorithm presented in this section provides a mapping from flavour structures to potentially allowed coupling powers.

First of all, we note that for any $2\to n$ tree-level process the total number of coupling powers of either $\alpha_s$ or $\alpha$ is given by

\begin{align}
n_{\text{tot}}\equiv p_s+p_e=n_{\text{legs}}-2=n
\end{align}

with $n_{\text{legs}}$ denoting the number of external legs. From this we can derive the complete set of possibilities for powers $l_s$ of the QCD coupling $\alpha_s$ and $l_e$ of the electromagnetic coupling $\alpha$, i. e.

\begin{align}
&\{l_s,l_e\}=\{n-k,k\}, & 0\leq k\leq n\quad.
\end{align}

In order to constrain this range of possibilities for arbitrary processes we can make the following naive observations.

Gluons as external states always couple to processes by the coupling $\alpha_s$ to the power of one. Counting the number of gluon external states $n_{g}$ gives a first constraint on the upper boundary of the range of $k$:

\begin{align}
0\leq k \leq n -n_{g}\quad.
\end{align}

Similarly, external legs associated with EW bosons $\gamma$, $W$, $Z$ and $H$ couple to the process each by $\alpha$ of order one which further leads to

\begin{align}
n_{W}\leq k \leq n -n_{g}\quad.
\end{align}

with $n_{W}$ the number of external EW bosons. Furthermore, leptons (including neutrinos) always couple as pairs with $\alpha$ to the power of one. The number of external leptons $n_{l}$ thus gives another constraint yielding

\begin{align}
n_{W}+\frac{n_l}{2}\leq k \leq n -n_{g}\quad.
\label{generallyallowedcouplings}
\end{align}

As quarks are particles of both theories, QCD and EW, these are the only SM particles which as external states may couple to a process in pairs by both $g_s$ and $e$ on the amplitude-level. At the level of interfering amplitudes this would lead to ill-defined coupling powers of $\alpha_s$ and $\alpha$.
As the coupling $g_s$ or $e$ is either due to a gluon or an EW boson identifiable intermediate and external states beyond the quarks further constrain the range of $k$. This concerns the following considerations:

* For the case of exactly one $q\bar{q}$ pair and only gluons as external states, i. e. $n_q+n_{g}=n_{\text{legs}}$, we find

\begin{align}
\end{align}

* For the case of exactly one $q\bar{q}$ pair and only EW bosons or leptons as external states, i. e. $n_q+n_{W}+n_{l}=n_{\text{legs}}$, we find

\begin{align}
\end{align}

* If the quarks as external states are all of different flavours which implies pure EW couplings to $W^{\pm}$ of the quarks, we find

\begin{align}
k=n -n_{g}=n_{W}+\frac{n_{l}}{2}+\frac{n_{q}}{2}\quad.
\label{interferenceCP}
\end{align}

Note that if none of the conditions of these bullet points are fulfilled the ambiguity in the coupling powers merely is due to two or more $q\bar{q}$ pairs as external particle states of the process. Thus, for the allowed coupling powers due to Eq. (<ref>), the flavour structure is still uniquely defined.

In the implemented algorithm a chosen input coupling power is checked on equality to any of those cases represented by the allowed $k$ according to the rules above for an arbitrary input flavour structure. In this way, for example a flavour structure can be rejected from further evaluations in the code if the corresponding tree-level process does not exist for a certain coupling order. Selecting flavour structures in this way for each part of the NLO calculation in addition to the user-defined coupling powers $\{p_s,p_e\}$ for the LO process yields a well-defined input for the OLP and the evaluation of FKS terms. This also includes subprocesses with two or more $q\bar{q}$ pairs as the ambiguity in the coupling powers purely influences the actual numerical value of the squared or interfering matrix elements. By means of the user-defined coupling powers $\{p_s,p_e\}$ and the associated ones with respect to the different terms and factors entering the NLO calculation the evaluation of OLP requests for these flavour structures is eventually well-defined.

§.§ IR-safe observables

The criterion for observables being infrared-safe is enforced by the KLN theorem. It demands that adding a collinear or soft particle to a process does not change the value of the observable of that process. In practice, if phase-space cuts are imposed on partonic events a jet clustering algorithm which classifies partons into jets guarantees IR-safety of observables.

In , jet clustering algorithms such as the $k_T$ and anti-$k_T$ algorithm <cit.> are provided by an interface to  <cit.>. These are inevitable concerning QCD corrections to processes with final-state jets. In order to guarantee IR-safety for QED corrections photon recombination algorithms as explained in Sec. <ref> have to be applied which act on all charged fermions.

For NLO QCD-EW mixed corrections the jet definition of processes has to be extended including particles beyond QCD partons. There are different prescriptions for the jet definition to be applied for NLO contributions $\delta \text{NLO}_{q_s,q_e}$ represented by the lower row of Fig. <ref>. In order to ensure IR-safe observables for QCD corrections to the process at leading $\alpha_s$ order, i. e the contribution $\delta \text{NLO}_{n+1,m}$, the definition of jets must include all coloured (light) partons, quarks and gluons.

For NLO EW corrections to the process at leading $\alpha_s$ order, i. e. inducing the contribution $\delta \text{NLO}_{n,m+1}$, photons must be recombined with quarks to fulfil the IR-safety criterion. This can be achieved by so-called `democratic' jets <cit.> which means to extend the particle set forming a jet object to account for photons in the same way as for gluons. Crucially, for real emissions of photons off quarks the jet clustering algorithm acts on both of these partons and according to a jet resolution criterion recombines them.
Furthermore, having both a gluon and a photon as final-state particles in the real emission event each of them possibly induces IR singularities. If the photon is included democratically in the jet definition, IR divergences of real squared matrix elements from emissions of soft gluons are rendered finite as the counter-event contains a Born configuration with the photon consistently treated as the jet fulfilling phase-space cut criteria.

For contributions $\delta\text{NLO}_{n-i,m+1+i}$ with $i>0$ the jet definition must be extended additionally to account for leptons. This is due to the fact that photons already occur at the LO-level for processes with jets at coupling powers $\{n-1,m+1\}$. For higher orders in $\alpha$ photons can split not only into $q\bar{q}$ pairs which yield hadronic jets but also into lepton-antilepton pairs, $l^{\pm}l^{\mp}$. Observables sensitive to the jet-energy for this reason would not be IR-safe if the leptons are not taken into account in the jet definition as light lepton pairs emerging from a splitting photon can induce collinear singularities.

As protons are similar hadronic particle containers as jets analogous particle set extensions can be made for them provided that for the used PDF set the additional parton is included in the fit. However, there are no IR-safety criteria which have to be imposed on protons representing initial-state particles. For that reason the proton particle set extension to account in addition for photons (and leptons) merely changes the numerical result due to the additional contributions.

QED PDFs in general are suppressed compared to quark and gluon PDFs as protons constitute a hadronic system with predominant QCD interactions. In fact, approximately $0.5\%$ of the proton's momentum is carried by the photon <cit.>. The actual contributions of photon-induced processes however highly depend on the process and the observable one is looking at. There are certain exceptions for which these can get very large such as contributions of about $5$-$10\%$ for high invariant masses $M_{ll}$ in Drell-Yan mass distributions <cit.>. Crucially, the contribution of the $q\gamma$ PDF channel can grow up to $40\%$ for $W^+W^+W^-x$ production in $W^-$ transverse momentum distributions <cit.>. In particular single photon-induced processes are represented by PDF channels opened up in a mixed coupling expansion of LO and NLO contributions such as $g\gamma\to t\bar{t}$ in top pair production. This in general implies that photon-induced processes for this purpose should not be neglected.

There are PDF sets including lepton PDFs <cit.> which in principle yield non-zero contributions. However, they are mostly numerically irrelevant for applications related to LHC processes as for example $l\gamma$ luminosities with respect to $\gamma\gamma$ ones are suppressed by a factor of about $200$-$300$ <cit.>.

Recommendations how to set  definitions  and  in  denoting the particle content of protons and jets for each NLO contribution indicated by blobs of the lower row in Fig. <ref> are shown in Table <ref>. The coupling order of an NLO contribution thereby is set by the second to fourth columns with the input  and  given through the coupling powers related to Eq. (<ref>). The suggestions for the proton and jet definitions in the last two columns follow the considerations above on IR-safety conditions and the contributions of photon (and lepton) PDFs. The following specifications for the SINDARIN files are used for the expressions  and  in Table <ref>:

\begin{align}
\begin{split}
&\texttt{alias q = u:d:s:c:b:U:D:S:C:B}\\
&\texttt{alias l = e1:e2:e3:E1:E2:E3}
\end{split}
\label{aliasdefql}
\end{align}

The definition of  for $pp \to X$ processes, where $X$ consists of leptons and/or massive particles, is different compared to that for $pp \to X+\text{jets}$ only with respect to EW corrections to subleading QCD processes (last two rows in Table <ref>). This is mandated by the IR-safety condition for photons existing at LO-level and splitting into lepton pairs in real emission diagrams as explained above. Furthermore, the definition of the proton , which for consistency reasons have to be set equal to , for some cases require PDF sets to include photons (or even leptons) where necessary from a phenomenological point of view. Technically, by adding photons and/or leptons to the particle sets  sets up corresponding PDFs with zero values if the PDF set itself does not contain PDFs for these added particles.

$\text{LO}_{p_s,p_e}+\delta \text{NLO}_{q_s,q_e}$                2l

(with $0<i<n$)               $pp\to X$    $pp\to X~\text{jets}$

Setting recommendations steering NLO computations in  with NLO contributions $\delta \text{NLO}_{q_s,q_e}$ corresponding to the blobs of Fig. <ref> in addition to contributions $\text{LO}_{p_s,p_e}$, where  denotes the $\alpha_s$ power and  the $\alpha$ power with respect to the leading QCD order contribution (upper left-most blob in Fig. <ref>). While the  definitions  and  for quarks and leptons are given in Eq. (<ref>),  denotes the gluon and  the photon, respectively.

If the final state of processes consists of jets as well as charged leptons for NLO EW calculations both measures to ensure IR-safe observables, jet clustering and dressing of leptons, must be undertaken.
This yields a non-trivial phase-space cut evaluation since photons have to be recombined with all charged fermions. On the one hand, photons render the charged leptons dressed. However, as they are part of the jet definition, they enter the jet clustering algorithm simultaneously to be recombined with QCD partons. This implies the condition to treat fermion dressing and jet clustering in a systematic way such that photons are always recombined with the closest charged fermion. This can be achieved in two ways <cit.>. One possibility is to recombine the photon with all charged fermions, i. e. light quarks and leptons, in the first step since the photon-recombination cone is typically defined more narrow as the jet clustering cone. If the photon fulfils the recombination criterion, i. e. it can be found within the cone with radius $R_0$ according to Eq. (<ref>), the photon is discarded for the subsequent step of clustering. For the jet clustering eventually only gluons, dressed quarks and non-recombined photons are taken into account. Alternatively, a jet-lepton separation criterion can be applied.

Validation and Results

CHAPTER: HADRON COLLISIONS IN NLO MIXED COUPLING EXPANSIONS

In this chapter the validation as well as illustrative results of NLO calculations in mixed coupling expansions for several benchmark processes at the LHC are shown. Total cross sections and differential distributions are obtained with the automated MC framework of  which has been extended to account for EW corrections. The methods employed to this end are explained in detail in the previous chapters: the phase-space construction for the case of massless initial-state radiation presented in Sec. <ref>, the FKS subtraction scheme in mixed coupling expansions described in Sec. <ref> and the technical background of an NLO computation performed with  given in Sec. <ref>.

Input parameters and further details on the setup used for the calculations of the results are given in Sec. <ref>. In Sec. <ref> NLO EW cross sections and differential distributions for pure electroweak LHC processes are presented. NLO mixed corrections are computed for processes for which the leading QCD coupling contributions are of $\mathcal{O}(\alpha_s)$ or higher, with results shown in Sec. <ref>. Dedicated cross-checks with other MC tools are provided for both of these process classes.

§ SETUP OF THE CALCULATION

For the results of NLO calculations related to $pp$ processes in this work, the following settings and input parameters are used.

Because $pp$-induced processes come along with large sets of subprocesses  is used for the matrix element computations. As explained in Sec. <ref> this OLP represents the most efficient tool concerning the interplay with  for this process class. The renormalisation of SM parameters due to UV divergences is performed by applying the complex-mass scheme (CMS) which is the default[For any other case the flag  can be set in .] in  and explained in detail in Sec. <ref>.

Furthermore, for the calculations presented here the $G_{\mu}$ input scheme is used for the computation of the electromagnetic coupling $\alpha$ for which the running of the coupling up to the EW scale is implicitly contained. This choice in  corresponds to the setting

\begin{align*}
\texttt{\$blha\_ew\_scheme = "GF"}\quad.
\end{align*}

The coupling $\alpha$ is defined by the input for the $G_{\mu}$ constant

\begin{equation}
G_{\mu}= 1.16639\cdot 10^{-5} \; \text{GeV}^{-2}
\end{equation}

and the pole masses and widths of $W$ and $Z$ bosons according to Eq. (<ref>). These are set through the transformation formulas

\begin{align}
&M_V={M_V^{\text{BW}}}/{\sqrt{1+\left(\Gamma_V^{\text{BW}}/M_V^{\text{BW}}\right)^2}} &\Gamma_V={\Gamma_V^{\text{BW}}}/{\sqrt{1+\left(\Gamma_V^{\text{BW}}/M_V^{\text{BW}}\right)^2}}
\end{align}

using the Breit-Wigner masses and widths

\begin{eqnarray}
\begin{split}
M_W^{\text{BW}} &=\; \phantom{1}80.385\phantom{0} \;\text{GeV} \quad\qquad
\Gamma_W^{\text{BW}} &=2.0897\;    \;\text{GeV} \quad~ \\
M_Z^{\text{BW}} &=\; \phantom{1}91.1876 \;\text{GeV} \quad\qquad
\Gamma_Z^{\text{BW}} &=2.4955\;    \;\text{GeV} \quad.
\end{split}
\label{WZmasseswidths}
\end{eqnarray}

For reasons of comparisons with reference MC tools, these are chosen according to those of Ref. <cit.>. The masses and widths in this study correspond to those of PDG <cit.> using a Breit-Wigner (OS-like) lineshape approach.
Other heavy particle masses and widths are set by the numerical values

\begin{eqnarray}
\begin{split}
M_H &=\; 125.0\phantom{000} \;\text{GeV} \quad\qquad
\Gamma_H &=0\;     \;\\
m_t &=\; 173.34\phantom{00} \;\text{GeV} \quad\qquad
\Gamma_t &=1.36918\;\text{GeV}    \;
\end{split}
\label{Htmasseswidths}
\end{eqnarray}

where light quarks and leptons are considered as massless, i. e.

\begin{eqnarray}
m_{q\neq t}  &=\;0 \quad\qquad m_l&=\;0 \quad.
\end{eqnarray}

Note that for processes with on-shell bosons or top quarks corresponding particle widths of Eqs. (<ref>) and (<ref>) are set to zero. The explicit settings for these cases are indicated in the discussion part below, where necessary.

