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+{"text":"\\section{Introduction}\n\\label{sec I}\nThe concept of frustration  plays an important role\nin the search for novel quantum states of condensed matter, see,\ne.g., \\cite{lnp04,buch2,moessner01,frust1,frust2}. The\ninvestigation of frustrating quantum spin systems is a challenging\ntask. Exact statements about the properties of  quantum spin\nsystem are known only in exceptional cases. The simplest known\nexact eigenstate is the fully polarized ferromagnetic state.\nFurthermore the one- and two-magnon excitations above the fully\npolarized ferromagnetic state  also can be calculated exactly,\nsee, e.g., \\cite{mattis81,Kuzian07,Zhitomirsky10,nishimoto2011}. An\nexample for non-trivial eigenstates is Bethe's famous solution for\nthe one-dimensional (1D) Heisenberg antiferromagnet (HAFM)\n\\cite{bethe}.\nThe investigation of strongly frustrated magnetic systems\nsurprisingly led to the discovery of several new exact\neigenstates. Some of the eigenstates found for  frustrated quantum\nmagnets are  of quite simple nature and for several physical\nquantities, e.g., the spin correlation functions, analytical\nexpressions can be found. Hence such exact eigenstates may play an\nimportant role either as groundstates of real quantum magnets or\nat least as groundstates of idealized models which can be used as\nreference states for more complex quantum spin systems.\nA well-known class of exact eigenstates are dimerized singlet\nstates, where a direct product of pair singlet states is an\neigenstate of the quantum spin system.  Such states become\ngroundstates for certain values\/regions of frustration. The most\nprominent examples are the Majumdar-Gosh state of the 1D $J_1-J_2$\nspin-half HAFM \\cite{majumdar} and the orthogonal dimer state of\nthe Shastry-Sutherland  model, see, e.g.,\n\\cite{shastry81,Mila,Miyahara,Lauchli,uhrig2004,darradi2005}. Many other frustrated spin\nmodels in one, two or three dimensions are known which have also\ndimer-singlet product states as groundstates, see, e.g.,\n\\cite{pimpinelli,ivanov97,japaner3d,koga,schul02}. A systematic\ninvestigation of systems with dimerized eigenstates can be found\nin  \\cite{schmidt05}. Note that these dimer-singlet product\ngroundstates  have gapped magnetic excitations and lead therefore\nto a plateau in the magnetization at $m=0$.\nRecently it has been demonstrated for the 1D counterpart of the\nShastry-Sutherland model \\cite{ivanov97,koga,schul02}, that more\ngeneral product eigenstates containing chain fragments of finite\nlength can lead to an infinite series of magnetization plateaus\n\\cite{schul02}.\n\n\nOther examples of product ground states are single-spin product\nstates of 1D XYZ model \\cite{mueller85} and the the highly\ndegenerate ground-state manifold of localized-magnon states found\nfor antiferromagnetic quantum spin systems on various frustrated\nlattices \\cite{lm}. Finally, we mention the so-called central-spin\nmodel or Heisenberg star where also exact statements on the\ngroundstate are known \\cite{starI}.\n\nAlthough, at first glance such singlet-product states seem to\nexist only for 'exotic' lattice models, it turned out that such\nmodels are not only a playground of theoreticians but may become\nrelevant for experimental research. The most prominent example is\nthe above mentioned Shastry-Sutherland model introduced in 1981\n\\cite{shastry81} for which only in 1999 the corresponding\nquasi-two-dimen\\-sional compound SrCu$_2$(BO$_3$)$_2$ was found\n\\cite{Kage,srcubo}. Other examples are the quasi-1D spin-Peierls\ncompound $CuGeO_3$, see, e.g., \\cite{cugeo},  or the star-lattice\ncompound\n[Fe$_3$($\\mu_3$-O)($\\mu$T-OAc)$_6$-(H$_2$O)$_3$][Fe$_3$($\\mu_3$-O)($\\mu$-OAc)$_{7.5}$]$_2\n\\cdot$7 H$_2$O.\\cite{star-exp,star-theor}\n\nIn the present paper we combine the ideas of Shastry and\nSutherland \\cite{shastry81} and our recent findings on exact\ntrimerized singlet product ground states (TSPGS's) for 1D\ninteger-spin Heisenberg systems \\cite{schmidt10} and discuss such\nTSPGS's on a  two-dimensional modified Shastry-Sutherland\nsquare-lattice model. Section \\ref{sec_egs} shortly recapitulates\nthe theory of TSPGS's and section \\ref{sec_model} defines the\nmodified  Shastry-Sutherland model and its finite realizations\nthat will be analyzed in what follows. We have concentrated in our\nnumerical studies on the size of the gap for the exact ground\nstate for finite lattices of $N=12$ (for spin  quantum numbers\n$s=1$, $s=2$), as well as  $N=18$ and $N=24$ (for $s=1$) and on\nthe magnetization curves for selected values of $J_2$, see section\n\\ref{sec_nr}. The analytical results in section \\ref{sec ar}\nmainly concern upper and lower bounds of the gap function. These\nresults depend on a slightly generalized statement and proof of\nthe gap theorem, first formulated in \\cite{schmidt10}, which is\ndone in appendix~\\ref{app}. Finally, appendix~\\ref{class} contains exact results on\nclassical magnetization curves for the model under consideration.\n\n\n\n\\section{Exact ground states}\n\\label{sec_egs} The anti-ferromagnetic uniform spin trimer\n\\begin{equation}\\label{egs1}\nH_1=J(\\op{\\bf{s}}_0\\cdot\\op{\\bf{s}}_1+\n\\op{\\bf{s}}_0\\cdot\\op{\\bf{s}}_2+\\op{\\bf{s}}_1\\cdot\\op{\\bf{s}}_2)\n\\end{equation}\nhas, for $J>0$ and integer $s$, a unique $S=0$ ground state, denoted\nby $[0,1,2]$, with ground state energy\n\\begin{equation}\\label{egs2}\nE_0=-\\frac{3}{2}J s(s+1) \\;.\n\\end{equation}\nThe corresponding product state\n\\begin{equation}\\label{egs3}\n\\Phi=\\bigotimes_{i=1}^{\\mathcal N}[i0,i1,i2]\n\\end{equation}\nwill be an eigenstate of a system of ${\\mathcal N}$ coupled spin\ntrimers indexed by $i=1,\\ldots,{\\mathcal N}$ with Hamiltonian\n\\begin{equation}\\label{egs4}\nH=\\sum_{i\\epsilon\nj\\delta}J_{i\\epsilon,j\\delta}\\,\\op{\\bf{s}}_{i\\delta}\\cdot\\op{\\bf{s}}_{j\\epsilon}\n\\;,\n\\end{equation}\nif and only if the coupling between different trimers is ``balanced\"\nin the following sense:\n\\begin{equation}\\label{egs5}\nJ_{i\\delta,j\\delta}+J_{i\\epsilon,j\\epsilon}=J_{i\\delta,j\\epsilon}+J_{i\\epsilon,j\\delta}\n\\end{equation}\nfor all $1\\le i<j\\le{\\mathcal N}$ and $\\delta,\\epsilon=0,1,2$, see\n\\cite{schmidt10}. Moreover, (\\ref{egs3}) will be a ground state of\n(\\ref{egs4}), a TSPGS, if the intra-trimer coupling is almost\nuniform and the inter-trimer coupling is not too strong\n\\cite{schmidt10}.\nThe domain of coupling constants where this is the case will be called the ``TSPGS-region\".\\\\\n\nIf the system of trimers has a periodic lattice structure, the\ndifference $\\Delta E$ between the energy of the first excited state\nand that of the ground state can be shown \\cite{schmidt10} to be\nbounded from below independently of the system size. In other\nwords, the TSPGS is ``gapped\".\n\n\n\\begin{figure}\n\\resizebox{0.9\\columnwidth}{!}{\n\\includegraphics{24_2d_trimer_a.eps}\n} \\caption{ The modified Shastry-Sutherland model on the decorated\nsquare lattice for  $N= 24$ sites (periodic conditions imposed) used for exact diagonalization.}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{12_18.eps}\n} \\caption{Two finite decorated square lattices of  $N= 12$  and\n$N=18$ sites used for exact diagonalization.} \\label{fig2}\n\\end{figure}\n\n\n\n\\section{The model}\n\\label{sec_model}\n\nWe consider the inter-spin Heisenberg Hamiltonian on a decorated\nsquare-lattice (see figure~\\ref{fig1}). It results from the\nwell-known Shastry-Sutherland model by replacing its diagonals by\nequilateral triangles with uniform intra-trimer interaction strength\n$J_1>0$. The set of triangles is divided in a bi-partite fashion\ninto two disjoint subsets of triangles of type I and type II,\ncorresponding to diagonals with positive slope resp.~negative ones,\nsee figure~\\ref{fig1}. Each triangle of, say, type I is surrounded\nby four\ntriangles of type II and connected to each of them with three bonds of strength $J_2$.\\\\\nIt follows that the inter-trimer coupling satisfies the balance\ncondition (\\ref{egs5}) and hence the theory of TSPGS's applies. In\nparticular, two questions arise which will be addressed in the\nfollowing sections: What is the size of the TSPGS-region and of what\nkind are the lowest excitations? The latter question is also\nconnected to the issue of magnetization plateaus which will be\nshortly discussed below.\n\n\n\n\n\\section{Results}\n\n\\subsection{Numerical results}\n\\label{sec_nr} In what follows we set $J_1=1$ and consider $J_2$ as\nthe variable bond strength. To study the region where the TSPGS is\nthe ground state of the model (\\ref{egs4}) we use the Lanczos exact\ndiagonalization (ED) technique. Since for spin quantum numbers $s >\n1\/2$ considered here the size of the Hamiltonian matrix grows much\nfaster with system size $N$ than for $s=1\/2$, we are restricted to\nfinite lattices of $N= 12,18$ and $24$ for $s=1$ and $N=12$ for\n$s=2$. The largest lattice is shown in figure~\\ref{fig1}, whereas\nthe smaller lattices are shown in figure~\\ref{fig2}. Although the\ncriterion for the existence of TSPGS's (see section \\ref{sec_model})\nare fulfilled, we have to mention that for the small lattices of\n$N=12$ and $N=18$ the exchange pattern of the $J_1$ diagonal bonds\nin the squares do not match to the infinite system. Nevertheless, we\nhave included the data for $N= 12$ and $18$ to get an impression on\nfinite-size effects and on the influence of the spin quantum number\n$s$.\n\nAccording to \\cite{schmidt10} the TSPGS is gapped. Hence we use the\nspin gap, see figure~\\ref{fig3}, to detect the critical points\n$J^{c1}_2$ and $J^{c2}_2$, where the  TSPGS gives way for other\nground states. We find for $s=1$ the values $J^{c1}_2\n=-0.570,-0.578$, and $-0.587$ and  $J^{c1}_2 =0.434, 0.446$, and\n$0.454 $  for $N=12,18$, and $24$, respectively (cf.\nfigure~\\ref{fig3}(a)). For $s=2$ and $N=12$ we have $J^{c1}_2\n=-0.400$ and  $J^{c2}_2 =0.322$, cf. figure~\\ref{fig3}(b). These\nvalues lie between the upper and lower bounds which will be derived\nfor $J^{c1}_2$ and  $J^{c2}_2$ in the next section for $N \\to\n\\infty$. The nature of the lowest excited state depends on $J_2$.\nAround  $J_2=0$ it is a triplet state with strong antiferromagnetic\ncorrelations along the trimer bonds and weak correlations between\nthe trimers. Near $J^{c1}_2$ the lowest excitation is a\nferrimagnetic state, i.e. the total spin is $S=Ns\/3$ and the system\nsplits into two ferromagnetically correlated sublattices containing\non the one hand the $2N\/3$ square-lattice sites (i.e. sites\n$0,1,\\ldots,15$ in figure~\\ref{fig1}) and on the other hand the\n$N\/3$ additional sites (i.e. sites $16,17,\\ldots,23$ in\nfigure~\\ref{fig1}). The spin correlations between both sublattices\nare anti-ferromagnetic. The ferrimagnetic state is the ground state\nfor $-1.5 < J_2 < J^{c1}_2$. Near $J^{c2}_2$ the lowest excitation\nis a collective singlet state with strong correlations along all\nbonds, and, this state becomes\nthe ground state at $J_2=J^{c2}_2$.\n\n\n\n\n\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.7}{\\includegraphics{gap_multi_mit_schranken_vert.eps}}\n\\caption{Numerical exact data for $N=12$, $18$, and\n$24$ (symbols) as well as upper (black solid line) and lower bounds\n(red solid line) for the excitation gap $\\Delta E $. (a) spin\nquantum number $s=1$;\n(b) spin quantum number $s=2$.\nNote that the labels $S=1$, $S=0$, $S=2N\/3$ (ferri), and $S=8$\n(ferri) characterize the total spin of the  excited state.\n}\n\\label{fig3}      \n\\end{figure}\n\n\n\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.7}{\\includegraphics{m_h_plat_multi_vertical.eps}}\n\\caption{(a) Magnetization curve $m(h)$ for selected\nvalues of $J_2$ and $s=1$ (thick lines $N=24$, thin lines $N=18$);\n(b) Plateau widths $\\Delta h$  of the $m=1\/3$ and the  $m=2\/3$ plateaus  as\na function of $J_2$ for $N=24$ and $N=18$ and $s=1$.\n}\n\\label{fig4}      \n\\end{figure}\n\n\n\nIt is well  known that the magnetization curve of the\nShastry-Sutherland model (as well as that of the corresponding\nmaterial SrCu$_2$(BO$_3$)$_2$) possesses a series of pla\\-teaus, see,\ne.g., \\cite{Kage,kodama,misguich,mila}. Motivated by this, we study\nnow briefly the  magnetization curve $M(h)$ (where $M$ is the total\nmagnetization  and  $h$ is the strength of the external magnetic\nfield) for the considered model for $s=1$ using ED\nfor $N=18$ and $N=24$ sites. ED results  for the\nrelative magnetization $m=M\/M_{sat}$ versus magnetic field $h$ for\n$N=18$ and $N=24$ sites are shown in figure~\\ref{fig4}a. Again the\nfinite-size effects seem to be small. Trivially, in the limit\n$J_2=0$ the $m(h)$ curve consists of three equidistant plateaus and\njumps according to the magnetization  curve of an individual\ntriangle. Switching on a ferromagnetic inter-triangle bond $J_2 < 0$\nthe general shape of the magnetization curve is preserved. However,\nthe  saturation field as well as the end points of the plateaus\ndecrease almost lineraly with $J_2$ and become zero at $J_2=1.5$,\nwhere the ground state becomes  the fully polarized ferromagnetic\nstate.\n\nIn case of a moderate antiferromagnetic inter-triangle bond $J_2 >\n0$ the plateaus at  $m=1\/3$ and $m=2\/3$ still exist, however the\ndiscontinuous transition between plateaus becomes smooth. Note that\na $m=1\/3$ plateau was also found for the standard Shastry-Sutherland\nmodel \\cite{misguich,mila}. The plateau widths $\\Delta h$ of the $m=1\/3$\nand $m=2\/3$ plateaus in dependence on $J_2$ is shown in\nfigure~\\ref{fig4}b. Obviously, both  widths shrink monotonously with\nincreasing of $|J_2|$.\nIf $J_2$ approaches the critical value   $J^{c1}_2$ we find\nindications for additional plateaus, e.g., at $m=5\/6$. Note, however,\nthat our finite-size analysis of the plateaus naturally could miss\nother plateaus present in infinite systems, see, e.g., the discussion\nof the ED data of the $m(h)$ curve of the standard\nShastry-Sutherland model in \\cite{wir04}. Hence, the study of\nthe magnetization process of the considered quantum spin model needs\nfurther attention based on alternative methods.\n\nOne might expect that the presence of these plateaus and jumps may\nbe linked purely to quantum effects because they are often not\nobserved in equivalent classical models at $T=0$\n\\cite{lm,kawamura,zhito,cabra}. However, for the present model the\nplateau at $m=1\/3$ survives in the classical limit for $J_2 < 0$ as\nwe will show in appendix~\\ref{class}.\n\n\n\n\\subsection{Analytical results}\\label{sec ar}\n\\subsubsection{$s=1$}\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.85}{\\includegraphics{HR69_vertical.eps}}\n\\caption{Two possible subsystems of the modified\nShastry-Sutherland lattice, see e.~g.~figure \\ref{fig1}. The upper\none, $H_6$, consists of two coupled triangles; the lower one, $H_9$\nof three triangles.\n}\n\\label{figH69}      \n\\end{figure}\nIn order to obtain analytical results about the TSPGS-region we have\nadapted theorem $3$ of \\cite{schmidt10} to the present situation. A\nslightly more general version of this theorem is stated and proven\nin appendix~\\ref{app}. It yields lower bounds for the gap $\\Delta E$ of the\nform $\\Delta E\\ge f(4J)$ and the TSPGS-region in terms of properties\nof simpler spin systems of which the lattice can be composed, see\nfigure~\\ref{fig3}. These subsystems are chosen here as systems\nisomorphic to $H_6$, see figure \\ref{figH69}, consisting of two\nneighboring triangles. For $s=1$ the gap function $x\\equiv \\delta_6\nE=f(J),\\,J\\equiv\\frac{J2}{J1}$ of $H_6$ is obtained as a special\ncase of equation (\\ref{ar2a}) given below.\nThis yields the corresponding bounds for the\nTSPGS-region $(J^{c1},J^{c2})$\n\\begin{equation}\\label{ar3}\nJ^{c1}<\\frac{3-\\sqrt{73}}{16}\\approx -0.3465<\\frac{1}{4}<J^{c2}\\quad\n\\mbox{for  }s=1\\;.\n\\end{equation}\nThe function $\\delta_6 E=f(J)$ according to (\\ref{ar2a}) also\nprovides an upper bound for the gap function of the lattice, since it\nrepresents the energy of a state orthogonal to the TSPGS, albeit not\nan eigenstate of $H$. This bound is very close to the numerically\ndetermined gap function in the case of $N=12$, see figure \\ref{fig3}a,\nbut considerably deviates in the cases of $N=18$ and $N=24$. This\nindicates that, in general, the lowest excitations of the lattice\nare different from the excitations of $H_6$.\\\\\n\n\\subsubsection{General $s$}\n\nIt is possible to analytically calculate the energy of the lowest\nexcitations of $H_6$ for general integer $s$. The corresponding gap\n$\\delta_6 E=x=f(J)$ is obtained as the lowest root of the following\ncubic equation\n\\begin{eqnarray}\\nonumber\n&-&(x-4) (x-2) (x-1 )\n -(x-1 ) (2x-5 ) J\\\\ \\label{ar2a}\n&+&( 1 - 3 r - x + r x)J^2 +r J^3=0\n\\end{eqnarray}\nwhere we have set $r\\equiv s(s+1)$. From this result one derives the\nlower bound\n\\begin{equation}\\label{ar2b}\n\\Delta E\\ge f(4J) \\quad \\mbox{for general }s\n\\end{equation}\nand a lattice of arbitrary size, see theorem $1$ in appendix~\\ref{app}\nadapted to the system under consideration.\nThe corresponding curves are shrinking in $J$-direction with\nincreasing $s$ and yield inner bounds for the TSPGS-region\n$(J^{c1},J^{c2})$ of the form\n\\begin{equation}\\label{ar2b1}\nJ^{c1} < J_{L}^{(1)}<0<J_{L}^{(2)}<J^{c2}\n\\;,\n\\end{equation}\nsee figure \\ref{figG1} (green curves).\nUpon scaling w.~r.~t.~the new variable\n$j\\equiv\\sqrt{r}J$ the graphs of (\\ref{ar2a}) asymptotically approach the curve given by\n\\begin{equation}\\label{ar2c}\nj^2=\\frac{(x-4)(x-2)(x-1)}{16(x-3)}\\;,\n\\end{equation}\nwith Taylor expansion\n\\begin{equation}\\label{ar2d}\nx=1-\\frac{32}{3}j^2+{\\mathcal O}(j^3)\\;,\n\\end{equation}\nsee figure \\ref{figG3b}.\nHence $J_{L}^{(i)}$ assumes for $s\\to\\infty$ asymptotically the form\n\\begin{equation}\\label{ar2d1}\n|J_{L}^{(i)}|\\sim\\frac{1}{\\sqrt{6s(s+1)}},\\;\\;i=1,2\n\\;.\n\\end{equation}\n\\\\\n\n\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGLB.eps}\n} \\caption{Lower bounds of the scaled gap function\n$\\delta_6 E(j),\\;j=\\sqrt{s(s+1)}J$ of the modified\nShastry-Sutherland spin lattice for $s=1,\\ldots,10$ (thin curves)\nobtained from Eq.~(\\ref{ar2a}). The curves approach the asymptotic\n(\\ref{ar2c}) for $s\\rightarrow\\infty$  (thick red curve) which has a\nsimple quadratic approximation (\\ref{ar2d}) (thick green curve).} \\label{figG3b}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGUB.eps}\n} \\caption{Upper bounds of the scaled gap function\n$\\delta_0 E(j),\\;j=\\sqrt{s(s+1)}J$ of the modified\nShastry-Sutherland spin lattice for $s=1,\\ldots,10$ (thin curves)\nobtained from Eq.~(\\ref{ar6}). The curves approach the asymptotic\n(\\ref{ar7}) for $s\\rightarrow\\infty$  (thick red curve) which has a\nsimple quadratic approximation (\\ref{ar8}) (thick green curve).} \\label{figG3a}\n\\end{figure}\n\n\nIn order to obtain close upper bounds $g(J)$ of the gap $\\Delta(E)$\nin the case $N\\ge 18$ we calculate\nthe energy of a certain (degenerate) state that involves three\ntriangles for arbitrary integer $s$, say, one triangle of type $I$\nand two neighboring triangles of type $II$, see figure \\ref{figH69}.\nThis state is obtained as an exact eigenstate of $H_0$, which is the\nfull Hamiltonian $H$, restricted to a $4^3=64-$dimensional subspace\nspanned by product states of the form\n\\begin{equation}\\label{ar4}\n\\phi_i\\otimes\\phi_j\\otimes\\phi_k,\\quad i,j,k=0,\\ldots,3\\;.\n\\end{equation}\nThe $\\phi_n$ live in the $(2s+1)^3$-dimensional Hilbert spaces\nbelonging to one of the three triangles. $\\phi_0=[0,1,2]$ denotes\nthe TSPGS of the corresponding triangle and\n\\begin{equation}\\label{ar5}\n\\phi_i\\equiv \\frac{\\op{s}_0^{(i)}\\phi_0}{||\n\\op{s}_0^{(i)}\\phi_0||},\\;i=1,2,3\\;,\n\\end{equation}\nwhere $\\op{s}_0^{(i)}$ is the $i$-th component of the spin operator\n$\\op{\\bf s}_0$ pertaining to the spin site number $0$, an\narbitrarily chosen spin site of the corresponding triangle. The gap\nfunction of $H_0$ will be denoted by $x\\equiv\\delta_0 E=g(J)$ and\nconstitutes an upper bound for $\\Delta(E)$.\nIt has the following implicit form, using $r\\equiv s(s+1)$:\n\\begin{eqnarray}\\nonumber\n0&=& -12(x-3)^2(x-2)(x-1)-6J(x-3)(x-1)(4x-9)\\\\ \\nonumber\n && +J^2\n(x-3)(9-9x+4r(7x-15))+16 J^3 r (2x-5)\\;.\\\\\n&& \\label{ar6}\n\\end{eqnarray}\nAgain, the function $g$ belongs to the lowest branch of (\\ref{ar6}).\nThe corresponding curves are shrinking in $J$-direction with\nincreasing $s$ and yield outer bounds for the TSPGS-region\n$(J^{c1},J^{c2})$ of the form\n\\begin{equation}\\label{ar6a}\nJ_{U}^{(1)}<J^{c1} < 0<J^{c2}<J_{U}^{(2)}\n\\;,\n\\end{equation}\nsee figure \\ref{figG1} (red curves).\nUpon scaling w.~r.~t.~the new variable\n$j\\equiv\\sqrt{r}J$ the graphs of (\\ref{ar6})\nasymptotically approach the curve given by\n\\begin{equation}\\label{ar7}\nj^2=\\frac{3(x-3)(x-2)(x-1)}{7x-15}\\;,\n\\end{equation}\nwith Taylor expansion\n\\begin{equation}\\label{ar8}\nx=1-\\frac{4}{3}j^2+{\\mathcal O}(j^3)\\;,\n\\end{equation}\nsee figure \\ref{figG3a}.\nHence $J_{U}^{(i)}$ assumes for $s\\to\\infty$ asymptotically the form\n\\begin{equation}\\label{ar8a}\n|J_{U}^{(i)}|\\sim\\sqrt{\\frac{6}{5s(s+1)}},\\;\\;i=1,2\n\\;.\n\\end{equation}\n\\\\\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGG1.eps}\n} \\caption{Exact bounds for the TSPGS-region $(J^{c1},J^{c2})$} for $s=1,\\ldots,10$\nof the form $J_{U}^{(1)}<J^{c1}< J_{L}^{(1)}< 0<J_{L}^{(2)}<J^{c2}<J_{U}^{(2)}$.\nThese are derived\nfrom (\\ref{ar2a}) (green curves, inner bounds) and (\\ref{ar6}) (red curves, outer bounds).\nIn the classical limit $s\\rightarrow\\infty$\nthe TSPGS-region shrinks to zero according to (\\ref{ar2d1}) and (\\ref{ar8a}).\n\\label{figG1}\n\\end{figure}\n\n\n\n\n\nAlthough these curves constitute only upper\nbounds of the true gap functions, the comparison with the numerical\nresults for $N=18$ and $N=24$ reveals a close approximation to both\ncurves, see figure \\ref{fig3}. This supports our conjecture that\n(\\ref{ar6}) indeed may serve as an analytical approximation of the\ngap functions for large $N$ and arbitrary integer $s$. This would\nmean that the excitations from the TSPGS can be viewed as local\nexcitations essentially concentrated on three neighboring triangles.\nNumerically determined spin correlation functions seem to be in accordance with\nthis conjecture. Of course, the corresponding excited state will be\nlargely degenerate due to the translational symmetry of the lattice.\nWe expect an almost flat $\\bf{k}$-dependance of the energy band\n$E(\\bf{k})$. This expectation is also supported by our numerical\nresults. We have found that the lowest excitations close to $J=0$\nhave the total spin quantum number $S=1$ in accordance with our\nmodel.\\\\\n\nIn the case $N=12$ where we have performed numerical calculations for\n$s=1$ and $s=2$ it is not possible to put a subsystem of type $H_9$  into the\nlattice and the above results do not apply. However,\nan analogous method can be applied to two\ncoupled triangles of type $H_6$ and yields an upper bound of the gap function of the form\n\\begin{equation}\\label{ar2e}\n\\Delta E\\le \\frac{1}{12} (18 - 3 J - \\sqrt{9(J-2)^2  + 96 J^2\ns(s+1)}).\n\\end{equation}\nThe numerically determined gap together with the bounds (\\ref{ar2b})\nand  (\\ref{ar2e}) is represented in figure \\ref{fig3} b.\\\\\n\n\n\\section*{Acknowledgement}\nThe numerical calculations were performed using J.~Schulenburg's\n{\\it spinpack}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Glossary}\n\n\\begin{itemize}[leftmargin=3cm]\n    \\item[Network]  A \\emph{network} is a collection of nodes (also called vertices) and edges (also called links) linking pair of nodes. Mathematically, it is represented by a graph ${G}=({V},{E})$ where ${V}$ is the set of nodes and ${E}\\subseteq V\\times V$ is the set of edges. Additional information can be attached to each node or edge, for example edges can have different weights. Edges can be undirected or directed.\n    \n    \\item[Adjacency matrix] The \\emph{adjacency matrix},  $\\matr{A}$,  of a network is a $N\\times N$ matrix ($N=\\left|{V}\\right|$) with element $A_{ij}=1$ if there is an edge from node $i$ and to node $j$ and $A_{ij}=0$ otherwise. If the network is weighted, $A_{ij}=w_{ij}$ where $w_{ij}\\in\\mathbb{R}$ is the weight associated with the edge between nodes $i$ and $j$ if it exists and $A_{ij}=0$ otherwise.\n    For undirected network, $A_{ij}=A_{ji}$, i.e. $\\matr{A}$ is symmetric.\n    \n    \\item[Degree of a node]  The \\emph{degree}, $k_i$, of node $i$ in an undirected network is equal to its number of connections, i.e. $k_i=\\sum_j A_{ij}$. For directed network, we differentiate the \\emph{in-degree}, $k^{in}_i=\\sum_j A_{ji}$ and the \\emph{out-degree}, $k^{out}_i=\\sum_j A_{ij}$, i.e. the number of edges in-coming to node $i$ and out-coming from node $i$, respectively. In weighted undirected networks, the degree of a node is replaced by its \\emph{strength}, $s_i=\\sum_j w_{ij}$ or \\emph{in-strength} and \\emph{out-strength} for weighted directed networks.\n    \n    \\item[Walk] A \\emph{walk} is an alternating sequence of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence.\n    \\item[Path] A \\emph{path} on a graph is a walk in which all vertices and edges are distinct.\n    \\item[Connected components] A \\emph{connected component} of an undirected graph $G(V,E)$ is a subgraph of $G$, made of a subset of $V$ and all the edges connecting nodes of the subset toghether,  where there exist a path between each pair of nodes. In directed graphs, we differentiate \\emph{strongly-connected components}, where there exist a path in both directions between all pairs of nodes, and \\emph{weakly-connected components}, where there exist a path in at least one direction between all pairs of nodes.\n\\end{itemize}\n\n\n\n\n\\section{Why study networks?}\n\n\nIn \\index{network science} complex systems are represented as a mathematical \\index{graphs} consisting of a set of nodes representing the components and a set of edges representing their interactions.\nThe framework of networks has led to significant advances in the understanding of the \nstructure, formation and function of complex systems\\,\\cite{newman2003structure,barrat2008dynamical,boccaletti2006complex,albert2002statistical}.\nSocial and biological processes \nsuch as the dynamics of epidemics\\,\\cite{pastor2015epidemic}, \nthe diffusion of information in social media\\,\\cite{zhang2016dynamics},\nthe interactions between species in ecosystems\\,\\cite{montoya2006ecological} or the  communication between neurons in our brains\\,\\cite{bullmore2009complex} are all actively studied using dynamical models on complex networks.\nIn all of these systems, the patterns of connections at the individual level play a fundamental role on the global dynamics and finding the most important nodes allows one to better understand and predict their behaviors.\n\n\n\n\n\n\n\\section{Definition of the Subject}\nReal-world \\index{complex networks} are characterized by a number of structural features differentiating them from regular networks, such as lattices, but also making them different than completely random graphs, such as Erd\\H{o}s-R\\'enyi graphs\\,\\cite{newman2003structure}.\nThe properties that can be found in complex networks include, for example, a modular organization, also called community structure\\,\\cite{Fortunato2010}, or the so-called small-world effect, i.e. the fact that most pairs of nodes are connected by a very short path compared to the sizes of the networks\\,\\cite{Watts1998}.\nA characteristic of real-world complex networks that interests us here and that has been intensively studied is the fact that their degree distributions are usually very heterogeneous\\,\\cite{albert2002statistical,newman2003structure}, indicating that there is usually large differences in the number of connections that nodes have.\nMany real world networks have been found to have degree distributions resembling power laws and are sometimes referred to as scale-free\\,\\cite{barabasi2009scale}.\nWhether these real-world systems are really scale-free and whether their degree distributions are really power laws is still disputed\\,\\cite{Holme2019}.\nHowever, what is undeniable is the fact that many real world networks have degree distributions that are heavy-tailed with a minority of the nodes concentrating the majority of the connections.\n\nIn these systems, a small set of essential nodes can shape the collective dynamics of the entire systems.\nFor example, during epidemic outbreaks of infectious diseases, some individuals, known as \\emph{super-spreaders}, infect disproportionately more secondary contacts, as compared to most others\\,\\cite{kitsak2010identification,stein2011super},\nkeystone species in ecology are responsible for the integrity and stability of ecosystems\\,\\cite{may1972will,scheffer2012anticipating,mills1993keystone,morone2018kcore}\nand specific regions in brain networks are more important than others in the formation of memory\\,\\cite{bullmore2009complex,del2018finding,reis2014avoiding,zamora2010cortical}.\nIn social networks, a small set of \\index{influencers} can drive the global dynamics of the system\\,\\cite{bovet2019influence} and opinion leaders are capable of influencing the public viewpoint on certain trending topics \\,\\cite{watts2007influentials}.\nAn important research effort in network science has therefore been dedicated to the development of methods allowing to find the most important nodes in networks. \nIntuitively, nodes with a large degree are likely to be more successful to trigger large-scale propagations or to control a large number of nodes. The degree centrality ranks nodes in terms of their degree and allows to identify highly connected \\emph{\\index{hubs}} present in most real-world complex networks that play an essential role in controlling their dynamics and maintaining their integrity\\,\\cite{barabasi1999emergence,albert2000error,cohen2001breakdown}.\nWhile the \\index{degree centrality} is arguably the simplest \\index{centrality measure}, it only uses local information about each node to rank its centrality and more complex centrality measures have been developed in order to capture the collective network effects impacting the influence of a node.\nMost centrality measures are based on a notion of distance between nodes or on a way to traverse the network that allows to compute how \"central\" a node is compared to other nodes while capturing the heterogeneous structural patterns of complex networks.\nIn the following, we describe centrality measures based on the notions of \\index{network traversal} they rely on.\nThis short entry aims at being an introduction to this extremely vast topic, with many contributions from several fields, and is by no means an exhaustive review of all the literature about network centralities (see, for example, \\,\\cite{freeman1978centrality,Borgatti2005,Perra2008,landherr2010critical,rodrigues2019network} for more exhaustive reviews).\n\n\n\\subsection{Centrality measures based on shortest paths}\n\n\\index{Closeness centrality}\\,\\cite{bavelas1950communication, beauchamp1965improved,sabidussi1966centrality,dangalchev2006residual} and \\index{betweenness centrality}\\,\\cite{freeman1978centrality,friedkin1991theoretical} are two dual measures of centrality initially developed in social sciences to assess the importance in terms of ease of access to others nodes (closeness) and brokering power (betweenness) of individuals in social networks.\nThese centralities have since then been used in many other contexts.\nThey are based on the assumption that information, or influence, propagates between two nodes in the most efficient way, i.e. by following the shortest path between them.\n\nConsidering an unweighted and undirected graph $G=(V,E)$, a walk on the graph $G$ is an alternating sequence of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence. A path on a graph is a walk in which all vertices and edges are distinct.\nA graph is said to be connected if there exists a path from any node node to any other node in the graph.