text:
\section{Introduction}\label{sec:intro}
It is well known that perturbative corrections in scattering processes have infrared (IR) sensitivity to soft and collinear emissions. For electroweak corrections involving massive $Z$ and $W$ bosons, the mass of the vector boson regulates the IR divergences, such that the IR sensitivity yields logarithmic dependence on the ratio of the vector boson mass over the partonic center of mass energy
\begin{align}
L_V \equiv \ln m_V^2 / s
\,.
\end{align}
Both virtual and real corrections give rise to such logarithmic corrections, and  as usual, their effect cancels in fully inclusive observables.

This situation is very similar to the more common situation of gauge theories with massless gauge bosons, such as QCD, for which both virtual and real corrections are IR divergent, and the divergence cancels in the sum that enters in any physical observable. However, there are two main differences: First, for massive gauge bosons, both virtual and real contribution are separately finite (albeit with logarithmic dependence on the ratio of the mass of the vector boson to the center of mass energy), such that there is no need to combine them for physical observables. Second, even if the measurement is completely inclusive over the final state, the initial beams of colliders are typically not SU(2) singlets, such that one can never perform a fully inclusive measurement. Thus, essentially any measurement has logarithmic sensitivity to the ratio $m_V^2 / s$, such that for high enough center of mass energies electroweak corrections become very large.

Much effort has been put into understanding the electroweak logarithms arising from virtual corrections~\cite{Kuhn:1999nn,Fadin:1999bq,Ciafaloni:1999ub,Beccaria:2000jz,Hori:2000tm,Ciafaloni:2000df,Denner:2000jv,Denner:2001gw,Melles:2001ye,Beenakker:2001kf,Denner:2003wi,Pozzorini:2004rm,Feucht:2004rp,Jantzen:2005xi,Jantzen:2005az,Jantzen:2006jv,Chiu:2007yn,Chiu:2008vv,Manohar:2012rs}. A general result for the logarithmic dependence at one-loop has been derived in~\cite{Denner:2000jv,Denner:2001gw}, and a general method for the resummation of these logarithms has been developed in~\cite{Chiu:2007yn,Chiu:2008vv}. This method uses SCET which has been developed in~\cite{Bauer:2001ct,Bauer:2000ew,Bauer:2000yr,Bauer:2001yt}. While real radiation of electroweak gauge bosons typically leads to different experimental signatures, in many cases some amount of real radiation is included in the event samples. For example, in many analyses missing energy is not vetoed, such that radiation of $Z$ bosons which decay to neutrinos is included, and any analysis that is inclusive over the number of jets includes real radiation of hadronically decaying vector bosons.

Logarithmic dependence from real emission, however, has been studied much less. The ${\cal O}(\alpha)$ correction from real emission can of course easily be included using phase space integrations over tree-level matrix elements, and a study of the effect of such real radiation on many experimental observables was performed in~\cite{Baur:2006sn}. It was found that the real corrections can have a sizable effect, albeit typically not quite as large as the virtual corrections. The effects of real radiation have  recently been studied in~\cite{Manohar:2014vxa,Bell:2010gi}.

In this paper we consider the Drell-Yan cross sections
$\sigma_{p p \to \ell_1 \ell_2}$  and $\sigma_{p p \to \ell_1 \ell_2 V} \,.$
Here $\ell$ denotes either an electron, positron or a neutrino, and $V$ denotes either a $Z$, a photon or a $W$ boson. For Drell-Yan production at NLO accuracy, the first cross section includes the tree-level and the one-loop virtual correction, while the second cross section denotes the real radiation of a gauge boson. Collectively, we will represent these two cross sections as
$\sigma_{p p \to \ell_1 \ell_2 X}\,.$

To calculate this hadronic scattering cross-sections one starts as always by factorizing the short distance partonic scattering from the non-perturbative physics describing the binding of partons into hadrons
\begin{align}
\sigma_{p p \to {\rm final}} &= \sum_{ab} \int \!\mathrm{d} x_a\mathrm{d} x_b \, f_{a/p}(x_a) f_{b/p}(x_b)
\,  \hat \sigma_{ab \to {\rm final}}(x_a,x_b)
\,,\end{align}
where $f_{i/p}(x_i)$ are the parton distribution functions to find parton $i$ in the proton $p$ with momentum fraction $x_i$, and $\hat \sigma_{ab \to {\rm final}}(x_a,x_b)$ denotes the partonic scattering cross-section. By Lorentz invariance, the dependence of the partonic cross-sections on the momentum fractions $x_i$ is only through the product $s = x_a x_b S$, and not through the rapidity $Y = \ln (x_a / x_b) / 2$. Thus, one can write
\begin{align}
\label{eq:hadronicDY}
\frac{\mathrm{d} \sigma_{p p \to {\rm final}} }{\mathrm{d} s} = \sum_{ab} {\cal L}_{ab} (s) \, \hat \sigma_{ab \to {\rm final}}(s)
\,,\end{align}
where the parton luminosity is defined as
\begin{align}
{\cal L}_{ab} (s) =  \int \!\mathrm{d} x_a\mathrm{d} x_b \, f_{a/p}(x_a) f_{b/p}(x_b) \, \delta(s - x_a x_b S)
\,.\end{align}