The centre-of-mass energy for the $pp$ collisions is chosen according to the collider energy of LHC Run II,

\begin{align}
\sqrt{s} = 13 \text{ TeV}\quad.
\end{align}

For the PDFs the  PDF set <cit.>, a fit including photon PDFs, from the library
(version 6.2.3) <cit.> is used. The running of the the QCD coupling constant $\alpha_s$ at three-loop accuracy is performed with  with reference value

\begin{align}
\alpha_s(M_Z)=0.118
\end{align}

associated with the chosen PDF set.

The renormalisation and factorisation scale corresponds to

\begin{align}
&\mu_R=\mu_F=\frac{H_T}{2} &H_T=\sum_i\sqrt{p_{T,i}^2+m_i^2}
\end{align}

where $H_T$ denotes the transverse mass with $p_{T,i}$ and $m_i$ the transverse momentum and masses of the final-state particles. By defining  in this way $\mu_R$ and $\mu_F$ are set simultaneously in .

For the photon recombination with massless fermions the parameter $R_0$ according to Eq. (<ref>) is set by

\begin{align*}
\texttt{photon\_rec\_r0 = 0.1}
\end{align*}

in the SINDARIN file. For processes with jets the anti-$k_T$ clustering algorithm <cit.> with jet radius $R=0.4$ is applied by specifying

\begin{align*}
\begin{split}
&\texttt{jet\_algorithm = antikt\_algorithm}\\
&\texttt{jet\_r = 0.4}
\end{split}
\end{align*}

which is used as an input to  <cit.>. Phase-space cut expressions acting on dressed fermions and clustered jets follow the conditions

* $p_{T,l^{\pm}}> 10$ GeV and $\lvert \eta_{l^{\pm}}\rvert<2.5$ on charged dressed leptons

* $\Delta R_{l^{+}l^{-}}>0.4$ and $M_{l^{+}l^{-}}> 30$ GeV on pairs of charged dressed leptons with same flavour and opposite charge

* $p_{T,j}> 30$ GeV and $\lvert \eta_{j}\rvert<4.5$ on clustered jets

with distance $\Delta R_{ij}$ given by Eq. (<ref>).

The process-specific settings for an NLO computation, i. e. the coupling powers defined by Eq. (<ref>), the correction type as well as  definitions of proton and jet objects, are consistently applied as prescribed by Table <ref>. By this photon-induced processes are taken into account automatically in the NLO computations where relevant with results presented in the subsequent sections.
An exemplary  file containing all settings for steering an NLO EW cross section computation for the process $pp\to e^+e^-j$ is attached in App. <ref>. All the NLO EW calculations are performed by using 's MPI parallelised setup. The computing time highly depends on the considered process. However, the estimated average time for cross section integrations is a couple of hours on around $\mathcal{O}(100)$ cores.

§ NLO EW CORRECTIONS TO PURE ELECTROWEAK PROCESSES AT THE LHC

In this section numerical results on NLO EW total cross sections and differential distributions to pure electroweak processes at the LHC are presented. These processes can be further classified into those involving on-shell and off-shell massive vector-bosons. Results and validation checks on the former are discussed in Sec. <ref> and those on the latter in Sec. <ref>, respectively.

In order to quantify the relative correction of the NLO EW with respect to the LO cross section in the following we define the quantity

\begin{align}
\delta\equiv\frac{\sigma^{\text{tot}}_{\text{NLO}}-\sigma^{\text{tot}}_{\text{LO}}}{\sigma^{\text{tot}}_{\text{LO}}}
\label{delta}
\end{align}

with $\sigma^{\text{tot}}_{\text{LO}}$ denoting the LO and $\sigma^{\text{tot}}_{\text{NLO}}$ the NLO total cross section, respectively.

§.§ On-shell boson production

In this section the production of EW massive bosons for $pp$ processes with two and three final states at NLO EW is studied. This involves neutral- and charged-current processes with $VV$, $VH$, $VVV$, $VVH$ and $VHH$ production for all possible combinations of $V=W^{\pm},Z$.
For the calculation of cross sections and fixed-order differential distributions for this class of processes the QCD coupling power $p_s$ is set to zero, and the electromagnetic coupling power $p_e$ to the number of final states, respectively.
Furthermore, the LHC setup of Sec. <ref> is adjusted with the additional requirement to set the vector-boson widths of Eq. (<ref>) to zero.

§.§.§ Total cross sections

In order to validate the NLO EW calculation for these processes the total cross sections are compared with twofold reference results of the MC framework  <cit.> applying both schemes, CS and $q_T$ subtraction. In Table <ref> results of the cross-checks[For technical reasons of the comparison the top-quark width is set to zero for all processes except for $W^+W^-$, $W^+W^-Z$ and $W^+W^-H$ production, for which top resonances can occur in real-emission diagrams. The width for these cases is set to $\Gamma_t=1.44262$ GeV.] with reference values using CS subtraction are shown. The checks can be assessed by evaluating the parameters

process      $\texttt{MUNICH}_{\text{(CS)}}\texttt{+OpenLoops}$     3c|       dev [%]     $\sigma^{\text{sig}}_{\text{NLO}}$

$pp \rightarrow X$    $\sigma_{\text{NLO}}^{\text{tot}}$  [fb]    $\sigma_{\text{LO}}^{\text{tot}}$  [fb]    $\sigma_{\text{NLO}}^{\text{tot}}$ [fb]    $\delta$ [%]

$ZZ$                  $ 1.05729(1) \cdot 10^4 $           $1.10367(1)\cdot 10^4 $     $ 1.0573(1) \cdot 10^4 $        $ -4.20 $     $ 0.0001 $     $ 0.01 $

$W^+Z$                  $ 1.71505(2) \cdot 10^4$          $ 1.71760(2)\cdot 10^4 $      $ 1.71507(2) \cdot 10^4 $        $-0.15 $     $ 0.001 $     $ 0.88 $

$W^-Z$           $ 1.08576(1) \cdot 10^4 $       $  1.084976(8)\cdot 10^4 $     $ 1.08574(1) \cdot 10^4 $     $ +0.07 $      $ 0.001 $     $ 0.90 $

$W^+W^-$               $ 7.93106(7) \cdot 10^4 $        $ 7.58598(4)\cdot 10^4 $          $ 7.9309(2)\cdot 10^4 $         $ +4.55 $     $ 0.002 $     $ 0.89 $

$ZH$             $ 6.18523(6) \cdot 10^2 $            $  6.53082(2)\cdot 10^2 $       $ 6.18533(6) \cdot 10^2 $         $ -5.29 $     $ 0.002 $    $ 1.17 $

$W^+H$        $ 7.18070(7) \cdot 10^2 $             $ 7.35035(4)\cdot 10^2 $       $ 7.18072(9) \cdot 10^2 $         $ -2.31 $     $ 0.0003 $    $ 0.18 $

$W^-H$              $ 4.59289(4) \cdot 10^2 $         $  4.69379(2)\cdot 10^2 $         $ 4.59299(5) \cdot 10^2 $        $ -2.15 $      $ 0.002 $    $ 1.62 $

$ZZZ$              $ 9.7429(2) \cdot 10^0 $           $  1.07605(8)\cdot 10^1 $      $ 9.742(1) \cdot 10^0 $        $-9.47 $     $ 0.012 $     $ 1.01 $

$W^+W^-Z$              $ 1.08288(2) \cdot 10^2 $      $ 1.00587(2)\cdot 10^2 $        $1.0829(1) \cdot 10^2 $         $ +7.67 $     $ 0.004 $     $ 0.45 $

$W^+ZZ$         $ 2.0188(4) \cdot 10^1 $     $  1.9875(2) \cdot 10^1 $    $  2.019(2) \cdot 10^1 $     $ +1.58 $     $ 0.0001 $     $ 0.01 $

$W^-ZZ$             $ 1.09844(2) \cdot 10^1 $      $  1.06546(8)\cdot 10^1 $         $ 1.0984(1) \cdot 10^1 $       $ +3.09 $     $ 0.006 $     $ 0.51 $

$W^+W^-W^+$           $ 8.7979(2) \cdot 10^1 $       $ 8.2867(7)\cdot 10^1 $        $ 8.7991(15) \cdot 10^1 $        $ +6.18 $     $ 0.014 $     $ 0.79 $

$W^+W^-W^-$           $ 4.9447(1) \cdot 10^1 $         $  4.6149(1)\cdot 10^1 $       $ 4.9441(2) \cdot 10^1 $        $ +7.13 $     $ 0.013 $     $ 2.52 $

$ZZH$            $ 1.91607(2) \cdot 10^0 $     $ 2.1006(1)\cdot 10^0 $    $ 1.91614(18) \cdot 10^0 $      $ -8.78 $     $ 0.004 $     $0.39$

$W^+ZH$                  $  2.48068(2) \cdot 10^0 $     $ 2.4409(2) \cdot 10^0 $     $ 2.48095(28) \cdot 10^0 $     $ +1.64 $     $ 0.011 $     $ 0.96 $

$W^-ZH$             $ 1.34001(1) \cdot 10^0 $          $ 1.30731(9) \cdot 10^0 $         $ 1.3402(1) \cdot 10^0 $        $ +2.51 $     $ 0.011 $     $ 1.02 $

$W^+W^-H$            $ 9.7012(2) \cdot 10^0 $          $ 8.8313(2) \cdot 10^0 $         $ 9.700(2) \cdot 10^0 $        $ +9.83 $     $ 0.014 $     $ 0.75 $

$ZHH$            $ 2.39350(2) \cdot 10^{-1} $          $ 2.6909(2) \cdot 10^{-1} $          $ 2.3934(3) \cdot 10^{-1} $        $ -11.06 $     $ 0.005 $     $ 0.41 $

$W^+HH$            $ 2.44794(2) \cdot 10^{-1} $         $  2.7827(2) \cdot 10^{-1} $          $  2.4478(2) \cdot 10^{-1} $       $ -12.04 $     $ 0.007 $     $ 0.74 $

$W^-HH$             $ 1.33525(1) \cdot 10^{-1} $            $ 1.5087(1)\cdot 10^{-1} $        $ 1.33471(19) \cdot 10^{-1} $        $ -11.53 $     $ 0.041 $     $ 2.80 $

[pure EW onshell]Comparison of NLO EW total cross sections for processes with two and three on-shell bosons at $\sqrt{s}=13$ TeV obtained with  and  using the OLP  with numbers $\delta$ defined in Eq. (<ref>) and dev and $\sigma^{\text{sig}}$ in Eq. (<ref>), respectively.

\begin{align}
&\textit{dev}\equiv  \frac{|\sigma^{\text{tot}}_{\texttt{WHIZARD}}-\sigma^{\text{tot}}_{\texttt{MUNICH}}|}{\sigma^{\text{tot}}_{\texttt{WHIZARD}}} & \sigma^{\text{sig}}\equiv\frac{|\sigma^{\text{tot}}_{\texttt{WHIZARD}}-\sigma^{\text{tot}}_{\texttt{MUNICH}}|}{\sqrt{\Delta_{\texttt{WHIZARD}}^2+\Delta_{\texttt{MUNICH}}^2}}
\label{MunWZparams}
\end{align}

for the NLO EW cross sections $\sigma^{\text{tot}}$ obtained with  and  and statistical MC uncertainties $\Delta_{\text{err}}$. For all processes in Table <ref>, agreement is found at a numerical accuracy level of relative MC errors well below sub-per-mille level. This can be concluded from $\textit{dev}\lesssim 0.01\%$ in the sixth column together with $\sigma^{\text{sig}}\lesssim3$ in the last column. Thus, the possibility for even minor bugs on the implemented subtraction schemes is ruled out.

Apart from these checks, the numerical values of the relative correction $\delta$ of the NLO EW with respect to the LO result, defined in Eq. (<ref>), can be understood by the following physical effects playing a role for the inclusion of EW corrections.

An inherent effect playing a role for all shown processes is the EW virtual massive vector-boson exchange between external state legs of a process which yields a negative contribution to the NLO correction. This effect is  enhanced by double logarithms of the form $\alpha \log^2 (s/M_W^2)$ for soft-collinear emission of EW vector bosons in the high-energy limit $s\gg M_W$. In general these can be approximated by EW Sudakov correction factors <cit.>. For di-boson processes this effect is dominant in particular for $ZZ$ and $ZH$ with suppressions of about $-5\%$. NLO EW cross sections to $ZZZ$, $ZZH$, $ZHH$ and $W^{\pm}HH$ with $\delta\sim- 10\%$ yield the largest suppressions for triple-boson production.

There are multiple effects from real-emission diagrams counteracting the EW Sudakov logarithmic suppression with a different impact depending on the process. The most prominent one is the enhancement due to single-top resonances in real-emission diagrams to the processes $pp\to W^+W^-$, $W^+W^-Z$ and $W^+W^-H$. The positive NLO contributions due to this overcompensate the EW Sudakov effects yielding enhancements up to $\delta \sim +10\%$ for $W^{+}W^-H$.
A positive relative correction of the NLO EW cross sections with respect to the LO ones can be caused as well by quasi-collinear splittings such as $q^*\to q'W^{\pm}$ or $q^*\to qZ$ in high-energy phase-space regions inducing large positive amplitudes <cit.>. This effect is enhanced especially if $q\gamma$ channels open up by the corresponding real configuration and the photon couples through the vertex $\gamma W^+W^-$. In addition to spin-1 exchange in the $t$-channel this vertex leads to large amplitudes for $W$ bosons propagating in forward direction <cit.>. These amplifying effects are present both in real-emission diagrams $q\gamma\to W^{\pm} Z q'~(+H/Z)$ with $q^*\to Zq$ and in $q\gamma \to W^+W^-q'~(+W^{\pm}/H/Z)$ with $q^*\to W^{\pm}q'$ quasi-collinear splittings. This twofold enhancement effect for the corresponding processes may be one of the reasons for their (over-) compensation of the EW Sudakov suppression to be seen from their values $\delta \gtrsim 0$ in Table <ref>. Note furthermore, that this NLO effect plays neither a role for the production of purely neutral bosons, where there is no $W$ boson, nor the Higgsstrahlung processes, $W^{\pm}H$ and $W^{\pm}HH$, with no significant enhancements due to quasi-collinear splittings at NLO. Dominant negative EW virtual contributions thus are reasonable for these processes explaining their negative relative correction $\delta$.

Another effect which can be relevant for the processes exhibiting $\delta \gtrsim 0$ is the so called `radiation zero' <cit.>. In general, this effect means that for $q\bar{q}^{\prime}\to W^{\pm}\gamma$ squared amplitudes for certain phase-space configurations, due to roots of the cross section differential in the photon-beam angle, vanish. The condition for this effect is based on the fact that the photon can couple to the $W^{\pm}$ inducing an $s$-channel in the same way as to the quarks inducing a $t$-channel.
In this way, LO contributions for this process become suppressed with respect to real-emission ones, leading to large K factors. Due to the similarity with respect to $W^{\pm}\gamma$ production the `radiation zero' effect and the consequential enhancement of NLO with respect to LO contributions may be also transferred in a mitigated form to $W^{\pm}Z$ production. In addition, it could play a role as well for the production of $W^{\pm}ZZ$, $W^{\pm}ZH$ and $W^+W^-W^{\pm}$ (due to $qq^{\prime}\to W^{\pm}\gamma^*(\to W^+W^-)$ diagrams). As contributions for the processes of $W^{\pm}H$ and $W^{\pm}HH$ merely come from $s$-channel diagrams at LO the `radiation zero' effect explicitly plays no role there.