\nConsidering the distance $\\text{dist}(s,r)$ between two nodes $s$ and $r$ in a connected undirected and unweighted graph $G=(V,E)$ as the number of edges in the \\index{shortest path} between them, the total distance of vertex $v$ is defined as the sum of its distance to all other vertices\n\n$$\\text{dist}(s) = \\sum_{r\\in V} \\text{dist}(s,r),$$\n\nwhich is larger for vertices that are the farther away from other vertices.\nTherefore, the \\emph{closeness centrality} of a node $s$ on a connected and undirected network is usually defined as\\,\\cite{bavelas1950communication, beauchamp1965improved}\n\n\\begin{equation}\n    c_C(s) = \\frac{1}{\\sum_{r\\in V} \\text{dist}(s,r)} = \\frac{1}{\\text{dist}(s)}.\n\\end{equation}\n\nThe betweenness centrality captures the brokering power of node as the opportunity it has to intercept or influence the communications happening between pairs of other nodes. \nLet $\\sigma(s,r)$ be the number of shortest paths between $s$ and $r$ and let $\\sigma(s,r|b)$ be the number of shortest paths between $s$ and $r$ passing by a brokering node $b\\in V\\setminus \\{s,r\\}$. We consider $\\sigma(s,s)=1$ and $\\sigma(s,r|b)=0$ if $b\\in\\{s,r\\}$.\nWe call the quantity $\\delta(s,b,r)=\\frac{\\sigma(s,r|b)}{\\sigma(s,r)}$ the \\emph{dependency} of a sender $s$ and a receiver $r$ on a broker $b$\\,\\cite{Brandes2016}.\nThe \\emph{betweenness centrality} in a connected and undirected graph $G=(V,E)$ is defined as the sum of dependencies of all communicating pairs on a broker $b$\\,\\cite{freeman1980gatekeeper}\n\n\\begin{equation}\n    c_B(b) = \\sum_{s,r\\in V} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)}= \\sum_{s,r\\in V} \\delta(s,b,r),\n    \\label{eq:betweenness}\n\\end{equation}\n\nand can be seen as the overall potential control of $b$ on the communications in G.\n\nBetweenness and closeness centrality can be seen as being dual to each other conceptually expressing either the independence from the control of others (closeness) or the potential control over others (betweenness)\\,\\cite{freeman1980gatekeeper,Brandes2016}.\nIndeed, by noting that when considering shortest paths between $s$ and $r$ with different brokers $b$ at a fixed distance $d$ from $s$ ($1 \\leq d < \\text{dist}(s,r)$) each shortest path pass through exactly one broker and therefore \n$\\sum_{b\\in V : \\text{dist}(s,b)=d} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)} = 1$.\nThis implies that the sum of dependencies between a sender $s$ and a receiver $r$ taken over of possible brokers $b$ is proportional to the distance between $s$ and $r$: $\\sum_b \\delta(s,b,r)= \\sum_{d=1}^{\\text{dist}(s,r)-1} \\sum_{b\\in V : \\text{dist}(s,b)=d} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)} =\\text{dist}(s,r)-1$.\nThis observation allows one to define the closeness centrality as $c_C(s)^{-1} = (N-1) + \\sum_{b,r\\in{V}}\\delta(s,b,r)$ revealing its mathematical duality with betweenness centrality as a different partial sum, over $b$ and $r$ instead of $s$ and $r$, of the dependencies $\\delta(s,b,r)$ showing its interpretation as a measure of lack of independence on others\\,\\cite{Brandes2016}.\nFigure \\ref{fig:closness_vs_betweenness} shows the kite graph introduced by  Krackhardt in 1990 as an illustration of the different ranking obtained by these centrality measures\\,\\cite{krackhardt1990assessing}.\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.5\\linewidth]{closness_vs_betweenness}\n \\caption{Krackhardt kite graph showing the different ranking obtained using degree, closeness and betweenness centralities\\,\\cite{krackhardt1990assessing}. \n The closeness of a node is represented by its size and the betweenness of a node is indicated by its color with nodes with larger betweenness being of a lighter shade.\n The node with largest degree centrality is 3.\n Nodes 5 and 6 have the largest closeness centrality and node 7 has the largest betweenness centrality.\n }\n \\label{fig:closness_vs_betweenness}\n\\end{figure}\n\nIn directed networks, closeness and betweenness centrality can be generalized by considering the notion of reachability instead of distance. In weighted networks where edge weights typically represent a distance or a lag in the connection between adjacent nodes, the length of a path is usually taken as being the sum of the weights of its edges.\nWe refer the reader to \\,\\cite{Brandes2016} for a discussion about the generalizations of the closeness and betweenness centrality to directed and weighted networks, and to their interpretations in these cases.\n\nMany variants of closeness and betweenness have been developed\\,\\cite{brandes2008variants}. The concept of betweenness centrality has also been extended to edges by considering the number of shortest paths containing an edge instead of a node in eq. (\\ref{eq:betweenness})\\,\\cite{brandes2008variants} or, for example, by considering the fraction of minimum spanning trees of a graph that contain a given edge\\,\\cite{teixeira2015not}.\nVersions of the closeness and betweenness centralities based on random walks instead of shortest paths have also been developed\\,\\cite{white2003algorithms,Newman2005}.\n\n\n\\subsection{Centrality measures based on walks}\n\nSeveral widely used centrality measures can be seen as being based on the concept of \\index{graph walks}. \nWalks, alternating sequences of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence, are useful to count the number of possible ways there is to reach a given node starting from another node.\nCentrality measures based on walks usually try to find the nodes from which there are the largest number of walks reaching other nodes.\nFor unweighted networks, the number of walks of length $\\ell$ existing between two nodes $i$ and $j$ is given by the element $(i,j)$ of the $l^{\\text{th}}$ power of the adjacency matrix: $\\left(\\matr{A}^\\ell\\right)_{ij}$.\nThe number of walks of length $\\ell$ starting from node $i$ is then given by $w(\\ell)_i=\\sum_j\\left(\\matr{A}^\\ell\\right)_{ij}$ or in matrix notation $\\matr{w}(\\ell)=\\matr{A}^\\ell \\matr{1}$, where $\\matr{1}$ is the unit vector.\nThe \\emph{degree centrality} of a node $i$ can be expressed as the number of walks of length 1 starting from it:\n\\begin{equation}\n    c_{\\text{deg}}(i) = w(1)_i = \\sum_j A_{ij}.\n\\end{equation}\n\nIn order to take into account information about the surrounding of a node in its centrality score, several researchers have developed centrality measures that consider longer walks. The idea being that if a node is close to a node with a high centrality, its centrality should also be high\\,\\cite{katz1953new,bonacich1972factoring}.\nStarting from an initial guess for the centrality of each node given by the vector $\\matr{x}(0)$ (e.g. $\\matr{x}(0)=\\matr{1}$), we can propagate this initial centrality through walks of a given length. \nThe centrality vector for walks of length $\\ell$ is given by \n\\begin{equation}\n    \\matr{x}(\\ell) = \\matr{A}^\\ell \\matr{x}(0).\n    \\label{eq:eig_cen_it}\n\\end{equation}\n\nWriting the initial centrality as a linear combination of the eigenvector of the adjacency matrix, one can see that as $\\ell\\xrightarrow{}\\infty$ the centrality will converge to a vector proportional to the leading eigenvector of $\\matr{A}$, the one corresponding to its largest eigenvalue, and that takes into account global information about the network structure\\,\\cite{Newman2010networks}.\nAs $\\matr{A}$ is non-negative and if $G$ is connected, the \\index{Perron-Frobenius theorem} ensures that the leading eigenvector is positive and that the associated eigenvalue is positive and simple.\nThe resulting vector is called the \\emph{\\index{eigenvector centrality}} whose element $i$ is defined as \n\n\\begin{equation}\n    c_\\text{eig}(i) \\propto \\sum_j A_{ij} c_\\text{eig}(j).\n\\end{equation}\n\nIn weighted networks, the eigenvector centrality is propagated along walks on the graph that are weighted by the edge weights.\nThe eigenvector centrality can be computed similarly on directed and undirected networks with the difference that for directed networks, the adjacency matrix is in general asymmetric and has therefore two different sets of left- and right-eigenvectors. Depending on the application, one may prefer to use the largest left-eigenvector or the largest right-eigenvector if one is more interested in nodes having a large number of incoming walks or outgoing walks, respectively.\nHowever, other types of centralities are usually preferred for directed networks as the eigenvector centrality of nodes outside of a strongly connected component (a subgraph where there exist a path in both directions between all pairs of nodes) may be equal to zero. In particular, in directed graphs without cycles (closed directed paths), the eigenvector centrality is zero for all nodes.\nA comparison of the degree centrality with the eigenvector centrality is shown in Fig. \\ref{fig:deg_vs_eigen} for an undirected network of American football teams\\,\\cite{girvan2002community,evans2010clique}.\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.8\\linewidth]{football_deg_vs_eigenvec}\n \\caption{Network of American football games between Division IA colleges during regular season Fall 2000\\,\\cite{girvan2002community,evans2010clique}.\n Degree centrality for this undirected graph is shown on the left and eigenvector centrality is shown on the right. Lighter shades indicate a higher centrality.\n Several nodes have the same degree in this network making the degree centrality unable to disantangle the importance of nodes with the same degree.\n Eigenvector centrality has a tendency of being localized in regions where many high degree nodes are close to each other.\n }\n \\label{fig:deg_vs_eigen}\n\\end{figure}\n\nThe \\emph{\\index{subgraph centrality}} is based on the idea of counting the number of times that a node takes part in the different connected subgraphs of a network, with smaller subgraphs having higher importance\\,\\cite{Estrada2005}.\nThe subgraphs are identified by counting closed walks of different lengths that start and end on a given node. The subgraph centrality of node $i$ is defined as\n\n\\begin{equation}\n    c_{\\text{Sub}}(i) = \\sum_{\\ell=0}^{\\infty}\\frac{\\left(\\matr{A}^\\ell\\right)_{ii}}{\\ell!}.\n\\end{equation}\n\nBy noticing that $c_{\\text{Sub}}(i)=\\left(e^\\matr{A}\\right)_{ii}$, where $e^\\matr{A}$ denotes the matrix exponential of $\\matr{A}$,\nthe subgraph centrality can be obtained from the spectrum of the adjacency matrix as\\,\\cite{Estrada2005}\n\n\\begin{equation}\n    c_{\\text{Sub}}(i) = \\sum_{i=1}^{N}(v_j^i)^2e^{\\lambda_j}, \n\\end{equation}\n\nwhere $v_1,v_2, \\ldots, v_N$ form an orthonormal basis of $\\mathbb{R}^N$ and are eigenvectors of $\\matr{A}$ associated with the eigenvalues $\\lambda_1, \\lambda_2, \\ldots, \\lambda_N$.\nThis centrality is more discriminative than the degree, betweenness, closeness or eigenvector centralities and gives a distinctly different ranking of nodes in real-world complex networks\\,\\cite{Estrada2005}.\n\nAnother centrality measures based on walks that tries to solve the issue of the eigenvector centrality in directed networks is the \\index{Katz centrality}\\,\\cite{katz1953new}.\nThe Katz centrality of node $i$ is the weighted sum of all walks emanating from $i$, with the count for walks of length $\\ell$ weighted by a factor $\\alpha^\\ell$ where $0<\\alpha<1$. In this way the importance of longer walks is diminished compared to shorter walks.\nThe \\emph{Katz centrality} of node $i$ can be written as\\,\\cite{katz1953new}\n\n\\begin{equation}\n    c_{\\text{Katz}}(i) = 1 + \\sum_{\\ell=1}^{\\infty}\\sum_{j} \\alpha^\\ell \\left(\\matr{A}^\\ell\\right)_{ij},\n\\end{equation}\n\nwhere the unit shift does not alter the ranking of the nodes and may be seen as arising from a single walk of length zero.\nIn order for the centrality to converge, the factor $\\alpha$ must be smaller than the reciprocal of the absolute value of the largest eigenvalue of $\\matr{A}$\\,\\cite{katz1953new,bonacich1972factoring,bonacich1987power}.\nIn this case, we can also note that $\\matr{I}+\\alpha\\matr{A}+\\alpha^2\\matr{A}^2+\\alpha^3\\matr{A}^3 + \\cdots = \\left(\\matr{I}-\\alpha\\matr{A}\\right)^{-1}$.\nThus, the vector $\\matr{x}$ giving the Katz centrality of each node is also solution of the matrix equation $\\matr{x}=\\left(\\matr{I}-\\alpha\\matr{A}\\right)^{-1}\\matr{1}$ which can be rewritten as $\\matr{x}=\\alpha\\matr{A}\\matr{x}+\\matr{1}$.\nThe Katz centrality can therefore be computed iteratively in a similar manner than the eigenvector centrality (with eq. (\\ref{eq:eig_cen_it})) using \n\\begin{equation}\n    \\matr{x}(t+1) = \\alpha\\matr{A}\\matr{x}(t)+1.\n    \\label{eq:katz_it}\n\\end{equation}\n\nA unit value is added at each iteration guaranteeing that the Katz centrality will never be equal to zero, even in directed networks.\nThe factor $\\alpha$ can also be seen as balancing the importance of the eigenvector term compared to the constant term added at each iteration.\nIn the limit as $\\alpha$ approaches the reciprocal of the absolute value of the largest eigenvalue of $\\matr{A}$, the Katz centrality approaches the eigenvector centrality on strongly connected graphs\\,\\cite{bonacich1972factoring,bonacich1987power}.\n\nWalk-based centralities have been generalized to hypergraphs (e.g. \\,\\cite{bonacich2004hyper,benson2019three}), multiplex networks (e.g. \\,\\cite{sola2013eigenvector,de2015ranking}) and temporal networks (e.g. \\,\\cite{nicosia2013graph,praprotnik2015spectral,taylor2017eigenvector}).\n\n\n\\subsection{Centrality measures based on random walks}\n\n\\index{Random walks} on graphs have been extensively studied and have many applications in the study of diffusive processes on networks and for the characterization of the structure of complex networks \\,\\cite{Rosvall2008,Delvenne2010,durrett2010some,Masuda2016}.\nA random walk on a graph is defined by the trajectory of a walker that, at each time step, jumps to a neighbors of $i$ with equal probability. On an unweighted and undirected network, the transition probability for a walker on node $i$ to jump to node $j$ is equal to \n\n\\begin{equation}\n    T_{ij} = \\frac{A_{ij}}{k_i},\n\\end{equation}\n\nor in matrix notation $\\matr{T}=\\matr{D}^{-1}\\matr{A}$, where $\\matr{D}=\\text{diag}(\\matr{k})$ is the diagonal degree matrix.\nIn directed networks, the degree is replaced by the out-degree and in weighted networks with positive weights, the probability transition is proportional to the weight of edge $(i,j)$.\nCentrality measures based on random walks exploit the fact that,\nif a centrality value propagates as a random walk, at each iteration, its value is divided across all out-going edges of a node.\nThis is different than for eigenvector and Katz centralities, based on walks, where the centrality tends to concentrate on a few hubs in the network, leaving other nodes almost indistinguishable with very low scores due to the repeated reflection of influence along the mutual connections between hubs during iterations of eqs. (\\ref{eq:eig_cen_it}) \\& (\\ref{eq:katz_it}) (see Figs. \\ref{fig:deg_vs_eigen} \\& \\ref{fig:pr_vs_katz}).\n\nThe distribution probability of finding a walker at time step $n$ on any node of the network can be written as a row-vector $\\matr{p}(n)$ with $\\sum_i p_i = 1$.\nThe evolution of the distribution probability is given by \n\n\\begin{equation}\n    \\matr{p}(n+1) = \\matr{p}(n)\\matr{T}.\n    \\label{eq:rw_update}\n\\end{equation}\n\nOn connected undirected network, the random walk reaches a stationary distribution satisfying $\\matr{p}^\\star=\\matr{p}^\\star\\matr{T}$ where the probability at each node is proportional to its degree:\n\\begin{equation}\n    p^\\star_i = \\frac{k_i}{\\sum_j k_j}.\n\\end{equation}\n\nOn undirected networks, the degree centrality can therefore also be seen as the stationary state of the random walk process.\nOn directed networks, random walks are not guaranteed to converge to a unique stationary state.\nA stationary distribution only exists on strongly connected components of a directed network\\,\\cite{aldous2002reversible,Masuda2016}.\n\n\n\n\nTo overcome this issue the \\emph{\\index{PageRank}} centrality modifies the classical random walk by introducing a \"teleportation\" probability, i.e. at each step, the walkers have a given probability to teleport uniformly at random to any other nodes of the network. The update equation for the PageRank probability density is given by \\,\\cite{brin1998anatomy,page1999pagerank,Masuda2016}\n\n\\begin{equation}\n    \\matr{p}(n+1) = \\alpha\\matr{p}(n)\\matr{S} + \\frac{1-\\alpha}{N}\\matr{1}^T,\n    \\label{eq:pg_it}\n\\end{equation}\n\nwhere $\\matr{S}$ is a transition matrix constructed from $\\matr{A}$ such that $S_{ij}=A_{ij}\/k_\\text{out}(i)$ when $k_\\text{out}(j)>0$.\nFor \"dangling nodes\", that have no out-going edges (i.e. $k_\\text{out}(i)=0$),\n$S_{ij}=1\/N$, meaning that a walker on such a node has a uniform probability to jump to any other nodes in the network.\nOn non-dangling nodes, at each step, walkers have a probability $\\alpha<1$ to follow an out-going edge and a probability $1-\\alpha$ to teleport to any node in the network.\nThis modified random walks is now ergodic also on networks that are not strongly connected and converges to the probability density vector giving the \\emph{PageRank} centrality of node $i$ as\n\n\\begin{equation}\n    c_\\text{PR}(i) = \\alpha \\sum_j c_\\text{PR}(j) S_{ji}  + \\frac{1-\\alpha}{N}\n\\end{equation}\n\nwhich is equal to the normalized eigenvector corresponding to the largest positive eigenvalue of the so-called Google matrix $\\matr{G}=\\alpha \\matr{S}+\\frac{1-\\alpha}{N}\\matr{1}\\matr{1}^T$\\,\\cite{ermann2015google}.\nPageRank was famously developed for ranking websites for the search engine Google considering a \"random surfer model\" navigating through webpages by randomly clicking on hyperlinks\\,\\cite{page1999pagerank}.\nPageRank also found many applications in other aspects, such as ranking scientists and academic papers \\,\\cite{ding2009pagerank}, images \\,\\cite{jing2008visualrank} and proteins \\,\\cite{ivan2010web}.\nFigure \\ref{fig:pr_vs_katz} shows a comparison of the PageRank and Katz centralities on a directed network of hyperlinks between weblogs\\,\\cite{adamic2005political}.\n\nThe similarities between equations (\\ref{eq:eig_cen_it}) and (\\ref{eq:rw_update}) as well as equations (\\ref{eq:katz_it}) and (\\ref{eq:pg_it}) reveal that the degree and the PageRank centralities can be seen as similar to the eigenvector and Katz centralities, but for random walks instead of regular graph walks. In the case of random walk based centralities, one looks for eigenvectors of transition matrices instead of adjacency matrices.\n\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.8\\linewidth]{polblogs_pagerank_vs_katz}\n \\caption{A directed network of hyperlinks between weblogs on US politics, recorded in 2005 by Adamic and Glance\\,\\cite{adamic2005political}.\n PageRank centrality is shown on the left and Katz centrality is shown on the right. Lighter shades indicate a higher centrality.\n The Katz centrality has a tendency of being \"localized\" in regions with many close-by hubs.\n }\n \\label{fig:pr_vs_katz}\n\\end{figure}\n\n\n\\subsection{Centrality measures based on non-backtracking walks}\n\nIn order to address the issue of \"localization\" of the eigenvector centrality mentioned above, i.e. when most of the weight of centrality vector concentrates around one or a few nodes in the network, several authors have focused on using \\index{non-backtracking walks} instead of regular or random walks\\,\\cite{Martin2014,Arrigo2018,Arrigo2020}.\nNon-backtracking walks are graph walks where backtracking steps, i.e. steps where the walk comes back to an immediately preceding node, are not permitted.\nUsing this type of walks can sometimes solve the problem of localization as they decrease reflections between hubs during iterations of the centrality\\,\\cite{Martin2014}.\nNon-backtracking walks on unweighted networks are usually described using the $2M\\times2M$ matrix $\\matr{B}$ where rows and columns correspond to the directed edges of the network ($M=\\left|{E}\\right|$).\nIf the network is undirected, an equivalent directed network is considered by replacing each edge by a pair of directed edges in both directions.\nElement ($i\\rightarrow{}j,\\ell\\rightarrow{}h$) of $\\matr{B}$ is equal to one only if $j=\\ell$ and $i\\ne h$, i.e. there is a non-backtracking path of length 2 $i\\rightarrow{}j\\rightarrow{}h$ in the network. \nMore succinctly, \n\n\\begin{equation}\n    B_{i\\rightarrow{}j,\\ell\\rightarrow{}h} = \\delta_{j\\ell}(1-\\delta_{ih}),\n\\end{equation}\n\nwhere $\\delta_{ij}$ is the Kronecker delta.\nThe matrix $\\matr{B}$ is referred to as the \\emph{\\index{non-backtracking matrix}} or the \\emph{\\index{Hashimoto matrix}}\\,\\cite{hashimoto1989zeta}.\nPowers of $\\matr{B}$ enumerate non-backtracking walks similarly to powers of the adjacency enumerate walks\\,\\cite{Arrigo2020}.\nThe non-backtracking matrix is in general asymmetric with eigenvalues that are in general complex, however, \nsince its entries are non-negatives, by the \\index{Perron-Frobenius theorem}, its leading eigenvalue, $\\lambda$, is real and non-negative, and there exists a corresponding leading eigenvector, $\\matr{v}$, whose elements are also non-negative real numbers.\nIf $G$ is connected and not a tree, i.e. it has at least one cycle, $\\lambda$ is positive\\,\\cite{lin2019non}.\nThey satisfy the eigenvector equation\n\n\\begin{equation}\n    \\lambda\\matr{v}=\\matr{B}\\matr{v}\n\\end{equation}\n\nand the \\emph{\\index{non-backtracking eigenvector centrality}} of node $j$ is defined by \\,\\cite{Martin2014}\n\n\\begin{equation}\n    c_\\text{NBeig}(j) = \\sum_i A_{ij}v_{i\\rightarrow{}j}.\n\\end{equation}\n\nFinding the leading eigenvector of $\\matr{B}$ can however be computationally demanding for large graphs as the size of $\\matr{B}$ is usually much larger than the size of $\\matr{A}$.\nHowever, the computation can be made much faster by computing directly $c_\\text{NBeig}$ as the first $N$ elements of a $2N\\times2N$ matrix called the Ihara-Bass matrix\\,\\cite{krzakala2013spectral}.\n\nNon-backtracking walks have also been used to modify other centrality measures, such as the Katz\\,\\cite{Arrigo2018}, random walk\\,\\cite{lin2019non} or PageRank\\,\\cite{aleja2019non} centralities.\nResearch on centrality measures based on non-backtracking walks is very active with recent results showing that they can also suffer from localization issues on some networks\\,\\cite{barucca2016centrality,pastor2020localization}.\\\\\n\nNon-backtracking walks are also used in the problem of \\index{influence maximization}: finding the minimal set of nodes, the \\emph{influencers},\nwhich, if activated, would cause the spread of information to the whole network, or, if immunized, would prevent the diffusion\nof a large scale epidemic\\,\\cite{kempe2003maximizing}.\nInfluence maximization can be mapped to the problem of \\index{optimal percolation} of random networks\\,\\cite{morone2015influence} which consists in identifying the minimal set of nodes whose removal would dismember the network in many disconnected and non-extensive components. The fragmentation of the network is measured by the size of the largest connected component, called the giant component of the network.\n\nThe intuition behind the usage of non-backtracking walks in the optimal percolation problem comes from the fact that the giant component is held together by long paths and that powers of the non-backtracking matrix allows to quickly find them.\nThe removal of nodes is represented with the vector $\\matr{n} =(n_1,\\ldots,n_N)$ where $n_i =0$ if $i$ is removed (influencer) or $n_i =1$ otherwise.\nConsidering undirected locally tree-like random graphs,\na modified $2M\\times 2M$ non-backtracking matrix $\\matr{M}$ is then defined as \n$M_{i\\rightarrow{}j,\\ell\\rightarrow{}h} = n_\\ell B_{i\\rightarrow{}j,\\ell\\rightarrow{}h}$.\nGiven an initial arbitrary positive vector $\\matr{w}(0)$, repeated iterations with $\\matr{M}$,\n\n\\begin{equation}\nw(n)_{i\\rightarrow{}j}=\\sum_{kl}M_{i\\rightarrow{}i,k\\rightarrow{}l}w(n-1)_{k\\rightarrow{}l},\n\\end{equation}\n\nincrease the norm of the vector as the influence of nodes on non-backtracking walks of larger and larger length \nare included.\nThe growth rate of the vector's norm is determined by the value of the largest eigenvalue, $\\lambda(\\matr{n})$ of the modified non-backtracking matrix.\nAs the influencer nodes are removed, the value of $\\lambda(\\matr{n})$ decreases until the giant component is reduced to a tree (a graph without cycles) plus only one cycle when $\\lambda(\\matr{n})$=1.\nThe removal of supplementary nodes quickly destroys the giant component and $\\lambda(\\matr{n})$ falls to zero\\,\\cite{morone2015influence}.\nThe optimal influence problem can therefore be rephrased as finding the optimal configuration $\\matr{n}$ that minimizes the largest eigenvalue of $\\matr{M}$\\,\\cite{morone2015influence}, which consists in removing the nodes that are on the most non-backtracking walks in the network and that are keeping the giant component connected.\nThe \\emph{\\index{collective influence}}\\,\\cite{morone2015influence} of a node, defined as \n\n\\begin{equation}\n    \\text{CI}_\\ell(i) = (k_i-1)\\sum_{j\\in \\partial \\text{Ball}(i,\\ell)}(k_j - 1),\n\\end{equation}\nwhere $\\partial \\text{Ball}(i,\\ell)$ is the set of the nodes at a distance $\\ell$ from $i$, is a measure of which node to remove in order to produce the largest diminution of the largest eigenvalue of non-backtracking matrix.\nThe ranking of the nodes in term of collective influence is produced by removing the node with the largest CI value, recomputing the CI values of its neighbors and repeating the operation until the giant component of the network disappears.\nAn advantage of CI is that it gives a high rank to seemingly \"weak nodes\" with a small number of connections that are in fact surrounded by highly connected nodes and therefore on the path of many non-backtracking walks.\nThe problem of influence maximization in networks is actively researched using many different approaches (see, for example, \\,\\cite{braunstein2016network,mugisha2016identifying,zdeborova2016fast,radicchi2016beyond,pan2016influence,pei2020influencer}).\n\n\n\n\n\n\n\n\n\n\n\\section*{Conclusion and Future Directions}\n\nWe have seen that there is not a single measure of centrality in networks and that one should carefully choose an appropriate measure depending on the subject of investigation. \nIn practice, centrality measures should be chosen depending on whether the network is directed or not and whether, for example, one is looking for highly connected nodes, nodes with the most brokering power, nodes that share the most influence with each others or the minimal set of nodes that can spread information to the whole network. The differences in ranking obtained from different centrality measures should also be examined in order to better understand the signification of each ranking in a given context.\n\nThe research landscape on centrality measures for complex networks is much vaster than what has been presented in this short entry.\nResearch is very active with recent and future directions including, for example, the properties of centrality measures based on non-backtracking walks\\,\\cite{pastor2020localization},\ncentralities in signed networks (networks where weights can be positive or negative)\\,\\cite{liu2020simple} and \nmultilayer networks\\,\\cite{sola2013eigenvector,de2015ranking,wang2018new,wu2019tensor}.\nResearch is also active on the development of centralities for specific applications based on network dynamics not necessarily captured by network traversals. Examples include centralities for social networks based on game theoretical concepts\\,\\cite{Shapley1953,Gomez2003,michalak2013efficient},  centralities for detecting vulnerabilities in power-grids based on networks of oscillators\\,\\cite{gutierrez2013vulnerability,tyloo2018robustness,Tyloo2019} or centralities for identifying key genes in gene regulatory networks based on the propagation of biological signals\\,\\cite{zotenko2008hubs,missiuro2009information,kim2011identifying,cowen2017network}.\n\nResearch is also ongoing about the development of centralities in network models that extend the concept of network as an ensemble of static pairwise relations, namely, centralities in temporal networks, i.e. time-evolving networks, \\,\\cite{nicosia2013graph,praprotnik2015spectral,taylor2017eigenvector,flores2018eigenvector,lv2019pagerank} and in hypergraphs or simplicial complexes, i.e. generalizations of graphs where an edge can join more than two nodes\\,\\cite{bonacich2004hyper,estrada2018centralities,benson2019three,aksoy2020hypernetwork,serrano2020centrality}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Renormalization constants for the leptonic vector-boson decay at \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)}}\n\\label{app:ff}\n\nIn this appendix we provide the expressions for the finite\n$\\mathcal{O}(\\alphas\\alpha)$ counterterms to the leptonic vertices of\nthe $\\PW$ and $\\PZ$ bosons in the on-shell renormalization\nscheme following the conventions of Ref.~\\cite{Denner:1991kt}.\n\n\n\\subsection{Vector-boson self-energies}\n\nThe transverse and longitudial parts of the vector-boson self-energy,\n$\\Sigma^{V_aV_b}_\\rT$ and $\\Sigma^{V_aV_b}_\\rL$, are defined by the decomposition of the\nirreducible two-point function $\\Gamma^{V_aV_b}_{\\mu\\nu}$ as\n\\begin{equation}\n  \\Gamma^{V_aV_b}_{\\mu\\nu}(q)=-\\ri \\delta_{ab} g_{\\mu\\nu} (q^2-M_{V_a}^2)-\n\\ri  \\left(g_{\\mu\\nu}-\\frac{q_\\mu q_\\nu}{q^2}\\right)\\Sigma^{V_aV_b}_\\rT(q^2)-\\ri\n  \\frac{q_\\mu q_\\nu}{q^2}\\Sigma^{V_aV_b}_\\rL(q^2),\n\\end{equation}\nwhere $q$ is the momentum carried by the vector bosons $V_{a,b}$.\nThe $\\mathcal{O}(\\alphas\\alpha)$ corrections to the vector-boson\nself-energies are given in Ref.~\\cite{Djouadi:1993ss} in  terms of\nscalar functions $\\Pi^{V,A}_\\rT$.\\footnote{%\nNote that in the expression given in Eq.~(5.4) of Ref.~\\cite{Djouadi:1993ss} for\n$\\Pi^{V,A}_\\rT$ in the special case of one vanishing fermion mass, the sign of the term \n$1\/3(2+\\alpha)(\\alpha-1)G(x)$ should be reversed in agreement with Ref.~\\cite{Djouadi:1987di}. We thank Paolo Gambino\nfor communication on this point.\n} \nTreating all quarks apart from the top quark as massless, the transverse parts of the vector-boson self-energies can be expressed as follows,\n\\begin{align}\n\\Sigma_\\rT^{WW, (\\alphas\\alpha)}(s) & =\\frac{\\alphas\\alpha}{8\\pi^2 \\sw^2}\n   \\left[2({\\Pi}^V_\\rT(s,0,0)+{\\Pi}^A_\\rT(s,0,0))\n   +   ({\\Pi}^V_\\rT(s,m_\\Pqt^2,0)+{\\Pi}^A_\\rT(s,m_\\Pqt^2,0))\n   \\right], \\nonumber\\\\\n\\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(s)&\n=\\frac{\\alphas\\alpha}{4\\pi^2 \\sw^2 \\cw^2}\n   \\left[\\left(\\frac{44}{9}\\sw^4-\\frac{14}{3}\\sw^2+\\frac{5}{4} \\right)\n     {\\Pi}^V_\\rT(s,0,0)+\\tfrac{5}{4}{\\Pi}^A_\\rT(s,0,0) \\right. \\nonumber\\\\\n&\\quad\\left.   +  \\left(\\frac{1}{2}-\\frac{4}{3} \\sw^2\\right)^2{\\Pi}^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\n+\\tfrac{1}{4}{\\Pi}^A_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\n   \\right], \\nonumber \\\\\n\\Sigma_\\rT^{AA,(\\alphas\\alpha)}(s)&\n=\\frac{\\alphas\\alpha}{\\pi^2}\n\\left[\\frac{11}{9}{\\Pi}^V_\\rT(s,0,0) + \\frac{4}{9} {\\Pi}^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\\right],\n\\nonumber \\\\\n\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(s)\n&=-\\frac{\\alphas\\alpha}{2\\pi^2\\sw\\cw}\\left[\n\\left(\\frac{7}{6}-\\frac{22}{9}\\sw^2\\right)\\Pi^V_\\rT(s,0,0)\n+\\left(\\frac{1}{3}-\\frac{8}{9} \\sw^2\\right)\\Pi^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\\right] ,\n\\end{align}\nwhere the explicit expressions for the scalar functions $\\Pi^{V,A}_\\rT$\ngiven in Ref.~\\cite{Djouadi:1993ss} include the colour factor $N_c=3$.\nThe top-quark mass renormalization is performed in the on-shell\nscheme.