It is the purpose of this paper to derive simple analytical expressions for the resummation of electroweak Sudakov logaritms at leading logarithmic (LL) accuracy. To achieve this, we combine knowledge of the resummation of the virtual corrections, together with the known IR structure of real radiation from parton showers. This allows us to obtain simple analytical results which we present here.
This approach is similar to the one used in \cite{Fadin:1999bq}, where these effects were discussed focussing on $e^+ e^-$ collider.
We then use these results to give expressions for the individual contributions to the hadronic cross-sections $\sigma_{p p \to \ell_1 \ell_2 X}$ as a function of the center of mass energy $\sqrt{s}$. We reproduce the well known fact that resummation has a large effect on the virtual contribution for $\sqrt{s} \gtrsim 2$ TeV. The resummation of the real corrections changes the fixed order results by more than 40\% for  $\sqrt{s} \gtrsim 2$ TeV, and this effect grows to about  200\% for $\sqrt{s} \sim 25$ TeV.

This paper is organized as follows: In \sec{structureOfLogs} we set up some notation and derive general expressions for the cross-sections which highlight the structure of the logarithmic corrections. In \sec{FO} we give the (well known) fixed order results of the virtual and real electroweak corrections which are enhanced by two powers of the logarithms $L_V$.  In \subsec{VirtLL} we give the resummed results for the virtual corrections, keeping only LL accuracy. In \subsec{RealInclLL}, we derive the resummation of the leading logarithmic terms for real corrections. In \sec{Results} we provide a numerical analyses of these results, both for the 13 TeV LHC and a 100 TeV $pp$ collider. We finish with Conclusions in \sec{Conclusions}.
\section{The structure of logarithmic terms in the perturbative expansion}\label{sec:structureOfLogs}
To first order in electroweak perturbation theory the partonic scattering cross-section can be written as
\begin{align}
\label{eq:RealVirtSplit}
\hat \sigma_{ij \to \ell_1 \ell_2 X} = \hat \sigma^{(0)}_{ij \to \ell_1 \ell_2} +  \hat \sigma^{(1)}_{ij \to \ell_1 \ell_2 X}
\,,\end{align}
where $\hat \sigma^{(0)}_{ij \to \ell_1 \ell_2}$ denotes the Born cross-section, and $\hat \sigma^{(1)}_{ij \to \ell_1 \ell_2 X}$ the ${\cal O}(\alpha)$ correction. This first order perturbative correction can be decomposed into a virtual and real contribution, and each of these two terms can be further separated by the flavor of the vector boson in the loop or real final state. This gives
\begin{align}
\label{eq:RealVirtSplit}
\hat \sigma^{(1)}_{ij \to \ell_1 \ell_2 X} = \sum_V \left[\hat \sigma^{V}_{ij \to \ell_1 \ell_2}  + \hat \sigma_{ij \to \ell_1 \ell_2 V}\right]
\,.
\end{align}
Here the first term describes the contribution from one-loop diagrams with $V = \{Z, W^\pm, \gamma\}$ in the loop, while the second term is given by the real radiation of an electroweak gauge boson $V$.

Each of the terms on the right hand side of \eq{RealVirtSplit}, contains double and single logarithmic terms of the form $\alpha \, L_V$ and $\alpha \, L_V^2$. By the KLN theorem \cite{Kinoshita:1962ur}, these logarithmically enhanced terms cancel when we sum over complete gauge multiplets. Thus, the completely inclusive cross-section has no logarithmically enhanced terms (called ``finite'' below)
\begin{align}
\label{eq:AllCancellation}
\sum_{a,b,\ell_1,\ell_2} \hat \sigma^{(1)}_{ab \to \ell_1 \ell_2 X} = {\rm finite}
\,.\end{align}
The sum of the initial state is over $a,b \in \{u, d\}$,  that of the final state is over $\ell_1,\ell_2 \in \{e^+, e^-, \nu\}$, and we sum over the virtual contributions, and well as the real contributions with either a $Z$, a photon or $W$ in the final state.
Since the logarithmic enhancement depends on the mass of the vector boson, the logarithmic corrections from $Z$ bosons, photons, and $W$ bosons cancel separately between real and virtual corrections. Furthermore, since the emission of a $Z$ boson or a photon does not change the flavor of a fermion, the cancellation of the terms enhanced by $L_Z$ happens for each flavor assignment separately. This gives
\begin{align}
\label{eq:ZCancellation}
\hat \sigma^{Z}_{ab \to \ell_1 \ell_2}  + \hat \sigma_{ab \to \ell_1 \ell_2 Z} = {\rm finite} \,, \qquad
\hat \sigma^{\gamma}_{ab \to \ell_1 \ell_2}  + \hat \sigma_{ab \to \ell_1 \ell_2 \gamma} = {\rm finite}
\,.
\end{align}
The emission of $W$ bosons, on the other hand, changes the flavor of the fermion. This implies that the cancellation of the terms enhance by $L_W$ only happens after summing over all flavors
\begin{align}
\label{eq:WCancellation}
\sum_{a,b,\ell_1,\ell_2} \left[ \hat \sigma^{W}_{ab \to \ell_1 \ell_2}  + \hat \sigma_{ab \to \ell_1 \ell_2 W}\right] = {\rm finite}
\,.
\end{align}

An important consequence of~\eq{WCancellation} is that a finite answer is only obtained if one sums over all initial state flavors. Thus, the hadronic cross-section given in \eq{hadronicDY} still contains contributions enhanced logarithmically by factors of $L_W$. This is because each initial state flavor is multiplied by a different parton luminosity, such that the sum over initial state flavors cannot be observed.