§.§.§ Fixed-order differential distributions

Illustratively, for the process $pp\to HHW^+$ fixed-order differential distributions at LO and NLO EW are presented in Figs. <ref> to <ref>. The histograms are produced with the analysis tool  <cit.> by means of simulated  event samples using .
Note that the distributions shown here represent fixed-order accurate results of the NLO calculation. For a realistic event-simulation of differential cross sections a proper matching to QED parton showers must be applied. As this is a topic beyond the scope of this thesis, in the following the focus is on understanding the physics effects occurring at NLO EW for the considered fixed-order observables. These apply in the same way for the charge-conjugated process, $pp\to HHW^-$.

In Fig. <ref> differential distributions of the transverse momentum of the $W^+$ boson and the Higgs boson with the largest transverse momentum, i. e. the hardest Higgs boson, is shown. First of all, the global behaviour of the two distributions at LO relative to each other can be interpreted in the following way. The distribution $d\sigma/dp_{T,W^+}$ for transverse momenta $p_{T,W^+}$ in the range $10$ - $100$ GeV only slightly increases, with a plateau around $40$ - $100$ GeV with a size of about $10^{-3}$ fb/GeV. In contrast, the distribution $d\sigma/dp_{T,H}$ has a much steeper increase for $p_{T,H}$ in this range of about two orders of magnitudes starting from about $10^{-5}$ fb/GeV. This behaviour can be traced back to the fact that there are much less events for the hardest Higgs with low $p_T$'s as for low $p_{T,W}$ as the $W$ recoils against both of the two Higgs bosons.

Differential distributions of  the transverse momentum of the $W^+$ boson (left) and the hardest Higgs (right) for the process $pp\to HHW^+$ at $13$ TeV

The behaviour of the distributions at NLO EW with respect to the corresponding LO ones for the $p_T$ of the two different final states is similar: The differential K factor, in fact, appears nearly identical for both of the histograms with a suppression from $1$ to about $0.65$ for the complete $p_T$ ranges shown in Fig. <ref>. This suppression can be directly related to negative EW Sudakov logarithms of the form $\sim-\alpha \log^2 p_T^2 / M_W^2$. In general, these double logarithms for an EW Sudakov approximation are defined with arguments $r_{kl}/M_W^2$ with $r_{kl}\gg M_W^2$ where $r_{kl}$ denotes the kinematic invariant $(p_k+p_l)^2$ for external states $k$ and $l$ <cit.>.
Large suppression factors of virtual loop with respect to Born amplitudes due to this effect can be related to large invariants of a process. For $pp\to HHW^+$ the largest suppressions can be found for high-$p_{T,W}$ regions for which all invariants $r_{kl}$ of the process are maximal. These can be quantified approximately by the invariant corresponding to the $W$ boson and the hardest Higgs. As this quantity grows symmetrically in both $p_{T,W^+}$ and $p_{T,H}$ the similarity of both differential K factors, in particular for high $W$ transverse momenta, is manifest.

Differential distributions of  the energy of the $W^+$ boson (left) and the hardest Higgs (right), respectively, for the process $pp\to HHW^+$ at $13$ TeV

In Fig. <ref>, the energy-distributions at LO and NLO EW for each, $W$ boson and hardest Higgs, are depicted. The K factor for the distribution differential in the $W$ energy can be considered as constant. This behaviour underlines the fact that the actual energy of the $W$ – which may vary at NLO EW in addition due to the real radiation of a photon – is not directly in correlation with the EW Sudakov suppression factor for which the double-logarithms depend purely on invariants of two external state legs of the (Born) process. As explained with respect to the total cross sections, this EW virtual effect is eventually overcompensating the effect of QED radiation.

The K factor of the differential distribution in  $E_H$, the energy of Higgs identified with the hardest $p_T$, is decreasing starting from the threshold until it stays constant at around $500$ GeV.
As explained above, the $p_T$ of the hardest Higgs can be seen as a measure for the kinematical invariant $r_{kl}$ on which the EW logarithmic Sudakov suppression factors depend on. This has an impact also on the variable $E_H$:
for small numbers of $E_H$ the correlation to the variable $p_{T,H}$ is enhanced as in general for very low energies also low transverse momenta are expected. The second-$p_{T}$-hardest Higgs due to its definition always receives transverse momenta below that of the $p_T$-hardest Higgs. Conversely, large energies $E_H$ do not necessarily induce large $p_{T,H}$ values.
To that end, EW Sudakov suppression factors indirectly depend on the variable $E_H$, in particular for small energies of the $p_T$-hardest Higgs.

The cross sections differential in the pseudorapidity of the $W^+$ and the hardest Higgs are shown in Fig. <ref>. From the distributions at LO as well as at NLO EW it can be concluded that the bulk of the total cross section comes from events with momenta of the final states distributed close to the plane perpendicular to the beam axis. This is due to the fact that contributions to the Born amplitudes purely come from the $s$-channel. At NLO EW, hard-photon ISR in the real-emission diagrams due to radiative return causes large real amplitudes if the photons are radiated in forward direction. This induces a boost of the $HHW$ system along the beam axis. This effect is visible in the distribution differential in $\lvert \eta_{W^+}\rvert$ where for large numbers of this variable the K factor amounts up to $1.5$. For pseudorapidities close to zero the K factor is around $0.8$ which can be related to large EW Sudakov logarithms which play a role purely for the virtual diagrams. These loop diagrams have Born-like phase-space configurations in the same way as the Born tree-level diagrams with enhanced amplitudes for $W^+$ scattered perpendicular to the beam axis.
For this reason and due to the shift of real events to phase-space regions with forward radiated $W^+$ bosons the suppression due to EW virtual logarithms is mostly visible for small pseudorapidities of the $W^+$ boson.

Differential distributions of the pseudorapidity of the $W^+$ boson (left) and the hardest Higgs (right), respectively, for the process $pp\to HHW^+$ at $13$ TeV

In contrast, the hardest Higgs pseudorapidity distribution, shown on the right hand side of Fig. <ref>, exhibits a relatively constant large suppression of the NLO EW with respect to the LO curves. By definition the `hardest Higgs'  corresponds to that Higgs boson final state with the largest $p_{T}$. Due to this, the latter explained effect of the boost of the Born final-state system in the case of ISR is less pronounced for the hardest Higgs than for the $W$ boson. In other words, the boost-energy is mainly deposited to the $W^+$ and second-$p_T$-hardest Higgs which per definition is the Higgs boson scattered closest to the beam axis.

Fig. <ref> shows the distribution differential in the invariant mass of the di-Higgs system at LO and NLO EW. The general behaviour of the curves can be understood as follows: The sharp edge at $250$ GeV indicates the threshold of the di-Higgs system which can be produced starting from an invariant mass of twice the Higgs mass.
From that mass the distributions decrease with $M_{HH}$ which is the typical behaviour as the centre-of-mass energy $\sqrt{\hat{s}}$ increases with this variable leading to an overall suppression of the amplitudes. The K factor related to these distributions can be considered as constant in $M_{HH}$ with the statistical precision. This can be related to the fact that real-emission and Born-like events in general are not kinematically separated in the di-Higgs mass, as e. g. in the case of the $W^+$ pseudorapidity. Photon radiation at NLO EW is merely due to initial-state or $W^+$ emitters but does not concern the di-Higgs system. The overall suppression of the LO distribution can be associated with  approximately $\delta\sim -12\%$ which corresponds to the value of the relative correction of the total cross section of this process in Table <ref>.

Differential distribution of the invariant mass of the di-Higgs system for the process $pp\to HHW^+$ at $13$ TeV

§.§ Off-shell vector boson production

Electroweak processes associated with off-shell vector bosons impose new technical challenges for a complete NLO EW computation. This is due to generally higher final-state multiplicities with respect to on-shell boson processes involving massless QCD partons and leptons. These massless final states require phase-space cut criteria as well as IR safety conditions. In this section the NLO EW corrections to several LHC benchmark processes are computed by using the setup defined in Sec. <ref>.

§.§.§ Total cross sections

The total cross sections at NLO EW for neutral- and charged-current processes (with and without associated Higgs) as well as vector-boson fusion (VBF) and single-top plus jet processes are computed with  with numerical results presented in Table <ref>. Corresponding reference results computed with  () <cit.> are added which serve as checks in order to validate the automated NLO EW framework of  for this class of processes.
For comparison we again define the quantity

\begin{equation}
\sigma^{\text{sig}}\equiv \frac{|\sigma^{\text{tot}}_{\texttt{WHIZARD}}-\sigma^{\text{tot}}_{\texttt{MG5}}|}{\sqrt{\Delta_{\texttt{WHIZARD}}^2+\Delta_{\texttt{MG5}}^2}}\quad.
\label{MG5WZsigma}
\end{equation}

with cross sections $\sigma^{\text{tot}}$ obtained with  and  and corresponding statistical MC uncertainties $\Delta$. In the same way as , the MC generator  uses the FKS subtraction scheme.

process      $\alpha^m$         3c|

$pp \rightarrow X~ (+\text{jets})$        $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]    $\sigma_{\text{LO}}^{\text{tot}}$ [pb]    $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\delta$ [%]

$e^+\nu_e$              $\alpha^2$          $ 5.2005(8)\cdot10^3 $          $5.2377(4)\cdot10^3$           $ 5.1994(4)\cdot10^3 $         $- 0.73 $
$ 1.24 $

$e^+e^-$           $\alpha^2$           $ 7.498(1)\cdot10^2 $        $7.535(1)\cdot10^2$        $ 7.498(1)\cdot10^2 $        $ -0.50 $
$ 0.004 $

$e^+\nu_e\mu^-\bar\nu_\mu$      $\alpha^4$         $ 5.2794(9)\cdot10^{-1} $     $5.0938(7)\cdot10^{-1}$       $ 5.2816(9)\cdot10^{-1} $     $ +3.69 $
$ 1.69 $

$e^+e^-\mu^+\mu^-$     $\alpha^4$              $ 1.2083(3)\cdot10^{-2} $      $1.2747(2) \cdot10^{-2}$         $ 1.2078(3) \cdot10^{-2}$         $- 5.25 $
$ 1.26 $

$H e^+\nu_e$           $\alpha^3$         $ 6.4740(17)\cdot10^{-2} $        $6.7488(6)\cdot10^{-2}$          $ 6.4763(6)\cdot10^{-2} $         $ -4.04 $
$ 1.24 $

$ H e^+e^-$      $\alpha^3$        $ 1.3699(2)\cdot10^{-2} $         $1.4552(1)\cdot10^{-2}$           $ 1.3699(1)\cdot10^{-2} $         $- 5.86 $
$ 0.32 $

$ H jj$          $\alpha^3$              $ 2.7058(4)\cdot10^{0} $       $2.8251(4)\cdot10^{0} $          $ 2.7056(6)\cdot10^{0} $        $ -4.23 $
$ 0.27 $

$tj$           $\alpha^2$           $ 1.0540(1)\cdot10^2 $       $1.0615(1)\cdot10^2$         $ 1.0538(1)\cdot10^2 $        $ -0.72 $
$ 0.74 $

[pure EW offshell]Comparison of NLO EW total cross sections for LHC benchmark processes associated with off-shell vector-bosons at $\sqrt{s}=13$ TeV obtained with  and  using the OLP  with numbers $\delta$ defined in Eq. (<ref>) and $\sigma^{\text{sig}}$ in Eq. (<ref>), respectively.

The NLO EW cross section results of both MC generators in Table <ref> agree for relative MC uncertainties of $\mathcal{O}(0.01\%)$ with respect to the corresponding absolute values at the level $\sigma_{\text{NLO}}^{\text{sig}}< 2$. In view of the relative correction $\delta$ with magnitudes $\geq0.5\%$ this yields a convincing check. It confirms the correctness of the implementation of NLO EW automated corrections in  for this class of processes requiring cut criteria on charged dressed fermions.

Towards the discussion of the results, the first observation which can be made with respect to relative corrections is that the only positive value appears for the charged-current process $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}$. This is due to the underlying process $pp\to W^{+}W^{-}$ which already in the case of on-shell $W$ bosons gets positive corrections to be seen in Table <ref>. The same argument for the enhancement of the NLO EW with respect to the LO cross sections holds for the off-shell process: Large real-emission amplitudes for the configuration $b\gamma\to e^+\nu_e\mu^-\bar{\nu}_{\mu}b$ (identically with $b\leftrightarrow\bar{b}$) are expected due to single-top resonances. This effect in addition to others causing large real amplitudes overcompensate the respective suppression effect due to negative EW virtual Sudakov logarithms.

Similar as for the production of on-shell bosons $ZZ$, $W^+H$ and $ZH$ in Table <ref> correction factors $\delta$ of a few negative percent can be observed as well for those containing diagrams with corresponding off-shell vector-bosons[Note that for the neutral-current processes the gauge-bosons $Z$ and $\gamma$ mix with each other. However, for high energies such as those at the LHC the photonic channels typically are suppressed.], i. e. $e^+e^-\mu^+\mu^-$, $He^+\nu_e$ and $He^+e^-$ production. As explained above in the context of the vector-boson on-shell processes for these diagrams the EW Sudakov suppression effect is dominant.

As the process of $Hjj$ production contains similar sub-graphs with additional interfering VBF diagrams a suppression factor of similar size as for the latter processes is evident.

That the magnitude of the relative correction $\delta$ for $e^+\nu_e$, $e^+e^-$ and $tj$ final states is below $1\%$ can be explained by the fact that these processes involve only two final states. EW Sudakov factors in general add up for all possible kinematical invariants $r_{kl}=(p_k+p_l)^2$ of the process. Thus, a small number of final states, and in this turn possible invariants, leads to a reduced size of the virtual suppression factor with respect to higher particle multiplicity processes.

§.§.§ Fixed-order differential distributions

For the charged-current process $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}$ cross section distributions differential in properties of  $e^+\nu_e$ and $e^+\mu^-$ systems at LO and NLO EW are shown in Figs. <ref> - <ref>. Apart from their qualitative discussion given below, for the validation of the event generation with  checks with  with respect to several differential distributions at NLO EW are provided in App. <ref>.

Differential distribution of the invariant mass of the $e^+\nu_e$ system for the process $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}$ at $13$ TeV

The distributions differential in the invariant mass of the $e^+\nu_e$ system, $M_{e^+\nu_e}$, at LO and NLO EW are presented in Fig. <ref>. The global behaviour of both curves corresponds to the typical shape of the invariant mass of final-states emerging from the decay of a $W$ boson with a peak at the $W$ mass. While for $M_{e^+\nu_e}$ above this mass the K factor is close to $1$, for values below this mass it amounts up to about $1.8$. This large K factor can be traced back to the effect which is called `radiative tail':
The bulk of the events contributing to the cross section at LO comes from phase-space configurations with invariant masses $M_{e^+\nu_e}$ at the peak around $M_W$. At NLO due to radiation of photons these events are shifted from the threshold to lower values of $M_{e^+\nu_e}$.

Differential distribution of the transverse momentum of the $e^+\nu_e$ system for the process $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}$ at $13$ TeV

In Fig. <ref> the distribution differential in $p_{T,e^+\nu_e}$, the transverse momentum of the $e^+\nu_e$ system, is shown. The behaviour of the K factor can be related to the superposition of the following physical effects.