\n\n\n\n\\subsection{Vertex counterterms}\nAt $\\order{\\alphas\\alpha}$, the vertex counterterms for the leptonic\nvector-boson decay receive only contributions from the vector-boson\nself-energies, \n\\begin{align}\n  \\delta_{\\PW\\Pl_1\\Pal_2}^{\\mathrm{ct},(\\alphas\\alpha)}&=\n  \\delta Z_e^{(\\alphas\\alpha)} -\\frac{\\delta \\sw^{(\\alphas\\alpha)}}{\\sw}\n  +\\frac{1}{2}\\delta Z_W^{(\\alphas\\alpha)},\\nonumber\\\\\n  \\delta^{\\mathrm{ct},\\tau_\\ell,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal} &=\n  \\frac{\\delta g_\\Pl^{\\tau_\\ell,(\\alphas\\alpha)}}{g_\\Pl^{\\tau_\\ell}}  \n  +\\frac{1}{2}\\delta Z^{(\\alphas\\alpha)}_{ZZ} - \\frac{Q_\\Pl}{2g_\\Pl^{\\tau_\\ell}}\n\\delta Z_{AZ}^{(\\alphas\\alpha)},\n \\end{align}\nwhere $\\tau_\\ell=\\pm$ denotes the lepton chirality.\nThe coupling constants entering the $\\PZ$-boson vertex are given by\n\\begin{equation}\n  \\begin{aligned}\n    g_\\Pl^+&=-\\frac{\\sw}{\\cw}Q_\\Pl, & \n    \\delta g_\\Pl^{+}&=-\\frac{\\sw}{\\cw}Q_\\Pl\n    \\left[\\delta Z_e\n        +\\frac{1}{\\cw^2}\\frac{\\delta \\sw}{\\sw}\\right],  \\\\\n    g_\\Pl^-&=\\frac{1}{\\sw\\cw}\\left(I^3_{\\rw,\\Pl}-\\sw^2 Q_\\Pl\\right),&\n    \\delta g_\\Pl^{-}&=\\frac{I^3_{\\rw,\\Pl}}{\\sw\\cw}\n    \\left[\\delta Z_e\n        +\\frac{\\sw^2-\\cw^2}{\\cw^2}\\frac{\\delta \\sw}{\\sw} \\right]+ \n\\delta g_\\Pl^{+},\n  \\end{aligned}\n\\end{equation}\nwhere $I^3_{\\rw,\\Pl}=-\\frac{1}{2}$ is the third component of the weak\nisospin of the charged lepton $\\Pl$.\n\nThe vector-boson wave-function renormalization constants can be expressed in terms of the self-energies as follows,\n\\begin{align}\n  \\delta Z_{AA}&=-\\left.\\frac{\\partial\\Sigma^{AA}_\\rT(k^2)}{\\partial k^2}\n  \\right|_{k^2=0}, &&\n  \\nonumber\\\\\n  \\delta Z_{W}&=-\\left.\n\\Re\\frac{\\partial \\Sigma_\\rT^{WW}(s)}{\\partial s}\\right|_{s=\\MW^2}, & \n \\delta Z_{ZZ}&=-\\left. \\Re \\frac{\\partial \\Sigma_\\rT^{ZZ}(s)}{\\partial s}\n   \\right|_{s=\\MZ^2},\n\\nonumber \\\\\n \\delta Z_{ZA}&=2\\frac{\\Sigma_\\rT^{AZ}(0)}{\\MZ^2}, &\n \\delta Z_{AZ}&=-2\\Re\\frac{\\Sigma_\\rT^{AZ}(\\MZ^2)}{\\MZ^2},\n\\end{align}\nwhere the $\\order{\\alphas\\alpha}$ contribution to $ \\delta Z_{ZA}$ vanishes~\\cite{Djouadi:1993ss}.\nThe charge-renormalization constant $\\delta Z_e$ in the  $\\alpha(0)$ input-parameter scheme\n is given by\n\\begin{align}\n\\label{eq:dZe0}\n  \\delta Z_e^{\\alpha(0)}\n&=-\\frac{1}{2}\\delta Z_{AA}-\\frac{\\sw}{\\cw}\\frac{1}{2}\\delta Z_{ZA}.\n\\end{align}\nThe transition to the  $G_\\mu$-scheme is performed according to\nEq.~\\eqref{eq:dZeGmu}. The renormalization constant for the weak mixing angle is given by\n\\begin{equation}\n  \\frac{\\delta \\sw}{\\sw}\n=-\\frac{\\cw^2}{2\\sw^2}\n\\left(\\frac{\\Re\\Sigma_\\rT^{WW}(\\MW^2)}{\\MW^2}-\\frac{\\Re\\Sigma_\\rT^{ZZ}(\\MZ^2)}{\\MZ^2}\\right).\n\\end{equation}\n\nThe final expressions of the  $\\order{\\alphas\\alpha}$ counterterms in the $G_\\mu$-scheme in terms of self-energies are then given by\n\\begin{align}\n\\delta_{\\PW\\Pl_1\\Pal_2}^{\\mathrm{ct},(\\alphas\\alpha)}=&\n  -\\frac{1}{2}\\left(\n    \\Re\\frac{\\partial \\Sigma_\\rT^{WW,(\\alphas\\alpha)}(s)}{\\partial s}|_{s=\\MW^2} \n    +\\frac{\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(0)\n      -\\Re \\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(\\MW^2)}{\\MW^2} \\right),\\nonumber \\\\\n   \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal} =&\n-\\frac{1}{2}\n    \\frac{\\Re\\partial \\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(s)}{\\partial s}|_{s=\\MZ^2}\n    -\\left(1-\\frac{2}{\\sw^2}\\right)\n    \\frac{\\Re\\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(\\MZ^2)}{2\\MZ^2}\n\\nonumber\\\\\n&\n-\\frac{\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(0)}{2\\MW^2}\n+ \\left(1-\\frac{1}{\\sw^2}\\right)\n  \\frac{\\Re\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(\\MW^2)}{\\MW^2}\n  -\\frac{\\cw}{\\sw}\\frac{\\Re\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(\\MZ^2)}{\\MZ^2},\\nonumber\\\\\n  \\delta^{\\mathrm{ct},-,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}=& \\,\n  \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}\n+\\frac{I^3_{\\rw,f}}{\\sw\\cw g_{\\ell}^-}\n\\left[-\\frac{2\\delta \\sw^{(\\alphas\\alpha)}}{\\sw}\n+\\frac{\\cw}{\\sw}\\frac{\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(\\MZ^2)}{\\MZ^2}\\right].\n\\label{eq:dZVNNLO}\n\\end{align}\n\n\n\\section{Explicit form of the IR-safe contributions to the factorizable initial--final corrections}\n\\label{app:ir-safe}\n\nIn this section we provide the explicit expressions for each of the contributions to the factorizable initial--final corrections in our master formula~\\eqref{eq:IF:final-master}.\n\nThe double-virtual corrections~\\eqref{eq:IF:master-VV} and the (virtual\nQCD)$\\times$(real photonic) corrections~\\eqref{eq:IF:master-VR}\nare obtained by dressing the virtual part of the NLO QCD corrections with the factorizable final-state EW corrections,\n\\begin{subequations}\n\\label{eq:IF:master-VX-qq}\n\\begin{align}\n \n  \\label{eq:IF:master-VV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} \n  &=\n  \\int_2 \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\;\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0\\;\n  \\diptimes\\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \n  \\\\\n \n  \\label{eq:IF:master-VR-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n  &=\n  \\iint\\limits_{2+\\Pgg}\n  \\biggl\\{ \n  \\rd\\sigma_\\dec^\\Rew \n  -4\\pi\\alpha \\biggl[Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\n  \\biggr] \n  \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n \n  \\biggr\\} ,\n\\end{align}\n\\end{subequations}\nwhere the integrated counterpart of the QED dipoles $\\Iew$ is defined in Eq.~\\eqref{eq:nlo:dec:master:Iew}.\nHere $\\lips[IJ]{2}$ denotes the set of momenta of the two-particle phase\nspace after applying the momentum mapping associated to  the dipole $\\gsub{IJ}$\nor $\\dsub{IJ}$. \nAs in the NLO QCD corrections, only the quark--anti-quark induced\nchannel receives a non-vanishing contribution.\n\n\nThe IR-regularized (real QCD)$\\times$(virtual EW) corrections~\\eqref{eq:IF:master-RV} and the double-real corrections~\\eqref{eq:IF:master-RR} are obtained by dressing the real-emission part of the NLO QCD cross sections with the final-state factorizable corrections.\nNote the two-fold application of the dipole subtraction formalism in case of of the double-real corrections.\nFor the quark--anti-quark channel, the explicit expressions are\n\\begin{subequations}\n\\label{eq:IF:master-RX-qq}\n\\begin{align} \n \n  \\label{eq:IF:master-RV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  &=\n  \\int_3 \\;\\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\nonumber\\\\\\MoveEqLeft\\times\\biggl\\{ \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}\n \n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}  \n  \\biggr\\}, \n  \\\\\n \n  \\label{eq:IF:master-RR-qq} \n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew} \n  &=\n  \\iint\\limits_{3+\\Pgg}\\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1V]{3}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1\\Pal_2]{3}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\nonumber\\\\&\\quad\n  +4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Paq_a\\Pg)\\Pq_b}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pq_b\\Pg)\\Paq_a}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Paq_a\\Pg)\\Pq_b}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pq_b\\Pg)\\Paq_a}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\} .\n\\end{align}  \n\\end{subequations}\nHere the phase-space kinematics obtained by the successive application\nof both QCD and EW dipole mappings  is denoted by $\\dtilde{\\Phi}^{(ab)c}_{2,IJ}$.  \nA detailed discussion of the behaviour of the individual terms of the double-real corrections in the various singular regions is given in Sect.~\\ref{sec:IF:master}.\nThe corresponding expressions for the quark--gluon-initiated subprocesses read\n\\begin{subequations}\n\\label{eq:IF:master-RX-gq}\n\\begin{align}\n \n  \\label{eq:IF:master-RV-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  &=\n  \\int_3 \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\biggl\\{ \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\biggr\\}, \n  \\\\\n \n  \\label{eq:IF:master-RR-gq} \n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew} \n  &=\n  \\iint\\limits_{3+\\Pgg}\\biggl\\{\n  \\rd\\sigma_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1V]{3}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1\\Pal_2]{3}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\nonumber\\\\&\\quad\n  +4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pg\\Pq_a)\\Pq_b}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pg\\Pq_a)\\Pq_b}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\} .\n\\end{align}\n\\end{subequations}\nThe contribution to the $\\Pg\\bar q_a$ channel is given in an analogous manner,\nbut is not spelled out explicitly.\n\nThe collinear counterterms with additional virtual EW~\\eqref{eq:IF:master-CV} \nand real-photonic~\\eqref{eq:IF:master-CR} corrections \nare constructed from the corresponding term of the NLO QCD corrections by dressing \nthem with the respective factorizable final-state corrections,\n\\begin{subequations}\n\\label{eq:IF:master-CX-qq}\n\\begin{align}\n \n  \\label{eq:IF:master-CV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_a,p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a} \n  \\nonumber\\\\&\\quad\n  +\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (p_a,xp_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b} ,\n  \\\\\n \n  \\label{eq:IF:master-CR-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes \\Rew}\n  &=\n  \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(xp_a,p_b)\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(xp_a,p_b)\\Bigr)\n  \\nonumber\\\\&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(xp_a,p_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a}  \n  \\nonumber\\\\&\n  + \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(p_a,xp_b)\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(p_a,xp_b)\\Bigr)\n  \\nonumber\\\\&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(p_a,xp_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b}  \n  .\n\\end{align}\n\\end{subequations}\nThe corresponding formulae for the gluon--quark channel read\n\\begin{subequations}\n\\label{eq:IF:master-CX-gq}\n\\begin{align}\n \n  \\label{eq:IF:master-CV-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_{\\Pg},p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a}  ,\n  \\\\\n \n  \\label{eq:IF:master-CR-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes \\Rew}\n  &=\n  \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(xp_{\\Pg},p_b)\n  \\nonumber\\\\*&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(xp_{\\Pg},p_b)\\Bigr)\n  \\nonumber\\\\*&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(xp_{\\Pg},p_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a}  \n\\end{align}\n\\end{subequations}\nand analogous expressions for the $\\Pg\\bar q_a$ channel.\nHere we have made the dependence on the momenta of the incoming partons explicit in order to indicate which particle undergoes a collinear splitting with the momentum fraction given by the convolution variable $x$.\n\n\n\n\\section{Non-collinear-safe observables}\n\\label{app:non-collinear-safe}\nIn order to treat non-collinear-safe observables with respect to the final-state leptons \n$i=\\Pl_1,\\Pal_2$ following Ref.~\\cite{Dittmaier:2008md}, \nthe $n$-particle kinematics in the phase space of the subtraction function\nis treated as an $(n+1)$-particle event with a collinear lepton--photon pair,\nwhere the momentum shared between the two collinear particles is controlled\nby the variable $z_{iJ}$,\n\n\\begin{subequations}\n\\label{eq:thetacut:kabelschacht:nnlo}\n\\begin{align}\n \n  \\rd\\sigma^0_{\\PA}\\Bigl(\\lips[iJ]{2}\\Bigr) \n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^0_{\\PA}\\Bigl(\\lips[iJ]{2}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\lips[iJ]{2} \\;\\Big\\vert\n  \\;k_{i}=z_{iJ}\\,\\tilde{k}_{i}, \n  \\;k=(1-z_{iJ})\\,\\tilde{k}_{i} \\Bigr) ,\n  \\\\\n \n  \\rd\\sigma^\\Rs_{\\PA}\\Bigl(\\lips[iJ]{3}\\Bigr) \n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^\\Rs_{\\PA}\\Bigl(\\lips[iJ]{3}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\lips[iJ]{3} \\;\\Big\\vert\n  \\;k_{i}=z_{iJ}\\,\\tilde{k}_{i}, \n  \\;k=(1-z_{iJ})\\,\\tilde{k}_{i} \\Bigr) ,\n  \\\\\n \n  \\rd\\sigma^0_{\\PA}\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ}\\Bigr)\n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^0_{\\PA}\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ} \\;\\Big\\vert\n  \\;\\tilde{k}_{i}=z_{iJ}\\,\\dtilde{k}_{i}, \n  \\;\\tilde{k}=(1-z_{iJ})\\,\\dtilde{k}_{i} \\Bigr) ,\n\\end{align}\n\\end{subequations}\nwhere we have made explicit the cut function for the computation of observables in the notation.\nThis modification induces additional convolution terms over the distribution $[\\cIbarew(z)]_+$ with\n\\begin{align}\n  \\label{eq:nlo:dec:master:cIbarew}\n  \\cIbarew(z)\n  &=\n  \\frac{\\alpha}{2\\pi}\\, Q_{\\Pl_1}\\biggl[\n  (Q_{\\Pl_1}-Q_{\\Pl_2})\\; \\cDbarsub{\\Pl_1 V}(z)\n  +Q_{\\Pl_2}\\; \\cGbarsub{\\Pl_1\\Pal_2}(z)\n  \\biggr]\n  \\nonumber\\\\&\\quad\\times\n  \\Theta_\\cut\\left(\\lips[]{2} \\;\\Big\\vert\n  \\;k_{\\Pl_1}=z\\,\\tilde{k}_{\\Pl_1}, \n  \\;k=(1-z)\\,\\tilde{k}_{\\Pl_1} \\right)\n \n  +(\\Pl_1\\leftrightarrow\\Pal_2) ,\n\\end{align}\nwhich we indicate by the label ``$\\Rbarew$''.\nThe contribution with virtual QCD corrections is given by\n\\begin{align}\n  \\label{eq:IF:master-VRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rbarew} \n  &=\n  \\int_0^1\\rd z \\int_2 \\left[\\cIbarew(z)\\right]_+ \n  \\;\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0\\;\n  \\diptimes\\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \n\\end{align}\nand the real-emission corrections for the quark--anti-quark and gluon--quark induced contributions read\n\\begin{align} \n  \\label{eq:IF:master-RRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rbarew}\n  &=\n  \\int_0^1\\rd z \\int_3 \\left[\\cIbarew(z)\\right]_+ \n  \\biggl\\{ \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\quad\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}  \n  \\biggr\\} ,\n\\end{align}\n\\begin{align}\n  \\label{eq:IF:master-RRbar-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rbarew}\n  &=\n  \\int_0^1\\rd z \\int_3 \\left[\\cIbarew(z)\\right]_+ \n  \\biggl\\{ \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\biggr\\} \n\\end{align}\nand analogous terms for the $\\Pg\\bar q_a$ channel.\nThe $\\CSK$ and $\\CSP$ operators contain an additional convolution over the momentum \nfractions~$x$ of the incoming partons and can be written in the following form,\n\\begin{align}\n  \\label{eq:IF:master-CRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Rbarew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2  \\left[\\cIbarew(z)\\right]_+\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_a,p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a} \n  \\nonumber\\\\&\\quad\n  +\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2 \\left[\\cIbarew(z)\\right]_+\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (p_a,xp_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b} ,\n\\end{align}\n\\begin{align}\n  \\label{eq:IF:master-CRbar-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Rbarew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2 \\left[\\cIbarew(z)\\right]_+ \n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_{\\Pg},p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a} \n\\end{align}\nand analogous terms for the $\\Pg\\bar q_a$ channel.\nNote that the $\\CSK$ and $\\CSP$ operators, in general, contain plus distributions with respect to the variable $x$, and the above equations need to be properly evaluated in combination with the plus distribution $[\\cIbarew(z)]_+$ that acts on the integration variable~$z$.\n\n\n\n\n\n\n\n\\section{PHOTOS settings}\n\\label{app:photos}\n\nThe results using the PHOTOS parton shower shown in Figs.~\\ref{fig:distNNLO-IF-FSR-Wp} and \\ref{fig:distNNLO-IF-FSR-Z} of Sect.~\\ref{sec:FSR} were obtained with version 2.15 of the program and the following options:\n\\begin{tabbing}\n  \\qquad\\=\\hspace{.35\\textwidth}\\=\\hspace{.35\\textwidth}\\=\\kill\n  \\> \\texttt{ISEC=.FALSE.}, \n  \\> \\texttt{ITRE=.FALSE.}, \n  \\> \\texttt{IEXP=.FALSE.}, \\\\\n  \\> \\texttt{IFTOP=.FALSE.}, \n  \\> \\texttt{XPHCUT=0.01D0} \\; (default value).\n\\end{tabbing}\nThese settings restrict the parton shower to \\emph{at most one} additional photon emission in order to simulate the impact of $\\order{\\alpha}$ corrections. \nFurther settings which differ for the charged-current and neutral-current processes are as follows:\n\\begin{tabbing}\n  \\qquad\\=\\hspace{.35\\textwidth}\\=\\hspace{.35\\textwidth}\\=\\kill\n  \\> \\textbf{\\boldmath{\\large \\PWpm production:}} \\> \\textbf{\\boldmath{\\large \\PZ production:}} \\\\\n  \\> \\texttt{IFW=.TRUE.},              \\> \\texttt{IFW=.FALSE.}, \\\\\n  \\> \\texttt{INTERF=.FALSE.},          \\> \\texttt{INTERF=.TRUE.}, \\\\\n  \\> \\texttt{ALPHA=}\\;$\\alpha_{G_\\mu}$, \\> \\texttt{ALPHA=}\\;$\\alpha(0)$.\n\\end{tabbing}\nNote that the electromagnetic coupling constant $\\alpha$ is adjusted to the respective value used in our calculation of the initial--final corrections as described in Sect.~\\ref{sec:IF:numerics}.\n\n\n\\section*{Acknowledgement}\nThis project is supported by the German Research Foundation (DFG) via grant DI 784\/2-1 and the German Federal Ministry for Education and Research (BMBF). \nMoreover, A.H.\\ is supported via the ERC Advanced Grant MC@NNLO (340983).\nC.S.\\ is supported by the Heisenberg Programme of the DFG.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nThe Drell--Yan-like $\\PW$- and $\\PZ$-boson production processes are among the most precise probes of the Standard Model and do \nnot only serve as key benchmark or ``standard candle'' processes, \nbut further allow for precision measurements of the $\\PW$-boson mass and the effective weak mixing angle.\nThis task of precision physics requires a further increase in the accuracy of the theoretical predictions, where the mixed QCD--electroweak corrections of \\order{\\alphas\\alpha} currently represent the largest unknown component of radiative corrections in terms of fixed-order predictions.\n\nIn our previous paper~\\cite{Dittmaier:2014qza} we have established a framework for evaluating the \\order{\\alphas\\alpha} corrections to Drell--Yan processes in the resonance region using the pole approximation and presented the calculation of the so-called non-factorizable corrections.  They turned out to be phenomenologically negligible, so that the \\order{\\alphas\\alpha} corrections almost entirely result from \nfactorizable corrections that can be separately attributed to production and decay of the $\\PW$\/$\\PZ$ boson (up to spin correlations).\n\n\nIn this paper we have presented the calculation of the \nso-called factorizable corrections of ``initial--final'' and ``final--final''\ntypes.  The latter were calculated in Sect.~\\ref{sec:calc-fact-nnlo-ff} and\nonly comprise finite  counterterm contributions which were found to be numerically very small ($< 0.1\\%$) and therefore can be safely neglected for all phenomenological purposes.  The former, on the other hand, combine large QCD corrections to the production with large EW corrections to the decay subprocesses and are expected to be the dominant contribution of the $\\order{\\alphas\\alpha}$ corrections.  Their calculation has been presented in Sect.~\\ref{sec:calc-fact-nnlo-if}, and we have shown numerical results in Sect.~\\ref{sec:if-results} for the most important observables for the $\\PW$-boson mass measurement: the transverse-mass and lepton-transverse-momentum distributions for \\PW production.  The results for the neutral-current process comprise the invariant-mass and the lepton-transverse-momentum distributions.  In the framework of the pole approximation, the only missing \\order{\\alphas\\alpha} corrections are now those of ``initial--initial'' type. Based on the results of the NLO electroweak calculation, these are expected to be numerically small.\n\nWe have used our results for the dominant $\\order{\\alphas\\alpha}$ corrections to test the validity of simpler approximate combinations of EW and QCD corrections: \nFirstly, we use a naive product ansatz multiplying the NLO QCD and EW\ncorrection\n factors, and secondly, we approximate the $\\order{\\alphas\\alpha}$ contribution by combining leading-logarithmic approximations of QED final-state radiation with the NLO QCD corrections.\n\nWe have demonstrated  in Sect.~\\ref{sec:if-results} that naive products fail to capture the factorizable initial--final corrections in distributions such as in\nthe transverse momentum of the lepton, which are sensitive to QCD\ninitial-state radiation and therefore require  a correct treatment of the double-real-emission part of the NNLO corrections. Naive products also fail to capture observables that are strongly affected by a\n redistribution of events due to final-state real-emission corrections,\n such as the invariant-mass distribution of the neutral-current process.  On the other hand, if an observable is less affected by such a redistribution of events or is only affected by it in the vicinity of the resonance, such as the transverse-mass distribution of the charged-current process, the naive products are able to reproduce the factorizable initial--final corrections to a large extent. \n\n In Sect.~\\ref{sec:FSR} we have investigated to which extent the factorizable initial--final corrections calculated in this paper  can be approximated by a combination of the NLO QCD corrections and a collinear approximation of  real-photon emission through a  QED structure function approach\nor a  QED parton shower such as PHOTOS.  For the invariant-mass distribution in $\\PZ$-boson production we observe a significant improvement in the agreement compared to the naive product ansatz,\nsince\nboth PHOTOS and the QED structure functions model  the redistribution of events due to final-state radiation, which\n is responsible for the bulk of the corrections in this observable. \nOur results can furthermore be used to validate Monte Carlo event generators where  $\\order{\\alphas\\alpha}$ corrections are approximated by  a combination of NLO matrix elements and parton showers. \n\nFinally, \nin Section~\\ref{sec:mw} we have illustrated the phenomenological impact of the $\\order{\\alphas\\alpha}$ corrections by estimating the mass shift induced by the \n factorizable initial--final corrections as  $\\approx-14~\\MeV$  for the case\n of bare muons  and $\\approx-4~\\MeV$ \nfor dressed leptons.\nThese corrections therefore have to be properly taken into account in the $\\PW$-boson mass measurements at the LHC, which aim at a precision of about $10~\\MeV$.\nIt will be interesting to investigate the impact of the $\\order{\\alphas\\alpha}$ corrections on the measurement of the effective weak mixing angle as well in the future.\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe class of Drell--Yan-like processes \nis one of the most prominent types of particle reactions\nat hadron colliders and describes the production of a lepton pair through an intermediate gauge-boson decay,\n\\begin{equation*}\n  \\Pp\\Pp \/ \\Pp\\Pap \\;\\to\\; V \\;\\to\\; \\Pl_1\\Pal_2 \\;+\\; X .\n\\end{equation*}\nDepending on the electric charge of the colour-neutral gauge boson $V$, the process can be further classified into the neutral-current ($V=\\PZ\/\\Pgg$) and the charged-current ($V=\\PWpm$) processes. \nThe large production rate in combination with the clean experimental signature of the leptonic vector-boson decay allows this process to be measured with great precision.\nMoreover, the Dell--Yan-like production of $\\PW$ or $\\PZ$ bosons is one of the theoretically best understood and most precisely predicted processes.\nAs a consequence, electroweak (EW) \ngauge-boson production is among the most important \n``standard-candle'' processes at the LHC (see, e.g.\\ Refs.~\\cite{Abdullin:2006aa,Gerber:2007xk}).\nIts cross section can be used as a luminosity monitor, and the measurement of\nthe mass and width of the $\\PZ$ boson represents a powerful tool for detector\ncalibration.\nFurthermore, the \\PW charge asymmetry and the rapidity distribution of the $\\PZ$ boson deliver important constraints in the fit of the \nparton distribution functions (PDFs)~\\cite{Boonekamp:2009yd}, which represent crucial ingredients for almost all predictions at the LHC.\n\nOf particular relevance for precision tests of the Standard Model \nis the potential of the Drell--Yan process at the LHC  for high-precision measurements in the resonance regions, where the effective weak mixing angle, quantified by $\\sin^2\\theta_\\text{eff}^{\\Pl}$,\nmight be extracted from data with LEP precision~\\cite{Haywood:1999qg}.\nThe $\\PW$-boson mass can be determined from a fit to the distributions of the lepton transverse momentum ($p_{\\rT,\\Pl}$) and the transverse mass of the lepton pair ($M_{\\rT,\\Pl\\Pgn}$) which exhibit Jacobian peaks around $\\MW$ and $\\MW\/2$, respectively, and allow for a precise extraction of the mass\nwith a sensitivity below \\SI{10}{\\MeV}~\\cite{Besson:2008zs,Baak:2013fwa} provided that PDF uncertainties can be reduced\\cite{Rojo:2013nia,Quackenbush:2015yra,Bozzi:2015hha,Bozzi:2015zja}.\n\nTo fully exploit the potential of the extraordinary experimental precision that is achievable for the Drell--Yan process, it is necessary to have theoretical predictions that match or even surpass the \nexpected accuracy.\nThe current state of the art includes\nQCD corrections at next-to-next-to-leading-order (NNLO)\naccuracy~\\cite{Hamberg:1990np,Harlander:2002wh,Anastasiou:2003ds,Melnikov:2006di,Melnikov:2006kv,Catani:2009sm,Gavin:2010az,Gavin:2012sy}\nsupplemented by leading higher-order soft-gluon\neffects~\\cite{Moch:2005ba,Laenen:2005uz,Idilbi:2005ky,Ravindran:2007sv}\n and soft-gluon resummation for  small transverse momenta~\\cite{Balazs:1997xd,Landry:2002ix,Bozzi:2010xn,Mantry:2010mk,Becher:2011xn,Guzzi:2013aja,Catani:2015vma}.\nFor event generation, next-to-leading-order (NLO) \ncalculations  have been matched to parton showers~\\cite{Frixione:2006gn,Alioli:2008gx,Hamilton:2008pd}, with a recent effort to include NNLO corrections in a parton-shower framework~\\cite{Hoeche:2014aia,Karlberg:2014qua,Alioli:2015toa}.\nConcerning EW effects, the \nNLO corrections~\\cite{Baur:1997wa,Zykunov:2001mn,Baur:2001ze,Dittmaier:2001ay,Baur:2004ig,Arbuzov:2005dd,CarloniCalame:2006zq,Zykunov:2005tc,CarloniCalame:2007cd,Arbuzov:2007db,Brensing:2007qm,Dittmaier:2009cr}\nas well as leading higher-order effects from multiple photon emission and\nuniversal weak effects~\\cite{Baur:1999hm,Placzek:2003zg,CarloniCalame:2003ux,Brensing:2007qm,Dittmaier:2009cr} are known. The sensitivity to the photon PDF through \nphoton-induced production channels has been studied in \nRefs.~\\cite{Arbuzov:2007kp,CarloniCalame:2007cd,Dittmaier:2009cr,Boughezal:2013cwa}.\n \nIn addition to the N$^3$LO QCD corrections, the next frontier in theoretical fixed-order computations is given by the calculation of the mixed QCD--EW corrections of \\order{\\alphas\\alpha}~\\cite{Andersen:2014efa}.\nThese corrections can affect observables relevant for the \\MW determination at the percent level~\\cite{Balossini:2009sa} and therefore must be under theoretical control.\nUp to now, QCD and EW corrections have been combined in various\napproximations~\\cite{Cao:2004yy,Richardson:2010gz,Bernaciak:2012hj,Barze:2012tt,Li:2012wna,Barze:2013yca}.  \nHowever, a full NNLO calculation at \\order{\\alphas\\alpha} is necessary \nfor a proper combination of  QCD and EW corrections without ambiguities.\nHere some partial results for two-loop amplitudes~\\cite{Kotikov:2007vr,Kilgore:2011pa,Bonciani:2011zz} as well as the  full \\order{\\alphas\\alpha} corrections to the \\PW and \\PZ decay widths~\\cite{Czarnecki:1996ei,Kara:2013dua} are known. \nA complete calculation of the \\order{\\alphas\\alpha} corrections requires to combine the double-virtual corrections with \n the \\order{\\alpha} EW corrections to $\\PW\/\\PZ+\\jet$ production~\\cite{Kuhn:2005az,Kuhn:2007cv,Hollik:2007sq,Denner:2009gj,Denner:2011vu,Denner:2012ts,Hollik:2015pja}, the \\order{\\alphas} QCD corrections to $\\PW\/\\PZ+\\Pgg$ \nproduction~\\cite{Smith:1989xz,Ohnemus:1992jn,Ohnemus:1994qp,Dixon:1998py,Campbell:1999ah,DeFlorian:2000sg,Hollik:2007sq,Campbell:2011bn,Denner:2014bna,Denner:2015fca}, and the double-real corrections using a method to regularize \ninfrared (IR) singularities.\n\n\n In a previous paper~\\cite{Dittmaier:2014qza}, we have initiated the\n calculation of the \\order{\\alphas\\alpha} corrections to Drell--Yan\n processes in the resonance region via the so-called \\emph{pole\n   approximation}~(PA)~\\cite{Stuart:1991xk}, which has been\n successfully applied to the EW corrections to \n $\\PW$~production~\\cite{Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\n and $\\PZ$~production~\\cite{Dittmaier:2014qza} at NLO.\n It is based on a systematic expansion of the cross section about the\n resonance pole and is suitable for theoretical predictions in the\n vicinity of the gauge-boson resonance, where the higher precision is\n especially relevant.  The PA splits the corrections into distinct\n well-defined subsets, which can be calculated separately. This allows\n to assess the numerical impact of different classes of corrections\n and to identify the dominant contributions.  More precisely, the\n contributions can be classified into two types: the factorizable and\n the non-factorizable corrections.  In the former, the\ncorrections can be separately attributed to the production and\nthe subsequent decay of the gauge boson, whereas in the latter \nthe production and decay subprocesses are\n linked by the exchange of soft photons.  \nAt \\order{\\alpha}, the PA shows agreement with the known \nNLO EW corrections up to fractions of $1\\%$ near the resonance, i.e.\\ at a phenomenologically satisfactory level~\\cite{Dittmaier:2014qza}.\nIn particular, the bulk of the NLO EW corrections near the resonance can be attributed to the factorizable corrections to the \\PW\/\\PZ~decay subprocesses, while\nthe factorizable corrections to the production process are mostly suppressed below the \npercent level,\nand the non-factorizable contributions being even smaller.\n\nBased on the quality of the PA at NLO \nwe are confident that this approach is suitable to calculate the  $\\order{\\alphas\\alpha}$ corrections with sufficient accuracy for the description of observables that are dominated by the resonances.\n The non-factorizable\n corrections comprise the conceptually most challenging contribution\n to the PA and have been computed at \\order{\\alphas\\alpha} in\n Ref.~\\cite{Dittmaier:2014qza}.  They turn out to be very small and,\n thus, demonstrate that for phenomenological purposes the\n \\order{\\alphas\\alpha} corrections can be factorized into terms\n associated with initial-state and\/or final-state corrections and combinations of the two types.\n In this paper we calculate the factorizable corrections of the type\n ``initial--final'', which combine large QCD corrections to the\n production with the large EW corrections to the decay of the $\\PW$\/$\\PZ$\n boson. \n Therefore we expect to capture  the dominant contribution at\n \\order{\\alphas\\alpha} to observables relevant for precision physics\n dominated by the \\PW and \\PZ resonances.  We also compute the corrections of ``final--final'' type, which are given only by finite counterterms to the leptonic vector-boson decay. \n The remaining factorizable ``initial--initial''\n corrections are expected to deliver only a small contribution \nand would further require $\\order{\\alphas\\alpha}$-corrected PDFs for a consistent evaluation, which are however not available.\nIt is all the more important to isolate this contribution in a well-defined\nmanner, as it is accomplished by the PA.