\section{Fixed order calculations of the virtual and real contributions}\label{sec:FO}
In this section we consider the fixed order expansion of the virtual corrections $\hat \sigma^{V}_{ab \to \ell_1 \ell_2}$ and the real contributions $\hat \sigma_{ab \to \ell_1 \ell_2 V}$. For both terms, we quote here only the dominant contributions that are enhanced by two powers of $L_V$.

For completeness, we start with the Born cross-sections, including the contributions from both photon and $Z$ exchange for the neutral initial and final states, and from $W$ exchange in the charged cases. They  are given by
\begin{align}
\label{eq:Born}
\hat \sigma^B_{u\bar u \to e^-e^+} &= N \,  \frac{85 \alpha_1^2 + 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54}
\nonumber \\
\hat \sigma^B_{u\bar u \to \nu\bar\nu} &= N \,  \frac{17 \alpha_1^2 - 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54}
\nonumber \\
\hat \sigma^B_{d\bar d \to e^-e^+} &= N \,  \frac{25 \alpha_1^2 - 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54}
\nonumber \\
\hat \sigma^B_{d\bar d \to \nu \bar\nu} &= N \,  \frac{5 \alpha_1^2 + 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54}
\nonumber \\
\hat \sigma^B_{u\bar d \to \nu e^+} &= N \,  \frac{2\alpha_2^2}{3}
\nonumber \\
\hat \sigma^B_{d\bar u \to e^-\nu} &= N \,  \frac{2\alpha_2^2}{3}
\,,\end{align}
where we have defined the constant
\begin{align}
N = \frac{\pi}{8 N_C s}
\,,
\end{align}
with $N_C = 3$ denoting the number of colors. To simplify the notation, we have written our results in terms of the coupling constants of the unbroken theory
\begin{align}
\alpha_1 = \frac{\alpha_{\rm em}}{\cos^2\theta_W} \,, \qquad \alpha_2 = \frac{\alpha_{\rm em}}{\sin^2\theta_W}
\,,\end{align}
where $\theta_W$ is the weak mixing angle and $\alpha_{\rm em}$ is the electromagnetic coupling constant, also known as the fine-structure constant. All coupling constant are evaluated at the scale $\sqrt{s}$, the center of mass energy, $\alpha_i=\alpha_i(\sqrt{s})$.