First of all, we observe a constant K factor of about $1.03$ up to $p_{T,e^+\nu_e}\sim 50$ GeV. The positive relative correction associated with it can be understood by the fact that in general contributions from the LO graphs $\gamma\gamma\to W^{+}W^{-}$ are enhanced for large momentum transfers. This is due to the momentum enhancement of the splitting $\gamma^*\to W^{+}W^{-}$, which e. g. is amplified compared to the $\gamma^*\to f\bar{f}$ vertex, and the additional $t$-channel nature of the process <cit.>. At NLO $\gamma q \to W^{+}W^{-}q$ channels open up containing this $\gamma\gamma\to WW$ sub-graph. These yield much larger contributions as the LO $\gamma\gamma\to W^{+}W^{-}$ channels since the photon PDFs in general are suppressed with respect to quark PDFs. In contrast to the $\gamma\gamma$ induced processes, the $\gamma q$ (and $\gamma\bar{q}$) channels also cover a large number of flavour and charge degrees of freedom for which all contributions add up for the complete real-emission part of the differential cross section. From the fact that this $\gamma\gamma\to WW$ sub-graph exists in the LO as well as in the NLO diagrams a constant behaviour of the K factor can be understood.

For $p_{T,e^+\nu_e}\gtrsim50$ GeV the K factor increases with the transverse momentum up to about $1.1$. The bulk of this effect can be explained by the fact that contributions due to single-top resonances in the real-emission diagrams $\gamma b\to W^{+}W^{-}b$ increase with $W^{\pm}b$ energies approaching the top mass. This explanation is in line with the observation that the largest K factor is reached for $p_{T,e^+\nu_e}$ values around the top mass apart from the low statistics for bins with large $p_{T,e^+\nu_e}$.
In general, this positive contribution in addition to enhancements of real amplitudes due to  $q^*\to Wq'$ quasi-collinear splittings in the high-$p_T$ regions overcompensate the suppressive effect of EW Sudakov virtual corrections.

Differential distribution of the invariant mass of the $e^+\mu^-$ system for the process $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}$ at $13$ TeV

The LO and NLO EW distribution differential in the invariant mass of the $e^+\mu^-$ system, $M_{e^+\mu^-}$, is depicted in Fig. <ref>. The shape of the K factor for these distributions originates from two effects which are directly related to $M_{e^+\mu^-}$. The first one is represented by large contributions of real amplitudes for $q\gamma$ PDF channels which are opened up with respect to the underlying Born diagrams $\gamma\gamma\to WW$, described in detail above. This effect unambiguously plays a role in the distribution differential in $M_{e^+\mu^-}$ as this variable crucially depends on the invariant mass of the intermediate $WW$ system. This property in turn is independent from momentum transfers in the $t$-channel diagrams which for large values, corresponding to small angles of the $W$ bosons to the beam axis, induce large amplitudes for $\gamma\gamma\to WW$ subgraphs. Events with these phase-space configurations thus contribute to any $M_{e^+\mu^-}$. This in particular leads to a constant high enhancement of the real with respect to the Born contributions.
It can be seen from the constant shift of the K factor of about $+5\%$ for low invariant masses where EW virtual Sudakov logarithmic effects can be neglected.
In fact, it can be seen that the K factor in Fig. <ref> decreases starting from masses $M_{e^+\mu^-}$ at $\mathcal{O}(100~\text{GeV})$. The most plausible explanation is the suppression due to EW Sudakov virtual corrections
which scale indeed with logarithms of the invariant $r_{kl}=M_{e^+\mu^-}$.

§ NLO MIXED CORRECTIONS TO PROCESSES AT THE LHC

For processes at the LHC with non-zero $\alpha_s$ power at Born level, EW corrections are less trivial to compute as for pure EW processes. Cancellations of IR-singularities of both gauge groups, QCD and QED, must be taken into account simultaneously. In order to achieve this for arbitrary processes and LO coupling powers, the formal description of the FKS scheme in mixed coupling expansions in Sec. <ref> and technical methods of Sec. <ref> are implemented in the framework of .
The checks which validate the automated computation of NLO mixed corrections are presented in the following. In Sec. <ref> the validation of LO and NLO contributions to all possible coupling powers for $t\bar{t}$, $t\bar{t}W$, $t\bar{t}Z$ and $t\bar{t}H$ production at the LHC is exhibited. Sec. <ref> includes the check for representative processes with massless final states which additionally require intricate criteria ensuring IR-safety as outlined in Sec. <ref>.

§.§ Top-quark pair production with associated $W/Z/H$

In general, regarding all contributions from the mixed coupling expansion, an NLO calculation yields an increased precision for theoretical predictions. For processes with LO contributions from more than one coupling order the interference of correction types for observables at NLO plays a role.
Top-quark pair production at the LHC is the simplest representative process of this class. Considering Fig. <ref> observables for this process entail contributions from three different coupling orders at LO, i. e. $\text{LO}_{20}$, $\text{LO}_{11}$ and $\text{LO}_{02}$, and from four different ones at NLO, i. e. $\delta\text{NLO}_{30}$, $\delta\text{NLO}_{21}$, $\delta\text{NLO}_{12}$ and $\delta\text{NLO}_{03}$. The contributions $\delta\text{NLO}_{21}$ and $\delta\text{NLO}_{12}$ thus require the consistent subtraction of IR-singularities of both QCD and QED splittings.
The whole tower of coupling power contributions are computed with  using the methods described in Sec. <ref> and the LHC setup of Sec. <ref>. Needless to say, as the top quark is identified in the final-state it is considered as on-shell with the top width set to zero.

All results for contributions to the cross sections per coupling power combinations are shown in Table <ref>. In addition, twofold reference results of  obtained with its NLO EW extension using CS <cit.> and $q_T$ subtraction <cit.> are included. The parameters $\sigma^{\text{sig}}$ and dev correspond to those defined in Eq. (<ref>). The relative MC uncertainties are chosen for each LO contribution at $\mathcal{O}(0.001\%)$ and for the NLO contribution at $\mathcal{O}(0.01\%)$ - $\mathcal{O}(0.1\%)$ with respect to the corresponding absolute result.

In view of these relative MC uncertainties the cross-checks can be considered as convincing. Furthermore, relative deviations dev of the results are found at the order of magnitude of the relative MC errors and the parameter $\sigma^{\text{sig}}$ takes values below $2$ for all contributions. This yields the proof of validity for the implemented FKS scheme in  and, simultaneously, for the two implemented subtraction schemes, CS and $q_T$, of  in the context of NLO mixed corrections. In order to prove the automation for computations of NLO corrections of this kind similar cross-checks are performed for the processes with associated Higgs, $Z$ and $W$ boson, i. e. $t\bar{t}H$, $t\bar{t}W^+$, $t\bar{t}W^-$ and $t\bar{t}Z$ production. For these processes, similarly to $t\bar{t}$ production, particle widths of the top quark and massive bosons are set to zero. Results on these checks are presented in Tables <ref> - <ref>. Likewise, agreement of all contributions of the different subtraction schemes with respect to each other are found using sufficiently high statistics.

The discussion of the underlying physics of the NLO cross section entailing all contributions from the mixed coupling expansion is as follows. For the process $pp\to t\bar{t}$, the largest contribution at LO comes from $\text{LO}_{20}$ at highest order in $\alpha_s$. The smallest, $\text{LO}_{02}$, is about 2 orders of magnitudes suppressed with respect to $\text{LO}_{20}$ which is clear from the approximate factor $\alpha^2/\alpha_s^2 \sim 1/100$ in the squared amplitudes. The contribution $\text{LO}_{11}$ is smaller than $\text{LO}_{02}$. This can be explained because of vanishing interfering amplitudes of diagrams  $q\bar{q}\to t\bar{t}$ with $q\neq b$ due to colour. Explicitly for these processes, the $s$-channel diagram at $\mathcal{O}(g_s^2)$ with internal gluon propagator interfering with that at $\mathcal{O}(g_e^2)$ with internal $\gamma/Z$ propagator is colour-forbidden. This fact clearly can be seen from Table <ref> containing the LO contributions split up into PDF channels, where the contribution of the $q\bar{q}$ channel (with $q\neq b$) vanishes. Non-zero contributions to $pp\to t\bar{t}$ at $\mathcal{O}(\alpha_s\alpha)$ hence merely come from $b\bar{b}$ and $g\gamma$ channels. Due to the relative size of bottom and photon PDFs with respect to those of other partons these channels yield rather small contributions.

1c        3c|$\sigma^{\text{tot}}$ [fb]       3c$\sigma^{\text{sig}}$ / dev

$pp \rightarrow t\bar{t}$     $\alpha_s^n\alpha^m$    2c|

$\text{LO}_{20}$           $\alpha_s^2$     2c|      $ 4.37969(3) \cdot 10^5 $              $ 4.37972(6) \cdot 10^5 $         3c$ 0.54$ / $ 0.001\%$

$\text{LO}_{11}$           $\alpha_s\alpha$            2c| $ 1.77303(4) \cdot 10^3 $             $ 1.77300(5) \cdot 10^3 $          3c$ 0.50 $ / $0.002\%$

$\text{LO}_{02}$           $\alpha^2$     2c| $ 2.76254(2) \cdot 10^3 $             $ 2.76253(4) \cdot 10^3 $        3c$ 0.20 $ / $0.000\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}$       2c|$4.42504(3) \cdot 10^5$    $4.42507(6) \cdot 10^5$    3c

qT   CS   FKS   ${\texttt{M}_{\text{qT}}}$-${\texttt{M}_{\text{CS}}}$   $\texttt{M}_{\text{qT}}$-$\texttt{W}_{\text{FKS}}$     ${\texttt{M}_{\text{CS}}}$-$\texttt{W}_{\text{FKS}}$

$\delta\text{NLO}_{30}$           $\alpha_s^3$     $ 2.0226(9) \cdot 10^5 $           $ 2.02154(3) \cdot 10^5 $             $ 2.0218(2) \cdot 10^5 $       $ 1.13 $ / $0.053\%$     $0.79$ / $0.038\%$    $ 1.20$ / $0.015\%$

$\delta\text{NLO}_{21}$           $\alpha_s^2\alpha$     -$ 4.7005(6) \cdot 10^3 $          -$ 4.7007(2) \cdot 10^3 $            -$ 4.7003(3) \cdot 10^3 $        $ 0.29 $ / $0.004\%$     $0.31$ / $0.004\%$    $ 1.27$ / $0.008\%$

$\delta\text{NLO}_{12}$           $\alpha_s\alpha^2$     $ 2.4058(4) \cdot 10^3 $           $ 2.40514(6) \cdot 10^3 $             $ 2.4053(3) \cdot 10^3 $        $ 1.80 $ / $0.027\%$     $1.09$ / $0.021\%$    $ 0.53$ / $0.006\%$

$\delta\text{NLO}_{03}$           $\alpha^3$    $ 2.0870(9) \cdot 10^1 $          $ 2.0874(5) \cdot 10^1 $            $ 2.090(1) \cdot 10^1 $        $ 0.41 $ / $0.019\%$     $1.95$ / $0.154\%$    $ 1.90$ / $0.135\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}+\sum_{\sigma^{\text{tot}}_{\text{NLO}}}$    1c|    $6.4249(9) \cdot 10^5$
$6.42384(4) \cdot 10^5$     $6.4242(3) \cdot 10^5$    $1.14$ /  $0.017\%$   $0.76$ / $0.012\%$   $1.32$ / $0.005\%$

Results of cross-checks of  and  for cross section contributions to the process $pp\to t\bar{t}$ at $\sqrt{s}=13$ TeV for each coupling power combination at LO and NLO

1c        3c|$\sigma^{\text{tot}}$ [fb]       3c$\sigma^{\text{sig}}$ / dev

$pp \rightarrow t\bar{t}H$     $\alpha_s^n\alpha^m$    2c|

$\text{LO}_{21}$           $\alpha_s^2\alpha$     2c|      $  3.44865(1)\cdot 10^2 $              $  3.4487(1)\cdot 10^2 $         3c$ 0.76$ / $ 0.003\%$

$\text{LO}_{12}$           $\alpha_s\alpha^2$            2c| $  1.40208(2)\cdot 10^0 $            $  1.4022(1)\cdot 10^0 $          3c$ 1.44 $ / $0.011\%$

$\text{LO}_{03}$           $\alpha^3$     2c| $  2.42709(1)\cdot 10^0 $             $ 2.4274(2) \cdot 10^0 $        3c$ 2.07 $ / $0.011\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}$       2c|$ 3.48694(1)\cdot 10^2$    $ 3.4870(1)\cdot 10^2$    3c

qT   CS   FKS   ${\texttt{M}_{\text{qT}}}$-${\texttt{M}_{\text{CS}}}$   $\texttt{M}_{\text{qT}}$-$\texttt{W}_{\text{FKS}}$     ${\texttt{M}_{\text{CS}}}$-$\texttt{W}_{\text{FKS}}$

$\delta\text{NLO}_{31}$           $\alpha_s^3\alpha$     $ 9.98(1) \cdot 10^1 $           $  9.9656(4)\cdot 10^{1} $             $ 9.968(4) \cdot 10^1 $       $ 1.05 $ / $0.108\%$     $ 0.79 $ / $ 0.086\%$    $ 0.62$ / $ 0.023\%$

$\delta\text{NLO}_{22}$           $\alpha_s^2\alpha^2$     $  6.210(4)\cdot 10^0 $           $ 6.209(1) \cdot 10^0 $            $  6.208(2)\cdot 10^0 $        $0.37 $ / $ 0.024\%$     $0.46 $ / $0.033 \%$    $ 0.20$ / $ 0.009 \%$

$\delta\text{NLO}_{13}$           $\alpha_s\alpha^3$     $ 1.7248(7) \cdot 10^0 $           $  1.7238(2)\cdot 10^0 $             $ 1.7232(5) \cdot 10^0 $        $ 1.33 $ / $0.058\%$     $1.92$ / $0.097\%$    $1.24 $ / $0.040\%$

$\delta\text{NLO}_{04}$           $\alpha^4$    $  1.5051(4)\cdot 10^{-1} $          $  1.5053(3)\cdot 10^{-1} $            $  1.5060(7)\cdot 10^{-1}  $        $ 0.28 $ / $0.010\%$     $1.10$ / $0.058\%$    $1.00 $ / $0.048\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}+\sum_{\sigma^{\text{tot}}_{\text{NLO}}}$    1c|    $4.565(1)\cdot 10^2$
$ 4.56433(4)\cdot 10^2 $     $ 4.5646(4)\cdot 10^2$    $1.07$ / $0.024\%$     $0.72$ / $0.017\%$     $0.82$ / $0.007\%$

[pure EW onshell]Results of cross-checks of  and  for cross section contributions to the process $pp\to t\bar{t}H$  at $\sqrt{s}=13$ TeV for each coupling power combination at LO and NLO

1c        2c|$\sigma^{\text{tot}}$ [fb]       1c$\sigma^{\text{sig}}$ / dev

$pp \rightarrow t\bar{t}W^+$     $\alpha_s^n\alpha^m$    $\texttt{MUNICH}_{\text{(CS)}}$

$\text{LO}_{21}$           $\alpha_s^2\alpha$     1c|      $  2.411403(1)\cdot 10^2 $              $  2.4114(1)\cdot 10^2 $         1c$ 0.72 $ / $ 0.003\%$

$\text{LO}_{12}$           $\alpha_s\alpha^2$            1c| $ 0.000 $            $ 0.000 $          1c$ 0.00 $ / $0.000\%$