\n\n\n\nA technical aspect of higher-order calculations involving massless particles is the proper treatment of \nIR singularities that are associated with configurations involving soft and\/or collinear particles.\nTo this end, we use the dipole subtraction \nformalism~\\cite{Catani:1996vz,Dittmaier:1999mb,Catani:2002hc,Dittmaier:2008md} \nand its extension for decay processes presented in Ref.~\\cite{Basso:2015gca}\nfor the analytic cancellation of all IR singularities.\nAlthough the cancellation of IR singularities in the \\order{\\alphas\\alpha} corrections presented in this work is accomplished by using a combined approach of the techniques developed for NLO calculations, it represents one of the main technical difficulties in the calculation and we devote special attention to its discussion.\n\n\nThis paper is organized as follows:\nIn Section~\\ref{sec:nnlo} we present the calculation of the initial--final and\nfinal--final factorizable corrections.\nWe discuss the construction of an IR-finite final result for the\ninitial--final corrections in detail with a special focus on the treatment of\nthe combined IR singularities of the QCD and EW corrections. \nOur numerical results are presented in Section~\\ref{sec:IF:numerics}, where we compare them to different versions of a naive product ansatz obtained by multiplying NLO QCD and EW correction \nfactors, and to a leading-logarithmic treatment of photon radiation \nas provided by the structure-function approach or QED parton showers such as PHOTOS~\\cite{Golonka:2005pn}.\nWe further perform a $\\chi^2$ fit in order to estimate the effect of the NNLO\n\\order{\\alphas\\alpha} corrections on the measurement of the  $\\PW$-boson mass.\nA summary is given in Sect.~\\ref{sec:concl}.\n\n\n\n\n\n\\section{Calculation of the dominant \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)} corrections in pole approximation}\n\\label{sec:nnlo}\n\n\nIn this section we identify and calculate the dominant $\\order{\\alphas\\alpha}$ corrections to the charged-current and neutral-current Drell--Yan processes in the vicinity of an intermediate vector-boson resonance. \nIn Sect.~\\ref{sec:method} we describe the classification of the  $\\order{\\alphas\\alpha}$ corrections in the framework of the PA~\\cite{Dittmaier:2014qza}. We identify  factorizable contributions of ``initial--final'' type---i.e.\\ the combination of QCD corrections to vector-boson production with EW corrections to vector-boson decay---as dominant source for corrections to distributions dominated by the vector-boson resonance.\nThe calculation of the building blocks contributing to \nthe initial--final factorizable corrections is performed in Sect.~\\ref{sec:calc-fact-nnlo-if}.\nIn Sect.~\\ref{sec:IF:master} the different building blocks of the initial--final contributions are combined into a formula suitable for numerical evaluation, where all IR singularities are cancelled explicitly.\nFinally, in Sect.~\\ref{sec:calc-fact-nnlo-ff} we calculate corrections of ``final--final'' type, which are given by pure counterterm contributions and are numerically small.\n\n\\subsection{Survey of types of \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)} corrections in pole approximation}\n\\label{sec:method}\n\n\n\\begin{figure}[b]\n  \\centering\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_ii_gen}\n    \\subcaption{Factorizable initial--initial corrections}\n  \\end{subfigure}\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_if_gen}\n    \\subcaption{Factorizable initial--final corrections}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_ff_gen}\n    \\subcaption{Factorizable final--final corrections}\n  \\end{subfigure}\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_nf_gen}\n    \\subcaption {Non-factorizable corrections}\n  \\end{subfigure}\n  %\n  \\caption{The four types of corrections that contribute to the mixed QCD--EW corrections in the PA illustrated in terms of generic two-loop amplitudes. Simple circles symbolize tree structures, double circles one-loop corrections, and triple circles two-loop contributions.}\n  \\label{fig:NNLOcontrib}\n\\end{figure}\nThe PA for Drell--Yan\nprocesses~\\cite{Stuart:1991xk,Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\nprovides a systematic classification of contributions to Feynman\ndiagrams that are enhanced by the resonant propagator of a vector\nboson~$V=\\PW,\\PZ$.\nThe leading corrections\nin the expansion around the resonance pole arise from factorizable\ncorrections to $\\PW$\/$\\PZ$ production and decay subprocesses, and\nnon-factorizable corrections that link production and decay by\nsoft-photon exchange.  The PA separates corrections to production and\ndecay stages in a consistent and gauge-invariant way.  \nThis is\nparticularly relevant for the charged-current Drell-Yan process, where\nphoton radiation off the intermediate $\\PW$~boson contributes\nsimultaneously to the corrections to production and decay of a\n$\\PW$~boson, and to the non-factorizable contributions.  Applications\nof different variants of the PA to NLO EW\ncorrections~\\cite{Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\nhave been validated by a comparison to the complete EW NLO\ncalculations and show excellent agreement at the order of some $0.1\\%$\nin kinematic distributions dominated by the resonance region.\n\nThe structure of the PA for \nthe \\order{\\alphas\\alpha} correction has\nbeen worked out in Ref.~\\cite{Dittmaier:2014qza}, where details of the\nmethod and our setup can be found.  \nThe corrections can be classified into the four types of\ncontributions shown in Fig.~\\ref{fig:NNLOcontrib} for the case of the\ndouble-virtual corrections. For each class of contributions with\nthe exception of the final--final corrections (c), also the\nassociated real--virtual and \ndouble-real corrections have to be\ncomputed, obtained by replacing one or both of the labels $\\alpha$ and $\\alphas$ in\nthe blobs in Fig.~\\ref{fig:NNLOcontrib} by a real\nphoton or gluon, respectively. The corresponding crossed partonic channels, e.g.\\ with\nquark--gluon initial states have to be included in addition.\n\nIn detail, the four types of corrections are characterized as follows:\n\\begin{enumerate}[(a)]\n\\item The initial--initial factorizable corrections are given by\n  two-loop \\order{\\alphas\\alpha} corrections to on-shell $\\PW\/\\PZ$\n  production and the corresponding one-loop real--virtual and\n  tree-level double-real contributions, i.e.\\ $\\PW\/\\PZ+\\jet$\n  production at \\order{\\alpha}, $\\PW\/\\PZ+\\Pgg$ production at\n  \\order{\\alphas}, and the processes $\\PW\/\\PZ+\\Pgg+\\jet$ at tree\n  level. Results for individual ingredients of the initial--initial\n  part are known, such as partial two-loop\n  contributions~\\cite{Kotikov:2007vr,Bonciani:2011zz} and the full\n  \\order{\\alpha} EW corrections to W\/Z+jet production including the\n  W\/Z decays~\\cite{Denner:2009gj,Denner:2011vu,Denner:2012ts}.\n  However, a consistent combination of these building blocks requires\n  also a subtraction scheme for IR singularities at ${\\cal\n    O}(\\alphas\\alpha)$ and has not been performed yet.  Note that\n  currently no PDF set including \\order{\\alphas\\alpha} corrections is\n  available, which is required to absorb IR singularities of the\n  initial--initial corrections from photon radiation collinear to the\n  beams.\n\n  Results of the PA at \\order{\\alpha} show that observables such as\n  the transverse-mass distribution in the case of \\PW production or\n  the lepton-invariant-mass distributions for \\PZ production are\n  extremely insensitive to initial-state photon\n  radiation~\\cite{Dittmaier:2014qza}. Since these distributions also\n  receive relatively moderate QCD corrections, we do not expect\n  significant initial--initial NNLO\n  \\order{\\alphas\\alpha}\n  corrections to such distributions. For observables sensitive to\n  initial-state recoil effects, such as the transverse-lepton-momentum\n  distribution, \nthe \\order{\\alphas\\alpha} corrections should be larger, but still\nvery small compared to the huge QCD \ncorrections.%\n\\footnote{Note that for such observables\n  a fixed-order QCD description is not adequate near the\n  Jacobian peak, so that in this case the initial--initial corrections need\n  to be combined with a resummation of multiple gluon emissions.  At\n  present, such a resummation is available in the POWHEG framework in\n  combination with an approximation to the double-real and\n  real--virtual part of the initial--initial corrections where the\n  first emitted photon or gluon is treated exactly, while further\n  emissions are generated in the collinear\n  approximation~\\cite{Barze:2012tt,Barze:2013yca}.}\n\n\\item  The factorizable initial--final corrections \nconsist of the \\order{\\alphas} corrections to $\\PW\/\\PZ$ production combined with the \\order{\\alpha} corrections to the leptonic  $\\PW\/\\PZ$ decay.\nBoth types of corrections are large and have a sizable impact on the shape of differential distributions at  NLO, so that we\n expect this class of the factorizable corrections to capture the dominant  \\order{\\alphas\\alpha} effects.\nThe computation of these contributions is the main result of this paper and is discussed in Sect.~\\ref{sec:calc-fact-nnlo-if}.  \n Preliminary numerical results of these corrections were presented in\n Refs.~\\cite{Dittmaier:2014koa,Huss:2014eea}.\n\n\\item Factorizable final--final corrections arise from the\n  $\\order{\\alphas\\alpha}$ counterterms of the\n  lepton--$\\PW\/\\PZ$-vertices, which involve only QCD corrections to\n  the vector-boson self-energies. There are no corresponding real\n  contributions, so that\nthe final--final corrections have practically no\n  impact on the shape of distributions. We compute these corrections\n  in Sect.~\\ref{sec:calc-fact-nnlo-ff} below and confirm the\n  expectation that they are phenomenologically negligible.\n\n\\item  The non-factorizable $\\order{\\alphas\\alpha}$ corrections are given by soft-photon corrections\nconnecting the initial state, the intermediate vector boson, and the\nfinal-state leptons, combined with QCD corrections to $V$-boson production. \nAs shown in detail in Ref.~\\cite{Dittmaier:2014qza}, these corrections\ncan be expressed in terms of soft-photon\ncorrection factors to squared tree-level or one-loop QCD matrix\nelements by using gauge-invariance arguments.\nThe numerical impact of these corrections was found to be below the\n$0.1\\%$ level and is therefore negligible for all phenomenological purposes.\n\\end{enumerate}\n\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{\\linewidth}\n    \\centering\n\\includegraphics[scale=1]{images\/diags\/DY_NNLO_R_propid_diag} \n\\caption{Decomposition of a diagram with photon emission off a $V$-boson line into initial--initial and initial--final corrections}\n\\label{fig:amp_NNLO:propagator}\n      \\end{subfigure}\n  \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=1.]{images\/diags\/DY_MIXif_Rfac_p_OS}\n  \\caption{Photon emission from the production subprocess}\n  \\label{fig:amp_NNLO:production}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}[b]{.475\\linewidth}\n  \\centering\n  \\includegraphics[scale=1.]{images\/diags\/DY_MIXif_Rfac_d_OS}\n  \\caption{Photon emission from the decay subprocess}\n    \\label{fig:amp_NNLO:decay}\n  \\end{subfigure}\\hfill\n  \\caption{\nDecomposition of the double-real corrections $\\Paq_a(p_a)\\Pq_b(p_b)\\to\\Pl_1(k_1)\\Pal_2(k_2)\\Pg(k_{\\Pg})\\Pgg(k)$  \ninto initial---initial~(\\subref{fig:amp_NNLO:production}) and initial--final~(\\subref{fig:amp_NNLO:decay}) parts,\nillustrated for an example in part~(\\subref{fig:amp_NNLO:propagator}). \n  The momentum $p_V$ of the intermediate vector boson $V$ is given by $p_V=p_a+p_b-k_{\\Pg}-k=k_1+k_2$.  A double line on a $V$ propagator indicates on-shellness while a gauge boson attached to an  encircled subdiagram indicates all possible insertions.}\n \\label{fig:amp_NNLO}\n\\end{figure}\nThe definition of the factorizable corrections and the separation of\ninitial- and final-state corrections is illustrated in\nFig.~\\ref{fig:amp_NNLO} for the case of the double-real\ncorrections. An example\ndiagram for the charged-current process is given in Fig.~\\ref{fig:amp_NNLO:propagator}, which\ncannot be attributed uniquely to the vector-boson production or decay\nsubprocess and displays an overlapping resonance structure due to the\npropagator poles at $p_V^2=\\mu_V^2$ and $(p_V+k)^2=\\mu_V^2$.\nHere $\\mu_V$ combines the real mass and width parameters of $V$,\n$M_V$ and $\\Gamma_V$, to a complex mass value,\n$\\mu_V^2=M_V^2-\\ri M_V\\Gamma_V$.\nHowever,\na simple partial-fractioning identity for the two $V$-boson\npropagators allows us\nto disentangle the two resonance structures and to\ndecompose such diagrams into contributions associated with photon\nemission from the production or decay subprocesses of an on-shell\n$V$~boson (see Eq.~(2.11) in Ref.~\\cite{Dittmaier:2014qza}). This is\nillustrated in Fig.~\\ref{fig:amp_NNLO:propagator}, where the double\nslash on a propagator line indicates that the corresponding momentum\nis set on its mass shell in the rest of the diagram (but not on the\nslashed line itself).  Using this decomposition, the double-real\ncorrections can be divided consistently into initial--initial and\ninitial--final contributions, as shown in\nFig.~\\ref{fig:amp_NNLO:production} and Fig.~\\ref{fig:amp_NNLO:decay},\nrespectively.  Here a diagrammatic notation is used where an encircled\ndiagram with an attached photon or gluon stands for all possibilities\nto attach the photon\/gluon to the fermion line and the gauge boson\n$V$ (see Eq.~(2.12) in Ref.~\\cite{Dittmaier:2014qza} for an\nexample). The initial--final (virtual QCD)$\\times$(real EW) corrections\nare treated analogously.\nAll different contributions to the\nfactorizable initial--final corrections are diagrammatically\ncharacterized in terms of interference diagrams in\nFig.~\\ref{fig:NNLO-if-graphs}.\n\n\\begin{figure}\n  \\centering\n  %\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vv1}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vv2}\n    \\caption{Factorizable initial--final double-virtual corrections}\n    \\label{fig:NNLO-if-graphs-vv}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rv1}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rv2}\n    \\caption{Factorizable initial--final (real QCD)$\\times$(virtual EW) corrections}\n    \\label{fig:NNLO-if-graphs-rv}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vr1_OS}\n    \\caption{Factorizable initial--final (virtual QCD)$\\times$(real photonic) corrections}\n    \\label{fig:NNLO-if-graphs-vr}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rr1_OS}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rr2_OS}\n    \\caption{Factorizable initial--final double-real corrections}\n    \\label{fig:NNLO-if-graphs-rr}\n  \\end{subfigure}\n  %\n  \\caption{Interference diagrams for the various contributions to the factorizable initial--final corrections of \\order{\\alphas\\alpha}, with blobs representing all relevant tree structures.\n  The blobs with ``\\alpha'' inside represent one-loop corrections of \\order{\\alpha}.}\n  \\label{fig:NNLO-if-graphs}\n\\end{figure}\n\n\n\n\\subsection{Calculation of the factorizable initial--final corrections}\n\\label{sec:calc-fact-nnlo-if}\n\nIn this section we calculate the various contributions to the\nfactorizable initial--final corrections of \\order{\\alphas\\alpha} shown\nin Fig.~\\ref{fig:NNLO-if-graphs}.  Most contributions can be expressed in terms of reducible products\nof NLO QCD and NLO EW building blocks. For details on the notation\nused for these NLO results we refer to Ref.~\\cite{Dittmaier:2014qza}.\n\n\n\n\n\\subsubsection{Double-virtual corrections}\n\\label{sec:IF:VV}\n\nThe double-virtual \\order{\\alphas\\alpha} initial--final corrections to the\nsquared $\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2$ amplitude are illustrated in\nFig.~\\ref{fig:NNLO-if-graphs}(a) in terms of interference diagrams.\nThey arise in two ways: from the interference of the tree amplitude\nwith the two-loop \\order{\\alphas\\alpha} amplitude \nand from the interference\nbetween the one-loop amplitudes with \\order{\\alphas} corrections to\n$V$-boson production and \\order{\\alpha} corrections to the decay, respectively,\n\\begin{align}\n    \\bigl\\lvert \\cM^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} \\bigr\\rvert^2 \n    \\;\\Bigr\\rvert_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} &= \n    \\hphantom{+} 2\\Re\\left\\{\n    \\delta\\cM_{\\Vs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n    \\left(\\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\\right)^* \\right\\}\n    \\nonumber\\\\&\\quad\n    + 2\\Re\\left\\{\n    \\delta\\cM_{\\Vew,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n    \\left(\\delta\\cM_{\\Vs,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\\right)^* \\right\\} .\n    \\label{eq:IF:NNLO-vv}\n\\end{align}\nThe LO amplitude in PA, $\\M_{0,\\PA}$, differs from the full LO\nmatrix element by the absence of the non-resonant photon diagram in\ncase of the neutral-current Drell--Yan process. \n The first term on the\nright-hand side in Eq.~\\eqref{eq:IF:NNLO-vv} involves the factorizable\ninitial--final contribution to the two-loop amplitude, which takes the form\nof reducible (one-loop)$\\times$(one-loop) diagrams and is defined\nexplicitly as\n\\begin{equation}\n  \\label{eq:IF:mix_vv1}\n    \\delta\\cM_{\\Vs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} =\n  \\sum_{\\lambda_V} \n  \\frac{\n    \\delta\\cM_\\Vs^{\\Paq_a\\Pq_b\\to V}({\\lambda_V})\\;\n    \\delta\\cM_\\Vew^{V\\to\\Pl_1\\Pal_2}({\\lambda_V})\n  }{p_V^2-\\mu_V^2} \n= \\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\; \\deltadec \\;  \n  \\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} ,\n\\end{equation}\nwhere a sum over the physical polarization states of the\nvector boson $V$, labelled by $\\lambda_V$, is performed. In the second\nstep in Eq.~\\eqref{eq:IF:mix_vv1} the fact is used that the one-loop\ncorrections to the production and decay factorize off the\ncorresponding LO matrix elements,\n\\begin{align}\n  \\label{eq:factQCD:pro:virt}\n   \\delta \\cM_{\\Vs}^{\\Paq_a\\Pq_b\\to V} &=\\delta_{\\Vs}^{V \\Paq_a\\Pq_b}\n \\;\\cM_{0}^{\\Paq_a\\Pq_b\\to V},  \\\\\n  \\label{eq:factEW:dec:virt}\n  \\delta \\cM_{\\Vew,\\dec}^{V\\to\\Pl_1\\Pal_2} &=\n  \\deltadec \n  \\;\\cM_{0,\\PA}^{V\\to\\Pl_1\\Pal_2} .\n\\end{align}\nThe virtual QCD corrections are well known and are quoted explicitly in \nEq.~(2.35) of Ref.~\\cite{Dittmaier:2014qza}.\nThe explicit expressions for the NLO EW correction factors can be found, \ne.g., in Refs.~\\cite{Dittmaier:2001ay,Dittmaier:2009cr}, and are quoted in \nAppendix B.2 of Ref.~\\cite{Huss:2014eea}.\nIn order to maintain gauge invariance, the NLO production \nand decay subamplitudes in Eq.~\\eqref{eq:IF:mix_vv1}, \nand in particular\nthe correction factor $\\deltadec$, are evaluated for on-shell $V$ bosons. \nWe keep the QCD correction factor $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ off shell,\ni.e.\\ without setting $s\\to M_V^2$ there to be closer to the full calculation,\nwhich is possible, because $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ does not depend on\n$M_V$ at all.\nThe on-shell projection $s\\to M_V^2$ in the EW correction\ninvolves some freedom, but numerical\neffects from different implementations are of the same order as the\nintrinsic uncertainty of the PA.  However, the choice of the mappings\nin the virtual and real corrections has to match properly in order to\nensure the correct cancellation of IR singularities. \n\nThe expressions~\\eqref{eq:factQCD:pro:virt}\nand~\\eqref{eq:factEW:dec:virt} also enter the one-loop interference\nterms in the second line of Eq.~\\eqref{eq:IF:NNLO-vv}.  The final\n  result for the double-virtual corrections to the cross section is\n  therefore given by\n\\begin{align}\n  \\label{eq:IF:mix_vv}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} &= \n  2 \\biggl[\n  \\Re\\Bigl\\{ \\deltadec \\; \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}\n  + \\Re\\Bigl\\{ \\deltadec \\Bigl(\\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr)^* \\Bigr\\}\n  \\biggr] \\; \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \n  \\nonumber\\\\&=\n  4\\Re\\Bigl\\{ \\deltadec \\Bigr\\}\n  \\Re\\Bigl\\{ \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}  \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 .\n\\end{align}\nBoth the EW and QCD correction factors contain soft and collinear\nsingularities, which take the form of $\\frac{1}{\\epsilon^2}$ poles for\nmassless fermions. Therefore, in principle, Eq.~\\eqref{eq:IF:mix_vv}\nrequires the evaluation of the correction factors up to\n\\order{\\epsilon^2} in order to obtain all finite \\order{\\epsilon^0}\nterms. %\nHowever, after applying the subtraction formalism, which we describe\nin detail in Sect.~\\ref{sec:IF:master}, the poles are cancelled before\nperforming the full expansion in $\\epsilon$ and, thus, the results up\nto order \\order{\\epsilon^0} turn out to be sufficient.  This result is obvious if\nthe soft and collinear singularities are not regularized in\n$D=4-2\\epsilon$ dimensions,\nbut by small mass parameters, where no rational terms \nfrom the multiplication of $1\/\\epsilon$ poles with $D$-dimensional quantities\nexist at all.\n\n\\subsubsection{(Real QCD)\\texorpdfstring{${\\times}$}{x}(virtual EW) corrections}\n\\label{sec:IF:RV}\n\nThe (real QCD)$\\times$(virtual EW) contributions to the factorizable initial--final corrections shown in Fig.~\\ref{fig:NNLO-if-graphs}(\\subref{fig:NNLO-if-graphs-rv}) arise by \nincluding the virtual corrections to the leptonic $\\PW$\/$\\PZ$ decays  to the various partonic subprocesses of $V+\\text{jet}$ production, \n\\begin{subequations}\n\\label{eq:rQCD}\n\\begin{align}\n  \\Paq_a(p_a) \\,+\\, \\Pq_b(p_b) &\\,\\to\\, V(p_V) \\,+\\, \\Pg(k_{\\Pg}) ,\\\\\n  \\Pg(p_{\\Pg}) \\,+\\, \\Pq_b(p_b) &\\,\\to\\, V(p_V) \\,+\\, \\Pq_a(k_a) ,\\\\\n  \\Pg(p_{\\Pg}) \\,+\\, \\Paq_a(p_a) &\\,\\to\\, V(p_V)  \\,+\\, \\Paq_b(k_b) .\n\\end{align}\n\\end{subequations}\nFor the quark-induced channel, the corrections are given by replacing the virtual QCD amplitude in Eq.~\\eqref{eq:IF:mix_vv1} by the corresponding amplitude for real-gluon emission,\n\\begin{align}\n  \\label{eq:IF:mix-rv-gen}\n  \\delta\\cM_{\\Rs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pg} &=\n  \\sum_{\\lambda_V} \n  \\frac{\n    \\cM_{\\Rs}^{\\Paq_a\\Pq_b\\to \\Pg V}({\\lambda_V})\\;\n    \\delta\\cM_\\Vew^{V\\to\\Pl_1\\Pal_2}({\\lambda_V})\n  }{p_V^2-\\mu_V^2} .\n\\end{align}\nAnalogously to the double-virtual case, the EW decay\nsubamplitude is evaluated for on-shell vector bosons,\nwhile the QCD correction is kept off shell.\nUsing the factorization property of the EW one-loop decay corrections~\\eqref{eq:factEW:dec:virt}, the (real QCD)$\\times$(virtual EW) correction to the cross section in the quark--anti-quark channel is proportional to the real NLO QCD corrections $\\rd\\sigma^{\\Rs}$,  \n\\begin{equation}\n\\label{eq:IF:mix-rv}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  = 2\\Re\\Bigl\\{ \\deltadec \\Bigr\\}\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}  .\n\\end{equation}\nAs for the Born amplitude, the label PA in the real-emission corrections indicates that all non-resonant terms, i.e.\\ the photon-exchange diagrams in case of the neutral-current process, are omitted in the QCD real-emission amplitudes. \nAnalogous expressions hold for the gluon--quark and gluon--anti-quark initiated subprocesses to $V+\\text{jet}$ production.\n\n\n\n\\subsubsection{(Virtual QCD)\\texorpdfstring{${\\times}$}{x}(real photonic) corrections}\n\\label{sec:IF:VR}\n\nThe (virtual QCD)$\\times$(real photonic) factorizable corrections of\ninitial--final type arise from the generic interference diagram shown\nin Fig.~\\ref{fig:NNLO-if-graphs}(\\subref{fig:NNLO-if-graphs-vr}). They\nare obtained by combining the real-photon corrections to on-shell\n$V$-boson decay with the virtual QCD corrections to $V$-boson\nproduction,\n\\begin{equation}\n\\cM_{\\Vs\\otimes\\Rew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}\n =\\sum_{\\lambda_V} \\frac{\\delta \\cM_\\Vs^{\\Paq_a\\Pq_b\\to V}({\\lambda_V})\\;\n  \\cM_{\\Rew}^{V\\to\\Pl_1\\Pal_2\\Pgg}({\\lambda_V})}{(p_V+k)^2-\\mu_V^2} = \n\\delta_{\\Vs}^{V \\Paq_a\\Pq_b}\\cM_{\\Rew,\\fact,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}.\n\\end{equation}\nIn the second step, Eq.~\\eqref{eq:factQCD:pro:virt} has been used to factorize the virtual QCD correction factor from the  matrix element $ \\cM_{\\Rew,\\fact,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}$ for the factorizable NLO decay corrections (see Eq.~(2.14) in Ref~\\cite{Dittmaier:2014qza}).\nAgain the matrix elements for the EW \ndecay subprocess is evaluated for on-shell vector bosons,\nwhile the QCD correction factor is kept off shell. \nAs illustrated in Fig.~\\ref{fig:amp_NNLO}\nand discussed in detail in Ref.~\\cite{Dittmaier:2014qza}, the splitting of photon-emission \neffects off the intermediate $V$-boson into parts corresponding to initial- or final-state\nradiation separates the two resonance propagator factors\n$1\/(p_V^2-\\MV^2)$ and $1\/[(p_V^2+k)^2-\\MV^2]$, respectively, where $p_V=k_1+k_2$.\nFor factorizable EW decay correction we, thus, have to perform the on-shell projection\n$(p_V^2+k)^2\\to \\MV^2$.\nThe resulting contribution \nof the (virtual QCD)$\\times$(real photonic) corrections to the cross section therefore assumes the form\n\\begin{align}\n  \\label{eq:IF:mix_vr}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rew} &= \n  2 \\Re\\Bigl\\{ \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}  \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew .\n\\end{align}\n\n\n\\subsubsection{Double-real corrections}\n\\label{sec:IF:RR}\n\nThe double-real emission corrections  are illustrated by interference diagrams in Fig.~\\ref{fig:NNLO-if-graphs}(d) and are defined by the real-emission matrix elements for the $V+\\text{jet}$ production subprocesses~\\eqref{eq:rQCD} with the subsequent decay $V\\to \\Pl_1\\Pal_2\\Pgg$,\n\\begin{align}\n  \\label{eq:IF:mix-rr-gen}\n \\cM_{\\Rs\\otimes\\Rew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg\\Pg}\n =\\sum_{\\lambda_V} \\frac{\\cM_\\Rs^{\\Paq_a\\Pq_b\\to V\\Pg}({\\lambda_V})\\;\n  \\cM_{\\Rew}^{V\\to\\Pl_1\\Pal_2\\Pgg}({\\lambda_V})}{(p_V+k)^2-\\mu_V^2}, \n\\end{align}\nwith analogous expressions for the $\\Pg q$ and $\\Pg\\bar q$ channels.\nThe non-resonant contribution arising from the case $V=\\Pgg$ in the neutral-current process is again not included.\nCompact explicit results for the helicity amplitudes of the double-real corrections can be found in Ref.~\\cite{Huss:2014eea}.\nThe double-real contribution to the cross section, $\\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}$, \nis defined in terms of the square of the matrix element~\\eqref{eq:IF:mix-rr-gen}\nwhere the decay subamplitudes are evaluated for on-shell $V$~bosons. \nDue to the spin correlations of the production and decay matrix elements and\nthe full kinematics of the $2\\to 4$ scattering process, the double-real\ncorrections do not factorize further into separate EW and QCD correction\nfactors, in contrast to the other classes of\n factorizable initial--final corrections.\n\n\n\n\n\\subsection{Treatment of infrared singularities for the factorizable initial--final corrections}\n\\label{sec:IF:master}\n\nThe NNLO \\order{\\alphas\\alpha} contributions to the cross section due\nto the factorizable initial--final corrections are\nobtained by integrating the four contributions discussed in the\nprevious section over the respective phase spaces,\n\\begin{align}\n  \\label{eq:IF:generic-master-first}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} \n  &=\n  \\int_{2} \\rd\\sigma_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}\n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n   + \\int_{3} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  + \\iint\\limits_{3+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  + \\int_{2} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew} \n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} ,\n\\end{align}\nwhere the additional QCD collinear counterterms in the last line were\nintroduced to absorb the collinear singularities associated with the\nquarks and gluons in the initial state into the NLO PDFs.  Note that\nthe EW corrections are completely confined to the decay subprocess,\nand consequently, there are \nno singularities from initial-state collinear\nquark--photon splittings.  This allows us\nto obtain the collinear\ncounterterms in the last line of\nEq.~\\eqref{eq:IF:generic-master-first} from the customary NLO QCD\ncollinear counterterms $\\rd\\sigma^{\\Cs}$~\\cite{Catani:1996vz} by\nreplacing the LO cross sections by the appropriate real or virtual EW\ndecay corrections in the PA. Using the results of\nSect.~\\ref{sec:calc-fact-nnlo-if} we can write\n\\begin{align}\n  \\label{eq:IF:generic-master}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} \n  &=\n  \\int_{2} 4\\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}\\;\n  \\Re\\Bigl\\{\\deltadec\\Bigr\\}\\;\n  \\rd\\sigma_\\PA^0\n  + \\iint\\limits_{2+\\Pgg} 2\\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}\\;\n  \\rd\\sigma_\\dec^{\\Rew} \n  \\nonumber\\\\&\\quad\n  + \\int_{3} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\}\\; \\rd\\sigma_\\PA^{\\Rs} \n  + \\iint\\limits_{3+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  + \\int_{2} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\; \\rd\\sigma_\\PA^{\\Cs} \n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} .\n\\end{align}\n\n\nApplying the QCD dipole subtraction formalism~\\cite{Catani:1996vz} \n in order to cancel the IR\nsingularities associated with the QCD corrections,\nEq.~\\eqref{eq:IF:generic-master} can be written in the following form,\n\\begin{align}\n  \\label{eq:IF:dipole-qcd-master}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} &= \n  \\int_{2} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\;\n  \\rd\\sigma_\\PA^0 \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n  \\nonumber\\\\&\\quad\n  + \\iint\\limits_{2+\\Pgg}  \n  \\rd\\sigma_\\dec^\\Rew \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n  \\nonumber\\\\&\\quad\n  + \\int_{3} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \n  \\biggl\\{ \\rd\\sigma_\\PA^{\\Rs} \n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\PA^0\\diptimes\\CSV \\biggr\\}\n  \\nonumber\\\\&\\quad\n  + \\iint\\limits_{3+\\Pgg} \\biggl\\{\n  \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\dec^\\Rew\\diptimes\\CSV \\biggr\\}\n  \\nonumber\\\\&\\quad\n  + \\int_0^1\\rd x\\int_2\n  2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\;\n  \\rd\\sigma_\\PA^0 \\diptimes(\\CSK+\\CSP) \n  \\nonumber\\\\&\\quad\n  + \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\rd\\sigma_\\dec^\\Rew\\diptimes(\\CSK+\\CSP) .\n\\end{align}\nThe explicit expressions of the dipole operators  $\\CSV$ and the insertion operators $\\CSI$, \n$\\CSK$, and $\\CSP$ can be found in Ref.~\\cite{Catani:1996vz}.\nThe symbol  $\\otimes$ denotes possible additional helicity and colour correlations,\nand it is implicitly assumed that the cross sections multiplying the dipole operators $\\CSV$ are evaluated on the respective dipole-mapped phase-space point.\nThe explicit expressions associated with the NLO QCD corrections were given in Ref.~\\cite{Dittmaier:2014qza}.\n\n\nAll individual integrals appearing in Eq.~\\eqref{eq:IF:dipole-qcd-master} are now free of QCD singularities, but remain IR divergent owing to the singularities contained in the EW corrections which still need to be cancelled between the virtual corrections and the corresponding real-photon-emission parts.\nFor this purpose we employ the dipole subtraction formalism for photon radiation~\\cite{Dittmaier:1999mb,Dittmaier:2008md}, in particular  the extension of the formalism to treat decay kinematics described in detail in Ref.~\\cite{Basso:2015gca}.