The virtual corrections from $W$ exchange that are enhanced by two powers of the logarithm are easily obtained for example using \cite{Denner:2000jv} and are given by
\begin{align}
\label{eq:FOVirtW}
\hat \sigma^{W^\pm}_{u\bar u \to e^-e^+} &= - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{11 \alpha_1^2 + 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{27}
\nonumber \\
\hat \sigma^{W^\pm}_{u\bar u \to \nu\bar\nu} &= - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{3 \alpha_1^2 -2 \alpha_1 \alpha_2 + 3 \alpha_2^2}{9}
\nonumber \\
\hat \sigma^{W^\pm}_{d\bar d \to e^-e^+} &= - N \,   \frac{\alpha_W \, L_W^2}{4\pi} \frac{5 \alpha_1^2 - 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{27}
\nonumber \\
\hat \sigma^{W^\pm}_{d\bar d \to \nu \bar\nu} &= - N \,   \frac{\alpha_W \, L_W^2}{4\pi} \frac{\alpha_1^2 + 2 \alpha_1 \alpha_2 + 3 \alpha_2^2}{9}
\nonumber \\
\hat \sigma^{W^\pm}_{u\bar d \to \nu e^+} &= - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{4 \alpha_2^2}{3}
\nonumber \\
\hat \sigma^{W^\pm}_{d\bar u \to e^-\nu} &= - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{4 \alpha_2^2}{3}
\,.\end{align}
Those from $Z$ exchange are given by
\begin{align}
\label{eq:FOVirtZ}
\hat \sigma^Z_{u\bar u \to e^-e^+} &= - N \,  \frac{\alpha_Z \, L_Z^2}{4\pi} \frac{(99-366s_W^2+2210s_W^4)\alpha_1^2 + (9 - 30s_W^2+26s_W^4) \left(6 \alpha_1 \alpha_2 + 9\alpha_2^2 \right)}{486}
\nonumber \\
\hat \sigma^Z_{u\bar u \to \nu\bar\nu} &= - N \,   \frac{\alpha_Z \, L_Z^2}{4\pi}
\frac{(81-12s_W^2+136s_W^4)\alpha_1^2 + (9 - 12s_W^2+8s_W^4) \left(-6 \alpha_1 \alpha_2 + 9\alpha_2^2 \right)}{486}
\nonumber \\
\hat \sigma^Z_{d\bar d \to e^-e^+} &= - N \,    \frac{\alpha_Z \, L_Z^2}{4\pi}
\frac{5(9-24s_W^2+100s_W^4)\alpha_1^2 + (9 - 24s_W^2+20s_W^4) \left(-6 \alpha_1 \alpha_2 + 9\alpha_2^2 \right)}{486}
\nonumber \\
\hat \sigma^Z_{d\bar d \to \nu \bar\nu} &= - N \,    \frac{\alpha_Z \, L_Z^2}{4\pi}
\frac{(27-6s_W^2+10s_W^4)\alpha_1^2 + (9 - 6s_W^2+2s_W^4) \left(6 \alpha_1 \alpha_2 + 9\alpha_2^2 \right)}{486}
\nonumber \\
\hat \sigma^Z_{u\bar d \to \nu e^+} &= - N \,  \frac{\alpha_Z \, L_Z^2}{4\pi} \frac{2(9-18s_W^2+14s_W^4) \alpha_2^2}{27}
\nonumber \\
\hat \sigma^Z_{d\bar u \to e^-\nu} &= - N \,  \frac{\alpha_Z \, L_Z^2}{4\pi} \frac{2(9-18s_W^2+14s_W^4) \alpha_2^2}{27}
\,.
\end{align}
The virtual corrections from photon exchange that are enhanced by two powers of the logarithm are easily obtained and are given by
\begin{align}
\label{eq:FOVirtPhoton}
\hat \sigma^\gamma_{u\bar u \to e^-e^+} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{85 \alpha_1^2 + 6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54} \frac{26}{9}
\nonumber \\
\hat \sigma^\gamma_{u\bar u \to \nu\bar\nu} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{17 \alpha_1^2 -6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54} \frac{8}{9}
\nonumber \\
\hat \sigma^\gamma_{d\bar d \to e^-e^+} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{25 \alpha_1^2 -6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54} \frac{20}{9}
\nonumber \\
\hat \sigma^\gamma_{d\bar d \to \nu \bar\nu} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{5 \alpha_1^2 -6 \alpha_1 \alpha_2 + 9 \alpha_2^2}{54} \frac{2}{9}
\nonumber \\
\hat \sigma^\gamma_{u\bar d \to \nu e^+} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{2 \alpha_2^2}{3} \frac{14}{9}
\nonumber \\
\hat \sigma^\gamma_{d\bar u \to e^-\nu} &= - N \,  \frac{\alpha_{em} \, L_\gamma^2}{4\pi} \frac{2 \alpha_2^2}{3} \frac{14}{9}
\,,\end{align}
where the logarithms depend on a scale $\Lambda$, below which a photon is no longer resolved,
\begin{align}
L_\gamma \equiv \ln \left( \frac{\Lambda^2}{s} \right)
\,.\end{align}
We have also defined $s_W = \sin(\theta_W)$ and
\begin{align}
\alpha_W = \alpha_2\,,  \qquad  \alpha_Z = \frac{\alpha_2}{\cos^2\theta_W}
\,.
\end{align}

The double-logarithmically enhanced real corrections with a $Z$ boson or a photon in the final state are related to the corresponding virtual corrections via
\begin{align}
\label{eq:FORealZ}
\hat \sigma_{q_1\bar q_2 \to \ell_1\ell_2 Z} = - \hat \sigma^Z_{q_1\bar q_2 \to \ell_1\ell_2}
\,,\end{align}
\begin{align}
\label{eq:RealPhoton}
\hat \sigma_{q_1\bar q_2 \to \ell_1\ell_2 \gamma} = - \hat \sigma^\gamma_{q_1\bar q_2 \to \ell_1\ell_2}
\,,\end{align}
which directly follow from \eq{ZCancellation},
for those with a $W^+$ boson in the final state we find
\begin{align}
\label{eq:RealWp}
\hat \sigma_{u \bar d \to e^- e^+ W^+} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{5\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{u \bar d \to \nu \bar \nu W^+} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{u \bar u \to e^- \bar \nu W^+} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{17\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{d \bar d \to e^- \bar \nu W^+} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{5\alpha_1^2+27\alpha_2^2}{54}
\,,
\end{align}
while those with a $W^-$ are given by
\begin{align}
\label{eq:RealWm}
\hat \sigma_{d \bar u \to e^- e^+ W^-} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{5\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{d \bar u \to \nu \bar \nu W^-} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{u \bar u \to \nu e^+ W^-} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{17\alpha_1^2+27\alpha_2^2}{54}
\nonumber \\
\hat \sigma_{d \bar d \to \nu e^+ W^-} &= N \,  \frac{\alpha_W \, L_W^2}{4\pi}
\frac{5\alpha_1^2+27\alpha_2^2}{54}
\,.\end{align}

From these results one can obtain the inclusive cross-sections, which have been summed over all final state flavors
\begin{align}
\label{eq:InclusivePartonic}
\hat \sigma_{u\bar u} = - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{\alpha_1^2-3\alpha_2^2}{9}
\nonumber \\
\hat \sigma_{d\bar d} = - N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{\alpha_1^2-3\alpha_2^2}{9}
\nonumber \\
\hat \sigma_{u\bar d} = N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{\alpha_1^2-3\alpha_2^2}{9}
\nonumber \\
\hat \sigma_{d\bar u} = N \,  \frac{\alpha_W \, L_W^2}{4\pi} \frac{\alpha_1^2-3\alpha_2^2}{9}
\,.\end{align}
From these expressions one can very easily validate that in $\sum_{ab} \hat \sigma_{ab}$ all double logarithms cancel.