$\text{LO}_{03}$           $\alpha^3$     1c| $ 2.31909(1) \cdot 10^0 $             $ 2.3193(1) \cdot 10^0 $        1c$ 1.76$ / $0.009\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}$       1c|$ 2.434594(9)\cdot 10^2$    $ 2.4347(1)\cdot 10^2$    1c$0.74$ / $0.003\%$

$\delta\text{NLO}_{31}$           $\alpha_s^3\alpha$           $  1.18993(2)\cdot 10^2 $             $  1.1905(5)\cdot 10^2 $       $ 1.06 $ / $0.048\%$

$\delta\text{NLO}_{22}$           $\alpha_s^2\alpha^2$          $- 1.09511(9) \cdot 10^1 $            $-  1.0947(3)\cdot 10^1 $        $ 1.13 $ / $ 0.035\%$

$\delta\text{NLO}_{13}$           $\alpha_s\alpha^3$          $  2.93251(3)\cdot 10^1 $             $2.9334(8)  \cdot 10^1 $        $ 1.14 $ / $0.030\%$

$\delta\text{NLO}_{04}$           $\alpha^4$           $ 5.759(3) \cdot 10^{-2} $            $  5.756(4)\cdot 10^{-2} $        $ 0.58 $ / $0.049\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}+\sum_{\sigma^{\text{tot}}_{\text{NLO}}}$    1c|    $ 3.80884(2)\cdot 10^2$     $ 3.8096(6)\cdot 10^2$    1c $1.40$ / $0.020\%$

[pure EW onshell]Results of cross-checks of  and  for cross section contributions to the process $pp\to t\bar{t}W^+$ at $\sqrt{s}=13$ TeV for each coupling power combination at LO and NLO

1c        2c|$\sigma^{\text{tot}}$ [fb]       1c$\sigma^{\text{sig}}$ / dev

$pp \rightarrow t\bar{t}W^-$     $\alpha_s^n\alpha^m$    $\texttt{MUNICH}_{\text{(CS)}}$

$\text{LO}_{21}$           $\alpha_s^2\alpha$     1c|      $ 1.217532(4) \cdot 10^2 $              $ 1.21756(5) \cdot 10^2 $         1c$ 0.44 $ / $ 0.002\%$

$\text{LO}_{12}$           $\alpha_s\alpha^2$            1c| $  0.000$            $  0.000 $          1c$ 0.00 $ / $0.000\%$

$\text{LO}_{03}$           $\alpha^3$     1c| $ 1.115922(4) \cdot 10^{0} $             $  1.11596(6)\cdot 10^0 $        1c$ 0.62 $ / $0.003\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}$       1c|$ 1.228691(4)\cdot 10^2$    $ 1.22871(5)\cdot 10^2$    1c $0.45$ / $0.002\%$

$\delta\text{NLO}_{31}$           $\alpha_s^3\alpha$           $ 6.18022(9) \cdot 10^1 $             $  6.178(2)\cdot 10^1 $       $ 0.72 $ / $0.029\%$

$\delta\text{NLO}_{22}$           $\alpha_s^2\alpha^2$          $ - 4.3881(4)\cdot 10^0 $            $-4.388(2)  \cdot 10^0 $        $ 0.10 $ / $ 0.004\%$

$\delta\text{NLO}_{13}$           $\alpha_s\alpha^3$          $  1.48300(2)\cdot 10^1 $             $  1.4822(4)\cdot 10^1 $        $ 1.74 $ / $0.052\%$

$\delta\text{NLO}_{04}$           $\alpha^4$           $  3.949(1)\cdot 10^{-2} $            $ 3.951(2) \cdot 10^{-2} $        $ 0.83 $ / $0.046\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}+\sum_{\sigma^{\text{tot}}_{\text{NLO}}}$    1c|    $ 1.95153(1)\cdot 10^2$     $ 1.9513(3)\cdot 10^2$    1c $0.90$ / $0.012\%$

[pure EW onshell]Results of cross-checks of  and  for cross section contributions to the process $pp\to t\bar{t}W^-$ at $\sqrt{s}=13$ TeV for each coupling power combination at LO and NLO

1c        2c|$\sigma^{\text{tot}}$ [fb]       1c$\sigma^{\text{sig}}$ / dev

$pp \rightarrow t\bar{t}Z$     $\alpha_s^n\alpha^m$    $\texttt{MUNICH}_{\text{(CS)}}$

$\text{LO}_{21}$           $\alpha_s^2\alpha$     1c|      $ 5.04648(2) \cdot 10^2 $              $  5.0463(2)\cdot 10^2 $         1c$ 0.82$ / $ 0.003\%$

$\text{LO}_{12}$           $\alpha_s\alpha^2$            1c| $  -3.48777(7)\cdot 10^0 $            $ -3.4876(2) \cdot 10^0 $          1c$ 0.71 $ / $0.004\%$

$\text{LO}_{03}$           $\alpha^3$     1c| $ 1.141785(6) \cdot 10^1 $             $  1.14179(4)\cdot 10^1 $        1c$ 0.008 $ / $0.000\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}$       1c|$ 5.16066(2)\cdot 10^2$    $ 5.1605(2)\cdot 10^2$    1c$0.82$ / $0.003\%$

$\delta\text{NLO}_{31}$           $\alpha_s^3\alpha$           $  2.26389(6)\cdot 10^2 $             $  2.264(1)\cdot 10^2 $       $ 0.26 $ / $0.017\%$

$\delta\text{NLO}_{22}$           $\alpha_s^2\alpha^2$        $ -4.231(2) \cdot 10^0 $             $  -4.228(3)\cdot 10^0 $        $ 1.02 $ / $0.084\%$

$\delta\text{NLO}_{13}$           $\alpha_s\alpha^3$            $ 4.2428(7) \cdot 10^0 $            $  4.245(3)\cdot 10^0 $        $ 0.76 $ / $ 0.062\%$

$\delta\text{NLO}_{04}$           $\alpha^4$           $  -4.111(1)\cdot 10^{-1} $            $ -4.111(2) \cdot 10^{-1} $        $ 0.21 $ / $0.012\%$

$\sum_{\sigma^{\text{tot}}_{\text{LO}}}+\sum_{\sigma^{\text{tot}}_{\text{NLO}}}$    1c|    $ 7.42056(7)\cdot 10^2$     $ 7.420(1)\cdot 10^2$    1c $0.31$ / $0.006\%$

[pure EW onshell]Results of cross-checks of  and  for cross section contributions to the process $pp\to t\bar{t}Z$ at $\sqrt{s}=13$ TeV for each coupling power combination at LO and NLO

For NLO contributions $\delta \text{NLO}_{q_s,q_e}$ to $pp\to t\bar{t}$, apart from $\delta \text{NLO}_{30}$ which is – about a factor of $100$ – enhanced with respect to the others due to the coupling powers, the origin of the numerical size and sign of a contribution is not obvious anymore. With respect to contributions $\delta\text{NLO}_{21}$, $\delta\text{NLO}_{12}$ and $\delta\text{NLO}_{03}$ this can be seen as well from the splitting of each into PDF channel contributions presented in Table <ref> which vary severely in the size and in the sign per channel. One observation can be deduced nevertheless: For $\delta \text{NLO}_{21}$ the largest size of PDF channel contributions comes from $gg$ initiated processes which is negative and about one order of magnitude enhanced with respect to the others. This can be concluded in first line from the amplification due to the PDFs. Secondly, as this contribution has a negative sign large EW Sudakov virtual corrections considered as dominant effect can be found as reasonable. This is underlined by the fact that EW corrections for this channel induce positive real-emission contributions only by photon radiation off the final-state. Virtual loop diagrams with EW boson exchange for which amplitudes can grow up to large values in high-energy phase-space regions in this way overcompensate the real contributions.

For this particular channel, EW virtual effects thus can be clearly detected as the coupling power of $\alpha_s$, which is two, at LO and NLO is associated with the coupling of two gluons to the hard process. This is not any more unambiguous considering quark-pair initiated processes, i. e. $q\bar{q}$ and $b\bar{b}$ channels, since QCD and EW corrections overlap in the virtual diagrams. This is depicted schematically in Fig. <ref> for exemplary $q\bar{q}\to q^{\prime}\bar{q}^{\prime}$ Feynman graphs. The diagrams on the left-hand side represent the EW corrections to the Born tree-level interfering graphs with pure QCD couplings. On the right-hand side the QCD corrections to the Born tree-level QCD-EW interfering graphs are shown.
Since the lower left loop-tree interfering diagrams can be considered equally as QCD corrections to the upper right interfering graphs at Born-level, the following fact becomes clear: In general, QCD and EW corrections cannot be distinguished from each other for a fixed coupling order, here $\mathcal{O}(\alpha_s^2\alpha)$, to which the process would be considered at NLO in either $\alpha$ or $\alpha_s$ with respect to a well-defined LO coupling order. This concerns the quark-pair channels of $pp\to t\bar{t}$ for both coupling order contributions, $\delta\text{NLO}_{21}$ and $\delta\text{NLO}_{12}$. Equally, this concerns contributions $\delta\text{NLO}_{22}$ and $\delta\text{NLO}_{13}$ of $t\bar{t}$ production with associated $W/Z/H$ boson.
For $\delta\text{NLO}_{03}$ with respect to $\delta\text{NLO}_{30}$ for $pp\to t\bar{t}$ in the same way as $\delta\text{NLO}_{04}$ with respect to $\delta\text{NLO}_{31}$, we observe a suppression of about 3 orders of magnitudes. This obviously can be traced back to the difference in the coupling powers corresponding to an approximate factor of $(\alpha/\alpha_s)^3\sim 1/1000$.

Interfering diagrams to the process $q\bar{q}\to q^{\prime}\bar{q}^{\prime}$ at the coupling orders $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s \alpha)$ at LO (upper row) and $\mathcal{O}(\alpha_s^2\alpha)$ (lower row) at NLO; cf. <cit.>

Considering the numerical size, two contributions of the processes $pp\to t\bar{t}W^+$ in Table <ref> and of $pp\to t\bar{t}W^-$ in Table <ref> are prominent. First of all, the $\text{LO}_{12}$ contributions in both processes vanish. This can be traced back to the colour-forbiddance, as described above, since only up-, down-, strange- and charm-quarks can occur in the initial-state at LO for the charged-current processes with one $W^{\pm}$ in the final-state. Obviously, at $\mathcal{O}(\alpha_s\alpha)$ contributions originate purely from the $\mathcal{O}(g_s^2)$ diagram with gluon $s$-channel propagator interfering with the one at $\mathcal{O}(g_e^2)$ involving a $\gamma/Z$ propagator which couples to the $t\bar{t}$ pair. From the colour structure – the interfering amplitudes lead to a trace over a single Gell-Mann matrix – these contributions vanish identically.

A second contribution to $pp\to t\bar{t}W^{\pm}$ which stands out corresponds to $\delta \text{NLO}_{13}$ leading to a relative correction with respect to $\text{LO}_{21}$ of about $+12\%$. This comes from real-emission diagrams involving subgraphs with $tW\to tW$ scattering with Higgs boson exchange. This leads to an enhancement of the amplitudes with these real configurations as outlined in Refs. <cit.>.

§.§ Jets and charged leptons in the final-state

For LHC processes with final-state jets additional challenges to achieve theoretical predictions with EW corrections arise. Similar as for top-pair production, these processes require the subtraction of both QCD and QED IR-singularities for NLO contributions apart from $\delta \text{NLO}_{n+1,m}$ and $\delta \text{NLO}_{0,m+n+1}$ (left- and right-most blob in Fig. <ref>). For contributions $\delta \text{NLO}_{n,m+1}$ overlapping QCD and EW corrections do not only occur in the virtual loop-tree interfering amplitudes, as it is the case e. g. for $t\bar{t}$ production, but also in real-emission diagrams. Treating the photon on a democratic basis with the gluon in the jet definition, as it is dictated by IR-safety conditions, IR divergences can occur from either the photon or gluon. For example, squared real-emission amplitudes to the process $pp\to Zj$ at $\mathcal{O}(\alpha_s\alpha)$ corresponding to flavour structures $q\bar{q}\to Z \gamma g$ for the subtraction of IR-divergences must be considered as both, EW corrections to $q\bar{q}\to Zg$ and QCD corrections to $q\bar{q}\to Z \gamma$ Born contributions.

Additional intricacies due to IR-safety conditions arise for the evaluation of phase-space cuts to massless final-states. Requirements to be met especially concern processes involving both massless charged leptons and jets. For these, photons must be included in the jet definition and recombined with leptons simultaneously in order to render observables IR-safe. In  this can be technically achieved for example, as suggested in Sec. <ref>, by first applying photon recombination to all charged fermions. This is followed by a jet clustering on dressed quarks, gluons and non-recombined photons. Finally, the cuts defined in Sec. <ref> are imposed on dressed leptons and clustered jets. An explicit example for  commands in order to compute NLO EW corrections to the process $pp\to e^+e^-j$ at $\mathcal{O}(\alpha_s\alpha^2)$ is given in App. <ref>.

process      $\alpha_s^n\alpha^m$    2c|     3c|

$pp \rightarrow X j$       $\sigma_{\text{LO}}^{\text{tot}}$ [pb]    $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]   $\sigma_{\text{LO}}^{\text{tot}}$ [pb]    $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\delta$ [%]    LO/NLO

$e^+\nu_ej$              $\alpha_s\alpha^2$      $914.81(6)$       $ 904.75(8) $             $914.74(7)$        $ 904.59(7) $         $ -1.11 $      $0.8$/$1.5 $

$e^+e^-j$           $\alpha_s\alpha^2$      $150.59(1)$        $ 149.09(2) $          $150.59(1)$      $ 149.08(2) $        $ -1.00 $     $0.05$/$ 0.4 $

Checks for cross sections at LO and NLO EW, $\sigma^{\text{tot}}_{\text{NLO}}=\text{LO}_{12}$ and $\sigma^{\text{tot}}_{\text{NLO}}=\text{LO}_{12}+\delta\text{NLO}_{13}$, for representative LHC processes with jets and leptons in the final state at $\sqrt{s}=13$ TeV; the parameters $\delta$ and $\sigma^{\text{sig}}$ are defined according to Eqs. (<ref>) and (<ref>)

Note that if jets are involved in the final-state, non-singular tree-level $(n+1)$-body contributions as explained in Sec. <ref> play a role. These result from the interference of amplitudes contributing to the NLO coupling order for which no common propagator exists from which two external states emerge and could induce an IR-divergence. Processes of this kind contributing to $pp\to e^+\nu_ej$ at NLO are interfering amplitudes at $\mathcal{O}(\alpha_s\alpha^3)$ for example of $uu\to e^+\nu_e ud$ and $\bar{c}\bar{s}\to e^+\nu_e \bar{c}\bar{c}$. Independently from the fact that some processes at this coupling order are suppressed due to colour such as $u\bar{d}\to e^+\nu_e b\bar{b}$ all real flavour structures possible from the coupling powers are taken into account in . This yields a consistent treatment of matrix elements to the desired coupling powers in the automated framework.
However, it causes a loss in computation efficiency due to the calling of the OLP  and matrix-element evaluation for zero-number contributions for a large number of real flavour structures. Even though this efficiency loss can be avoided in a technical sense a more inherent efficiency issue remains: The FKS phase-space is constructed such that there is merely one set of FKS variables, $\xi$ and $y$, to be integrated over for all FKS regions. As the flavour-space for the radiated parton for this class of processes is extended accounting for the photon, in addition, the cut evaluation acting on jets and dressed leptons becomes highly complicated. The adaption of the integration over merely two FKS variables thus is affected by a complicated topology of non-vetoed phase-space configurations.