\nAs a result, we are able to arrange the six contributions in Eq.~\\eqref{eq:IF:dipole-qcd-master} into a form where all IR divergences are cancelled in the integrands explicitly,\n\\begin{align}\n   \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}\n  &=\\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} ,\n \\label{eq:IF:final-master}\n\\end{align}\nwhere each term is an IR-finite object and its phase-space integration can be performed numerically in four dimensions.\nEquation~\\eqref{eq:IF:final-master} is our master formula for the numerical evaluation discussed in Sect.~\\ref{sec:IF:numerics}.\nExplicit expressions for all contributions for the quark--anti-quark\nand quark--gluon induced channels are given in\nAppendix~\\ref{app:ir-safe}.\n\n\nThe first two terms in Eq.~\\eqref{eq:IF:final-master} arise from the sum of the double-virtual and the (virtual QCD)$\\times$(real photonic) corrections, including the insertion operators from the QCD dipole formalism, and correspond to the sum of the first two lines in Eq.~\\eqref{eq:IF:dipole-qcd-master}.\nApplying the dipole formalism to rearrange the IR singularities of photonic origin between the virtual and real EW corrections, we obtain the following expressions for the IR-finite virtual QCD contributions to the cross section, \n\\begin{align} \n  \\label{eq:IF:master-VV}\n  \\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}&= \n  \\int_{2} \n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr] \\;\n  \\rd\\sigma_\\PA^0 \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \\\\\n  \\label{eq:IF:master-VR}\n  \\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}&=\n  \\iint\\limits_{2+\\Pgg}  \\biggl\\{ \\rd\\sigma_\\dec^\\Rew \n  -\\sum_{\\substack{I,J\\\\I\\ne J}} \\rd\\sigma_\\PA^0 \\diptimes \\dVew[IJ] \\biggr\\}\n  \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] ,\n\\end{align}\nwhere the sum over the emitter--spectator pairs ($I,J$) in Eq.~\\eqref{eq:IF:master-VR} extends over all particles of the decay subprocess, i.e.\\ $I,J=\\Pl_1,\\Pal_2,V$.\nWe have introduced a compact notation for the QED dipoles,\n\\begin{align}\n  \\label{eq:definition:dVew}\n  \\dVew[IJ] &= 4\\pi\\alpha\\; \\eta_IQ_I\\,\\eta_JQ_J\\,\n  \\begin{cases}\n  \\dsub{IV} , & \\text{for } (I=\\Pl_1,\\Pal_2) \\land (J=V) , \\\\\n  \\gsub{IJ} , & \\text{for } (I=\\Pl_1,\\Pal_2) \\land (J=\\Pl_1,\\Pal_2) , \\\\\n  0 , & \\text{for } I=V ,\n  \\end{cases}\n\\end{align}\nwhere $\\eta_i=1$ for incoming particles and outgoing antiparticles and $\\eta_i=-1$ for incoming antiparticles and outgoing particles. \nThe corresponding endpoint contributions are given by\n\\begin{align}\n  \\label{eq:nlo:dec:master:Iew}\n  \\Iew\n  &=\n  \\frac{\\alpha}{2\\pi}\\, Q_{\\Pl_1}\\biggl[\n  (Q_{\\Pl_1}-Q_{\\Pl_2})\\; \\Dsub{\\Pl_1 V}\n  +Q_{\\Pl_2}\\; \\Gsub{\\Pl_1\\Pal_2} \n  \\biggr]\n  +(\\Pl_1\\leftrightarrow\\Pal_2) ,\n\\end{align}\nwhere the functions $\\gsub{}$ and $\\Gsub{}$ are given in Ref.~\\cite{Dittmaier:1999mb}, while  $\\dsub{}$ and $\\Dsub{}$ are the decay dipoles and their integrated counterparts constructed in Ref.~\\cite{Basso:2015gca}.\nWhenever we write $\\Pl_1\\leftrightarrow\\Pal_2$, this implies the interchange\n$Q_{\\Pl_1}\\leftrightarrow Q_{\\Pl_2}$ of the electric charges of the respective fermions,\nirrespective of their particle or antiparticle nature.\n\nAs anticipated in Sect.~\\ref{sec:IF:VV}, all IR singularities contained in $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ cancel exactly against the corresponding poles of the $\\CSI$ operator within the second square bracket of Eq.~\\eqref{eq:IF:master-VV}.\nSimilarly, all singularities in $\\deltadec$ cancel against the corresponding poles in $\\Iew$ in the first square bracket of Eq.~\\eqref{eq:IF:master-VV}.\nAs a consequence, it is sufficient to use the correction factors $\\deltadec$ and $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ up to \\order{\\epsilon^0}.\nFurthermore, we recall that the correction factors $\\deltadec$ \nare evaluated at the on-shell point $p_V^2=\\MV^2$ and, thus, are\nindependent of the phase-space kinematics.%\n\n\nThe contributions involving real QCD corrections are given by the third and forth term in Eq.~\\eqref{eq:IF:final-master}.\nThey are obtained by applying the QED dipole subtraction formalism to the sum of the third and forth line of Eq.~\\eqref{eq:IF:dipole-qcd-master} and result in the following expressions for the IR-finite real-gluon contributions to the cross section,\n\\begin{align}\n  \\label{eq:IF:master-RV}\n  \\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}&= \n  \\int_{3} \n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\biggl\\{ \\rd\\sigma_\\PA^{\\Rs} \n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\PA^0\\diptimes\\CSV \\biggr\\} ,\n  \\\\\n  \\label{eq:IF:master-RR}\n  \\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}&=\n  \\iint\\limits_{3+\\Pgg} \\biggl\\{\n  \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}} \\rd\\sigma_\\dec^\\Rew\\diptimes\\CSV \n  - \\sum_{\\substack{I,J\\\\I\\ne J}}\\rd\\sigma_\\PA^\\Rs \\diptimes \\dVew[IJ]\n  \\nonumber\\\\&\\qquad\n  +\\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\n  \\sum_{\\substack{I,J\\\\I\\ne J}}\n  \\rd\\sigma_\\PA^0 \\diptimes\\CSV \\diptimes \\dVew[IJ]\n  \\biggr\\}  .\n\\end{align}\n\nIt is instructive to  examine the local cancellation of the IR singularities in Eq.~\\eqref{eq:IF:master-RR} in more detail.\nThe second term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR} acts as a local counterterm to the double-real emission cross section $\\rd\\sigma^{\\Rs\\otimes\\Rew}$ in all regions of phase space where the additional QCD radiation becomes unresolved, i.e.\\ soft and\/or collinear to the beam. \nThe third term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR} analogously  ensures the cancellation of IR singularities in the phase-space regions where the photon becomes soft and\/or collinear to a final-state lepton.\nA subtlety arises in the double-unresolved cases, where the cross sections $\\rd\\sigma_\\dec^\\Rew$ and $\\rd\\sigma_\\PA^\\Rs$ become singular as well, and both subtraction terms above will simultaneously act as a local counterterm, leading to the twofold subtraction of the IR singularities.\nThis disparity in the double-unresolved limits is exactly compensated by the last term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR}, which therefore has the opposite sign.\nNote that the evaluation of this last term  involves the successive application of two dipole phase-space mappings.\nOwing to the property of the factorizable initial--final corrections where the emissions in the production and decay stages of the $V$ boson proceed independently, the two dipole mappings do not interfere with each other and\nthe order in which they are applied is irrelevant.\nA related property is the factorization of the dipole phase space, where the two one-particle subspaces associated with the two unresolved emissions can be isolated simultaneously.  \nThis has the important consequence\nthat the analytic integration over the \ngluon and photon momenta can be carried out in the same manner as  at\nNLO, which allows \nus to reuse the known results for the integrated\ndipoles without modification.\n\n\n\nFinally, we consider the convolution terms with additional virtual or real EW corrections given by the last two terms in Eq.~\\eqref{eq:IF:final-master}.\nSince these contributions are essentially given by the lower-order (in \\alphas) cross sections, convoluted with the insertion operators $\\CSK$ and $\\CSP$, they pose no additional complications, and the resulting IR-finite contributions to\nthe cross section can be written as\n\\begin{align}\n \\label{eq:IF:master-CV}\n \\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}&=\n \\int_0^1\\rd x\\int_2\n \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\\;\n \\rd\\sigma_\\PA^0 \\diptimes(\\CSK+\\CSP)  , \\\\\n \\label{eq:IF:master-CR}\n \\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} &= \n \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n \\biggl\\{ \\rd\\sigma_\\dec^\\Rew \n -\\sum_{\\substack{I,J\\\\I\\ne J}} \n \\rd\\sigma_\\PA^0 \\diptimes \\dVew[IJ] \\biggr\\}\n \\diptimes(\\CSK+\\CSP) .\n\\end{align}\nOwing to the Lorentz invariance of the dipole formalism, no special treatment is required in contrast to our calculation of the non-factorizable corrections discussed in Ref.~\\cite{Dittmaier:2014qza}, which was carried out with the slicing method to isolate soft-photon singularities.\n\n\nThe results presented so far are appropriate for the case of IR-safe observables, i.e.\\ for the case where\ncollinear photons and leptons are recombined to a ``dressed'' lepton carrying their total momentum.\nFor non-collinear-safe observables with respect to the final-state leptons, i.e.\\ the treatment of bare muons without photon recombination, we use the method of Ref.~\\cite{Dittmaier:2008md} and its extension to decay kinematics described in Ref.~\\cite{Basso:2015gca}.\nThe required modifications are described \nin Appendix~\\ref{app:non-collinear-safe}.\n\n\n\n\n\n\n\\subsection{Factorizable final--final corrections}\n\\label{sec:calc-fact-nnlo-ff}\n\nThe factorizable  NNLO corrections of final--final type arise purely from the counterterms to \nthe $V\\Pl_1\\Pal_2$ vertex and therefore factorize from the LO matrix element,\n\\begin{equation}\n  \\label{eq:FF:mix_vv1}\n  \\delta\\cM_{\\Vs\\otimes\\Vew,\\dec\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n  = \\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}\\;  \\;  \n  \\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} .\n\\end{equation}\nThe counterterms for the leptonic vector-boson decay only receive\ncontributions from the vector-boson self-energies at\n$\\order{\\alphas\\alpha}$~\\cite{Chang:1981qq,Djouadi:1987gn,Djouadi:1987di,Kniehl:1988ie,Kniehl:1989yc,Djouadi:1993ss},\nwhich enter the counterterms through the vector-boson wave-function\nrenormalization constants and through the renormalization constants of the\nelectromagnetic coupling and the weak-mixing angle. \nThere is only one type of\ncontribution from one-loop diagrams with insertions of one-loop\n$\\order{\\alphas}$ or $\\order{\\alpha}$ counterterms.\nIt results from massive quark loops in the vector-boson self-energies \nwhere the QCD mass renormalization\nconstant has to be taken into account. \nWe make use of the expressions\nfor the vector-boson self-energies of Ref.~\\cite{Djouadi:1993ss},\nwhich include the QCD quark mass counterterm in the on-shell scheme.  \nThe expressions  for the self-energies in terms of the scalar integrals computed in Ref.~\\cite{Djouadi:1993ss} are given in Appendix~\\ref{app:ff}.\n\nThe vertex counterterms in the on-shell renormalization scheme are\nobtained from the expressions for the corresponding NLO EW\ncounterterms~\\cite{Denner:1991kt} upon replacing the one-loop\nvector-boson self-energies by the two-loop\n$\\mathcal{O}(\\alphas\\alpha)$ results and dropping lepton wave-function\nrenormalization constants, which receive no correction at this\norder.   \nWe employ the $G_\\mu$ input-parameter scheme where the\n electromagnetic\n coupling constant\nis derived from the Fermi constant $G_\\mu$ via the relation\n\\begin{equation}\n  \\label{eq:G_mu-scheme}\n  \\alpha_{G_\\mu} \\;=\\; \\frac{\\sqrt{2}}{\\pi} G_\\mu \\MW^2 \n  \\left( 1-\\frac{\\MW^2}{\\MZ^2} \\right) .\n\\end{equation}\nThe counterterm $\\delta Z_e$ for\nthe electromagnetic charge in the $G_\\mu$ scheme  is related to the one\nin the $\\alpha(0)$ input-parameter scheme as follows,\n\\begin{equation}\n\\label{eq:dZeGmu}\n   \\delta Z_e^{G_\\mu}= \\delta Z_e^{\\alpha(0)}\n-\\frac{1}{2}\\Delta r .\n\\end{equation}\nThe quantity $\\Delta r$ comprises all higher-order corrections to muon\ndecay excluding the contributions that constitute QED corrections in\nthe Fermi model, which are included in the definition of the muon\ndecay constant $G_\\mu$~\\cite{Sirlin:1980nh},\n\\begin{align}\n\\Delta r &=\n\\left.\\frac{\\partial\\Sigma^{AA}_{\\rT}(k^2)}{\\partial k^2}\\right\\vert_{k^2=0} \n- 2\\frac{\\delta \\sw}{\\sw}+\n2\\frac{\\cw}{\\sw}\\;\\frac{\\Sigma_{\\rT}^{AZ}(0)}{\\MZ^2} +\n\\frac{\\Sigma^{WW}_{\\rT}(0)- \\Re\\, \\Sigma^{WW}_{\\rT}(\\MW^2)}{\\MW^2} +\\delta r ,\n\\end{align}\nwith the renormalization constant $\\delta \\sw$ \nof the weak-mixing angle and the transverse parts of the vector-boson self-energies,\n$\\Sigma^{VV'}_{\\rT}$.  The $\\mathcal{O}(\\alphas\\alpha)$\ncontribution to $\\Delta r$ simplifies due to the fact that there is no contribution\nto the finite remainder\n$\\delta r$ at this order and the photon--$\\PZ$-boson mixing\nself-energy $\\Sigma_\\rT^{AZ}$ vanishes at zero\nmomentum~\\cite{Djouadi:1993ss}.  \nMoreover, since there are no loop corrections\nto the leptonic vector-boson decay at $\\mathcal{O}(\\alphas\\alpha)$,\nthe vertex counterterms are finite. The expressions for the\ncounterterms $\\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}$\nin terms of vector-boson self-energies are explicitly given in\nAppendix~\\ref{app:ff}.  \n\n\nThe contribution of the final--final corrections to the cross-section prediction are obtained by a simple phase-space integration over the Born kinematics,\n\\begin{equation}\n  \\sighat_{\\dec\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} = \n  \\int_{2} 2  \\Re\\Bigl\\{\\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}\\Bigr\\}\\;\n  \\rd\\sigma_\\PA^0 .\n\\end{equation}\n\n Using the values\nof the input parameters given in Eq.~\\eqref{eq:params} below, the\nnumerical value of the counterterm for the  $\\PW\\Pgnl\\Pal$ vertex\nis given by\n\\begin{equation}\n  \\delta_{\\PW\\Pgnl\\Pal}^{\\mathrm{ct},(\\alphas\\alpha)}=\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times 0.93.\n\\end{equation}\nThe final--final correction to the cross section for the\ncharged-current cross section is therefore below the $0.1\\%$ level and\nphenomenologically negligible.  This can be partially attributed to the choice of the $G_\\mu$-scheme where universal corrections to charged-current processes are absorbed in the value of $\\alpha_{G_\\mu}$.\nThe numerical values of the counterterms $\\delta^{\\mathrm{ct},\\tau,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}$\nfor the $\\PZ\\Pl\\Pal$ vertices with lepton chiralities $\\tau=\\pm$\nare somewhat larger, but of opposite sign:\n\\begin{subequations}\n\\begin{align}\n   \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}& =\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times (-49.3), \\\\\n   \\delta^{\\mathrm{ct},-,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}&  =\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times (+31.8).\n\\end{align}\n\\end{subequations}\nThe resulting corrections to the neutral-current Drell--Yan\nprocess are, however, suppressed\nfar below the $0.1\\%$ level due to cancellations between the right-\nand left-handed production channels and are therefore also negligible for all phenomenological purposes.\n\n\n\n\n\n\\section{Numerical results}\n\\label{sec:IF:numerics}\n\n\nIn this section we present the numerical results for the \ndominant mixed QCD--EW corrections to the Drell--Yan process at the LHC for a centre-of-mass energy of $\\sqrt{s}=14~\\TeV$. We consider the two processes \n\\begin{eqnarray}\n  \\label{eq:nlo-num-procs}\n\\nonumber\n  \\Pp+\\Pp \\;\\to&\\; \\PWp \\;&\\to\\; \\Pgnl+\\Plp +X, \\\\\n  \\Pp+\\Pp \\;\\to&\\; {\\PZ} \\;&\\to\\; \\Plm+\\Plp +X\n\\nonumber\n\\end{eqnarray}\nwith electrons or muons in the final state ($\\Pl=\\Pe,\\Pgm$). \nWe further distinguish two\nalternative treatments of photon radiation: In the ``dressed-lepton''\ncase, collinear photon--lepton configurations are treated inclusively\nusing a photon-recombination procedure.  As a result, the numerical\npredictions do not contain large logarithms of the lepton mass, which\ncan be set to zero. The dressed-lepton results are appropriate\nmostly for electrons in the final state. In the ``bare-muon'' case, no such\nrecombination is performed, reflecting the experimental situation\nwhich allows for the detection of isolated muons.  We perform a comparison\nto naive factorization prescriptions of QCD and EW corrections, as\nwell as to a modelling of photonic final-state radiation~(FSR) by\nstructure functions or a photon shower. Moreover, we estimate the impact of the NNLO QCD--EW\ncorrections on the measurement of the $\\PW$-boson mass.\n\n\\subsection{Input parameters and event selection}\n\\label{sec:input-cuts}\n\nThe setup for the calculation is analogous to the one used in\nRef.~\\cite{Dittmaier:2014qza}.  \nThe choice of input parameters closely follows\nRef.~\\cite{Beringer:1900zz},\n\\begin{equation}\n\\label{eq:params}\n\\begin{aligned}\n  \\MW^\\OS \\;=&\\; 80.385~\\GeV ,\n  &\\GW^\\OS \\;=&\\; 2.085~\\GeV , \\\\\n  \\MZ^\\OS \\;=&\\; 91.1876~\\GeV ,\n  &\\Gamma_\\PZ^\\OS \\;=&\\; 2.4952~\\GeV , \\\\\n  M_\\PH \\;=&\\; 125.9~\\GeV , \n  &m_\\Pqt \\;=&\\; 173.07~\\GeV , \\\\\n  G_\\mu \\;=&\\; 1.1663787\\times 10^{-5} ~\\GeV^{-2} , \n  &\\alpha(0) \\;=&\\; 1\/137.035999074 , \\\\\n  \\alphas(\\MZ) \\;=&\\; 0.119 .\n\\end{aligned}\n\\end{equation}\n We convert the on-shell masses\nand decay widths of the vector bosons to the corresponding pole masses\nand widths as spelled out in Ref.~\\cite{Dittmaier:2014qza}.\n\nThe electromagnetic coupling constant used in the LO predictions is\nobtained from the Fermi constant by Eq.~\\eqref{eq:G_mu-scheme}.\nIn the charged-current process, all relative electroweak corrections are computed using $\\alpha_{G_\\mu}$.\nIn the neutral-current process, however, we follow  Ref.~\\cite{Dittmaier:2009cr} and use $\\alpha(0)$  consistently in the relative photonic corrections while the remaining relative weak corrections are proportional to $\\alpha_{G_\\mu}$.\nThe same prescription is applied to the relative $\\order{\\alpha_s\\alpha}$ corrections. \n\n\nThe masses of the light quark flavours (\\Pqu, \\Pqd, \\Pqc, \\Pqs, \\Pqb) and of the leptons are neglected throughout, with the only exception in case of non-collinear-safe observables, where the final-state collinear singularity is regularized by the mass of the muon,\n\\begin{equation}\n  \\label{eq:muon-mass}\n  m_{\\Pgm} \\;=\\; \\SI{105.658369}{\\MeV} .\n\\end{equation}\nThe CKM matrix is chosen diagonal in the third generation and the mixing between the first two generations is parametrized by the following values for the entries of the quark-mixing matrix,\n\\begin{equation}\n  \\label{eq:ckm}\n  \\lvert V_{\\Pqu\\Pqd} \\rvert \\,=\\, \n  \\lvert V_{\\Pqc\\Pqs} \\rvert \\,=\\, 0.974, \\qquad\n  \\lvert V_{\\Pqc\\Pqd} \\rvert \\,=\\, \n  \\lvert V_{\\Pqu\\Pqs} \\rvert \\,=\\, 0.227. \n\\end{equation}\nFor the PDFs we consistently use the NNPDF2.3 sets~\\cite{Ball:2012cx}, where the NLO and NNLO \nQCD--EW corrections are evaluated using the NNPDF2.3QED NLO set~\\cite{Ball:2013hta}, which also includes \\order{\\alpha} corrections.\nThe value of the strong coupling $\\alphas(\\MZ)$ quoted in Eq.~\\eqref{eq:params} is dictated by the choice of these PDF sets.\nFor the evaluation of the full NLO\nEW corrections entering the naive products below, we employ the DIS factorization scheme to absorb the mass singularities into the PDFs.\nThe renormalization and factorization scales are set equal, with a fixed value given by the respective gauge-boson mass, \n\\begin{equation}\n  \\label{eq:scale}\n  \\mur \\;=\\; \\muf \\;\\equiv\\; \\mu \\;=\\; \\MV ,\n\\end{equation}\nof the process under consideration.\n\nFor the experimental identification of the Drell--Yan process we impose the following cuts on the transverse momenta and rapidities of the charged leptons,\n\\begin{align}\n  \\label{eq:cut-lep}\n  p_{\\rT,\\Plpm} &> 25~\\GeV , &\n  \\lvert y_{\\Plpm} \\rvert &< 2.5 , \n\\end{align}\nand an additional cut on the missing transverse energy\n\\begin{equation}\n  \\label{eq:cut-miss}\n  E_\\rT^\\miss > 25~\\GeV ,\n\\end{equation}\nin case of the charged-current process.\nFor the neutral-current process we further require a cut on the invariant mass of the lepton pair,\n\\begin{equation}\n  \\label{eq:cut-mll}\n  M_{\\Pl\\Pl} > 50~\\GeV ,\n\\end{equation}\nin order to avoid the photon pole at $M_{\\Pl\\Pl}\\to0$.\n\nFor the dressed-lepton case, in addition, a photon recombination procedure analogous to the one used in Refs.~\\cite{Dittmaier:2001ay,Dittmaier:2009cr} is applied:\n\\begin{enumerate}\n\\item Photons close to the beam with a rapidity $\\lvert \\eta_{\\Pgg} \\rvert > 3$ are treated as beam remnants and are not further considered in the event selection.\n\\item For the photons that pass the first step, the angular distance to the charged leptons $R_{\\Plpm\\Pgg}=\\sqrt{(\\eta_{\\Plpm}-\\eta_{\\Pgg})^2 + (\\phi_{\\Plpm}-\\phi_{\\Pgg})^2}$ is computed, where $\\phi$ denotes the azimuthal angle in the transverse plane. \n  If the distance $R_{\\Plpm\\Pgg}$ \nbetween the photon and the closest lepton is smaller than \n$0.1$, the photon is recombined with the lepton by adding the respective four-momenta, $\\Plpm(k_i)+\\Pgg(k)\\to\\Plpm(k_i+k)$.\n\\item Finally, the event selection cuts from Eqs.~\\eqref{eq:cut-lep}--\\eqref{eq:cut-mll} are applied to the resulting event kinematics.\n\\end{enumerate}\n\n\n\n\\subsection{Results for the dominant factorizable corrections}\n\\label{sec:if-results}\n\nThe NNLO QCD--EW corrections to the hadronic Drell--Yan cross section \nare dominated by\nthe factorizable initial--final \\order{\\alphas\\alpha} corrections,\n$\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}$,\nwhich are obtained\nby convoluting the corresponding partonic corrections  \n$\\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}$ calculated in\nSection~\\ref{sec:nnlo} with the PDFs.\n Our default prediction for Drell--Yan processes is then obtained by\n adding these NNLO corrections to  the sum of \nthe full NLO QCD and EW corrections, \n\\begin{align} \n\\label{eq:def:nnlo:if}\n  \\sigma^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  \\sigma^0 +\\Delta\\sigma^{\\NLO_\\rs} + \\Delta\\sigma^{\\NLO_\\rew}\n  +\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} ,\n\\end{align}\nwhere all terms are consistently evaluated with the NNPDF2.3QED NLO PDFs.\nThe non-factorizable corrections computed in \nRef.~\\cite{Dittmaier:2014qza} were found to have a negligible impact on the cross section and are therefore not included here.\nSimilarly, the factorizable corrections of ``final--final'' type discussed in Sect.~\\ref{sec:calc-fact-nnlo-ff} turn out to have a negligible impact on the cross-section prediction and are therefore not included in Eq.~\\eqref{eq:def:nnlo:if} either.\n\nOur result allows to validate estimates of the NNLO QCD--EW corrections based on a naive product ansatz. For this purpose, we define the naive product of the NLO QCD cross section and the relative EW corrections,\n\\begin{align}\n  \\sigma_{\\text{naive fact}}^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  \\sigma^{\\NLO_\\rs} (1+\\delta_{\\alpha})\n  \\nonumber\\\\&=\n  \\sigma^0 +\\Delta\\sigma^{\\NLO_\\rs} + \\Delta\\sigma^{\\NLO_\\rew}\n  +\\Delta\\sigma^{\\NLO_\\rs}\\;\\delta_{\\alpha}  ,\n  \\label{eq:def:nnlo:naive:fact}\n\\end{align}\nwhere the relative EW corrections are defined as the ratio of the NLO EW contribution $\\Delta\\sigma^{\\NLO_\\rew}$ with respect to the LO contribution $\\sigma^0$ according to\n\\begin{align}\n  \\label{eq:def:deltaew}\n  \\delta_{\\alpha} &\\equiv \n  \\frac{\\Delta\\sigma^{\\NLO_\\rew}}{\\sigma^0} ,\n\\end{align}\nwhere both denominator and numerator are evaluated with the same NLO PDFs, so that the EW correction factors are practically independent of the PDFs.\nIn order to compare the factorized expression to the NNLO corrections, we define two different versions of the NLO EW corrections in Eq.~\\eqref{eq:def:deltaew}: First, based on the full \\order{\\alpha} correction ($\\delta_\\alpha$), and second, based on the dominant EW final-state correction of the PA ($\\delta_\\alpha^\\dec$).\n\nDefining the correction factors,%\n\\footnote{\n  Note that the correction factor $\\delta_{\\alphas}'$ differs from that in the standard QCD $K$~factor $K_{\\mathrm{NLO}_{\\mathrm{s}}}=\\sigma_{\\mathrm{NLO}_{\\mathrm{s}}}\/\\sigma_{\\mathrm{LO}}\\equiv 1+\\delta_{\\alpha_s}$ due to the use of different PDF sets in the Born contributions.\n  See Ref.~\\cite{Dittmaier:2014koa} for further discussion.\n} \n\\begin{align}\n  \\label{eq:def:deltas'}\n  \\delta^{\\pro\\times\\dec}_{\\alphas\\alpha} &\\equiv\n  \\frac{\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}}{\\sigma^\\LO} , &\n  \\delta_{\\alphas}' &\\equiv \n  \\frac{\\Delta\\sigma^{\\NLO_\\rs}}{\\sigma^\\LO} ,\n\\end{align}\nwe can cast the relative difference of our best prediction~\\eqref{eq:def:nnlo:if} and the \nproduct ansatz~\\eqref{eq:def:nnlo:naive:fact} into the following form,\n\\begin{align}\n\\label{eq:diff-naive}\n  \\frac{\\sigma^{\\NNLO_{\\rs\\otimes\\rew}}\n    -\\sigma_{\\text{naive fact}}^{\\NNLO_{\\rs\\otimes\\rew}}}\n  {\\sigma^\\LO}\n  &=\n  \\delta^{\\pro\\times\\dec}_{\\alphas\\alpha} - \\delta_{\\alphas}' \\delta_\\alpha ,\n\\end{align}\nwhere the LO prediction $\\sigma^\\LO$ in the denominators is evaluated with the LO PDFs.\nThe difference of the relative NNLO\ncorrection $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and the naive\nproduct $\\delta_{\\alphas}' \\delta_\\alpha^{(\\dec)}$ therefore allows to\nassess the validity of a naive \nproduct ansatz.  As observed in\nSect.~\\ref{sec:calc-fact-nnlo-if}, most contributions to the\nfactorizable initial--final corrections take the reducible form of a\nproduct of two NLO corrections, with the exception of the double-real\nemission corrections which are defined with the full kinematics of the\n$2\\to 4$ phase space.\nNote that the double-real contributions are the only ones where the final-state leptons receive recoils from both QCD and photonic radiation, an effect that cannot be captured by naively multiplying NLO QCD and EW corrections. \nAny large deviations between\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and\n$\\delta_{\\alphas}'\\delta_\\alpha^{(\\dec)}$ can therefore be attributed\nto this type of contribution. The difference of the naive product\ndefined in terms of $\\delta_\\alpha^{\\dec}$ and $\\delta_\\alpha$\nallows us to assess the  impact of the missing $\\order{\\alphas\\alpha}$ corrections beyond the initial--final\ncorrections considered in our calculation and therefore also provides\nan error estimate of the PA, and in particular of the omission of the\ncorrections of initial--initial type.\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIF+rec}\n  \\end{minipage}\n  \\caption{Relative factorizable corrections of \\order{\\alphas\\alpha} induced by initial-state QCD and final-state EW contributions to the transverse-mass~(left) and transverse-lepton-momentum~(right) distributions for \\PWp production at the LHC.\n  The naive products of the NLO correction factors $\\delta_{\\alphas}'$ and $\\delta_\\alpha$ are shown for comparison.}\n  \\label{fig:distNNLO-IF-Wp}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIF+rec}\n  \\end{minipage}\n  \\caption{Relative factorizable corrections of \\order{\\alphas\\alpha} induced by initial-state QCD and final-state EW contributions to the lepton-invariant-mass distribution~(left) and a transverse-lepton-momentum \ndistribution~(right) for \\PZ production at the LHC.\n  The naive products of the NLO correction factors $\\delta_{\\alphas}'$ and $\\delta_\\alpha$ are shown for comparison.}\n  \\label{fig:distNNLO-IF-Z}\n\\end{figure}\n\nFigure~\\ref{fig:distNNLO-IF-Wp} shows the numerical results for the\nrelative \\order{\\alphas\\alpha} initial--final factorizable corrections $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ to the transverse-mass ($M_{\\rT,\\Pgn\\Pl}$) and the\ntransverse-lepton-momentum ($p_{\\rT,\\Pl}$) distributions for \\PWp\nproduction at the LHC.  For \\PZ production,\nFigure~\\ref{fig:distNNLO-IF-Z} displays the results for the\nlepton-invariant-mass ($M_{\\Pl\\Pl}$) distribution and a\ntransverse-lepton-momentum ($p_{\\rT,\\Pl^+}$) distribution.  In both\nfigures, the upper\nplots show the results for bare muons, the lower panels correspond to\nthe corrections with photon recombination. \nIn Figs.~\\ref{fig:distNNLO-IF-Wp} and~\\ref{fig:distNNLO-IF-Z} we\nalso compare to the two different implementations of a naive\nproduct of correction factors discussed after\nEq.~\\eqref{eq:diff-naive}.  \n In the following, we\nmainly focus on the results for bare muons.  The respective results\nwith photon recombination display the same general features as those\nfor bare muons, but the relative corrections\nare reduced by approximately a factor of two.\nThis reduction is familiar from NLO EW results and is induced by the\ncancellation of the collinear singularities by restoring the level of\ninclusiveness required for the KLN theorem.\nOne observes that the NNLO\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ corrections are in general better\napproximated by the simple product ansatz for the case of bare muons\nthan for dressed leptons. This can\nbe understood from the fact that the dominant part of the corrections\nstem from the collinear logarithms $\\ln(m_{\\Pgm})$ which are known to\nfactorize.\n\n\nFor the $M_{\\rT,\\Pgn\\Pl}$ distribution for $\\PWp$ production (upper left plot in\nFig.~\\ref{fig:distNNLO-IF-Wp}), the mixed NNLO QCD--EW corrections\nfor bare muons are moderate and amount to approximately $\\SI{-1.7}{\\%}$ around the\nresonance, which is about an order of magnitude smaller than the NLO\nEW corrections.\\footnote{The structure observed in the correction\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ around\n$M_{\\rT,\\Pgn\\Pl}\\approx\\SI{62}{\\GeV}$ can be attributed to the\ninterplay of the kinematics of the double-real emission\ncorrections and the event selection. It arises close to the kinematic\nboundary $M_{\\rT,\\Pgn\\Pl}>\\SI{50}{\\GeV}$ for the back-to-back kinematics of the non-radiative process implied by the  cut\n$p_{\\rT,\\Plpm},E_{\\rT}^\\text{miss}>\\SI{25}{\\GeV}$.} \nBoth variants of the naive product provide a good approximation to the\nfull result in the region around and below the Jacobian peak, which is\ndominated by resonant $\\PW$ production.  For larger values of\n$M_{\\rT,\\Pgn\\Pl}$, the product $\\delta_{\\alphas}' \\delta_\\alpha $\nbased on the full NLO EW correction factor deviates from the other curves, which signals\nthe growing importance of effects beyond the PA. However, the\ndeviations amount to only few per-mille for\n$M_{\\rT,\\Pgn\\Pl}\\lesssim\\SI{90}{\\GeV}$.\nThe overall good agreement between the\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ corrections and both naive\nproducts can be attributed to  well-known insensitivity of the\nobservable $M_{\\rT,\\Pgn\\Pl}$ to initial-state radiation effects\nalready seen in the case of NLO corrections in Ref~\\cite{Dittmaier:2014qza}.\n \nFor the $p_{\\rT,\\Pl}$ distributions in the case of bare muons (upper right plots\nin Figs.~\\ref{fig:distNNLO-IF-Wp} and \\ref{fig:distNNLO-IF-Z},\nrespectively) we observe corrections that are small far below the\nJacobian peak, but which rise to about $15\\%$ ($20\\%$) on the Jacobian\npeak at $p_{\\rT,\\Pl}\\approx\\MV\/2$ for the case of the \\PWp boson (\\PZ\nboson) and then display a steep drop reaching almost $-50\\%$ at\n$p_{\\rT,\\Pl}=\\SI{50}{\\GeV}$.  This enhancement stems from the large\nQCD corrections above the Jacobian peak familiar from the NLO QCD\nresults (see e.g.\\ Fig.~8 in Ref.~\\cite{Dittmaier:2014qza}) where\nthe recoil due to real\nQCD radiation shifts events with resonant $\\PW$\/$\\PZ$ bosons above the\nJacobian peak.\nThe\nnaive product ansatz fails to provide a good description of the full\nresult $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and deviates by\n5--10\\% at the Jacobian peak, where the PA is expected to be the most\naccurate.  This can be attributed to the strong influence of the\nrecoil induced by initial-state radiation on the transverse momentum,\nwhich implies a larger effect of the double-real emission corrections\non this distribution that are not captured correctly by the naive\nproducts.  