\section{LL resummation of the leading logarithms}
\subsection{Virtual corrections}\label{subsec:VirtLL}
To resum the electroweak Sudakov logarithms for for the virtual contributions we follow the work of~\cite{Chiu:2008vv,Chiu:2009mg}, which uses renormalization group equations in SCET. The calculation proceeds
in three steps. At the scale $\mu_Q^2 \sim s$, one matches the full theory onto 4-fermion
operators in SCET, where each of the fermion is represented by a different collinear sector in SCET. One then runs the theory from the scale $\mu_Q$ to the scale $\mu_V \sim m_V$, at which point the massive gauge bosons are integrated out of the theory and electroweak symmetry is broken.
As long as $s \gg m_V^2$, the mass of the vector boson can be set to zero in the matching onto SCET at the scale $\mu_Q$, and in the calculation of the anomalous dimension which governs the running from $\mu_Q$ to $\mu_V$. This implies that one can use an unbroken SU(2), simplifying these calculations substantially.

In~\cite{Chiu:2008vv,Chiu:2009mg}, this resummation was carried out in full generality to NLL$^\prime$ accuracy. For the purposes of this work, we only work to LL accuracy, which simplifies the structure significantly. The first simplification is that only tree level matching is required both at the high and low scale. Furthermore, no operator mixing arises in the running from the high to the low scale. This will allow us to write relatively simple analytical formulae.

As already mentioned, in the effective theory between $\mu_Q$ and $\mu_V$ one can use unbroken electroweak symmetry, such that there are a total of 7 operator that  contribute
\begin{align}
\label{eq:4fermionopsAbove}
{\cal L} = &  C_{QLT} \, Q^T L^T + C_{QLS} \, Q^S L^S +C_{ULS} \, U^S L^S
+ C_{DLS} \, D^S L^S
\nonumber\\
& \qquad
 + C_{QES} Q^S E^S
 + C_{UES} \, U^S E^S + C_{DES} \, D^S E^S
\,,\end{align}
where we have defined the fermion bilinears in either triplet or singlet representation
\begin{align}
\label{eq:bilinears}
F^{S} = \bar F \gamma^\mu F\,,
\qquad
F^{T} = \bar F \tau^a \gamma^\mu F
\,.
\end{align}
Here $Q$ and $L$ denote left-handed quark and lepton fields, respectively, while $U$, $D$ and $E$ denote the right handed up-type quarks, down-type
quarks and electron fields.
The tree level matching is given by
\begin{align}
s \, C^{(0)}_{QLT} & = 4 \pi \alpha_2
\nonumber\\
s \, C^{(0)}_{IFS} & = 4 \pi \alpha_1 \, Y_I Y_F
\,,\end{align}
where $I$ and $F$ denote the initial and final fermions of each singlet operator, and $Y_i$ is the hypercharge of particle $i$. The hypercharge normalization used is
\begin{align}
Y_i = Q_i - T^3_i
\label{eq:hypercharge}
\,,\end{align}
where $Q_i$ is the electromagnetic charge of the fermion $f=q / \ell$ and $T^3_i$ is the weak isospin.

In general, the renormalization group can mix these operators, however this mixing only arises starting at NLL. Since we are only interested in LL accuracy we can write
\begin{align}
\mu \frac{\mathrm{d}}{\mathrm{d} \mu} O_i(\mu) =\Gamma_i(\mu) \ln \frac{\mu^2}{s} O_i(\mu)
\,,
\end{align}
where the cusp anomalous dimension is
\begin{align}
\Gamma_i(\mu) = \sum_j\left[ \frac{\alpha_1(\mu)}{2 \pi} Y_j^2 + \frac{\alpha_2(\mu)}{2 \pi}T_j^2\right]_i
\,.
\end{align}
Here $[T_j]_i^2$ is the SU(2) Casimir of the j'th fermion (3/4 for left-handed fermions and 0 for right-handed fermions) in $O_i$.

The solution to this RGE can easily be written down analytically, and one finds for the Wilson coefficients at the low scale
\begin{align}
C_i(\mu_V) = U^{LL}_i(\mu_V, \mu_Q) C_i(\mu_Q)
\,.
\end{align}
Working in the limit $\alpha_i \ln m_V^2 / s \sim 1$, one obtains the simple result
\begin{align}
U^{LL}_i(\mu_V, \mu_Q; s) = \frac{\exp\left[ - \Gamma_i(\sqrt{s}) \ln^2\frac{\mu_V}{\sqrt{s}}\right]}{\exp\left[ - \Gamma_i(\sqrt{s}) \ln^2\frac{\mu_Q}{\sqrt{s}}\right]}
\,.\end{align}
For $\mu_Q = \sqrt{s}$ and $\mu_V = m_V$ this result simplifies to
\begin{align}
U_i \equiv U^{LL}_i(m_V, \sqrt{s}; s) = \exp\left[ - \Gamma_i(\sqrt{s}) \ln^2 \frac{m_V}{\sqrt{s}}\right]
\,.\end{align}
Note that the anomalous dimension and therefore the evolution Kernel only depends on the type of initial and final state particles. Thus, we have
\begin{align}
U_{IFS} = U_{IFT} \equiv U_{IF}
\,.
\end{align}