In Table <ref> results for checks[For these checks, slightly different $W$ and $Z$ boson widths as stated in Sec. <ref> are used, i. e. $\Gamma_W=2.4952$ GeV and $\Gamma_Z=2.0850$ GeV. These correspond to the PDG widths  <cit.>.] with  <cit.> on the total cross section at LO and NLO EW for the processes $pp\to e^+\nu_ej$ and $pp\to e^+e^-j$ at $\mathcal{O}(\alpha_s\alpha^2)$, i. e. including $\text{LO}_{12}$ and $\delta\text{NLO}_{13}$ contributions, are shown. The MC uncertainty of each calculation is chosen at $\mathcal{O}(0.01\%)$ of the absolute cross section result which makes the check convincing with respect to a relative NLO correction $\delta$ parametrically of $\mathcal{O}(1\%)$. As the parameter $\sigma^{\text{sig}}$ is below $2$ for both processes the NLO EW computation performed with  can be considered as reliable for this class of processes.

CHAPTER: LEPTON COLLISIONS AT NLO EW

For making precise predictions for lepton collider physics  represents a suitable tool which provides a broad set of beam structure features as outlined in Sec. <ref>. The extension of this framework accounting for NLO EW corrections in arbitrary lepton collision processes requires the following considerations.

For a universal treatment of collinear ISR effects for the precision level of an NLO EW calculation NLO-NLL lepton PDFs must be applied. By this, logarithms of the ratio $\mu^2/m^2$, where $\mu$ denotes the factorisation scale and $m$ the initial-state mass, are resummed and factorised into the PDFs as explained in Sec. <ref>. For $\mu^2\sim s$ the perturbative series of the expansion of partonic cross sections in $\alpha$ is thus valid for arbitrary lepton collider processes with respect to QED corrections.

Formally, this would yield the most suitable approach for precise predictions in this context. However, the technical achievement of results at this precision level in a MC setup highly depends on the applied integration methods as the PDFs contain a singular structure. This is the case already for LO calculations applying LL PDFs for which the ISR structure function mapping presented in Sec. <ref> is implemented in  <cit.> and has proven itself sufficiently stable. An extension of this phase-space mapping with respect to NLO calculations applying NLL electron PDFs is suggested in the same section. At this time, the analytical solution for the NLL PDF's functions have been implemented and validated in . Numerically exact same results as obtained with the code  <cit.>, using its default settings, have been reproduced.

The implementation of the  phase-space mappings adjusted for NLO computations overcoming the extremely peaked behaviour of the electron PDFs is completed for separate components, i. e. the Born and virtual part with Born-like phase-space configurations as well as the DGLAP remnant. For the latter, the PDFs entering FKS terms with ISR in the collinear limit are rescaled with a parametrisation of the respective variable according to Eq. (<ref>).
This extended mapping is probed together with the grid interpolation method proposed in Sec. <ref> using numerical NLL PDF values obtained with  as grid points. A sufficiently high integration efficiency and adaption quality for these separate components is achieved. In addition, internal consistency checks are performed showing the correctness of the new mapping procedure. For the purpose of these checks, the electron PDFs are set to constant value $1$, and the integration is performed once with the original and once with the adjusted parametrisation of the NLO phase-space variables. Both integrations yield the same numerical results.
The component of the real-emission and counterterms requires a twofold mapping as the rescaling of PDFs follows the FKS requirements of Sec. <ref>. The proposed parametrisation according to Eqs. (<ref>) - (<ref>) is based on a similar mapping as for the DGLAP remnant component. Their implementation in  is under development.

Another approach to achieve a sufficiently good performance of a MC integration applying electron PDFs within the  framework is presented in Refs. <cit.>. This however will not be discussed in the following as the preferred choice of methods always depends on the fundamental structure of the MC tool itself. The checks in  applying NLL electron PDFs to NLO calculations, described above, are merely on partial terms of an NLO observable which separately have no physical meaning. For this reason, the complete validation of the setup in , including also cross-checks with , will be deferred to a future study.

Apart from this approach, treating collinear ISR effects in a resummed way, NLO EW corrections to lepton collider observables can be computed at fixed order by keeping the initial-state mass-dependencies explicit in the matrix elements. This ansatz inevitably depends on the collider setup as for some cases the ratios $s/m^2$ might lead to large logarithms such that the NLO QED fixed-order perturbative expansion would give an unreliably imprecise result. However, by considering kinematic cases where terms involving collinear logarithms of a perturbative order beyond NLO are sufficiently suppressed by $\alpha$, e. g. by means of Eq. (<ref>), fixed NLO EW corrections can be considered as reliable for certain collider setups.
The FKS phase-space construction for ISR in this approach follows the considerations on radiation off massive initial-state emitters according to Sec. <ref>.

With the implementation of this phase-space construction in  NLO EW calculations for lepton collisions can be performed by treating the leptons as massive. By this, collinear radiation off the initial state is regularised by the mass itself and neither PDFs nor collinear counterterms are needed. The OLP which can be used for this purpose is  as it can account for full mass dependence of leptons in the amplitudes.

While in Sec. <ref> the validation of this framework is presented, Sec. <ref> contains NLO EW results on cross sections and differential distributions on multi-boson processes at a future muon collider obtained with it.

For the results presented in this chapter the parameter

\begin{align}
\delta_{\text{EW}}=\frac{\sigma_{\text{NLO}}^{\text{tot}}}{\sigma_{\text{LO}}^{\text{tot}}}-1
\end{align}

is used describing the relative NLO EW correction of the cross section at NLO, $\sigma_{\text{NLO}}^{\text{tot}}$, with respect to that at LO, $\sigma_{\text{LO}}^{\text{tot}}$, for a lepton collision process.

§ VALIDATION FOR $E^+E^-$ PROCESSES

In this section explicit checks on total cross sections at LO and NLO EW are shown which validate the framework of . These are performed besides internal FKS sanity checks. Automated NLO EW corrections to lepton collider processes are thus proven reliable for the massive initial-state fixed-order approximation.

$e^+e^-\to HZ$   2c| <cit.>   3c|   1c$\sigma^{\text{sig}}$

$\sqrt{s}$ [GeV]    $\sigma_{\text{LO}}^{\text{tot}}$ [fb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [fb]     $\sigma_{\text{LO}}^{\text{tot}}$ [fb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [fb]     $\delta_{\text{EW}}$ [%]    (LO/NLO)

$\phantom{0}250$           $225.59(1)$           $ 206.77(1)$             $225.60(1)$        $207.0(1)$    $-8.25$    $0.4$/$2.1$

$\phantom{0}500$           $53.74(1)$           $62.42(1) $             $53.74(3)$        $62.41(2)$    $+16.14$    $0.2$/$0.3$

$1000$      $12.05(1)$         $14.56(1)$         $12.0549(6)$     $14.57(1)$   $+20.84$   $0.5$/$0.5$

[muon collider NLO EW]Comparison of LO and NLO total inclusive cross sections results of the  and  setup for the process $e^+e^-\rightarrow HZ$ with unpolarised beams

The cross-checks are performed using reference results of  <cit.>. In Table <ref> cross section results at LO and NLO EW for both tools with respect to the process $e^+e^-\to HZ$ for unpolarised beams are shown. In addition, the relative correction $\delta_{\text{EW}}$ and the parameter

\begin{align}
\sigma^{\text{sig}}\equiv \frac{|\sigma^{\text{tot}}_{\texttt{WHIZARD}}-\sigma^{\text{tot}}_{\texttt{MCSANCee}}|}{\sqrt{\Delta_{\texttt{WHIZARD}}^2+\Delta_{\texttt{MCSANCee}}^2}}
\end{align}

using the MC uncertainties $\Delta_{\texttt{WHIZARD}}$ and $\Delta_{\texttt{MCSANCee}}$ are included. The cross sections are obtained using the input parameters of the reference results in Ref. <cit.> which corresponds to an ILC-like setup with centre-of-mass energies of $\sqrt{s}=250$, $500$ and $1000$ GeV, respectively. Moreover, it involves the application of the $\alpha(0)$ input and on-shell renormalisation scheme for which all particle widths are set to zero.

From the $\sigma^{\text{sig}}$ values in Table <ref> we conclude that the results of both MC frameworks agree very well for MC uncertainties below the permille-level of the absolute cross section values. For relative corrections $\delta_{\text{EW}}$ at $\mathcal{O}(10\%)$ this is sufficient to say that the framework  yields reliable predictions including NLO EW corrections for such processes.

$e^+e^-\to \mu^+\mu^-$   2c| <cit.>    3c|   1c$\sigma^{\text{sig}}$

$\sqrt{s}$ [GeV]    $\sigma_{\text{LO}}^{\text{tot}}$ [pb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\sigma_{\text{LO}}^{\text{tot}}$ [pb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\delta_{\text{EW}}$ [%]    (LO/NLO)

$\phantom{0}5$           $2978.6(1)$           $3434.2(1) $             $2978.7(1)$        $ 3433.5(3)$    $+15.27$    $0.3$/$2.2$

$\phantom{0}7$           $1519.6(1)$           $ 1773.8(1) $             $1519.605(4)$        $1773.1(2) $    $+16.68$    $0.05$/$3.0$

[muon collider NLO EW]Comparison of LO and NLO total cross sections results of the  and  setup for the process $e^+e^-\to \mu^+\mu^-$ with unpolarised beams

$e^+e^-\to \tau^+\tau^-$   2c| <cit.>    3c|   1c$\sigma^{\text{sig}}$

$\sqrt{s}$ [GeV]    $\sigma_{\text{LO}}^{\text{tot}}$ [pb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\sigma_{\text{LO}}^{\text{tot}}$ [pb]     $\sigma_{\text{NLO}}^{\text{tot}}$ [pb]     $\delta_{\text{EW}}$ [%]   (LO/NLO)

$\phantom{0}5$           $2703.3(1)$           $2816.7(1) $             $2703.353(5)$        $ 2816.7(2)$    $+4.19$    $0.5$/$0.2$

$\phantom{0}7$           $1503.0(1)$           $1648.8(1) $             $1503.066(4)$        $1648.5(1)$    $+9.68$    $0.7$/$1.6$

[muon collider NLO EW]Comparison of LO and NLO total cross sections results of the  and  setup for the process $e^+e^-\to \tau^+\tau^-$ with unpolarised beams

A similar check can be done for off-shell neutral-current processes including final-state radiation.
For the processes $e^+e^-\to \mu^+\mu^-$ and $e^+e^-\to \tau^+\tau^-$ at low collider energies, $\sqrt{s}=5$ and $7$ GeV, LO and NLO EW cross sections computed with  are compared with those of  from Ref. <cit.>. Results of these checks, for which settings and parameters are described in the latter reference, are displayed in Tables <ref> and <ref>.
In both of these, agreement between the shown cross sections from the two generators can be found for MC uncertainties parametrically below $\mathcal{O}(0.01\%)$ of the absolute results.

Due to this, the framework of  can be considered validated for NLO EW precision predictions for these kind of processes. This is useful with respect to the collider energies which are of the typical order of magnitude of those at contemporary $e^+e^-$ collider experiments such as Belle II at SuperKEKB or BES III at IHEP in Beijing. Note that for these low collider energies collinear ISR effects inducing the logarithmic terms as described above are much less pronounced. However, the effects of pure soft photon radiation beyond the NLO in general may play an increased role depending highly on the process and the collider setup. For example, $e^+e^-\to HZ$ at $250$ GeV or $e^+e^-\to \tau^+\tau^-$ at $5$ GeV, due to the final-state mass thresholds, yield the phase-space for the radiated photon at NLO much more restricted compared to higher collider energies. This could imply the need for soft photon resummation which is captured for example by the YFS approach <cit.>. As this is beyond the scope of this thesis it will not be discussed here.

§ MULTI-BOSON PROCESSES AT A MUON COLLIDER

In this section NLO EW cross sections and distributions for multi-boson
processes  $\mu^+\mu^- \to V^n H^m$ with $V \in \{W^\pm,Z\}$
and $n+m \leq 4$ are computed with results and discussions presented in Ref. <cit.>.

For this purpose, one-loop EW virtual contributions are provided by  accounting for the full mass dependence of fermions and bosons.  Additionally, for NLO QED corrections to $HZ$ and $ZZ$ production the interface to the OLP  is used.
The FKS phase-space construction with massive particles in the initial-state, i. e. the mapping between the Born and real-radiation phase-space parameterisations, follows the considerations of Sec. <ref>.
The integration proceeds via numerical phase-space sampling with
multi-channel adaption using 's MPI-based
parallelisation <cit.>.

Further details on the setup for the numerical results achieved in this section are as follows:
The $G_{\mu}$ scheme is applied for the computation of the electromagnetic coupling $\alpha$ for which light fermion mass logarithms are resummed in the
incorporated running of $\alpha$.
Regarding massive vector bosons, on-shell renormalisation conditions are imposed and particle widths set to zero.
Throughout the calculation, non-zero masses for all particles except for photon and neutrinos are applied, and the corresponding Yukawa couplings are included. For light fermions in loops this is merely a technical detail without phenomenological significance.
However, keeping the muon mass non-zero regulates IR singularities from collinearly radiated photons off the initial-state.  In fact, QED corrections beyond NLO due to ISR at a muon collider are parametrically of order $(\alpha/\pi)^2\log^2(s/m_\mu^2) \sim 0.1\;\%$ according to the considerations with respect to Eq. (<ref>). Hence, they can be considered as sufficiently suppressed in the particular context of this study. This allows us to treat the colliding $\mu^+\mu^-$ system perturbatively without the need for higher-order resummation by applying leptonic PDFs.

The following numerical input parameters are used for the calculations with results presented in this section:

\begin{equation*}
G_{\mu}= 1.166379\cdot 10^{-5} \; \text{GeV}^{-2} \\
\end{equation*}

\begin{eqnarray*}
m_u &=\; \phantom{17}0.062           \;\text{GeV}\qquad\qquad
m_d &=\; 0.083           \;\text{GeV} \\
m_c &=\; \phantom{17}1.67\phantom{0} \;\text{GeV}\qquad\qquad
m_s &=\; 0.215           \;\text{GeV} \\
m_t &=\;          172.76\phantom{0}  \;\text{GeV}\qquad\qquad
m_b &=\; 4.78\phantom{0} \;\text{GeV}
\end{eqnarray*}

\begin{eqnarray*}
M_W &=\; \phantom{1}80.379\phantom{0} \;\text{GeV} \quad\qquad
m_e &=\;  0.0005109989461  \;\text{GeV}  \\
M_Z &=\; \phantom{1}91.1876 \;\text{GeV}  \quad\qquad
m_{\mu} &=\; 0.1056583745\phantom{000} \;\text{GeV} \\
M_H &=\; 125.1\phantom{000} \;\text{GeV} \quad\qquad
m_{\tau} &=\; 1.77686\phantom{00000000} \;\text{GeV}\qquad.
\end{eqnarray*}

For ensuring the correctness of the calculations, in addition to the checks of Sec. <ref> with an analogous beam setup, technical sanity checks on the implemented FKS subtraction scheme for muon collider processes are performed. These include numerical checks for the IR cancellation in the soft limit and consistency checks. For the latter, integrated results using the FKS real phase-space parameterisation to an underlying Born process $\mu^+\mu^- \to X$ and the LO parameterisation of $\mu^+\mu^- \to X + \gamma$ with a well-defined photon, i. e. using a (technical) energy cut $E_{\gamma}>10$ GeV, are compared.
The derived $\mu^+\mu^- \to HZ$ EW Sudakov correction factor of App. <ref> as well as reference results of 's automated evaluation of EW Sudakov factors <cit.> served as additional cross-checks on the presented differential distributions.