The two versions of the naive products display larger\ndeviations than in the $M_{\\rT,\\Pgn\\Pl}$ distribution discussed above,\nwhich signals a larger impact of the missing \\order{\\alphas\\alpha}\ninitial--initial corrections.  However, these deviations should be\ninterpreted with care, \nsince a fixed-order\nprediction is not sufficient to describe this distribution around the\npeak region $p_{\\rT,\\Pl}\\approx\\MV\/2$, which corresponds to the\nkinematic onset for $V+\\text{jet}$ production and is known to\nrequire QCD resummation for a proper description.\n\nIn case of the $M_{\\Pl\\Pl}$\ndistribution for $\\PZ$ production (left-hand plots in\nFig.~\\ref{fig:distNNLO-IF-Z}),  corrections up to $10\\%$  are\nobserved below the resonance for the case of bare muons. This is consistent with the large EW\ncorrections at NLO in this region, which arise from final-state photon\nradiation that shifts the reconstructed value of the invariant\nlepton-pair mass away from the resonance to lower values.\nThe naive product approximates the\nfull initial--final corrections\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ reasonably well at the resonance\nitself ($M_{\\Pl\\Pl}=\\MZ$) and  above, but completely fails already a\nlittle below the resonance where the naive products do not even\nreproduce the sign of the full\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ correction.\nThis deviation occurs although  the invariant-mass distribution is \nwidely unaffected by initial-state radiation effects. \nThe fact that we obtain almost identical corrections \nfrom the two versions of the product $\\delta_{\\alphas}'\\delta_\\alpha^\\dec$ \nand $\\delta_{\\alphas}'\\delta_\\alpha$ demonstrates the insensitivity of this\nobservable to photonic initial-state radiation.\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relNLOZ}\n  \\caption{}\n  \\label{fig:distNNLO-IFZ:NLO-breakdown}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFZ}\n  \\caption{}\n  \\label{fig:distNNLO-IFZ:more}\n  \\end{subfigure}\n  \\caption{(\\subref{fig:distNNLO-IFZ:NLO-breakdown})~The correction factors \nto Z-boson production (as shown in the upper left plot of Fig.\\ref{fig:distNNLO-IF-Z})\nentering the naive product ansatz broken down into its individual contributions with the QCD corrections further divided into the $\\Pq\\Paq$- and $\\Pq\\Pg$-induced contributions.\n  (\\subref{fig:distNNLO-IFZ:more})~A comparison of the corrections shown in the upper left panel of Fig.~\\ref{fig:distNNLO-IF-Z} with the modified product using the value of the QCD corrections at the resonance $\\delta_{\\alphas}'(M_{\\Pl\\Pl}=\\MZ)\\approx 6.5\\%$.}\n  \\label{fig:distNNLO-IFZ}\n\\end{figure}\n\nIn order to locate the source \nof this large discrepancy we examine\nthe individual correction factors in the naive product in more detail.\nWe restrict ourselves to the case of bare muons and  the full NLO EW\ncorrection factor $\\delta_\\alpha$ for definiteness, which does not\naffect our conclusions.\nIn Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown}),\n we separately\n plot the two correction factors that enter the naive \nproduct $\\delta_{\\alphas}'\\times\\delta_\\alpha$ and further divide the QCD corrections into the \n$\\Pq\\Paq$- and the $\\Pq\\Pg$-induced contributions.\nWe observe that the two different $\\Pq\\Paq$- and $\\Pq\\Pg$-induced channels individually receive large QCD corrections, however, they differ in sign, \nso that large cancellations take place in the sum $\\delta_{\\alphas}'$.\nA small mismatch in the corrections of the individual channels can therefore quickly lead to a large effect in the QCD corrections which is then further enhanced by the large EW corrections in the product ansatz $\\delta_{\\alphas}' \\times \\delta_{\\alpha}$.\nMoreover,\nFigure~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown})\nreveals that the QCD correction factor $\\delta_{\\alphas}'$ is\nresponsible for the sign change at $M_{\\Pl\\Pl}\\approx\\SI{83}{\\GeV}$\nwhich is the most striking  disagreement of the naive product ansatz\nwith the full factorizable initial--final corrections.\nThis zero crossing happens more than three widths below the resonance where the cross section is reduced by almost two orders of magnitudes compared to the resonance region and, furthermore, will be very sensitive to event selection cuts, since $\\delta_{\\alphas}'$ arises from the cancellation of two large corrections as we have seen above.\nIn Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown}) we\nobserve the large EW corrections below the resonance mentioned above, which \narise due to the redistribution of  events near the $\\PZ$\npole to lower lepton invariant masses by final-state photon radiation.\nThe similar form of the factorizable NNLO initial--final\ncorrections indicates that they mainly stem from an analogous mechanism.\nThis suggests that it is more appropriate to  replace\nthe QCD correction factor $\\delta_{\\alphas}'$ in the naive product by its value at the resonance\n$\\delta_{\\alphas}'(M_{\\Pl\\Pl}=\\MZ)\\approx 6.5\\%$, which corresponds to\nthe location of the events that are responsible for the bulk of the\nlarge EW corrections below the resonance. \nIn contrast, the naive product ansatz simply multiplies the\ncorrections locally on a bin-by-bin basis. This causes a mismatch in the correction factors and fails to account for the\nmigration of events due to FSR.\nThe comparison of the previous results and the modified product is shown in Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:more}) and clearly shows an improvement despite its very crude construction.\n\nContrary to the lepton-invariant-mass distribution, the\ntransverse-mass distribution is dominated by events with resonant \\PW\nbosons even in the range below the Jacobian peak,\n$M_{\\rT,\\Pgn\\Pl}\\lesssim\\MW$, so it is less sensitive to the\nredistribution of events to lower $M_{\\rT,\\Pgn\\Pl}$.\nThis explains why the naive product can provide a good approximation\nof the full initial--final NNLO corrections.  It should be emphasized,\nhowever, that even in the case of the $M_{\\rT,\\Pgn\\Pl}$ distribution\nany event selection criteria that deplete events with resonant \\PW bosons\nbelow the Jacobian peak will result in increased sensitivity to the effects of\nFSR and can potentially lead to a failure of a naive\nproduct ansatz. \n\nIn conclusion, simple approximations in terms of products of\ncorrection factors have to be used with care and require a careful\ncase-by-case investigation of their validity.\n\n\\subsection{Leading-logarithmic approximation for final-state photon radiation}\n\\label{sec:FSR}\n\nAs is evident from Figs.~\\ref{fig:distNNLO-IF-Wp} and~\\ref{fig:distNNLO-IF-Z}, a naive \nproduct of QCD and EW correction factors~\\eqref{eq:def:nnlo:naive:fact} is not adequate to approximate the NNLO QCD--EW corrections for all observables.  \nA promising approach to a factorized approximation for the dominant initial--final corrections can be obtained by combining the full NLO QCD corrections to vector-boson production with the leading-logarithmic~(LL) approximation for the final-state corrections.\nThe benefit in this approximation lies in the fact that the interplay of the recoil effects from jet and photon emission is properly taken into account.\nOn the other hand, the logarithmic approximation neglects certain (non-universal) finite contributions, which are, however, suppressed with respect to the dominating radiation effects.\n\nIn the structure-function approach~\\cite{Kuraev:1985hb}, the\nleading-logarithmic approximation of the {pho\\-to\\-nic} decay\ncorrections is\ncombined with the NLO QCD corrections to the production by a convolution,\n\\begin{align}\n  \\label{eq:LL1FSR}\n  \\Delta\\sigma_{\\Pp\\Pp, \\llog\\FSR}^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  %\n  \\int \\rd\\sigma^{\\mathrm{NLO}_\\rs}(p_1,p_2;k_1,k_2) \n  \\int^1_0 \\rd z_1 \\, \\int^1_0 \\rd z_2 \\,\n  \\;\\Theta_\\cut\\bigl(z_1k_1\\bigr)\n  \\;\\Theta_\\cut\\bigl(z_2k_2\\bigr)\n  \\nonumber\\\\* &\\quad\\times \n\\biggl[\\,\\Gamma_{\\Pl_1\\Pl_1}^{\\llog}(z_1,Q^2)\\Gamma_{\\Pl_2\\Pl_2}^{\\llog}(z_2,Q^2)\n-\\delta(1-z_1)\\,\\delta(1-z_2)\\biggr]\n  \\nonumber\\\\ &=\n  %\n  \\int \\rd\\sigma^{\\mathrm{NLO}_\\rs}(p_1,p_2;k_1,k_2) \\int^1_0 \\rd z_1 \\, \\int^1_0 \\rd z_2 \\,\n  \\;\\Theta_\\cut\\bigl(z_1k_1\\bigr)\n  \\;\\Theta_\\cut\\bigl(z_2k_2\\bigr)\n  \\nonumber\\\\*\n  &\\quad \\times\\biggl[\n \\delta(1-z_2)\\,\\Gamma_{\\Pl_1\\Pl_1}^{\\llog,1}(z_1,Q^2)\n  +\\delta(1-z_1)\\,\\Gamma_{\\Pl_2\\Pl_2}^{\\llog,1}(z_2,Q^2)\n  +\\order{\\alpha^2} \\biggr] \n  ,\n\\end{align}\nwhere  $\\rd\\sigma^{\\mathrm{NLO}_\\rs}$ includes the virtual and real QCD corrections.\nThe step function $\\Theta_{\\cut}(z_i k_i)$ is equal to 1 if the event\npasses the cut on the rescaled lepton momentum $z_i k_i$ and 0\notherwise.  The variables $z_i$ are the momentum fractions describing the\nrespective \nlepton energy loss by collinear photon emission. For the charged-current process only one of the convolutions is present.\nThe $\\order{\\alpha}$ contribution to the structure function $\\Gamma_{\\Pl\\Pl}^{\\llog}$ reads\n\\begin{eqnarray}\n\\label{eq:LLll}\n  \\Gamma_{\\Pl\\Pl}^{\\llog,1}(z,Q^2) &=&\n  Q_{\\Pl}^2\\,\\frac{\\beta_\\Pl}{4} \\left(\\frac{1+z^2}{1-z}\\right)_+ ,\n\\end{eqnarray}\nwhere the large mass logarithm appears in the variable\n\\begin{equation}\n\\label{eq:beta_l}\n\\beta_\\Pl = \\frac{2\\,\\alpha}{\\pi}\n\\left[\\ln\\biggl(\\frac{Q^2}{\\Ml^2}\\biggr)-1\\right]\n\\end{equation}\nand \n$Q_{\\Pl}$ denotes the relative electric charge of the lepton $\\ell$. \nIn order to be consistent in the comparison with our calculation as described in Sect.~\\ref{sec:input-cuts}, the electromagnetic coupling constant $\\alpha$ appearing in Eq.~\\eqref{eq:beta_l} is set to $\\alpha_{G_\\mu}$ and $\\alpha(0)$ for the charged-current and neutral-current processes, respectively.\nThe scale $Q$ is chosen as the gauge-boson mass,\n\\begin{equation}\n\\label{eq:FSR_scale}\nQ = M_{V} ,\n\\end{equation}\nand the scale-variation bands shown in the numeric results are obtained by varying the scale by a factor of two up and down from the central scale choice,\n\\begin{equation}\nQ = \\xi \\cdot M_V , \\qquad \\xi = \\tfrac{1}{2}, 1, 2 .\n\\label{eq:ll1fsr:variation}\n\\end{equation}\nSince the mass logarithms cancel in observables where photon emission collinear to the final-state charged leptons is treated fully inclusively, \nthe structure-function approach is only applicable to non-collinear-safe observables, i.e.\\ to the bare-muon case. \n\nIn contrast, in \nparton-shower approaches to photon radiation (see e.g.\\ Refs.~\\cite{Placzek:2003zg,CarloniCalame:2003ux,CarloniCalame:2005vc})\nthe photon momenta transverse to the lepton momentum are \ngenerated as well, following the differential factorization formula, so that\nthe method is also applicable to the case of collinear-safe observables,\ni.e.\\ to the dressed-lepton case. For this purpose, we have\nimplemented the combination of the exact NLO QCD prediction for\nvector-boson production with the simulation of final-state photon radiation\nusing PHOTOS~\\cite{Golonka:2005pn}. Since we are interested in\ncomparing  to the\n\\order{\\alphas\\alpha} corrections in our setup, we only generate a\nsingle photon emission using PHOTOS and use the same scheme for\n$\\alpha$ as described in Sect.~\\ref{sec:input-cuts}. \nDetails on the specific settings within the PHOTOS parton shower are given in Appendix~\\ref{app:photos}.\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIFCompGF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIFCompGF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIFCompGF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIFCompGF+rec}\n  \\end{minipage}\n  \\caption{Comparison of the approximation obtained\n    from PHOTOS for the relative \\order{\\alphas\\alpha} initial-state\n    QCD and final-state EW corrections to our best prediction\n    $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ for the case of the\n    transverse-mass~(left) and transverse-lepton-momentum~(right)\n    distributions for \\PWp production at the LHC, as in\n    Fig.~\\ref{fig:distNNLO-IF-Wp}.  In the bare-muon case, the\n    result~\\eqref{eq:LL1FSR} of the structure-function approach is\n    also shown.  }\n  \\label{fig:distNNLO-IF-FSR-Wp}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFComp} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFComp+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIFComp} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIFComp+rec}\n  \\end{minipage}\n  \\caption{Comparison of the approximation obtained from PHOTOS for the\n    relative \\order{\\alphas\\alpha} initial-state QCD and final-state\n    EW corrections to our best prediction\n    $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ for the case of the\n    lepton-invariant-mass distribution~(left) and a\n    transverse-lepton-momentum distribution~(right) for \\PZ production at\n    the LHC, as in Fig.~\\ref{fig:distNNLO-IF-Z}. In the bare-muon case, the\n    result~\\eqref{eq:LL1FSR} of the structure-function approach is\n    also shown.}\n  \\label{fig:distNNLO-IF-FSR-Z}\n\\end{figure}\n\nIn Figs.~\\ref{fig:distNNLO-IF-FSR-Wp} and \\ref{fig:distNNLO-IF-FSR-Z}\nwe compare our best prediction~\\eqref{eq:def:nnlo:if} for the\nfactorizable initial--final \\order{\\alphas\\alpha} corrections to the\ncombination of NLO QCD corrections with the approximate FSR\nobtained from PHOTOS for the case of $\\PWp$ production and $\\PZ$\nproduction, respectively. For the bare-muon case also the result of\nthe structure-function approach according to Eq.~\\eqref{eq:LL1FSR} is\nshown. The combination of the NLO QCD corrections and \napproximate FSR leads to a clear improvement compared to the\nnaive product approximations investigated in\nSection~\\ref{sec:if-results}.  This is particularly apparent in the\nneutral-current process where the $M_{\\Pl\\Pl}$ distribution is correctly modelled\nby both FSR approximations, whereas the naive products shown in\nFigs.~\\ref{fig:distNNLO-IF-Z} and~\\ref{fig:distNNLO-IFZ} completely failed to\ndescribe this distribution. \n In the $M_{\\rT,\\Pgn\\Pl}$ spectrum of the charged-current\nprocess in Fig.~\\ref{fig:distNNLO-IF-FSR-Wp} one also finds good\nagreement of the different results below the Jacobian peak and an\nimprovement over the naive product approximations in\nFig.~\\ref{fig:distNNLO-IF-Wp}. \nThe description of the $p_{\\rT,\\Pl}$ distributions is also improved compared to the naive product approximations, but some differences remain in the charged-current process.\n\n\nIn spite of the good agreement of the two versions of incorporating final-state-radiation effects, the intrinsic uncertainty of the leading-logarithmic approximations should be kept in mind.\nFor the structure-function approach, this uncertainty is illustrated by the band width resulting from the variation~\\eqref{eq:ll1fsr:variation} of the QED scale $Q$.\nWe remark that the multi-photon corrections obtained by employing the\nun-expanded\nstructure-functions $\\Gamma_{\\Pl\\Pl}^{\\llog}(z,Q^2)$  in Eq.~\\eqref{eq:LL1FSR}\nlie well within the aforementioned scale bands, which shows that a\nproper matching to the full NLO EW calculation is needed to\nremove the dominant uncertainty of the LL approximation and to \n predict the higher-order effects reliably.\nFor PHOTOS the intrinsic uncertainty is not shown and not easy to\nquantify.\nThe good quality of the PHOTOS approximation results from the fact that the finite terms in the photon emission probability are specifically adapted to $\\PW\/\\PZ$-boson decays.\nThe level of agreement with our ``full prediction'', thus, cannot be taken over to other processes.\n\n\n\\subsection{Impact on the \\texorpdfstring{$\\PW$}{W}-boson mass extraction}\n\\label{sec:mw}\n\nIn order to estimate the effect of the \\order{\\alphas \\alpha}\ncorrections on the extraction of the $\\PW$-boson mass at the LHC we\nhave performed a $\\chi^2$ fit of the $M_{\\rT,\\nu \\ell}$ distribution.\nWe treat the $M_{\\rT,\\nu \\ell}$ spectra calculated in various\ntheoretical approximations for a reference mass $\\MW^\\OS= 80.385~\\GeV$\nas ``pseudo-data'' that we fit with \n ``templates'' calculated\nusing the LO predictions $\\sigma^0$ (with NLO PDFs)\nfor different values of $\\MW^\\OS$.\nSpecifically, we have generated results for $27$ transverse-mass bins in the\ninterval $M_{\\rT,\\nu \\ell}= [64,\\,91]~\\GeV$ in steps of $1~\\GeV$, varying the $\\PW$-boson mass in the interval $\\MW=[80.085,\\,80.785]~\\GeV$ with steps of $\\Delta \\MW=10~ \\MeV$ (steps of $\\Delta\n\\MW=5~\\MeV$ in the interval $\\MW=[80.285,\\,80.485]~\\GeV$).  \nUsing a linear interpolation between neighbouring $\\MW$ values, we obtain the integrated cross sections in the $i$th $M_{\\rT,\\nu \\ell}$ bin, $\\sigma_i^0(\\MW)$, as a continuous function of $\\MW$. \nThe best-fit value $\\MW^{\\mathrm{fit}, \\mathrm{th}}$ \nquantifying the impact of a higher-order correction\nin the theoretical cross section $\\sigma^{\\mathrm{th}}$ is\nthen obtained from the minimum of the function\n\\begin{equation}\n\\label{eq:chi2}\n  \\chi^2(\\MW^{\\mathrm{fit},\\mathrm{th}})=\\sum_{i}\n  \\frac{\\left[\\sigma_i^{\\mathrm{th}}(\\MW^\\OS)\n      -\\sigma_i^0(\\MW^{\\mathrm{fit},\\mathrm{th}})\\right]^2}{\n    2 \\Delta\\sigma_i^2}\\,,\n\\end{equation}\nwhere the sum over $i$ runs over the transverse-mass bins.\nHere $\\sigma_i^{\\mathrm{th}}$ and $\\sigma_i^0$ are the\nintegrated cross sections in the $i$-th bin, uniformly rescaled so\nthat the sum over all $27$ bins is identical for all considered cross\nsections.  We assume a statistical error of the pseudo-data and take\n$\\Delta\\sigma_i^2\\propto \\sigma_i^{\\mathrm{th}}$.  We have also\nperformed a two-parameter fit where the normalization of the templates\nis fitted simultaneously, leading to identical results.  Similarly,\nallowing the $\\PW$-boson width in the templates to float and fitting\n$\\MW$ and $\\GW$ simultaneously does not\nsignificantly affect the estimate of the effect of the\n\\order{\\alphas\\alpha} corrections on the \\MW measurement.\n\n \\begin{figure}\n   \\centering\n   \\includegraphics[width=.48\\linewidth]{images\/mwfit\/plot_dchi2_nlo_int}\n   \\includegraphics[width=.48\\linewidth]{images\/mwfit\/plot_dchi2_nnlo_int}\n   \\caption{The $\\Delta\\chi^2=\\chi^2-\\chi^2_{\\mathrm{min}}$\n     distributions obtained from the fit of the NLO EW corrections\n     (left) and the NNLO production--decay corrections (right) to LO\n     templates in arbitrary units for $\\chi^2$. The NLO mass shift $\\Delta\\MW^{\\mathrm{NLO}_\\rew}$ is given relative to the reference mass  $\\MW^\\OS= 80.385~\\GeV$, the NNLO shift $\\Delta\\MW^{\\mathrm{NNLO}}$ is given relative to the output mass of the fit to the sum of the EW and QCD corrections as defined in Eq~\\eqref{eq:dmwnnlo}.}\n   \\label{fig:chi2}\n \\end{figure}\n\nIn the experimental measurements of the transverse-mass distribution,\nthe Jacobian peak is washed out due to the finite energy and momentum\nresolution of the detectors. In our simple estimate of the impact of\nhigher-order corrections on the extracted value of the $\\PW$-boson\nmass, we do not attempt to model such effects.  We expect the detector\neffects to affect the different theory predictions in a similar way\nand to cancel to a large extent in our estimated mass shift, which is\nobtained from a difference of mass values extracted from pseudo-data\ncalculated using different theory predictions.  This assumption is\nsupported by the fact that our estimate of the effect of the NLO EW\ncorrections is similar to the one obtained in\nRef.~\\cite{CarloniCalame:2003ux} using a Gaussian smearing of the\nfour-momenta to simulate detector effects.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{@{} l r@{\\enspace}l r@{\\enspace}l @{}}\n  \\toprule\n  & \\multicolumn{2}{c}{bare muons} & \\multicolumn{2}{c}{dressed leptons} \\\\\n  \\cmidrule(r){2-3} \\cmidrule(l){4-5}\n  & $M^\\mathrm{fit}_\\mathrm{W}\\;[\\mathrm{GeV}]$ & $\\qquad\\Delta M_\\mathrm{W}$ \n  & $M^\\mathrm{fit}_\\mathrm{W}\\;[\\mathrm{GeV}]$ & $\\qquad\\Delta M_\\mathrm{W}$ \\\\\n  \\midrule\n  LO          & $80.385$ & \\multirow{2}*{$\\bigg\\}\\;-90~\\mathrm{MeV}$} \n              & $80.385$ & \\multirow{2}*{$\\bigg\\}\\;-40~\\mathrm{MeV}$} \\\\\n         $\\NLO_\\rew$ & $80.295$ &                   \n              & $80.345$ &                   \\\\\n  \\midrule\n   $\\NLO_{\\rs\\oplus\\rew}$\n        & $80.374$ & \\multirow{2}*{$\\biggr\\}\\;-14~\\mathrm{MeV}$} \n              & $80.417$ & \\multirow{2}*{$\\biggr\\}\\;-4~\\mathrm{MeV}$} \\\\\n  NNLO        & $80.360$ &                   \n              & $80.413$ &                   \\\\\n  \\bottomrule\n\\end{tabular}\n\\caption{Values of the $\\PW$-boson Mass in  \\GeV obtained from the $\\chi^2$ fit\nof the  $M_{\\rT,\\nu \\ell}$ distribution in  different theoretical\napproximations  to LO templates and the resulting mass shifts.}\n  \\label{tab:mass_shift}\n\\end{table}\n\nThe  fit results for several NLO approximations and our best NNLO prediction~\\eqref{eq:def:nnlo:if} are given in Table~\\ref{tab:mass_shift}.\nTo validate our procedure we estimate the mass shift due to the NLO EW\ncorrections by using the prediction\n$\\sigma^{\\NLO_\\rew}=\\sigma^0+\\Delta\n\\sigma^{\\NLO_\\rew}$\n as the pseudo-data $\\sigma^{\\mathrm{th}}$\nin~\\eqref{eq:chi2}. The $\\chi^2$ distribution is shown on the left-hand side of\nFig.~\\ref{fig:chi2} as a function of the mass shift\n$\\Delta\\MW^{\\NLO_\\rew}$ for the dressed-lepton and bare-muon cases.\nFrom the minima of the distributions one finds a mass shift of\n$\\Delta \\MW^{\\NLO_\\rew}\\approx -90~\\MeV$ for bare muons and $\\Delta\n\\MW^{\\NLO_\\rew}\\approx -40~\\MeV$ for dressed muons.  These values are\ncomparable to previous results reported in \nRef.~\\cite{CarloniCalame:2003ux}.\\footnote{\n  In Ref.~\\cite{CarloniCalame:2003ux} the values $\\Delta\\MW=110~\\MeV$\n  ($20~\\MeV$) are obtained for the bare-muon (dressed-lepton)\n  case. These values are obtained using the\n  $\\order{\\alpha}$-truncation of a LL shower and for\n  lepton-identification criteria appropriate for the Tevatron taken\n  from Ref.~\\cite{Baur:1998kt}, so they cannot be compared directly to\n  our results. In particular, in the dressed-lepton case, a looser\n  recombination criterion $R_{\\Plpm\\Pgg}< 0.2$ is applied, which is\n  consistent with a smaller impact of the EW corrections.  Note that\n  the role of pseudo-data and templates is reversed in\n  Ref.~\\cite{CarloniCalame:2003ux} so that the mass shift has the\n  opposite sign.}  \nAlternatively, the effect of the EW corrections can\nbe estimated by comparing the value of $\\MW$ obtained from a fit to\nthe naive product of EW and QCD\ncorrections~\\eqref{eq:def:nnlo:naive:fact} to the result of a fit to\nthe NLO QCD cross section. The results are consistent with the shift\nestimated from the NLO EW corrections alone. \n\nWe have also estimated the effect of multi-photon radiation on the\n$\\MW$ measurement in the bare-muon case using the\nstructure-function approach given in  Eq.~\\eqref{eq:LL1FSR}. As\ndiscussed in detail in Ref.~\\cite{Brensing:2007qm} we match the exponentiated LL-FSR\ncorrections evaluated in the  $\\alpha(0)$-scheme to\nthe NLO calculation in the $\\alpha_{G_\\mu}$-scheme, avoiding double-counting. \nWe obtain a mass shift $\\Delta\\MW^{\\mathrm{FSR}}\\approx 9~\\MeV$\nrelative to the result of the fit to the NLO EW prediction, which\nis in qualitative agreement with the result of\nRef.~\\cite{CarloniCalame:2003ux}.\n\nTo estimate the impact of the  initial--final \\order{\\alphas\\alpha}\ncorrections  we consider the mass shift relative to the full NLO result,\n\\begin{equation}\n  \\label{eq:dmwnnlo}\n  \\Delta\\MW^{\\NNLO}=\\MW^{\\mathrm{fit},\\NNLO_{\\rs\\otimes\\rew}^{\\pro\\times\\dec}}\n  -\\MW^{\\mathrm{fit},\\NLO_{\\rs\\oplus\\rew}}\n\\end{equation}\nwhere $\\MW^{\\mathrm{fit},\\NNLO_{\\rs\\otimes\\rew}^{\\pro\\times\\dec}}$ is the\nresult of using our best\nprediction~\\eqref{eq:def:nnlo:if} to generate the pseudo-data, while\nthe sum of the NLO QCD and EW corrections is used for \n$\\Delta\\MW^{\\mathrm{fit},\\NLO_{\\rs\\oplus\\rew}}$. The resulting $\\Delta\\chi^2$\ndistributions for the mass shift are shown in the right-hand plot in\nFig.~\\ref{fig:chi2}.  \nIn the bare-muon case, we obtain\na mass shift due to \\order{\\alphas\\alpha} corrections of $\\Delta\n\\MW^{\\mathrm{NNLO}}\\approx -14~\\MeV$ while for the dressed-lepton case we\nget $\\Delta \\MW^{\\mathrm{NNLO}}\\approx -4~\\MeV$.  \n\nIdentical shifts result from replacing the NNLO prediction by the\nnaive product~\\eqref{eq:def:nnlo:naive:fact}, which is expected from\nthe good agreement for the $M_{\\rT,\\nu \\ell}$-spectrum in\nFig.~\\ref{fig:distNNLO-IF-Wp}.  Using instead the leading-logarithmic\napproximation of the final-state photon radiation obtained using\nPHOTOS to compute the \\order{\\alphas\\alpha} corrections, we obtain a\nmass shift of $\\Delta\\MW^{\\NNLO}=-11~\\MeV$ ($-4~\\MeV$) for the\nbare-muon (dressed-lepton) case.  The effect of the\n\\order{\\alphas\\alpha} corrections on the mass measurement is therefore\nof a similar or larger magnitude than the effect of multi-photon\nradiation.\n  We emphasize that \nthe result $\\Delta\\MW^{\\NNLO}\\approx -14~\\MeV$ is a simple estimate of the\n impact of the full \\order{\\alphas\\alpha} corrections\non the $\\MW$ measurement.  The order of magnitude shows that these\ncorrections must \nbe taken into account properly in order to reach the $10~\\MeV$ accuracy\ngoal of the LHC experiments. \nIt is beyond the scope of this paper to\nvalidate the accuracy of the \nprevious and current theoretical modelling used by the\nexperimental collaborations in the $\\MW$ measurements, which includes\nthe \\order{\\alphas\\alpha} corrections in some approximation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection*{Singularities of energy minimizing harmonic maps}\n\nEnergy minimizing harmonic maps between manifolds may have singularities if the domain dimension is $3$ or higher. The most well-known example is the map \n\\[\n{\\ensuremath{\\mathbb R}}^3 \\times {\\ensuremath{\\mathbb R}}^{n-3} \\ni (x,y) \\mapsto x\/|x| \\in {\\ensuremath{\\mathbb S}}^2. \n\\]\nIn general, any energy minimizer $u$ is smooth outside the closed singular set $\\operatorname{sing} u$ of Hausdorff dimension $n-3$ or less, $n$ being the dimension of the domain (Schoen, Uhlenbeck \\cite{SchUhl82,SchUhl83cor}). The phenomenon of singularities is now well-understood in dimension $3$, when singularities form a discrete set. In recent years, there has been a~substantial progress concerning the case $n \\ge 4$. Naber and Valtorta \\cite{NabVal17} have proved that the singular set has locally finite $(n-3)$-dimensional Hausdorff measure and is $(n-3)$-rectifiable, i.e., can be essentially covered by countably many Lipschitz images of ${\\ensuremath{\\mathbb R}}^{n-3}$; the latter was already known (due to Simon \\cite{Sim95}) in the case when the target manifold is real-analytic. \n\nThe results cited above are mostly concerned with the size of the singular set, but do not imply \\textit{lower bounds} on the singular set. In particular, the possibility that the singular set is an arbitrary subset of an $(n-3)$-dimensional manifold (with many small gaps) is not excluded by \\cite{NabVal17,SchUhl82,SchUhl83cor,Sim95}. \n\nLower bounds on the size are indeed possible in the presence of a~topological obstruction; the following example is simple but instructive. \n\n\\begin{ex}\nConsider the smooth boundary map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\times {\\ensuremath{\\mathbb S}}^1 \\to {\\ensuremath{\\mathbb S}}^2$ given by $\\varphi(x,y) = x$ and (some) $u \\colon \\B^3 \\times {\\ensuremath{\\mathbb S}}^1 \\to {\\ensuremath{\\mathbb S}}^2$ minimizing the energy in the class of maps equal to $\\varphi$ on the boundary. Restricting $u$ to a slice $\\B^3 \\times \\{ y \\}$ and applying Brouwer's theorem, we see that each such slice contains a singular point. This shows that $\\mathcal{H}^1(\\operatorname{sing} u) \\ge \\mathcal{H}^1({\\ensuremath{\\mathbb S}}^1) = 2 \\pi$. In this particular case one can actually prove that $u(x,y)=x\/|x|$, but the presented reasoning applies also to any $\\varphi$ in the same homotopy class. \n\\end{ex}\n\nIn the special case of maps $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$, Hardt and Lin \\cite{HarLin90} obtained the following remarkable result. \n\\begin{thm}\n\\label{thm:HarLin90}\nThe singular set of an energy minimizer $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$ is locally a~union of a~finite set and a~finite family of H{\\\"o}lder continuous closed curves with a~finite number of crossings. \n\\end{thm}\nThe same claim was obtained also for maps $u \\colon \\B^5 \\to {\\ensuremath{\\mathbb S}}^3$ (Lin-Wang \\cite{LinWan06}). To the author's knowledge, these are the only two cases where $\\operatorname{sing} u$ was shown to be essentially a~manifold.  \n\nThe above theorem relies on the classification of tangent maps from ${\\ensuremath{\\mathbb R}}^3$ into ${\\ensuremath{\\mathbb S}}^2$ carried out by Brezis, Coron and Lieb \\cite{BreCorLie86}; for ${\\ensuremath{\\mathbb S}}^3$, a similar classification was obtained by Nakajima \\cite{Nak06}. These maps describe the infinitesimal behavior of $u$ at a typical point of $\\operatorname{sing} u$. \n\n\\subsection*{Main results}\n\nThe present paper aims to extract the topological obstruction responsible for preventing gaps in the singular set of maps into ${\\ensuremath{\\mathbb S}}^2$. To this end, we distinguish particular homotopy classes of tangent maps ${\\ensuremath{\\mathbb R}}^3 \\to {\\ensuremath{\\mathcal N}}$ (called here \\textit{indecomposable classes}) for any closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$. \n\nTo each homotopy class $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ we assign its lowest energy level $\\Theta(\\alpha)$ and call $\\alpha$ indecomposable if $\\Theta(\\alpha) < \\infty$ and $\\alpha$ cannot be represented as a sum of homotopy classes $\\alpha_j \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with strictly smaller energy levels $\\Theta(\\alpha_j)$. We then restrict our attention to singularites with fixed topological type $\\alpha$ -- we define $\\operatorname{sing}_\\alpha u$ to be the set of points at which some tangent map of $u$ has type $\\alpha$. Rigorous definitions are given in Section~\\ref{ch:HCMs}. \n\nAnother goal is to generalize the result of Hardt and Lin \\cite{HarLin90} to higher dimensional domains. The difficulty lies in the fact that the singular set is stratified -- it decomposes into parts of different dimensions. For $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$, there are only two strata: one is formed by H{\\\"o}lder continuous curves and the other by their crossing points and some isolated points. In the theorem below, we were only able to study the top-dimensional part $\\operatorname{sing}_* u$ of the singular set. Again, the necessary notions are introduced in Section \\ref{ch:tangent-maps}. \n\nFor simplicity, we only consider the standard Euclidean ball $\\B^n$ as the domain, but the results hold true for any manifold. This is due to the fact that we only consider the infinitesimal behavior of maps. A detailed explanation can be found in \\cite{NabVal17} and \\cite[Sec.~8]{Sim95}. \n\n\\begin{thm}\n\\label{thm:Holder-regularity}\nLet $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ be an energy minimizing map into a closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$, $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ be an indecomposable homotopy class, and $\\Theta(\\alpha)$ be its lowest energy level. Then for each exponent $0 < \\gamma < 1$ there is $\\delta(\\gamma,n,\\alpha,{\\ensuremath{\\mathcal N}}) > 0$ such that the set \n\\[\n\\left \\{\nx \\in \\operatorname{sing}_\\alpha u : \n\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 < \\Theta(\\alpha) + \\delta\n\\right \\}\n\\]\nforms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$. \n\\end{thm}\n\nIn the case when ${\\ensuremath{\\mathcal N}}$ is a real-analytic manifold, Simon \\cite[Lemma~4.3]{Sim95} showed that the set of possible energy densities $\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2$ is discrete. This allows us to slightly strenghten the statement above. The same conclusion holds also if ${\\ensuremath{\\mathcal N}}$ satisfies the integrability assumption introduced in \\cite[Ch.