The SU(2) $ \otimes $ U(1) gauge structure is broken at $\mu_V=m_V$ to the U(1)$_{\rm em}$, and below that scale only the photon remains as gauge degree of freedom. Thus, below the scale $m_V$ one continues to run to a scale $\mu_\Lambda$, at which point the photon becomes unresolved. This running is determined by the cusp anomalous dimension
\begin{align}
\Gamma^{\rm em}(\mu) = \frac{\alpha_{\rm em}(\mu)}{2 \pi} Q_{\rm tot}^2
\,,
\end{align}
where
\begin{align}
Q_{\rm tot}^2 \equiv \sum_i Q_i^2
\end{align}
is the sum of the square of the electromagnetic charges of all particles in the operator. The evolution Kernel between $\mu_V$ and $\mu_\Lambda$ is
\begin{align}
U^{\rm em}_{Q_{\rm tot}^2}\left(\mu_\Lambda, \mu_V; s \right)= \frac{\exp \left[ - \frac{\alpha_{\rm em}}{2\pi} Q_{\rm tot}^2 \ln^2 \frac{\mu_\Lambda}{\sqrt{s}} \right]}{\exp \left[ - \frac{\alpha_{\rm em}}{2\pi} Q_{\rm tot}^2 \ln^2 \frac{\mu_V}{\sqrt{s}} \right]}
\,.
\end{align}

We obtain the simple results for the resummed virtual corrections,
\begin{align}
\label{eq:virtualResummed}
\hat \sigma^{\rm LL}_{u\bar u \to e^-e^+} &= N \,  \frac{4\left(4\,U_{UL}^2+U_{QE}^2+16\,U_{UE}^2\right) \alpha_1^2 + U_{QL}^2 \left(\alpha_1+ 3 \alpha_2\right)^2}{54} \, \left[ U^{\rm em}_{26/9}\left(\Lambda, m_V; s \right) \right]^2
\nonumber \\
\hat \sigma^{\rm LL}_{u\bar u \to \nu\bar\nu} &= N \,  \frac{16 \, U_{UL}^2 \alpha_1^2 + U_{QL}^2 \left(\alpha_1- 3 \alpha_2\right)^2}{54} \, \left[ U^{\rm em}_{8/9} \left(\Lambda, m_V; s \right) \right]^2
\nonumber \\
\hat \sigma^{\rm LL}_{d\bar d \to e^-e^+} &= N \,  \frac{4 \left(U_{DL}^2 + U_{QE}^2 + 4 \, U_{DE}^2\right) \alpha_1^2 + U_{QL}^2 \left(\alpha_1- 3 \alpha_2\right)^2}{54} \, \left[ U^{\rm em}_{20/9}\left(\Lambda, m_V; s \right) \right]^2
\nonumber \\
\hat \sigma^{\rm LL}_{d\bar d \to \nu \bar\nu} &= N \,  \frac{4 \, U_{DL}^2  \alpha_1^2 + U_{QL}^2 \left(\alpha_1+ 3 \alpha_2\right)^2}{54} \, \left[ U^{\rm em}_{2/9} \left(\Lambda, m_V; s \right) \right]^2
\nonumber \\
\hat \sigma^{\rm LL}_{u\bar d \to \nu e^+} &= N \,  \frac{2 \, U_{QL}^2 \alpha_2^2}{3} \, \left[ U^{\rm em}_{14/9}\left(\Lambda, m_V; s \right) \right]^2
\nonumber \\
\hat \sigma^{\rm LL}_{d\bar u \to e^-\nu} &= N \,  \frac{2 \, U_{QL}^2 \alpha_2^2}{3} \, \left[ U^{\rm em}_{14/9}\left(\Lambda, m_V; s \right) \right]^2
\,.\end{align}
Note that since $\alpha_1 / \alpha_2 = \tan^2(\theta_W) \sim 0.32$, the term proportional to $\alpha_1^2$ (which depends on  various different evolution Kernels) is numerically suppressed compared to the term proportional to $(\alpha_1 \pm 3 \alpha_2)^2$. Thus, to a good approximation, each leptonic final state gets the same suppression factor $U_{QL}$ from the resummation.

A simple check of our results is that they reproduce the Born results given in~\eq{Born} if we set all resummation Kernels to unity, and that they reproduce the fixed order results in Eqs.(\ref{eq:FOVirtW}), (\ref{eq:FOVirtZ}), and (\ref{eq:FOVirtPhoton}) if we use the expansion $U_i^2 = 1 - 2\Gamma_i \ln^2 \frac{m_V}{\sqrt{s}} + \ldots$ and $\left[ U^{\rm em}_{Q_{\rm tot}^2}\left(\Lambda, m_V; s \right) \right]^2 = 1 + \frac{\alpha_{\rm em}(\mu)}{\pi} Q_{tot}^2 \left( \ln^2 \frac{m_V}{\sqrt{s}} - \ln^2 \frac{\Lambda}{\sqrt{s}} \right) + \ldots$.