In Sec. <ref> cross section scans in the collider energy for the multi-TeV range for $HZ$ and $ZZ$ production are shown. Sec. <ref> contains total cross section results of two-, three- and four-boson processes at a muon collider for proposed collider energies, $\sqrt{s}= 3$, $10$ and $14$ TeV. For the process $\mu^+\mu^- \to HZ$ distributions differential in Higgs properties are presented and discussed in Sec. <ref>.

§.§ Collider energy scans for cross sections of $HZ$ and $ZZ$ production

The simplest process class from which fundamental EW higher order
perturbative effects can be understood is neutral di-boson production. These processes, by neglecting contributions associated with muon-Higgs Yukawa couplings, have a clear kinematical structure at both LO and NLO EW. The LO dominant contributions come from the $s$-channel diagram for $HZ$ and from $t/u$-channel diagrams for $ZZ$ production which are depicted in the upper row in Fig. <ref>. This classification of the two kind of processes is useful in order to distinguish different kinds of effects which are enhanced at NLO EW as explained below. Furthermore, for the neutral di-boson processes QED radiation in the corresponding real-emission diagrams exclusively comes from the initial state at NLO in $\alpha$.
The lower row of Fig. <ref> depicts typical one-loop diagrams, underlining the
fact that with respect to the virtual one-loop diagrams these processes are closely related.

Upper row: tree-level diagrams for the processes
$\mu^+\mu^-\rightarrow ZZ$ and to $\mu^+\mu^-\rightarrow HZ$
(omitting contributions from the muon-Yukawa coupling). Lower row: representative one-loop
diagrams for the virtual contribution to $\mu^+\mu^-\rightarrow
ZZ$ and to $\mu^+\mu^-\rightarrow HZ$; cf. <cit.>

LO and NLO inclusive cross section scans in
$\sqrt{s}$ for $HZ$ and $ZZ$ production in the upper plot;
relative NLO correction $\delta_{\text{EW}}$ in the lower

The relative corrections $\delta_{\text{EW}}$ (lower plot) for both processes can be interpreted by scans over $\sqrt{s}$ for LO and NLO inclusive cross sections of $HZ$ and $ZZ$ production at the muon collider, shown in
Fig. <ref> (upper plot).

First, consider the Higgsstrahlung process $\mu^+\mu^-\to HZ$.  A large suppression in Fig. <ref> which increases in size starting from the peak of the cross section at $\sqrt{s}\sim 0.8$ TeV can be observed which can be attributed to large virtual effects as will be shown in the following.

In order to understand the behaviour in the regime of high centre-of-mass energies, in general the approximation of EW Sudakov logarithmic correction factors can be applied for which pioneering works have been done in
In the Sudakov limit

\begin{align}
r_{kl}= (p_k+p_l)^2 \sim s \gg M_W^2 % \sim M^2_Z, M^2_H
\label{sudakovlimit}
\end{align}

with external states $k$ and $l$ and by using a fictitious photon mass $\lambda=M_W$ these correction factors effectively correspond to the EW virtual contributions in addition to real corrections with photon transverse momenta smaller than a cut-off scale at the order of $M_W$. They represent form factors in terms of double and single logarithms of the ratio $r_{kl}/M_W^2$ which are factorised in the soft and/or collinear limit.

The exact treatment of QED IR subtraction, which is mandated for FKS subtraction, naturally demands photons to be massless. The virtual loop contributions in this way contain diagrams with massless photon exchange regulated by the IR scheme. In contrast, virtual contributions with massive weak vector boson exchange are still regularised by their masses which are at the EW scale $M_W$. Therefore, for large $\sqrt{s}$, these contributions are implicitly contained in the EW next-to-leading logarithmic (NLL) Sudakov factors.
Combining FKS QED subtraction and these NLL EW Sudakov factors in order to obtain inclusive results comparable with those shown in Fig. <ref> in general requires a consistent matching of these for an NLO calculation:
The subtraction of QED IR poles in the FKS virtual terms is defined via $\overline{\text{MS}}$ subtraction at the scale $\mu=\mu_R$ such that remnant finite terms from  virtual loop contributions receive a logarithmic dependence on this scale. More details on this are given within the description of virtual subtraction terms in the FKS formalism presented in Sec. <ref>.
Using an EW Sudakov approximation with the cut-off scale of the effective photon mass $\lambda=M_W$ in place of these virtual (finite) loop contributions however yields missing terms of order $\alpha\log(\mu_R/M_W)$ for the complete NLO computation <cit.>.

Because of the technical challenges of this combined treatment of FKS QED subtraction and virtual EW Sudakov approximation factors, it is considered as beyond the scope of this work. For a completely automated treatment of EW Sudakov correction factors in the FKS framework see Ref. <cit.>. The EW Sudakov factor to $\mu^+\mu^-\to HZ$, discussed in the following, thus merely serves for purposes of explanations.

In the following, it will be shown that the effective EW virtual loop contributions of $HZ$, implicitly contained in the respective results of Fig. <ref>, are quantitatively approximated well enough by the Sudakov approach.

Sudakov factors $\Lambda^{\kappa}_{\lambda}$ for muon chiralities $\kappa=L,R$ and $Z$ polarisations $\lambda=T,L$ and
estimated unpolarised correction factor
$\Lambda_{\text{est}}^{\text{unpol}}$ at
$\theta_{H}=90^{\circ}$ as well as
$\Lambda^{\text{unpol}}_{\text{est,c}}$ (without angular dependent terms) for $HZ$ production
at the muon collider as a function of the collider
energy, $\sqrt{s}$.

To this end, the NLL Sudakov form factor to $\mu^+\mu^- \to HZ$ is derived in
App. <ref> by using analytical results to the process $q\bar{q}\rightarrow HZ$ <cit.> in analogy. For the final result we arrive at the estimate of
Eq. (<ref>) with all technical details and considerations explained in App. <ref>.
This Sudakov factor depending on
$\sqrt{s}$ for a fixed polar angle $\theta_H=90^{\circ}$ of the Higgs as
shown in Figure <ref> exhibit large suppressions in particular
for left-handed muons in the initial state due to the larger
coupling to $SU(2)$ interactions. Note, that the results in Fig. <ref> are not representing Sudakov approximations for inclusive cross sections (integrated over $\theta_{H}$). This would either require the technical implementation of these factors in  in order to multiply them with the corresponding Born contributions on amplitude-level or a completely analytical approach. In addition, results according to this would also miss terms related to the matching with QED FKS subtraction, as explained above, in order to give a completely accurate description of the virtual EW corrections approximated for results of Fig. <ref>. As the factors of Fig. <ref> are presented merely for purposes of clarifications the angular integration is left out here and the following considerations are made.

In a further approach we approximated the unpolarised correction factor $\Lambda^{\text{unpol}}_{\text{est,c}}$ for which the angular dependent part of Eq. (<ref>) is excluded, shown as black dotted curve. This correction factor corresponds to the amount to which at least the inclusive result due to the virtual loop corrections is suppressed.
According to Eq. (<ref>), the angular-dependent part of the Sudakov factor is negative, of subleading logarithmic type and amounts to about $-17\%$ at $\sqrt{s}=16
\text{ TeV}$ for angles $\theta_H$ in the perpendicular plane,
i. e. for $\theta_H\sim90^{\circ}$.
For these angles also the Born process is
enhanced according to

\begin{align}
\left(\frac{d\sigma}{d\Omega}\right)_{\text{Born}}\propto
\frac{\beta
\quad.
\label{bornHZcrosssec}
\end{align}

which can be observed as well in the differential
cross sections for the Born case presented in
Sec. <ref> within Fig. <ref>.

The	estimated unpolarised correction factor, $\Lambda^{\text{unpol}}_{\text{est}}$,  given by Eq. (<ref>) including the angular-dependent terms with a chosen polar angle $\theta_{H}=90^{\circ}$, is shown as black dashed curve in Fig. <ref>. This decreases the cross section with a largest suppression of about $-65\%$ for high energies.
Concerning the magnitude of $\Lambda^{\text{unpol}}_{\text{est}}$ at
this angle relatively to $\Lambda^{\text{unpol}}_{\text{est},c}$ as
well as the similar enhancement behaviour at angles around $\theta_{H}=90^{\circ}$ for the Born differential in $\theta_{H}$ the relative inclusive virtual corrections to $HZ$ production can be estimated to be of the order of $\Lambda^{\text{unpol}}_{\text{est}}$. The counteracting effects of QED radiation in an inclusive calculation  will be discussed in the following.

Relative QED corrections
to $HZ$ and $ZZ$ production at the muon collider as a function of
the collider energy, $\sqrt{s}$.

For including only pure NLO QED corrections in the calculation the resulting relative correction $\delta_{QED}$ as depicted in Fig. <ref> is positive and growing with $\sqrt{s}$. About the
contributing NLO parts we can make the following qualitative statements:
In general, virtual loop-tree interfering amplitudes are supposed to give negative contributions such that a positive overall NLO correction factor can be explained by dominating real corrections. In particular, since the main contribution for $HZ$ production at Born-level is $s$-channel induced, large QED real corrections are expected due to large amplitudes for hard photons radiated in forward
direction, the effect of which is enhanced with growing $\sqrt{s}$. This explains the bulk of the correction factor to be seen in Figure <ref> as the blue curve.
From the superposition of these two effects, the negative EW Sudakov logarithms overcompensating the QED real corrections, a reduction of $\delta_{{HZ}}$ down to a range of about $ -20~\%$ to $-10~\%$ at high energies in Figure <ref> is found to be reasonable.

For $\mu^+\mu^-\to ZZ$ a detailed discussion about the composition of pure weak and QED parts of the NLO inclusive corrections is left out here since the process compared to $HZ$ production is more intricate from its angular dependency at LO as well as at NLO. Explicitly, the result for the Sudakov
correction factors of the analogous process $e^+e^-\rightarrow ZZ$ given in <cit.> cannot be related straightforwardly to an estimation for the Sudakov suppression of the inclusive cross section.
This is due to the completely different
angular dependence of the Sudakov factor depicted in Fig. 7 within Ref. <cit.> which is minimal for $Z$ polar angles around $90^{\circ}$ compared to the Born amplitudes with the largest contributions at angles around the beam axis.
Moreover, considering the suppression  $\delta_{QED,ZZ}$ relative to $\delta_{QED,HZ}$ in Fig. <ref> leads to the conclusion that the impact of real emission amplitudes with hard photon radiation on the relative NLO correction is reduced compared to that of $HZ$ production.
This can be traced back to the effect that for $ZZ$ production tree-level amplitudes due to the forward scattering of the $Z$ bosons at high energies are enhanced already at Born-level due to the dominant $t/u$-channel process. The positively contributing QED real corrections due to hard photon radiation thus yield a mitigated enhancement of the cross section compared to the $s$-channel Higgsstrahlung process.
In Fig. <ref> an overall positive correction $\delta_{ZZ}$ can be seen. This can be explained by an overcompensation of the EW Sudakov effects by the QED real corrections in the inclusive results.

§.§ Total cross sections for multi-boson processes

In the same way as for $\mu^+\mu^-\to HZ$ and $\mu^+\mu^-\to ZZ$ NLO EW corrections can be computed for all other possible combinations of final-state bosons. Numerical results of the LO and NLO inclusive cross sections for all these combinations in two-, three- and four-boson production processes at $\sqrt{s}= 3$ TeV are presented in Table <ref>. Table <ref> and <ref> contain the corresponding results for two- and three-boson production at $\sqrt{s}= 10$ and $14$ TeV.
For very high centre-of-mass energies as well as high EW boson multiplicities, the fixed-order perturbative approximation breaks down.
To this end, the computations for four-boson production at $\sqrt{s}= 10$ and $14$ TeV are omitted which render meaningful results only by taking appropriate EW higher order resummation approaches, e. g. by soft-collinear effective field theory, into account.

In order to investigate collinear ISR effects influencing the presented NLO EW results from a kinematical point of view, the LL lepton PDF with its analytical form given in Eq. (<ref>) is applied to the LO cross sections.
Note, that for the purpose of computing inclusive fixed-order cross sections, higher-order QED contributions beyond NLO are parametrically subleading, as explained above.  It can be seen that the PDF convolution captures the dominant QED-radiation effect within the NLO cross section, to be compared with the complete NLO EW results.

For each collider energy, the cross sections computed by using ISR structure functions are displayed in
Table <ref>, <ref> and <ref>. The relative correction $\delta_{\text{ISR}}$ is
defined as

\begin{align}
\delta_{\text{ISR}} =
\frac{\sigma_{\text{LO+ISR}}^{\text{incl}}}{\sigma_{\text{LO}}^{\text{incl}}}-1
\qquad .
\end{align}

As mentioned above, for inclusive observables QED effects beyond NLO are suppressed compared to those of NLO EW corrections to the hard scattering process. However, resummation methods in general improve the numerical accuracy of predicted cross sections. They become essential, in particular, when computing exclusive observables. Beyond collinear resummation via the application of lepton PDFs, higher-order soft-photon radiation effects are captured by YFS resummation methods <cit.>.
In addition, for high-energy muon colliders EW PDFs might play a phenomenological role <cit.>. Their effects are not taken into account here.

A few general remarks on the discussion of the LO and NLO EW results:
First of all, we observe that for all processes the absolute value of the cross section decreases with $\sqrt{s}$ and with the number of bosons in the final state. Apart from the processes with pure Higgs final-state, these range from $\sim10^{-4}$ to $\sim10^2$ fb. As the
cross sections for the tree-level processes of $ HH$ and $HHH$ production are numerically below $\mathcal{O}(10^{-7}\text{ fb})$, owing to the mass suppressed muon-Higgs couplings at the collider energies used for this simulation, a detailed discussion on the respective results is left out here.
The abnormally  large corrections to cross sections labeled with `$*$' in in Table <ref>, <ref> and <ref> at this fixed order can be related to missing squared one-loop amplitudes with e. g. $W^{\pm}$
exchange. These phenomenologically have to be taken into account but require cross section calculations beyond fixed coupling powers if considered in the predictions. However, although not
relevant with respect to the small size of the cross sections at tree-level, the LO and formal NLO $\mu^+\mu^-\to
HH$ and $\mu^+\mu^-\to HHH$ numerical cross section results are included in these tables for completeness.