~3.13]{Sim96}. \n\n\\begin{cor}\n\\label{cor:discrete-levels}\nIf $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ is an energy minimizing map into a real-analytic manifold ${\\ensuremath{\\mathcal N}}$ and $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ is an indecomposable homotopy class, then \n\\[\n\\left \\{\nx \\in \\operatorname{sing}_\\alpha u : \n\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 = \\Theta(\\alpha) \n\\right \\}\n\\]\nforms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$ with any $0 < \\gamma < 1$. \n\\end{cor}\n\nSpecializing to the case ${\\ensuremath{\\mathcal N}} = {\\ensuremath{\\mathbb S}}^2$ and recalling the classification of tangent maps \\cite{BreCorLie86}, we obtain a partial generalization of Theorem \\ref{thm:HarLin90} \\cite{HarLin90} to arbitrary dimensions: \n\n\\begin{cor}\n\\label{cor:sphere-target}\nIf $u \\colon \\B^n \\to {\\ensuremath{\\mathbb S}}^2$ is an energy minimizing map, then the top-dimensional part $\\operatorname{sing}_* u$ forms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$ with any $0 < \\gamma < 1$. \n\\end{cor}\n\n\\subsection*{An outline}\n\nSection \\ref{ch:preliminaries} recalls all notions and results needed in the sequel. These mostly come from the seminal work of Schoen and Uhlenbeck \\cite{SchUhl82}, but the presentation here follows Simon's lecture notes \\cite{Sim96}. We also define indecomposable homotopy classes of maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$.\n\nSince Theorem \\ref{thm:Holder-regularity} only concerns the singularities of indecomposable types, it is worthwhile to investigate the existence of such classes, which we do in Section \\ref{ch:good-classes-exist}. Indeed, we show that for any ${\\ensuremath{\\mathcal N}}$ the second homotopy group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ is generated (up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$ on $\\pi_2({\\ensuremath{\\mathcal N}},p)$) by indecomposable homotopy classes. This is very close to the classical (slightly weaker) result due to Sacks and Uhlenbeck \\cite{SacUhl81} (see also \\cite{Str85}): smooth harmonic maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$ generate the whole group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ (up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$). \n\nTo obtain bi-H{\\\"o}lder-equivalence with a Euclidean ball, we employ Reifenberg's topological disc theorem \\cite{Rei60} (see also~\\cite{Sim-notes}). We recall its statement and the so-called Reifenberg flatness condition in Section \\ref{ch:flatness}. We also introduce a flatness condition for an energy minimizer $u$ which includes Reifenberg flatness for $\\operatorname{sing} u$, but also forces $u$ to be close to a tangent map. \n\nThe main results are proved in Section \\ref{ch:proof}. The difficulty in applying Reifenberg's theorem to $\\operatorname{sing} u$ lies in showing that this set has no gaps. This is done in Lemma \\ref{lem:no-drop}; this is also the point where our topological assumptions play a role. Then we are able to show that if $u$ satisfies the flatness condition on the ball $\\B_2(0)$, it also satisfies the same condition on each smaller ball $\\B_r(0)$ (Corollary \\ref{cor:all-scales}) and on each ball $\\B_1(z)$ centered at a point $z \\in \\B_1$ with enough energy density (Proposition \\ref{prop:all-balls}). Combining these results, we check the hypotheses of Reifenberg's theorem and establish Theorem \\ref{thm:Holder-regularity}.\n\nSome interesting observations not needed for the proof of Theorem \\ref{thm:Holder-regularity} are gathered in Section \\ref{ch:additional}. \n\n\\section{Preliminaries}\n\\label{ch:preliminaries}\n\n\\subsection*{Regularity of energy minimizers}\n\nIn what follows, $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ is an energy minimizing map into a closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$. Here we recall the basic properties of such maps \\cite{SchUhl82}. \n\nDenote the rescaled energy \n\\[\n\\theta_u(x,r) := r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 \n\\quad \\text{for } \\B_r(x) \\subseteq \\B_1, \n\\]\nwhich is monotone in $r$: \n\\[\n\\tfrac{\\partial}{\\partial r} \\theta_u(x,r) = 2 \\int_{\\partial \\B_r(x)} \\frac{|\\nabla u \\cdot (y-x)|^2}{|y-x|^n} \\ge 0.\n\\]\nThis enables us to define the energy density at $x$: \n\\[\n\\theta_u(x,0) := \\lim_{r \\to 0} \\theta_u(x,r),\n\\]\nwhich is by definition an upper semicontinuous function (in both $x \\in \\B^n$ and $u \\in W^{1,2}$) \\cite[2.11]{Sim96}. Obviously, $\\theta_u(x,0)=0$ at regular points. \n\nThe main regularity statement is the following $\\varepsilon$-regularity theorem: \n\\begin{align}\n\\label{eq:eps-regularity}\n\\text{there is } \\varepsilon(n,{\\ensuremath{\\mathcal N}})>0 \\text{ s.t. } \\theta_u(x,2r) < \\varepsilon \n& \\Rightarrow u \\text{ is smooth on } \\B_r(x), \\\\\n\\nonumber\n\\text{in particular } \\theta_u(x,0) < \\varepsilon & \\Rightarrow x \\notin \\operatorname{sing} u.\n\\end{align}\n\nWe also note two compactness theorems for a sequence $u_k$ of energy minimizers: \n\\begin{itemize}\n\\item\nif $u_k \\rightharpoonup u$ in $H^1$, then $u$ is an energy minimizer and the convergence is actually strong in $H^1$ \\cite{Luc88}, \n\\item\nif $u_k \\to u$ as above, then the convergence is uniform on compact sets disjoint from $\\operatorname{sing} u$ \\cite{SchUhl82}. \n\\end{itemize}\n\n\\subsection*{Tangent maps}\n\\label{ch:tangent-maps}\n\nGiven an energy minimizer $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ and a point $x \\in \\B^n$, consider the family of rescaled maps $u_r(y) = u(x+ry)$. By the results from the previous section, each sequence $r_j \\to 0$ has a subsequence for which $u_{r_j}$ converges in $W^{1,2}_\\mathrm{loc}({\\ensuremath{\\mathbb R}}^n)$ to some energy minimizer $\\varphi$, called a tangent map of $u$ at $x$ (possibly non-unique). This limit map is homogeneous, i.e., $\\varphi(\\lambda x) = \\varphi(x)$ for all $\\lambda > 0$, $x \\in {\\ensuremath{\\mathbb R}}^n$. \n\nFor a homogeneous energy minimizer $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$, the energy density $\\theta_\\varphi(y,0)$ is maximal at $y=0$; moreover, equality $\\theta_\\varphi(y) = \\theta_\\varphi(0)$ at some other point $y$ leads to higher symmetry: $\\varphi(x+ty) = \\varphi(x)$ for all $t \\in {\\ensuremath{\\mathbb R}}$, $x \\in {\\ensuremath{\\mathbb R}}^n$. Let $S(\\varphi)$ be defined by \n\\[\nS(\\varphi) = \\left\\{ y \\in {\\ensuremath{\\mathbb R}}^n : \\theta_\\varphi(y) = \\theta_\\varphi(0) \\right\\}. \n\\]\nThen $S(\\varphi)$ is a linear subspace of ${\\ensuremath{\\mathbb R}}^n$ describing the symmetries of $\\varphi$: \n\\[\n\\varphi(x+y) = \\varphi(x) \\quad \\text{for all } x \\in {\\ensuremath{\\mathbb R}}^n, \\ y \\in S(\\varphi). \n\\]\nFor non-constant $\\varphi$, we have $S(\\varphi) \\subseteq \\operatorname{sing} \\varphi$. If $\\dim S(\\varphi) = n-3$, this is necessarily an equality. \n\n\\subsection*{Top-dimensional part of the singular set}\n\\label{ch:HCMs}\n\nIf $u$ is an energy minimizer, for each $j = 0,1,2,\\ldots,n-1$ we define \n\\[\nS_j = \\left\\{ y \\in \\operatorname{sing} u : \\dim S(\\varphi) \\le j \\text{ for all tangent maps $\\varphi$ of $u$ at $y$} \\right\\},\n\\]\nwhich leads to the classical stratification of the singular set \n\\[\nS_0 \\subseteq S_1 \\subseteq \\ldots \\subseteq S_{n-3} = S_{n-2} = S_{n-1} = \\operatorname{sing} u.\n\\]\nIt is known \\cite{SchUhl82} that each $S_j$ has Hausdorff dimension at most $j$, in particular \n\\[\n\\dim_H \\operatorname{sing} u \\le n-3.\n\\] \n\n\\medskip\n\nIn what follows, we are interested in the top-dimensional part of the singular set: \n\\begin{align*}\n\\operatorname{sing}_* u & = S_{n-3} \\setminus S_{n-4} \\\\\n& = \\left\\{ y \\in \\operatorname{sing} u : \\dim S(\\varphi) = n-3 \\text{ for some tangent map $\\varphi$ of $u$ at $y$} \\right\\}. \n\\end{align*}\nNote that \n\\[\n\\dim_H (\\operatorname{sing} u \\setminus \\operatorname{sing}_* u) \\le n-4. \n\\]\nFollowing \\cite{Sim96}, we shall call any homogeneous energy minimizing $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $\\dim S(\\varphi) = n-3$ a \\textit{homogeneous cylindrical map} (abbreviated HCM). \n\n\\medskip\n\nConsider now such a HCM $\\varphi_0 \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $S(\\varphi_0) = {\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$. This map actually depends only on $3$ variables, i.e. $\\varphi_0(x,y) = \\varphi_1(y)$ for some homogeneous $\\varphi_1 \\colon {\\ensuremath{\\mathbb R}}^3 \\to {\\ensuremath{\\mathcal N}}$. By \\cite[Lemma~2.1]{HarLin90}, the map $\\varphi_1$ defined in this way is energy minimizing if and only if $\\varphi_0$ is. Since $\\varphi_1$ is homogeneous, it is uniquely determined by its restriction to the unit sphere $\\varphi_2 \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$, which is a smooth harmonic map. \n\nFrom now on, we shall abuse the notation and use the same symbol for all three maps $\\varphi_0,\\varphi_1,\\varphi_2$; the precise meaning should be clear from the context. Note that their energies differ by a multiplicative constant: \n\\[\n\\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi_2|^2 = \\int_{\\B_1^3} |\\nabla \\varphi_1|^2 = C(n) \\int_{\\B_1^n} |\\nabla \\varphi_0|^2, \n\\]\nso energy comparison does not lead to confusion. \n\n\\textit{Homotopy type} of a HCM always refers to the map $\\varphi_2 \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ (as $\\varphi_0,\\varphi_1$ are discontinuous and defined on contractible domains). For a general HCM $\\varphi_0$ we may choose a rotation $q$ that maps $S(\\varphi_0)$ to ${\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$ and thus reduce to the previous case. We then say that $\\varphi_0$ has homotopy type $\\alpha$ if $\\varphi_0 \\circ q^{-1}$ restricted to $\\mathbf{0} \\times {\\ensuremath{\\mathbb S}}^2$ has type $\\alpha$. \n\n\\begin{rem}\nThere is a subtle ambiguity here. Depending on the choice of $q$, we may obtain two homotopy types that differ by a composition with the antipodal map, i.e. both $[\\varphi_2(x)]$ and $[\\varphi_2(-x)]$. \n\\end{rem}\n\nUsing this terminology, singular points in $\\operatorname{sing}_* u$ can be classified according to their energy density and the homotopy type of a tangent map. Since we only consider basepoint-free homotopies, we denote by $\\pi_2({\\ensuremath{\\mathcal N}})$ the set of homotopy classes of continuous maps ${\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$. Note that in general it does not carry a group structure, as it is the quotient of the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$ on $\\pi_2({\\ensuremath{\\mathcal N}},p)$. \n\\begin{df}\n\\label{df:notation}\nFor any homotopy type $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ we let \n\\[\n\\operatorname{sing}_{\\alpha} u = \n\\left\\{ y \\in \\operatorname{sing} u : \\text{ some tangent map of $u$ at $y$ is a HCM of type } \\alpha \\right\\}. \n\\]\nWe also denote its lowest energy level by \n\\[\n\\Theta(\\alpha) := \\inf \\left \\{ \\int_{\\B_1^n} |\\nabla \\varphi|^2 : \\varphi \\text{ is a HCM of type } \\alpha \\right \\}.\n\\]\nA simple compactness argument shows that this infimum is either infinite (if no HCM has type $\\alpha$) or achieved by some minimal HCM. We also let \n\\[\n\\operatorname{sing}_{\\ge \\Theta} u = \n\\left\\{ y \\operatorname{sing} u : \\theta_u(y,0) \\ge \\Theta \\right\\}. \n\\]\nwhich is a closed set by upper semicontinuity of $\\theta_u(\\cdot,0)$. \n\\end{df}\n\nAt this point we cannot exclude the case when there are many homotopically different tangent maps at one point. However, this cannot happen under an additional assumption described below (see Remark \\ref{rem:unique-type}). Again, since $\\pi_2({\\ensuremath{\\mathcal N}})$ is not necessarily a group, the decomposition in Definition \\ref{df:decomposition} is to be understood up to the action of $\\pi_1({\\ensuremath{\\mathcal N}})$, as described in Section \\ref{ch:good-classes-exist} (see also the formulation of \\cite[Thm.~5.9]{SacUhl81}). \n\n\\begin{df}\n\\label{df:decomposition}\nConsider $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with $\\Theta(\\alpha) < \\infty$. This homotopy class is called decomposable if there is a decomposition \n\\[\n\\alpha = \\alpha_1 + \\ldots + \\alpha_k \\quad \\text{in } \\pi_2({\\ensuremath{\\mathcal N}}),\n\\]\nwhere $\\Theta(\\alpha_j) < \\Theta(\\alpha)$ for each $j=1,\\ldots,k$. Otherwise $\\alpha$ is called indecomposable. \n\\end{df}\n\nNote that the above criterion does not depend on the dimension $n$, but only on the manifold ${\\ensuremath{\\mathcal N}}$. \n\nAs a special case, $\\alpha$ is indecomposable if $\\Theta(\\alpha)$ is the smallest among all non-trivial homotopy types. In this case the proof is much easier (see the remark below Lemma \\ref{lem:no-drop}).\n\n\\medskip\n\nSimilar decompositions of this type appear naturally as a result of the \\textit{bubbling phenomenon} when one tries to minimize the energy in a given homotopy class. More precisely, recall that by \\cite{SacUhl81} (see also \\cite{Str85}) any smooth map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ can be decomposed as $[\\varphi] = [\\varphi_1] + \\ldots + [\\varphi_k]$, where each $\\varphi_j$ is a harmonic map and \n\\[\n\\sum_{j=1}^k \\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi_j^2| \\le \\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi^2|.\n\\]\nMotivated by these decompositions, one could replace the condition $\\Theta(\\alpha) > \\max_j \\Theta(\\alpha_j)$ in Definition \\ref{df:decomposition} by $\\Theta(\\alpha) \\ge \\sum_j \\Theta(\\alpha_j)$, thus enlarging the set of indecomposable classes. \nA natural conjecture here would be that Theorem \\ref{thm:Holder-regularity} continues to hold in this case, but the author was not able to verify it. \n\n\\begin{ex}\nBy the classification from \\cite{BreCorLie86}, the only HCMs into the sphere ${\\ensuremath{\\mathbb S}}^2$ are isometries $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathbb S}}^2$. Thus, for $\\alpha \\in \\pi_2({\\ensuremath{\\mathbb S}}^2)$ we have \n\\[\n\\Theta(\\alpha) = \n\\begin{cases}\n0 & \\text{ for } \\alpha = 0, \\\\\n4 \\pi & \\text{ for } \\alpha = [\\pm \\operatorname{id}], \\\\\n\\infty & \\text{ otherwise.}\n\\end{cases}\n\\]\nBy Definition \\ref{df:decomposition}, the indecomposable classes here are $0,[\\operatorname{id}],[-\\operatorname{id}]$. Note that these classes generate the whole group $\\pi_2({\\ensuremath{\\mathbb S}}^2)$ (see Proposition \\ref{prop:indecomposable-types} for the general case). \n\\end{ex}\n\n\\section{Existence of indecomposable homotopy classes}\n\\label{ch:good-classes-exist}\n\nWe show that the set of all indecomposable homotopy classes generates $\\pi_2({\\ensuremath{\\mathcal N}})$. Similarly to \\cite[Thm.~5.9]{SacUhl81}, we only consider basepoint-free homotopies, so this statement should be understood as generating $\\pi_2({\\ensuremath{\\mathcal N}},p)$ up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$. In other words, for any $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ there are indecomposable homotopy classes $\\alpha_1,\\ldots,\\alpha_k \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and a continuous map \n\\[\nu \\colon \\B^3 \\setminus \\bigcup_{j=1}^k \\B_j \\to {\\ensuremath{\\mathcal N}} \n\\]\nsuch that $u|_{\\partial \\B} \\in \\alpha$ and $u|_{\\partial \\B_j} \\in \\alpha_j$, where $\\B_j \\Subset \\B$ are smaller disjoint balls. \n\n\\medskip\n\nThis can be divided into two steps as follows. \n\n\\begin{prop}\n\\label{prop:indecomposable-types}\nLet ${\\ensuremath{\\mathcal N}}$ be a closed Riemannian manifold. Then \n\\begin{enumerate}[(a)]\n\\item\nthe set of all HCMs $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ generates $\\pi_2({\\ensuremath{\\mathcal N}})$, \n\\item\neach HCM $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ as an element of $\\pi_2({\\ensuremath{\\mathcal N}})$ can be decomposed into indecomposable homotopy classes. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nTo show part (a), fix $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and choose a smooth map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ of this type. Then there exists (possibly non-unique) $u \\in W^{1,2}(\\B_1^3,{\\ensuremath{\\mathcal N}})$ such that \n\\[\n\\int_{\\B_1} |\\nabla u|^2 = \\min \\left \\{ \\int_{\\B_1} |\\nabla v|^2 : v \\in W^{1,2}(\\B_1,{\\ensuremath{\\mathcal N}}), \\ v = \\varphi \\text{ on } {\\ensuremath{\\mathbb S}}^2 \\right \\}.\n\\]\nSuch a minimizer has at most a~finite number of interior singularities $p_1,\\ldots,p_k \\in \\B_1$. At each $p_j$ there is a~(possibly non-unique) tangent map $\\varphi_j$, which is necessarily a~HCM; by uniform convergence away from the singularity, $u$ restricted to $\\partial \\B_r(p_j)$ is homotopic to $\\varphi_j$ for some arbitrary small $r$ (in consequence, also for all sufficiently small $r$). This yields the decomposition \n\\[\n[\\varphi] = [\\varphi_1] + \\ldots + [\\varphi_k] \\quad \\text{ in } \\pi_2({\\ensuremath{\\mathcal N}}). \n\\]\n\nPart (b) follows from the definition by a compactness argument, which allows us to exclude infinite decompositions. Consider any homotopy type $\\alpha \\in {\\ensuremath{\\mathcal N}}$ represented by a HCM, i.e. with $\\Theta(\\alpha) < \\infty$. First let us show that there are only finitely many homotopy types $\\beta \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with $\\Theta(\\beta) \\le \\Theta(\\alpha)$. Indeed, otherwise we would have an infinite sequence of HCMs $\\varphi_k \\colon \\B_1^3 \\to {\\ensuremath{\\mathcal N}}$ with distinct homotopy types and uniformly bounded energy. Without loss of generality, $\\varphi_k$ converges to some HCM $\\varphi$ in $W^{1,2}(\\B_1)$, but also in $C^0({\\ensuremath{\\mathbb S}}^2)$. This shows that almost all $\\varphi_k$ have the same homotopy type as $\\varphi$, which is a contradiction. \n\nIf $\\alpha$ is decomposable, we have $\\alpha = \\alpha_1 + \\ldots + \\alpha_k$, where $\\Theta(\\alpha_j) < \\Theta(\\alpha)$ for each $j$. Decomposing further each $\\alpha_j$ whenever possible, and iterating this procedure until all obtained homotopy types are indecomposable, we arrive at claim (b). One only needs to note that this procedure stops after at most $N$ steps, where $N$ is the number of homotopy types from the last paragraph. Indeed, any branch of the decomposition tree is a sequence $\\beta_0, \\beta_1, \\beta_2, \\ldots$ with $\\beta_0 = \\alpha$ and $\\Theta(\\beta_{j+1}) < \\Theta(\\beta_j)$ for each $j$, so it contains at most $N$ elements. \n\\end{proof}\n\nWe remark that a similar decomposition was obtained by Sacks and Uhlenbeck \\cite{SacUhl81} (see also \\cite{Str85}): smooth harmonic maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$ generate the whole group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$. It may be that some homotopy classes in $\\pi_2({\\ensuremath{\\mathcal N}})$ do not contain any harmonic map. Since here we only consider harmonic maps $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ for which the homogeneous extension $\\varphi \\colon \\B^3 \\to {\\ensuremath{\\mathcal N}}$ is energy minimizing, our result is a slight generalization. \n\n\\section{Notions of flatness and Reifenberg's topological disc theorem}\n\\label{ch:flatness}\n\nH{\\\"o}lder regularity of the singular set will be obtained by an application of Reifenberg's topological disc theorem \\cite{Rei60} (see also~\\cite{Sim-notes}). To state it, we first need the following notion of flatness (for our purposes restricted to codimension $3$). \n\n\\begin{df}\n\\label{df:set-flatness}\nA set $A \\subseteq {\\ensuremath{\\mathbb R}}^n$ is said to be $\\varepsilon$-Reifenberg flat in the ball $\\B_r(x)$ (with respect to $L$) if \n\\[\nA \\cap \\B_r(x) \\subseteq \\B_{r \\varepsilon} L \n\\quad \\text{and} \\quad\nL \\cap \\B_r(x) \\subseteq \\B_{r \\varepsilon} A \n\\]\nfor some $(n-3)$-dimensional affine plane $L$ through $x$.\n\\end{df}\n\nThe above condition means exactly that the normalized Hausdorff distance on $\\B_r(x)$ from $A$ to some $(n-3)$-dimensional affine plane through $x$ is not larger than~$\\varepsilon$. \n\n\\begin{thm}[Reifenberg's topological disc theorem]\n\\label{thm:Reifenberg}\nFor each H{\\\"o}lder exponent $0 < \\gamma < 1$ there is $\\varepsilon(n,\\gamma) > 0$ such that the following holds. If a closed set $A \\subseteq {\\ensuremath{\\mathbb R}}^n$ containing the origin is $\\varepsilon$-Reifenberg flat in each ball $\\B_r(x)$ with $x \\in A \\cap \\B_1$ and $r < 1$, then the set $A \\cap \\B_1$ is bi-H{\\\"o}lder equivalent to the closed unit ball $\\B^{n-3} \\subseteq {\\ensuremath{\\mathbb R}}^{n-3}$ with exponent $\\gamma$. \n\\end{thm}\n\n\\medskip\n\nWe shall also make repeated use of the following condition for energy minimizing maps. \n\n\\begin{df}\n\\label{df:flatness}\nFix an indecomposable homotopy class $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and let $\\Theta = \\Theta(\\alpha)$ be its lowest energy level (as in Definition \\ref{df:notation}). We say that an energy minimizer $u$ is $\\delta$-flat in the ball $\\B_r(x)$ (of type $\\alpha$) if \n\\begin{enumerate}\n\\item\n\\label{df:flatness-energy}\n$x$ is a singular point of $u$ and $\\Theta \\le \\theta_u(x,0) \\le \\theta_u(x,r) \\le \\Theta + \\delta$, \n\\item\n\\label{df:flatness-Reifenberg}\n$\\operatorname{sing} u$ is $\\frac{1}{10}$-Reifenberg flat in $\\B_r(x)$ with respect to some $L$, and $u$ restricted to $(x+L^\\perp) \\cap \\partial \\B_r(x)$ has homotopy type $\\alpha$. \n\\end{enumerate}\n\\end{df}\n\nFrom now on, we consider a non-trivial indecomposable class $\\alpha$ and its lowest energy level $\\Theta = \\Theta(\\alpha)$ to be fixed. \n\nThe main feature of this definition is that $\\delta$-flatness in a ball trivially ensures that condition \\ref{df:flatness-energy} is satisfied in all smaller concentric balls, and one only needs to check condition \\ref{df:flatness-Reifenberg} (see Corollary \\ref{cor:all-scales}). This is why the constant $\\tfrac{1}{10}$ in condition \\ref{df:flatness-Reifenberg} was chosen reasonably weak. \n\n\\section{Regularity of the singular set}\n\\label{ch:proof}\n\n\\subsection*{Stability of indecomposable singularities}\n\nAs a first step, we show that if an energy minimizer $u$ restricted to some sphere has homotopy type $\\alpha$, then $\\operatorname{sing} u$ satisfies the flatness condition of Definition \\ref{df:flatness} and the energy density of $u$ cannot drop in a smaller ball. Note that the claim of Lemma \\ref{lem:no-drop} is essentially stronger than the condition $L \\cap \\B_1 \\subseteq \\B_{\\varepsilon} (\\operatorname{sing}_{\\ge \\Theta} u)$ from Definition \\ref{df:set-flatness}. \n\n\\medskip\n\nObserve that some tubular neighborhood $\\B_{\\eta} {\\ensuremath{\\mathcal N}} \\subseteq {\\ensuremath{\\mathbb R}}^M$ admits a continuous retraction $\\pi_{\\ensuremath{\\mathcal N}}$ onto ${\\ensuremath{\\mathcal N}}$. As a consequence, if two continuous functions $f,g$ into ${\\ensuremath{\\mathcal N}} \\subseteq {\\ensuremath{\\mathbb R}}^M$ are close enough in supremum norm, then \n\\[\n(t,x) \\mapsto \\pi_{{\\ensuremath{\\mathcal N}}}(tf(x) + (1-t)g(x)) \n\\]\nyields a homotopy between them. \n\n\\begin{lem}\n\\label{lem:no-drop}\nAssume that $\\operatorname{sing} u \\cap \\B_1 \\subseteq \\B_{\\varepsilon} L$ for some $0 < \\varepsilon < \\tfrac 12$ and some $(n-3)$-dimensional plane $L$ through $0$. Assume further that $u$ restricted to $L^\\perp \\cap \\partial \\B_1$ has homotopy type $\\alpha$. Then \n\\[\nL \\cap \\B_{1-\\varepsilon} \\subseteq \\pi_L( \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1 ),\n\\]\nwhere $\\pi_L$ denotes the orthogonal projection onto $L$. In particular, $\\operatorname{sing}_{\\ge \\Theta} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$. \n\\end{lem}\n\nBefore giving the full proof, let us consider the special case when $\\Theta(\\alpha)$ is the lowest among all non-trivial homotopy types. In this case, the proof is simpler and does not depend on the deep results of Naber and Valtorta \\cite{NabVal17}.\n\nFor each $y \\in L \\cap \\B_{1-\\varepsilon}$, $u$ restricted to the sphere $(y + L^\\perp) \\cap \\partial \\B_\\varepsilon(y)$ has homotopy type $\\alpha$, hence cannot be continuously extended to the ball $(y + L^\\perp) \\cap \\B_\\varepsilon(y)$. This shows the weaker inclusion $L \\cap \\B_{1-\\varepsilon} \\subseteq \\pi_L( \\operatorname{sing} u \\cap \\B_1 )$. Recall that $\\mathcal{H}^{n-3}$-a.e. point $z \\in \\operatorname{sing} u$ belongs to $\\operatorname{sing}_* u$ and hence $\\theta_u(z,0) \\ge \\Theta$ due to our additional assumption. Since $\\operatorname{sing}_{\\ge \\Theta} u$ is a closed set, we obtain the stronger inclusion. \n\n\\begin{proof}[Proof of Lemma \\ref{lem:no-drop}]\nFor simplicity, let us rotate so that $L = {\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$. Assume for the contrary that $L \\cap \\B_{1-\\varepsilon}$ is not covered by the projection of $\\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$. Since the latter is a compact set, it has to be disjoint with some cylinder $\\B^{n-3}_\\delta(z) \\times {\\ensuremath{\\mathbb R}}^3$. \n\nRecall that by the recent important work of Naber and Valtorta \\cite{NabVal17}, the set $\\operatorname{sing} u \\cap \\B_1$ has finite $\\mathcal{H}^{n-3}$ measure. Moreover, the set $\\operatorname{sing} u$ is $(n-3)$-rectifiable and for $\\mathcal{H}^{n-3}$-a.e. $y \\in \\operatorname{sing} u$ there exists an $(n-3)$-dimensional tangent plane $\\operatorname{Tan}(\\operatorname{sing} u,y)$ coinciding with $S(\\varphi)$ for every tangent map $\\varphi$ of $u$ at $y$. Let us temporarily assume that these tangent planes are transversal to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^{n-3}$, i.e. \n\\begin{equation}\n\\label{eq:transverse}\n\\operatorname{Tan}(\\operatorname{sing} u,y) \\pitchfork \\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3 \\quad \\text{for $\\mathcal{H}^{n-3}$-a.e. } y \\in \\operatorname{sing} u \\cap \\B_1. \n\\end{equation}\n\nWe shall need Eilenberg's inequality, a Fubini type inequality valid for any $\\mathcal{H}^{n-3}$-measurable set $A$ with finite measure (see \\cite[7.7,~7.8]{Mat95}): \n\\[\n\\int_{\\B^{n-3}_\\delta(z)} \\mathcal{H}^0(A \\cap \\pi_L^{-1}(y)) \\phantom{.}\\mathrm{d} y \\le \\omega_{n-3} \\mathcal{H}^{n-3}(A) \n\\]\nApplying the above inequality twice -- once with $A$ as the singular set and once with $A$ as its exceptional part of measure zero -- we learn that for a.e. $y \\in \\B^{n-3}_\\delta(z)$ the slice $\\operatorname{sing} u \\cap \\B_1 \\cap \\pi_L^{-1}(y)$ consists of finitely many points, at each of them the tangent plane exists and is transverse to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$ (i.e. the direction of slicing). \n\nLet us choose one such $y$ and denote these singular points by $p_1,\\ldots,p_k$. Let also \n\\[\nL_j := \\operatorname{Tan}(\\operatorname{sing} u, p_j), \n\\quad \nr_0 := \\tfrac 12 \\min_{i,j} \\left ( |p_i-p_j|, \\varepsilon - |p_i-y| \\right ).\n\\]\nFor each $j=1,\\ldots,k$, there is a HCM $\\varphi_j \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $S(\\varphi_j) = L_j$ such that the sequence of rescaled maps $u_{r_i}(x) = u(p_j+r_i x)$ converges to $\\varphi_j$ in $W^{1,2}(\\B_1)$ for some sequence $r_i \\to 0$. Note that by our assumption, $\\varphi_j$ has energy density strictly less than $\\Theta$. Since this convergence is uniform away from $L_j$, maps $u_{r_i}$ and $\\varphi_j$ are homotopic on $L_j^\\perp \\cap \\partial \\B_1$ for large enough $i$. Tilting $L_j^\\perp$ to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$ and rescaling, we get that maps $u$ and $\\varphi_j$ are homotopic on $\\pi_L^{-1}(y) \\cap \\partial \\B_{r_j}(p_j)$ for some small $r_j < r_0$. Recalling that $u$ restricted to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3 \\cap \\partial \\B_1$ (and hence also to $\\pi_L^{-1}(y) \\cap \\partial \\B_\\varepsilon(y)$) has homotopy type $\\alpha$, we conclude \n\\[\n\\alpha = [\\varphi_1] + \\ldots + [\\varphi_k] \\quad \\text{in } \\pi_2({\\ensuremath{\\mathcal N}}),\n\\]\nwhere each $\\varphi_j$ has energy density smaller than $\\Theta$, which is a contradiction with the assumption that $\\alpha$ is indecomposable. \n\nTo finish the proof, we need to get rid of the additional assumption \\eqref{eq:transverse}. This is done by using the following simple transversality lemma. \n\n\\begin{lem}\n\\label{lem:transverse}\nLet $n = a+b$, consider the Grassmannian $G(n,a)$ with the standard volume measure $\\lambda$ and $G(n,b)$ with a finite positive Borel measure $\\mu$. Then the set \n\\[\n\\{ E \\in G(n,a) : \\mu(\\{ F \\in G(n,b) : E \\not\\pitchfork F \\}) > 0 \\})\n\\]\nhas zero $\\lambda$ measure. \n\\end{lem}\n\nPostponing its proof for the moment, we complete the reasoning as follows. Choose $a = 3$, $b = n-3$, and let $\\mu$ be the measure $\\mathcal{H}^{n-3} \\llcorner \\operatorname{sing} u \\cap \\B_1$ pushed-forward by the map $\\operatorname{Tan}(\\operatorname{sing} u, \\cdot)$, i.e.\n\\[\n\\mu(U) = \\mathcal{H}^{n-3} (\\{ y \\in \\operatorname{sing} u \\cap \\B_2 : \\operatorname{Tan}(\\operatorname{sing} u, y) \\in U \\}). \n\\]\nThen the set in Lemma \\ref{lem:transverse} has measure zero, in particular its complement is dense. Hence we can choose $E \\in G(n,3)$ so that $E \\pitchfork F$ for $\\mu$-a.e. $F \\in G(n,n-3)$, with $E$ arbitrarily close to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$. This amounts to satisfying \\eqref{eq:transverse} with a slightly tilted direction of slicing. Recall that $\\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ is disjoint with the cylinder $\\B^{n-3}_\\delta(z) \\times {\\ensuremath{\\mathbb R}}^3$, in consequence it is also disjoint with a smaller cylinder in direction $E^\\perp$. It is easy to see that the rest of the proof remains unchanged. \n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{lem:transverse}]\nFirst note that for each $F \\in G(n,b)$ the set of all $E \\in G(n,a)$ non-transversal to $F$ is a finite sum of smooth submanifolds of $G(n,a)$ of positive codimension \n\\[\n\\{ E \\in G(n,a) : E \\not\\pitchfork F \\} = \\bigcup_{c=1}^{\\min(a,b)} \\{ E \\in G(n,a) : \\dim E \\cap F = c \\},\n\\]\nhence it has zero $\\lambda$ measure. Applying Fubini theorem, we get \n\\begin{align*}\n\\int_{G(n,a)} \\mu(\\{ F \\in G(n,b) : E \\not\\pitchfork F \\}) \\phantom{.}\\mathrm{d} \\lambda(E) \n& = \\int_{G(n,a)} \\int_{G(n,b)} \\mathbf{1}_{E \\not\\pitchfork F} \\phantom{.}\\mathrm{d} \\mu(F) \\phantom{.}\\mathrm{d} \\lambda(E) \\\\\n& = \\int_{G(n,l)} \\lambda(\\{ E \\in G(n,a) : E \\not\\pitchfork F \\}) \\phantom{.