\subsection{Real corrections}\label{subsec:RealInclLL}

In this section we will calculate the resummation of the real emissions. We first give the results for the case of a single SU(2) gauge symmetry, and then extend the results to the case of the broken SU(2) $\otimes$ U(1) of the standard model.
\subsubsection{Simple SU(2)} \label{subsec:Simple SU(2)}
For a single SU(2) symmetry, the virtual results can be obtained from \eq{virtualResummed} by setting $\alpha_1 = \alpha_{\rm em} = 0$.
It will be useful to rewrite them in a slightly different form, separating the contributions from the different helicities
\begin{align}
\label{eq:virtualAsShower}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} &= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)
\,.\end{align}
The resummed logarithms are now contained in the factor $\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)$.
The superscript $H$ denotes that each fermion has a fixed helicity.
The Born cross-sections are given by
\begin{align}
\hat \sigma^{B}_{q^H q^H \to \ell^H \ell^H} &= N \,  \frac{8\, \alpha_2^2 \left( \, T^3_{q^H} T^3_{\ell^H}\right)^2}{3}
\nonumber \\
\hat \sigma^{B}_{q_1^L q_2^L \to \ell_1^L \ell_2^L} &= N \,  \frac{2\alpha_2^2 }{3}
\,,\end{align}
where $T^3_{q^H}$ denotes the weak isospin of the fermion $q / \ell$ with helicity $H$. The factor $\Delta^{\rm SU(2)}$resums the leading logarithms and is given by
\begin{align}
\label{eq:DeltaSU2}
\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) = \exp\left[ - \frac{A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H}}{2}\ln^2 \frac{m_V^2}{s}\right]
\,,\end{align}
where
\begin{align}
\label{eq:AfHdef}
A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H} = \frac{\alpha_2}{2\pi} \sum_iT_{i}^2
\,,\end{align}
and the sum over $i$ runs over all particles $i \in \{q_1^H, q_1^H, \ell_1^H, \ell_2^H$\}.
Summing \eq{virtualAsShower} over all possible helicity structures, we reproduce \eq{virtualResummed} in the limit $\alpha_1 = \alpha_{\rm em} = 0$.

By rewriting our result as in \eq{virtualAsShower}, one notices that it can be interpreted as the exclusive cross-section for the scattering process $q_1^H q_2^H \to \ell_1^H \ell_2^H$, where $\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m^2, s; s)$ is a Sudakov factor describing the probability of not having an emission of electroweak gauge bosons between the scales $s$ and $m_V^2$ for a process with center of mass energy $s$. Since of course the emission of a massive gauge boson always gives rise to a scale above $m_V^2$, this exclusive cross-section is by definition equal to the virtual result.

\eq{virtualAsShower} is precisely the result that a parton shower would predict for the exclusive cross-section\footnote{Note that our Sudakov factor for the initial state particles does not include the ratios of PDFs that usually arise in backward evolution. This ratio of PDFs only contributes to NLL.}, and one can use insight from parton shower evolution to derive the expressions for real gauge boson radiation. The real emission of a gauge boson is given in a parton shower by the product of Altarelli-Parisi splitting functions, which describe the emission with a given transverse momentum $k_T^2$, multiplied by a Sudakov factor, which gives the no-branching probability above the value of $k_T^2$ as explained in \cite{Sjostrand:2006za}.
Thus, the total inclusive real radiation cross-section (the cross section with one or more extra gauge bosons in the final state) is given by
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + nV} &= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \int_{m_V^2}^{s} \! \mathrm{d} k_T^2 \, \frac{\mathrm{d}}{\mathrm{d} k_T^2}
\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(k_T^2, s; s)
\nonumber\\
&=
\hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \left[ 1 - \Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} (m_V^2, s; s)\right]
\,.\end{align}
Such an inclusive cross-section makes sense only if the measurement is not breaking the SU(2) symmetry. This is because the inclusive cross section is defined at a scale $\mu \sim k_T$, while the SU(2) symmetry is only broken at the scale $\mu \sim m_V$. This implies that the flavor structure one would obtain at the scale $k_T$ can be changed by the further emissions of extra gauge bosons, making an inclusive measurement with definite flavor structure (which is what breaks the symmetry) ill defined.