$\mu^+\mu^-\to X, \sqrt{s}=3$ TeV
$\sigma_{\text{LO}}^{\text{incl}}$ [fb]
$\sigma_{\text{NLO}}^{\text{incl}}$ [fb]
$\delta_{\text{EW}} $ [%]

$W^+W^-$                  $  4.6591(2)\cdot 10^2 $            $ 4.847(7) \cdot 10^2 $        $ +4.0(2) $

$ZZ$                $ 2.5988(1) \cdot 10^1$           $ 2.656(2) \cdot 10^1 $        $ +2.19(6) $

$HZ$           $ 1.3719(1) \cdot 10^0 $         $ 1.3512(5) \cdot 10^0 $     $  -1.51(4)$

$HH$         $ 1.60216(7) \cdot 10^{-7} $
$ 5.66(1)\cdot 10^{-7}~^* $         $  $

$ W^+W^-Z$        $ 3.330(2) \cdot 10^1 $               $ 2.568(8) \cdot 10^1 $         $ -22.9(2) $

$W^+W^-H$              $ 1.1253(5) \cdot 10^0 $             $ 0.895(2)\cdot 10^{0} $        $ -20.5(2) $

$ZZZ$             $ 3.598(2) \cdot 10^{-1} $          $ 2.68(1) \cdot 10^{-1} $        $ -25.5(3)$

$HZZ$             $  8.199(4)\cdot 10^{-2} $        $ 6.60(3)\cdot 10^{-2} $         $ -19.6(3) $

$ HHZ$       $ 3.277(1) \cdot 10^{-2} $      $ 2.451(5) \cdot 10^{-2} $     $ -25.2(1)$

$ HHH$           $ 2.9699(6) \cdot
10^{-8} $        $ 0.86(7) \cdot 10^{-8}~^* $       $  $

$W^+W^-W^+W^-$         $ 1.484(1) \cdot 10^0 $         $  0.993(6)\cdot 10^0 $        $ -33.1(4) $

$W^+W^-ZZ$           $  1.209(1)\cdot 10^0 $      $ 0.699(7) \cdot 10^0 $      $ -42.2(6) $

$W^+W^-HZ$                $ 8.754(8)  \cdot 10^{-2} $     $ 6.05(4) \cdot 10^{-2} $     $ -30.9(5) $

$W^+W^-HH$          $ 1.058(1) \cdot 10^{-2} $            $ 0.655(5)\cdot 10^{-2} $        $ -38.1(4) $

$ZZZZ$           $ 3.114(2) \cdot 10^{-3} $            $ 1.799(7)\cdot 10^{-3} $        $ -42.2(2) $

$HZZZ$           $ 2.693(2)\cdot 10^{-3} $            $ 1.766(6)\cdot 10^{-3} $        $ -34.4(2) $

$HHZZ$           $ 9.828(7) \cdot 10^{-4} $            $ 6.24(2)  \cdot 10^{-4} $       $ -36.5(2)$

$HHHZ$           $ 1.568(1) \cdot 10^{-4} $            $ 1.165(4) \cdot 10^{-4} $        $ -25.7(2) $

[muon collider NLO EW]Total inclusive cross sections
at LO and NLO EW with corresponding relative corrections
$\delta_{\text{EW}}$, for two-, three- and four-boson
production at $\sqrt{s}= 3$ TeV. For (*), see remarks in the text.

$\mu^+\mu^-\to X, \sqrt{s}=10$ TeV
$\sigma_{\text{LO}}^{\text{incl}}$ [fb]
$\sigma_{\text{NLO}}^{\text{incl}}$ [fb]
$\delta_{\text{EW}}$ [%]

$W^+W^-$                  $ 5.8820(2) \cdot 10^1 $         $ 6.11(1) \cdot 10^1 $        $ +3.9(2) $

$ZZ$                  $ 3.2730(4) \cdot 10^0$            $ 3.401(4)\cdot 10^0 $        $+3.9(1) $

$HZ$           $ 1.22929(8) \cdot 10^{-1} $        $  1.0557(8)\cdot 10^{-1} $     $ -14.12(7) $

$HH$              $ 1.31569(5) \cdot
10^{-9} $          $ 42.9(4)\cdot 10^{-9}~^{*} $

$ W^+W^-Z$         $ 9.609(5) \cdot 10^{0} $              $  5.86(4)\cdot 10^0 $         $ -39.0(2) $

$W^+W^-H$                   $ 2.1263(9)\cdot 10^{-1} $             $ 1.31(1)\cdot 10^{-1} $        $ -38.4(5)  $

$ZZZ$              $ 8.565(4) \cdot 10^{-2} $       $ 5.27(8) \cdot 10^{-2} $        $ -38.5(9)$

$HZZ$              $  1.4631(6)\cdot 10^{-2} $       $0.952(6)\cdot 10^{-2} $         $ -34.9(4) $

$ HHZ$       $ 6.083(2) \cdot 10^{-3} $     $  2.95(3)\cdot 10^{-3} $     $ -51.6(5) $

$ HHH$           $ 2.3202(4) \cdot 10^{-9} $        $ -1.0(2) \cdot 10^{-9}~^{*}  $

[muon collider NLO EW]Total inclusive cross sections
at LO and NLO with corresponding relative correction
$\delta_{\text{EW}}$ for di- and tri-boson production at
$\sqrt{s}= 10$ TeV. For (*), see remarks in the text.

$\mu^+\mu^-\to X, \sqrt{s}=14$ TeV
$\sigma_{\text{LO}}^{\text{incl}}$ [fb]
$\sigma_{\text{NLO}}^{\text{incl}}$ [fb]
$\delta_{\text{EW}}$ [%]

$W^+W^-$                  $  3.2423(1)\cdot 10^1 $           $  3.358(8)\cdot 10^1 $        $ +3.6(2) $

$ZZ$                 $ 1.80357(9) \cdot 10^0$           $ 1.872(4)\cdot 10^0 $        $ +3.8(2)$

$HZ$           $  6.2702(4)\cdot 10^{-2} $        $  5.097(6)\cdot 10^{-2} $     $ -18.7(1) $

$HH$               $ 3.4815(1) \cdot 10^{-10} $          $
217(2)\cdot 10^{-10}~^{*} $         $ $

$ W^+W^-Z$         $ 6.369(3) \cdot 10^{0} $               $  3.51(3)\cdot 10^0 $         $ -45.0(4) $

$W^+W^-H$                   $ 1.2846(6)\cdot 10^{-1} $            $ 0.73(1)\cdot 10^{-1} $        $ -43.3(9) $

$ZZZ$              $  5.475(3)\cdot 10^{-2} $         $  3.06(3)\cdot 10^{-2} $        $ -44.2(6)$

$HZZ$             $  8.754(4)\cdot 10^{-3} $        $5.28(3)\cdot 10^{-3} $         $ -39.7(4) $

$ HHZ$        $  3.668(1)\cdot 10^{-3} $      $  1.49(1)\cdot 10^{-3} $     $ -59.4(3)$

$ HHH$           $  1.1701(2)\cdot 10^{-9} $        $  -0.739(8)\cdot 10^{-9}~^{*} $       $  $

[muon collider NLO EW]Total inclusive cross sections
at LO and NLO with corresponding relative correction
$\delta_{\text{EW}}$ for di- and tri-boson production at
$\sqrt{s}= 14$ TeV. For (*), see remarks in the text.

For the di-boson processes $ZZ$ and $HZ$ production a discussion on inclusive cross sections is given already in detail in the analysis of the previous chapter. However, the different kinematical effects of these processes are highlighted by several aspects given below.

$W^+W^-$ production with dominant contributions from $t$-channel diagrams at Born level and high energies is similar to $ZZ$ exept for the possibility for photon radiation off the final-state at NLO. This can induce quasi-collinear effects in $W^{\pm}\to W^{\pm}\gamma$ splittings which increase with the energy scale of the process.
However, this is a minor effect compared to large real-emission amplitudes due to hard photon quasi-collinear radiation off the light massive muons. Another difference to $ZZ$ is that the $W$ bosons can have longitudinal gauge boson polarisations corresponding to two charged Goldstone bosons in the final-state for which the $s$-channel process is dominant. This contribution is suppressed with $1/s$ <cit.>.
These considerations may explain the similarity of the relative corrections $\delta_{\text{EW}}$ of the two gauge boson pair production processes at high energies $10$ and $14$ TeV and their deviation at $3$ TeV.

Quantitatively, the influence of the kinematics which causes the main difference in $\delta_{\text{EW}}$ of gauge boson pair production $WW$ and $ZZ$ compared to Higgsstrahlung $HZ$ at the considered collider energies can be seen as well in their different $\delta_{\text{ISR}}$ factors listed in Table <ref>, <ref> and <ref>. According to these the correction due to ISR resummation for $HZ$ is approximately twice as big as for $ZZ$ and $WW$ and grows with the energy. This originates from the same effect as explained for the comparison of NLO QED cross sections with respect to Fig. <ref>:
We note, that NLO QED as well as NLO EW corrections implicitly involve terms at $\mathcal{O}(\alpha)$ of the LL PDF expansions containing the leading logarithms which are dominant with respect to ISR effects.
These positive $\mathcal{O}(\alpha\ln (Q^2/m_{\mu}^2))$ contributions get large due to hard collinear photon ISR inducing a boost of the photon recoil-system along the beam axis which is enhanced with the collider energy.
For $HZ$ production this induces a radiative return back to the threshold. In this way, the cross section at NLO gets largely enhanced as the Higgsstrahlung cross sections at Born-level receive dominant contributions from the $s$-channel diagrams which fall off with $1/s$. For $VV$ production however this $1/s$ fall-off at Born-level gets damped by large contributions from amplitudes with small scattering angles of the gauge bosons. Hence, at NLO the radiative return leads to a less pronounced enhancement of the cross sections for gauge-boson pair production.

Upper row: exemplary tree-level diagrams for the processes
$\mu^+\mu^-\rightarrow W^+W^-Z$ (left) and to
$\mu^+\mu^-\rightarrow W^+W^-HH$ (right). Lower row: one-loop diagrams for the virtual contribution to $\mu^+\mu^-\rightarrow
W^+W^-Z$ (left) and to $\mu^+\mu^-\rightarrow W^+W^-HH$ (right); cf. <cit.>

For the results of three- and four-boson production processes, for which exemplary Feynman diagrams at tree- and one-loop level are depicted in Fig. <ref>, the following observations can be made.
For all triple-boson processes at $3$ TeV except for pure Higgs final-state production, the NLO EW cross sections are suppressed by $-25\%$ to $-20\%$ relatively to the LO results, to be seen in Table <ref>.
That the absolute values of $\delta_{\text{EW}}$ of the di-boson processes in general are smaller than those of the triple-boson processes can be explained by the fact that negative Sudakov logarithm factors add up for all kinematic invariants of external states <cit.>.
For four-boson final-state processes, this effect is enhanced, seen by $\delta_{\text{EW}}$ ranging from $-26~\%$ to $ - 42~\%$. In addition, Table <ref>, <ref> and 	<ref> show that with increasing number of bosons in the final-state the enhancement due to ISR effects gets damped.
This originates from the multi-peripheral kinematics of these hard processes. Similar to the discussion above on the di-boson processes: the radiative return is not so much pronounced as there is not a single dominating threshold to return to, except for the Higgs- and multi-Higgsstrahlung processes. For the latter class of processes a larger enhancement can be seen from their $\delta_{\text{ISR}}$ values in comparison with those of other processes at fixed number of bosons and beam energies.
Consequently, concerning the NLO EW results the dominant contribution to $\delta_{\text{EW}}$ of the three-boson and nearly the complete size of $\delta_{\text{EW}}$ of the four-boson processes can be led back to negative EW virtual final-state correction factors.

$\mu^+\mu^-\to X, \sqrt{s}=3$ TeV      $\sigma_{\text{LO}}^{\text{incl}}$ [fb]     $\sigma_{\text{LO+ISR}}^{\text{incl}}$ [fb]      $\delta_{\text{ISR}} $ [%]

$W^+W^-$                 $  4.6591(2)\cdot 10^2 $          $5.303(2)\cdot 10^2$        $ +13.82(4) $

$ZZ$                  $ 2.5988(1) \cdot 10^1$           $3.007(1) \cdot 10^1 $        $ +15.71(4) $

$HZ$          $ 1.3719(1) \cdot 10^0 $       $1.7868(4)
\cdot 10^0 $      $ +30.24(3) $

$ W^+W^-Z$        $ 3.330(2) \cdot 10^1 $          $ 3.427(2)\cdot 10^1 $           $ +2.90(9) $

$W^+W^-H$             $ 1.1253(5) \cdot 10^0 $        $1.2052(7)\cdot 10^0$            $ +7.10(8) $

$ZZZ$            $ 3.598(2) \cdot 10^{-1} $         $3.786(2)\cdot 10^{-1} $        $+5.24(8) $

$HZZ$             $  8.199(4)\cdot 10^{-2} $      $ 8.887(5)\cdot 10^{-2} $        $ +8.39(8) $

$ HHZ$       $ 3.277(1) \cdot 10^{-2} $     $3.525(2)\cdot 10^{-2}$      $ +7.58(7)$

$W^+W^-W^+W^-$         $ 1.484(1) \cdot 10^0 $      $1.465(1)\cdot 10^0$         $ -1.3(1) $

$W^+W^-ZZ$           $  1.209(1)\cdot 10^0 $     $1.187(1)\cdot 10^0$      $ -1.8(1) $

$W^+W^-HZ$               $ 8.754(8)  \cdot 10^{-2} $     $8.742(8)\cdot 10^{-2}$      $ -0.1(1) $

$W^+W^-HH$          $ 1.058(1) \cdot 10^{-2} $          $1.076(1)\cdot 10^{-2}$        $ +1.7(1) $

$ZZZZ$           $ 3.114(2) \cdot 10^{-3} $        $ 3.139(3) \cdot 10^{-3} $         $ +0.8(1) $

$HZZZ$            $ 2.693(2)\cdot 10^{-3} $        $2.730(2)\cdot 10^{-3}$        $ +1.4(1) $

$HHZZ$           $ 9.828(7) \cdot 10^{-4} $        $10.042(8)\cdot 10^{-4}$       $ +2.2(1) $

$HHHZ$            $ 1.568(1) \cdot 10^{-4} $        $1.657(1)\cdot 10^{-4}$         $ +5.7(1) $

[muon collider NLO EW]Total inclusive cross sections
at LO with LL ISR photon resummation, and including relative correction
$\delta_{\text{ISR}}$ for two-, three- and four-boson production at
$\sqrt{s}= 3$ TeV.

$\mu^+\mu^-\to X, \sqrt{s}=10$ TeV      $\sigma_{\text{LO}}^{\text{incl}}$ [fb]      $\sigma_{\text{LO+ISR}}^{\text{incl}}$ [fb]      $\delta_{\text{ISR}}$ [%]

$W^+W^-$                 $ 5.8820(2) \cdot 10^1 $      $7.295(7)\cdot 10^1$          $ +24.0(1) $

$ZZ$                $ 3.2730(4) \cdot 10^0$       $4.119(4)\cdot 10^0$          $+25.8(1) $

$HZ$          $ 1.22929(8) \cdot 10^{-1} $     $1.8278(5)\cdot 10^{-1}$      $ +48.69(4) $

$ W^+W^-Z$        $ 9.609(5) \cdot 10^{0} $           $10.367(8)\cdot 10^{0}$          $ +7.9(1) $

$W^+W^-H$             $ 2.1263(9)\cdot 10^{-1} $        $2.410(2)\cdot 10^{-1}$        $ +13.3(1)  $

$ZZZ$           $ 8.565(4) \cdot 10^{-2} $       $9.431(7)\cdot 10^{-2}$          $+10.1(1) $

$HZZ$             $  1.4631(6)\cdot 10^{-2} $      $1.677(1)\cdot 10^{-2}$           $ +14.62(8) $

$ HHZ$      $ 6.083(2) \cdot 10^{-3} $     $6.916(3)\cdot 10^{-3}$      $ +13.68(6) $