}\\mathrm{d} \\mu(F) \\\\\n& = 0, \n\\end{align*}\nso the integrand has to be zero for $\\lambda$-a.e. $E \\in G(n,a)$. \n\\end{proof}\n\n\\subsection*{Propagation of $\\delta$-flatness to finer scales}\n\nIn this section we investigate some important consequences of Definition \\ref{df:flatness}. Assuming that an energy minimizing map $u$ is $\\delta$-flat in $\\B_1$ (with small $\\delta > 0$), we shall see that $\\operatorname{sing} u$ is actually more flat than a priori assumed (Lemma \\ref{lem:fund}), $u$ is also $\\delta$-flat in all smaller concentric balls (Corollary \\ref{cor:all-scales}), and that $0 \\in \\operatorname{sing}_\\alpha u$ (Corollary \\ref{cor:top-stratum}). \n\n\\begin{lem}\n\\label{lem:fund}\nFor every $\\varepsilon > 0$ there is $\\delta_1(\\varepsilon) > 0$ such that if $u$ is $\\delta_1$-flat in $\\B_1$, then $\\operatorname{sing} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$ and $\\| u - \\varphi \\|_{W^{1,2}(\\B_1)} \\le \\varepsilon$ for some HCM $\\varphi$ of homotopy type $\\alpha$ with energy density $\\Theta$. Moreover, $\\operatorname{sing} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$ with respect to the $(n-3)$-dimensional plane $S(\\varphi)$. \n\\end{lem}\n\n\\begin{proof}\nWe employ the usual contradiction argument. Let $u_k$ be a sequence of minimizing harmonic maps such that $u_k$ is $1\/k$-flat in $\\B_1$, with $\\operatorname{sing} u$ $\\tfrac{1}{10}$-Reifenberg flat with respect to a fixed plane $L$. \nChoosing a subsequence, we have $u_k \\to \\varphi$ in $W^{1,2}(\\B_1)$ for some energy minimizing $\\varphi$. By condition \\ref{df:flatness-energy} in Definition \\ref{df:flatness}, $\\varphi$ is homogeneous with energy density $\\Theta$. By Lemma \\ref{lem:no-drop}, for each $k$ the set $\\operatorname{sing}_{\\ge \\Theta} u_k$ is $\\frac{1}{10}$-Reifenberg flat in $\\B_1$ with respect to $L$. Taking the limit and exploiting the upper semicontinuity of $\\theta_\\cdot(\\cdot,0)$ with respect to both the map and the point, we conclude that the set \n\\[\nS(\\varphi) \\equiv \\operatorname{sing}_{\\ge \\Theta} \\varphi\n\\]\nis not contained in any $(n-4)$-dimensional plane. On the other hand, it is itself a~linear subspace of dimension at most $n-3$, so we learn that $\\varphi$ is a HCM of homotopy type $\\alpha$ (by uniform convergence away from $L$). For large enough $k$, $u_k$ is $\\varepsilon$-close to $\\varphi$ in $W^{1,2}(\\B_1)$ and its singular set is contained in $\\B_\\varepsilon S(\\varphi)$ (this is a~consequence of upper semicontinuity of $\\theta_\\cdot(\\cdot,0)$ and $\\varepsilon$-regularity \\eqref{eq:eps-regularity}), which finishes the proof by another application of Lemma \\ref{lem:no-drop}. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:all-scales}\nIf $\\delta \\le \\delta_1(\\frac{1}{20})$ and $u$ is $\\delta$-flat in $\\B_1$, then $u$ is also $\\delta$-flat in any smaller ball $\\B_r$ centered at $0$ with $0 < r \\le 1$. \n\\end{cor}\n\n\\begin{proof}\nCondition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} is trivially satisfied. As for condition \\ref{df:flatness-Reifenberg}, it follows from Lemma \\ref{lem:fund} that $\\operatorname{sing} u$ is $\\frac{1}{20}$-Reifenberg flat in $\\B_1$, hence $\\frac{1}{10}$-flat in any ball $\\B_r$ with $\\frac 12 \\le r \\le 1$. In consequence, $u$ is $\\delta$-flat in each of these balls. Then the claim follows by iteration of Lemma \\ref{lem:fund} rescaled to smaller and smaller balls. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:top-stratum}\nIf $\\delta \\le \\delta_1(\\frac{1}{20})$ and $u$ is $\\delta$-flat in $\\B_1$, then every tangent map to $u$ at $0$ is a HCM of type $\\alpha$. In particular, $0 \\in \\operatorname{sing}_{\\alpha} u$. \n\\end{cor}\n\n\\begin{proof}\nLet $\\varphi$ be any tangent map to $u$ at $0$, i.e. a $W^{1,2}(\\B_1)$-limit of rescaled functions $u_k(x) = u(r_k x)$ for some sequence $r_k \\to 0$; any thus obtained $\\varphi$ is homogeneous. By Corollary \\ref{cor:all-scales}, each $u_k$ is $\\delta$-flat in $\\B_1$ (with $\\operatorname{sing} u_k$ $\\tfrac{1}{10}$-Reifenberg flat with respect to some $L_k$), so the claim follows from Lemma \\ref{lem:no-drop} as in the proof of Lemma \\ref{lem:fund}. The only difference is that the planes $L_k$ may change, but without loss of generality $L_k \\to L$ in $G(n,n-3)$, which is enough to conclude that $\\operatorname{sing}_{\\ge \\Theta} \\varphi$ spans an $(n-3)$-dimensional plane. \n\\end{proof}\n\n\\subsection*{Moving the ball center}\n\n\\begin{prop}\n\\label{prop:all-balls}\nFor every $\\varepsilon > 0$ there is $\\delta_2(\\varepsilon) > 0$ such that if $u$ is $\\delta_2$-flat in $\\B_2$, then $u$ is $\\delta_1(\\varepsilon)$-flat in each of the balls $\\B_r(z)$ with $z \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ and $0 < r \\le 1$. Moreover, the sets $\\operatorname{sing}_\\alpha u$ and $\\operatorname{sing}_{\\ge \\Theta} u$ restricted to the ball $\\B_1$ coincide. \n\\end{prop}\n\n\\begin{proof}\nChoose $\\delta_2 := \\min(\\delta_1(\\varepsilon),\\delta_1(\\eta\/2))$ according to Lemma \\ref{lem:fund}, where $\\eta>0$ is to be fixed in a moment. Applying Lemma \\ref{lem:fund} rescaled to the ball $\\B_2$, denote by $\\varphi$ the approximating HCM and let $L = S(\\varphi)$. To obtain the first claim, we first show that $\\theta_u(z,1) \\le \\Theta + \\delta_1(\\varepsilon)$ for each $z \\in \\B_1 \\cap \\B_{\\eta} L$. First, \n\\[\n\\int_{\\B_1(z)} |\\nabla u|^2 \\le \\int_{\\B_1(z)} |\\nabla \\varphi|^2 + C \\eta^2,  \n\\]\nby Lemma \\ref{lem:fund}. If $z' = \\pi_L(z)$, then $|z-z'| < \\eta$ and \n\\[\n\\int_{\\B_1(z)} |\\nabla \\varphi|^2 \\le \\int_{\\B_{1+\\eta}(z')} |\\nabla \\varphi|^2 = (1+\\eta)^{n-2} \\Theta\n\\]\nby $L$-invariance of $\\varphi$ in $z'$-direction. If $\\eta$ is chosen small enough (depending on $\\delta_1(\\varepsilon)$), we obtain $\\theta_u(z,1) \\le \\Theta + \\delta_1(\\varepsilon)$. \n\n\\medskip\n\nSince each point $z \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ lies in $\\B_\\eta L$ by Lemma \\ref{lem:fund}, the above reasoning shows that condition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} holds for the ball $\\B_1(z)$. Condition \\ref{df:flatness-Reifenberg} is satisfied by our assumptions, this ball is $\\delta_1(\\varepsilon)$-flat. Then Corollary \\ref{cor:all-scales} implies $\\delta_1(\\varepsilon)$-flatness of $u$ also in all smaller balls $\\B_r(z)$. \n\nBy Corollary \\ref{cor:top-stratum} we now have $z \\in \\operatorname{sing}_\\alpha u$ for each $z \\in \\operatorname{sing}_{\\ge \\Theta} \\cap \\B_1$. The inverse inclusion $\\operatorname{sing}_\\alpha u \\subseteq \\operatorname{sing}_{\\ge \\Theta} u$ is evident from the definition of $\\Theta(\\alpha)$. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:only-top}\nUnder the assumptions of Proposition \\ref{prop:all-balls}, the whole singular set $\\operatorname{sing} u$ restricted to the ball $\\B_{1\/2}$ coincides with $\\operatorname{sing}_{\\ge \\Theta} u$ (and hence with $\\operatorname{sing}_\\alpha u$). \n\\end{cor}\n\n\\begin{proof}\nAssume that the ball $\\B_{1\/2}$ contains a point $p \\in \\operatorname{sing} u \\setminus \\operatorname{sing}_{\\ge \\Theta} u$. We may choose a point $z \\in \\operatorname{sing}_{\\ge \\Theta} u$ closest to $p$ (as it is a closed set) and set $r = 2|p-z|$. Clearly $z \\in \\B_1$ and $0 < r \\le 1$, so $u$ is $\\delta_1(\\varepsilon)$-flat in $\\B_r(z)$. Choose $L = L(z,r)$ according to Definition \\ref{df:set-flatness}. Then by Lemma \\ref{lem:no-drop} there is a point $z' \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_r(z)$ such that $\\pi_L(z')=\\pi_L(p)$. Since both $|\\pi_L(p)-p|$ and $|\\pi_L(z')-z|$ are less than $\\tfrac{r}{10}$, the triangle inequality yields a~contradiction with minimality of $z$. \n\\end{proof}\n\nIn order to apply the above results, one needs to know that $u$ is $\\delta$-flat in at least one ball. \n\n\\begin{lem}\n\\label{lem:some-flatness}\nLet $\\delta > 0$. If $0 \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(0,0) < \\Theta + \\delta$, then there is $r > 0$ such that $u$ is $\\delta$-flat in $\\B_r$. \n\\end{lem}\n\n\\begin{proof}\nNote that condition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} is trivially satisfied for small enough~$r$. \n\nBy definition of $\\operatorname{sing}_\\alpha u$, some sequence of rescaled functions $u_k(x) = u(r_k x)$ converges in $W^{1,2}(\\B_1)$ to a HCM $\\varphi$ of homotopy type $\\alpha$ for some sequence $r_k \\to 0$. For large enough $k$, we have $\\operatorname{sing} u_k \\cap \\B_1 \\subseteq \\B_{1\/10} S(\\varphi)$. Since the convergence is uniform away from $S(\\varphi)$, $u_k$ restricted to $S(\\varphi)^\\perp \\cap \\partial \\B_1$ has homotopy type $\\alpha$, so condition \\ref{df:flatness-Reifenberg} follows from Lemma \\ref{lem:no-drop}. Rescaling, we see that $u$ is $\\delta$-flat in $\\B_{r_k}$ for large enough $k$. \n\\end{proof}\n\n\\begin{rem}\n\\label{rem:unique-type}\nCombining Lemma \\ref{lem:some-flatness} with Corollary \\ref{cor:top-stratum}, we see that \\textit{some} can be changed to \\textit{any} in the definition of $\\operatorname{sing}_\\alpha$, if only we restrict ourselves to points with energy density close to optimal. \nThat is, if $y \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(y,0) < \\Theta + \\delta_1(\\tfrac{1}{20})$, then every tangent map of $u$ at $y$ is a HCM of type $\\alpha$.\n\\end{rem}\n\nWe are now ready to prove the main theorem. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:Holder-regularity}]\nFix the H{\\\"o}lder exponent $0 < \\gamma < 1$ and choose $\\varepsilon = \\varepsilon(\\gamma,n) > 0$ according to Reifenberg's topological disc theorem (Theorem \\ref{thm:Reifenberg}), then fix $\\delta$ to be $\\delta_2(\\varepsilon)$ from Proposition \\ref{prop:all-balls}. \n\nChoose a point $p \\in \\operatorname{sing}_\\alpha u$ such that $\\theta_u(p,0) < \\Theta + \\delta$. According to Lemma \\ref{lem:some-flatness}, $u$ is $\\delta_2(\\varepsilon)$-flat in some ball $\\B_{2r}(p)$. By Proposition \\ref{prop:all-balls}, the set $\\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ is closed and $\\varepsilon$-flat in each ball $\\B_s(z)$ centered at $z \\in \\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ with radius $0 < s < r$. Applying Theorem \\ref{thm:Reifenberg}, we conclude that $\\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ is bi-H{\\\"o}lder equivalent (with exponent $\\gamma$) to an $(n-3)$-dimensional ball. \n\nBy upper semicontinuity of $\\theta_u(\\cdot,0)$ we can ensure $\\theta_u(y,0) < \\Theta + \\delta$ for all $y \\in \\B_{r}(p)$ (jsut by taking $r$ small enough), which together with Corollary \\ref{cor:only-top} shows that the set in question forms an open subset of $\\operatorname{sing} u$. \n\\end{proof}\n\n\\section{Additional results}\n\\label{ch:additional}\n\nIn this subsection we discuss two elementary observations that give a better description of $\\delta$-flatness, but were not needed for the proof of Theorem \\ref{thm:Holder-regularity}. We fix an indecomposable homotopy type $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and its lowest energy level $\\Theta = \\Theta(\\alpha)$. \n\n\\medskip\n\nThe following lemma shows that condition \\ref{df:flatness-Reifenberg} in Definition \\ref{df:flatness} can be dropped if one assumes a priori that $x \\in \\operatorname{sing}_\\alpha u$. This gives us an equivalent condition for $\\delta$-flatness. \n\n\\begin{lem}\n\\label{lem:only-energy-assumption}\nAssume that $0 \\in \\operatorname{sing}_\\alpha u$. If $\\delta \\le \\delta_1(\\frac{1}{20})$ and $\\theta_u(0,1) \\le \\Theta + \\delta$, then $u$ is $\\delta$-flat in $\\B_1$. \n\\end{lem}\n\n\\begin{proof}\nInspecting the proof of Lemma \\ref{lem:fund}, we see that condition \\ref{df:flatness-Reifenberg} of Definition \\ref{df:flatness} was only needed to ensure required symmetry of approximating homogenous minimizer $\\varphi$. Hence it would be enough to assume condition \\ref{df:flatness-Reifenberg} of Definition \\ref{df:flatness} in a~smaller ball $\\B_{1\/2}$, and $\\delta$-flatness in $\\B_1$ follows as in Lemma \\ref{lem:fund}. \n\nBy Lemma \\ref{lem:some-flatness}, there is $r > 0$ (possibly very small) such that $u$ is $\\delta$-flat in $\\B_r$. Applying the reasoning above, we see it is also $\\delta$-flat in every ball $\\B_s$ with $r \\le s \\le \\min(1,2r)$. An iteration of this argument (as in Corollary \\ref{cor:all-scales}, but in the opposite direction) leads to the claim. \n\\end{proof}\n\n\\medskip\n\nThe last lemma gives a uniform bound (independent of $u$) for the rate of convergence $\\theta_u(x,r) \\to \\theta_u(x,0)$ when $r \\to 0$, assuming $\\theta_u(x,r)$ is already close to $\\theta_u(x,0)$. This assumption cannot be dropped, if only there exist tangent maps $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $\\dim_H \\operatorname{sing} \\varphi = n-3$ which are not HCMs. \n\nAn additional assumption is needed to ensure that the energy density is not greater than $\\Theta$. This assumption is automatically satisfied if ${\\ensuremath{\\mathcal N}}$ is real-analytic of integrable in the sense of \\cite[Ch.~3.13]{Sim96}; see the remark preceding Corollary \\ref{cor:discrete-levels}. \n\n\\begin{lem}\n\\label{lem:rate-of-convergence}\nAssume additionally that $\\Theta$ is an isolated energy level for HCMs of type $\\alpha$. Assume that $0 \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(0,1) \\le \\Theta + \\delta_4$ with $\\delta_3(n,\\alpha,{\\ensuremath{\\mathcal N}}) > 0$ sufficiently small. Then for every $\\delta > 0$ there is $r(\\delta,n,{\\ensuremath{\\mathcal N}}) > 0$ such that $\\theta_u(0,r) \\le \\Theta + \\delta$ (in consequence, $u$ is $\\delta$-flat in $\\B_r$). \n\\end{lem}\n\n\\begin{proof}\nWe choose $\\delta_3 > 0$ smaller than $\\delta_1(\\frac{1}{20})$ from Lemma \\ref{lem:fund} and such that \n\\[\n\\int_{\\B_1^n} |\\nabla \\varphi|^2 \\notin (\\Theta,\\Theta+\\delta_3]\n\\]\nfor each HCM $\\varphi$ of type $\\alpha$. \n\nFor the sake of contradiction, assume there is a sequence of such energy minimizing maps $u_k$ with \n\\[\n\\Theta + \\delta \\le \\theta_{u_k}(0,1\/k) \\le \\theta_{u_k}(0,1) \\le \\Theta + \\delta_3.\n\\]\nTaking a subsequence, we obtain a limit map $u$ such that \n\\[\n\\Theta + \\delta \\le \\theta_u(0,0) \\le \\theta_{u}(0,1) \\le \\Theta + \\delta_3.\n\\]\nIt follows from Lemma \\ref{lem:only-energy-assumption} that each $u_k$ is $\\delta_1(\\frac{1}{20})$-flat in $\\B_1$, hence so is $u$ and by Corollary \\ref{cor:top-stratum} we infer $0 \\in \\operatorname{sing}_\\alpha u$. In particular the energy density $\\theta_u(0,0)$ is either $\\Theta$ or greater than $\\Theta+\\delta_3$, a contradiction.  \n\\end{proof}\n\n\\section*{Acknowledgments}\n\nThe author would like to thank Maciej Borodzik for fruitful discussions and helpful suggestions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\nTo build applications that can cope with a continuous and dynamic threat landscape, it is vital to empower developers with methodologies and tools for the development of effective defences \\cite{green2016}. It is also essential that these methodologies and tools can handle the fast-paced nature of modern development practices, such as Continuous Integration (CI) and Continuous Delivery (CD).  These practices are often applied within a DevOps environment in which collaboration between software development and IT operations, including security, is encouraged\\cite{ebert2016}. If the current tools and methodologies remain unchanged, security will be perceived as slowing down the development process, making it less appreciated by developers and thus leading to insecure software applications.  \n\nOrganizations which utilize a DevOps culture typically have an overarching view regarding what is deployed in their environment.  However, this usually focuses on the performance of applications, or errors caused within applications. To gain similar insights on security-related events, it is necessary to integrate attack awareness into an application. Otherwise, ongoing attacks may be missed, and there is the potential for a compromise or data breach to go unnoticed. Data breaches can have a substantial financial and reputational impact on an organization- according to IBM Security\\cite{ibmsecurity2019}, a breach which takes 200 days to resolve can cost a company \\$4.56 million. \n\nThis paper discusses a taxonomy of approaches regarding how to integrate attack awareness into applications, providing a guide for developers and security researchers.  The remainder of the paper is structured as follows: Section \\ref{problemstatement} describes the need for attack aware applications.  Current approaches are described in Section \\ref{relatedwork}.  Section \\ref{analysis} provides an analysis of these solutions before conclusions are drawn in Section \\ref{conclusion}.\n \n\\section{Problem Statement} \\label{problemstatement}\nMany existing applications cannot provide real-time intelligence regarding their security state \\cite{Watson2011}. This lack of information means developers do not know whether an attack has taken place, or which parts of an application attracts the most attention from those with malicious intentions.  Integrating attack awareness can address this knowledge gap, providing developers with actionable insights based on the context of the application. \n\nIntegration is generally performed by the developer or by utilizing an agent which makes an application attack-aware at runtime. While a fully automated approach is desirable, it is only feasible for application components developed using established practices and technologies \\cite{zhu2016}.  Additionally, off-the-shelf protection methods implemented by agents are limited to specific attack classes \\cite{noderasp20182}\\cite{hawkins2017} and known attacks \\cite{openrasp20182}. A taxonomy of approaches can guide researchers and developers in their choice of an appropriate integration method, taking into consideration the usability of the integration method.\n\n\\section{Related Work} \\label{relatedwork}\nApproaching intrusion detection from within an application has been hypothesized by Sielken \\cite{Sielken1999} to detect attackers that abuse target applications without exposing anomalous behavior, and by avoiding the use of detectable attack patterns. Detection of these attacks can be achieved by utilizing contextual information such as the methods invoked while using a certain application feature. There are two approaches proposed to collect contextual application information.  The first approach regularly scans internal values of the target application, such as the authorization level of an user, and documents any changes of these values. The second approach embeds code triggers directly into the target applications' code. These can be conditional checks such as whether a predefined option that has been selected by a user has been modified.  Work by Kerschbaum et al. \\cite{kerschbaum2002} provides an example of code triggers usage to detect network-based attacks within the OpenBSD kernel, and application-specific attacks from within Sendmail. The AppSensor framework \\cite{Watson2011}, and BlackWatch \\cite{Hall2019} are further examples which apply the code trigger variant.\n\nThe concept of reference monitoring, as defined by Anderson \\cite{Anderson1972}, enforces an access policy upon user programs in execution that access references such as other programs, data or peripherals. Originally, reference monitors were implemented at operating system (OS) level to enforce a security policy on applications interacting with the OS kernel. However, there are two further approaches- the reference monitor is either embedded in an interpreter that runs the target application or it is embedded in the target application directly \\cite{Erlingsson2003}.\n\nSelf-protecting software systems are another closely related field of research, focusing on systems which autonomously defend themselves against attacks, as envisioned by Kephart and Chess\\cite{Kephart2003}. Furthermore, these systems can also anticipate security risks and mitigate them by proactively enabling countermeasures. In the nine dimensional taxonomy of self-protecting software systems by Yuan and Malek \\cite{Yuan2012}, the authors dedicate one dimension for research which focuses on achieving self-protection at a specific phase within the Software Development Lifecycle (SDLC). These phases are classified as design, development, testing and implementation, but are simplified in Yuan and Malek's \\cite{Yuan2012} work as self-protection achieved at development-time and runtime. \n\n\\section{Analysis} \\label{analysis}\nThis section provides an analysis of previous research and existing solutions, summarized in Figure \\ref{fig:intfig}, creating a taxonomy of approaches for integrating attack-awareness in applications. Based on the research in section \\ref{relatedwork}, the integration is performed by a developer or by an agent.  In the proposed taxonomy, it is assumed there is access to the source code of the target application required for the developer-driven approaches.\n\n\\textbf{Developer-Driven: }Developers of an application can integrate attack awareness by implementing security controls in the target application, or by configuring existing security controls within the target application. The main body of work in developer-driven attack awareness is conducted manually, and more detail is provided in subsections \\ref{manualintegration} and \\ref{aspectorientedprogramming}.\n\n\\textbf{Agent-Driven: }An agent (a software component or a software application which autonomously acts on behalf of its user) can also be designed to integrate attack awareness. Though the focus is on autonomous integration, the agent can provide an interface to enable manual configuration, generating security controls at runtime. An agent that is part of the interpreter can provide attack awareness through the runtime environment and thus to any application running in this environment. Subsection \\ref{runtimeinstrumentation} describes agents integrating attack awareness through instrumentation. \n\n\\begin{figure*}\n    \\center\n    \\includegraphics[height=4.5cm]{integration-approaches.pdf}\n    \\caption{Developer and agent-driven approaches to integrate attack awareness into an application}\n    \\label{fig:intfig}\n\\end{figure*} \n\n\\subsection{Manual Integration} \\label{manualintegration}\nSolutions like the AppSensor framework \\cite{Watson2011} and BlackWatch \\cite{Hall2019} rely on the hypothesis that an attacker or malicious behavior can be detected by knowing the `normal' behavior of the target application, and by being able to monitor deviations from such behavior. Application developers are therefore ideal candidates to apply this integration approach as they have specified, designed and implemented the application, and know where to strategically place security controls.\n\nWhile this approach is effective against attackers targeting the business logic of an application, it is less so when it comes to attacks such as injection attacks, which require developers to have expertise in their execution. This, however, is often not the case for security expertise among developers in general as argued by Wurster and Oorschot \\cite{Wurster2008} - developers may lack security expertise.\n\nIt is the developers responsibility to put the security controls in place, however there is a chance that the placement is neglected due to a lack of priority or time spent on non-security tasks, leading to areas of the application which are not covered. In the worst case, this approach will not be accepted by the developers as it can be seen as an additional effort which needs to be completed on top of others tasks \\cite{Hall2019}\\cite{zhu2016}. Due to this issue, integration approaches should utilize proven development techniques e.g. Kim et al. \\cite{kim2012} utilizes the dependency injection technique to make the integration of attack awareness scalable and reusable. This technique is part of modern application frameworks and while it is familiar to many developers, it can also address the issue of accidentally omitting areas of the application. From the perspective of framework developers, it is suggested to build the security controls into the framework.  According to an empirical study by Peguero et al. \\cite{peguero2018}, applications using built-in security controls derived from a framework are less susceptible to vulnerabilities than applications where developers were responsible for implementing security controls. \n\n\\subsection{Aspect-Oriented Programming} \\label{aspectorientedprogramming}\nAspect-Oriented Programming (AOP) is a technique to separate cross-cutting concerns, such as logging and monitoring, which are used throughout an application but which do not represent any business functionality \\cite{Kiczales1997}. These cross-cutting concerns, also referred to as aspects, are weaved in to the target functionality of the application by intercepting function or method calls at runtime.\n\nSecurity controls can also be defined as aspects as they are not tied to the target application's business functionality. Serme et al.\\cite{Serme2014} present an AOP approach on integrating input validation controls, whereas Phung et al. \\cite{Phung2009} demonstrate aspectized security policies which prevent malicious behaviour in the target application.  \n\nAOP reduces the manual effort required when initially implementing the security policies that are processed by aspects. An aspect can be implemented as code in the programming language of the target application \\cite{Serme2014}\\cite{Phung2009} but it can also be implemented in a modeling language such as Unified Modeling Language (UML)\\cite{zhu2009}, or in a Program Query Language (PQL) \\cite{livshits2006}. The various implementation methods can address the usability requirements of the different roles involved with the development of an application. A developer may prefer to implement an aspect as code whereas an architect may rather implement an aspect in a modeling language. Regardless of the role, the implementation method of choice should be usable for its target audience as highlighted by Viega et al.\\cite{Viega2001}, to prevent developer-induced errors and to reduce the required expertise to implement security controls.  \n\nInstead of manually implementing an aspect from scratch, it can also be generated using, for example, the output of a static code analysis tool \\cite{simic2013}. While this method seems to automate the integration process at first, in reality it shifts the manual effort to the usage of a tool or application which generates the output required for the aspect generation. Whether the integration can be fully automated depends on the degree of automation supported by the tool or application in question.  \n\nIraqi and Bakkali \\cite{iraqi2019} developed a framework that can learn to detect outlier behavior in method invocations without supervision. The method invocation features required for the learning process are implemented as a feature extractor aspect, which demonstrates another use case for integrating attack awareness with AOP.  \n\n\\subsection{Runtime Environment and Binary Instrumentation} \\label{runtimeinstrumentation}\nThe approaches analysed in the previous sections are effective when integrating security controls into a single application at one time. However, integrating security controls from within the runtime environment can affect any application running in the environment. Existing solutions are based on a modified interpreter as in ZenIDS \\cite{hawkins2017} or Node RASP \\cite{noderasp2018}. Another variant is also known as a Runtime Application Self-Protection (RASP) agent \\cite{gartner2020}, a library which can be loaded by an interpreter at runtime e.g. OpenRASP \\cite{openrasp2018} or Sqreen \\cite{jeanbaptiste2019}.\n\nIn the ideal case, RASP agents can be deployed in a plug and play manner, requiring only an initial configuration as Haupert et al. \\cite{haupert2018} describes regarding the deployment of Promon SHIELD RASP \\cite{promon2020}. In cases where an agent does not require any configuration or a learning phase, attacks are detected using techniques that, e.g., combine taint-tracking with lexical analysis \\cite{noderasp20182} or that monitor common input sinks and output sources for known malicious behavior and signatures \\cite{openrasp20182}. Depending on the RASP implementation, the aforementioned techniques may only cover a limited set of sinks and sources, and the detection of known attacks relies on the completeness of the malicious behavior or signature database available. As RASP solutions are platform specific by design, it can also be the case that specific technologies used within a target application, or the entire application, might not be able to benefit from this integration approach.\n\nWhile an instrumented runtime environment may be suitable for applications written in interpreted languages, those written in compiled languages require individual instrumentation.  With binary instrumentation, security controls can be integrated into an applications binary code to guarantee control-flow integrity (CFI) at runtime \\cite{abadi2009}. The effectiveness of CFI is limited to attacks attempting to hijack the control flow of an application, such as buffer overflows.  Although it is feasible to integrate CFI in closed source applications, many available CFI implementations require access to source code\\cite{goktas2014}, making it impractical for legacy or third-party applications.\n\n\\subsection{Summary}\nThe previous subsections have shown that all attack awareness approaches require some form of manual interaction, either at the setup or configuration phase. This is particularly pertinent for the detection of attacks against the business logic of an application, as these are unique for every application. The runtime environment and binary instrumentation approaches are no exception and also require manual intervention by the developers in the form of placing function calls which can track authenticated users\\cite{hawkins2017} or custom events\\cite{sqreen2020}. As with AOP approaches, it is then up to the agent implementation to provide a method of specifying policies, acting on the tracked events. This will influence how usable that method is in the context of the application.\n\nAlthough the agent-driven approach has the most potential for automation, there are other aspects which make manual integration worthwhile, e.g. the insertion of code triggers can be completed in a few lines of non-fragmented code\\cite{kerschbaum2002}. Compared to instrumented monitoring, code triggers are only called when they lie in the execution path of an attack and thus cause no performance overhead during non-malicious interactions. The code triggers are an inherent part of the application, providing constant attack-awareness in any environment in which the application is deployed.\n\nThe programming language can also be relevant when choosing AOP for attack awareness integration. To implement AOP in JavaScript, one can overwrite built-in functions, which makes self-protection possible \\cite{Phung2009}. In contrast, PHP does not support the same functionality for built-in functions- a language extension must be installed first. Extension usage might not be feasible in certain environments due to restrictions in what can be deployed, or the lack of control over the environment. This would also be an exclusion criteria for choosing a RASP agent in the previously described environment. \n\nThe approaches thus far focused on using attack awareness for application protection. However, the same concept can be utilized for security testing e.g. the insights provided by an agent could be used for the generation of security tests. Since the tester is an established role in modern development environments, enhancing the capabilities of such individuals could be a method to improve the adoption of attack awareness. \n\n\\section{Conclusion and Future Work} \\label{conclusion}\nWork presented in this paper explored different methods of including attack awareness within applications. The taxonomy highlights that while there are a number of potential solutions, they do not necessarily meet the needs of developers. \nResearch currently being undertaken by \u00dcnl\u00fc \\cite{Unlu2019} seeks to enhance the field of attack awareness and software security. As part of this process, the work will involve the investigation of techniques which aim to enhance the process by which developers include attack awareness within applications. By achieving an enhancement in the adoption of attack awareness techniques, the proposed research will contribute to bringing application security closer to developers by creating the necessary tools for them.  Such tools will be tailored to the needs of developers, empowering them to build secure applications.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}