Continuing to work in an unbroken SU(2) theory, one can also define the exclusive real radiation cross section (the cross section with exactly one extra gauge boson in the final state). This requires adding an extra no-branching probability from the scale $k_T^2$ to the scale $m_V^2$, which accounts for the fact that no extra gauge bosons are emitted from the fermions and the extra gauge boson with lower $k_T^2$. This extra factor is given by
\begin{align}
\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^HV}(m_V^2, k_T^2; s) \equiv \Delta_V(m_V^2, k_T^2; \hat k_T^2) \Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, k_T^2; s)
\,,
\end{align}
where the term $\Delta_V$ gives the probability of not emitting extra gauge bosons off the emitted vector boson
\begin{align}
\Delta_V(m_V^2, k_T^2; k_T^2) = \exp\left[ - \frac{\alpha_2 \, C_A }{4\pi}\ln^2 \frac{m_V^2}{k_T^2} \right] \,,
\end{align}
while the second term describes the no-emissions probability below $k_T$ off the fermions
\begin{align}
\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, k_T^2; s) \equiv \frac{\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)}{\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(k_T^2,  s; s)}
\,.
\end{align}
Combining everything together, one therefore finds
\begin{align}
& \hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + V}
\nonumber \\
&\qquad =  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
 \int_{m_V^2}^{s} \! \mathrm{d} k_T^2 \, \frac{\mathrm{d}}{\mathrm{d} k_T^2} \left[\Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(k_T^2, s; s)\right] \Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^HV}(m_V^2, k_T^2; s)
\nonumber\\
&\qquad =  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
\, A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H}\, \Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)
\int_{m_V^2}^{s} \!\!\! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2} \Delta_{V}(m_V^2, k_T^2; k_T^2)
\,.\end{align}
The integral can be performed easily, and we write a general result
\begin{align}
\label{eq:Idef}
I_\beta(m_V^2, s) &\equiv \int_{m_V^2}^{s} \! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2} \,\left[\Delta_{V}(m_V^2, k_T^2; k_T^2) \right]^\beta
\\
&= \frac{2\pi}{\alpha_2 \, \beta \, C_A} \left[ \frac{\sqrt{\alpha_2 \, \beta \, C_A}}{2} \ln\frac{m_V^2}{s} \, {\rm Erf}\left( \sqrt{\frac{\alpha_2 \, \beta \, C_A}{4\pi}} \, \ln\frac{m_V^2}{s}\right)   + \left[\Delta_V(m_V^2, s; s)\right]^\beta - 1
\right]
\,.
\nonumber
\end{align}
With this result, the exclusive cross section for a single emission is given by
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + V}
&=  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
\, A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H}\, \Delta^{\rm SU(2)}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)  \, I_1(m_V^2,s)
 \,.
 \label{eq:Simple SU(2) cross section}
\end{align}

\subsubsection{Full SU(2) $\otimes$ U(1)}
We now extend the results of \subsec{Simple SU(2)} to include the full SU(2) $\otimes$ U(1) gauge structure. The Born cross-section is now given by
\begin{align}
\hat \sigma^{B}_{q^H q^H \to \ell^H \ell^H} &= N \,  \frac{8\, \left( \, \alpha_2 T^3_{q^H} T^3_{\ell^H} + \alpha_1 Y_{q^H} Y_{\ell^H} \right)^2}{3}
\nonumber \\
\hat \sigma^{B}_{q_1^L q_2^L \to \ell_1^L \ell_2^L} &= N \,  \frac{2\alpha_2^2 }{3}
\,,\end{align}
where as before $T^3_{f^H}$ denotes the weak isospin of the fermion $f=q / \ell$ with helicity $H$ and $Y_{f^H}$ denotes the hypercharge of the fermion $f=q / \ell$ with helicity $H$. The hypercharge normalization used is given by \eq{hypercharge}.

As in \subsec{Simple SU(2)}, we can write the LL cross section as the Born cross section times a Sudakov factor. However, contrary to the case of a single SU(2) symmetry, in the broken SU(2)$\otimes$U(1) symmetry of the standard model, below the scale $\mu = m_V$ one needs to continue to evolve the operators with the electromagnetic running. This gives
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} &= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
\Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)
\,.\end{align}
The Sudakov factor from $s$ to $m_V^2$ factors into
two pieces, one for the SU(2) symmetry and one for the U(1)
\begin{align}
\Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) &=\Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}^{\rm SU(2)}(m_V^2, s; s) \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}^{\rm U(1)}(m_V^2, s; s)
\,.\end{align}
The SU(2) contribution was given in \eq{DeltaSU2}, while the term coming from the U(1) symmetry is given by
\begin{align}
 \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}^{\rm U(1)}(m_V^2, s; s)) &=  \exp\left[ - \frac{A_{q_1^Hq_2^H\ell_1^H\ell_2^H}^{U(1)}}{2}\ln^2 \frac{m_V^2}{s} \right]
\,,\end{align}
with
\begin{align}
\label{eq:AfHdefU1}
A^{\rm U(1)}_{q_1^H q_1^H\ell_1^H\ell_2^H} = \frac{\alpha_1}{2\pi} \sum_iY_{i}^2
\,.\end{align}
The running below $m_V$ is determined only by the total charge of the operator, and one finds
\begin{align}
\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s) =  \exp\left[ - \frac{\alpha_{\rm em} Q_{\rm tot}^2}{4\pi}\left(\ln^2\frac{\Lambda^2}{s} - \ln^2\frac{m_V^2}{s} \right)\right]
\,.\end{align}
Summing over all possible helicity structures, we reproduce the resummed results of \subsec{VirtLL}.

To obtain the resummation of the real radiation, we follow the steps of \subsec{Simple SU(2)}, taking into account the full SU(2)$\otimes$U(1) structure above $m_V^2$ and the running due to the photon below $m_V^2$. For the $W^\pm$ bosons, the U(1) symmetry does not contribute, but one needs to be careful about the flavor structure when breaking the electroweak symmetry.
One finds
\begin{align}
&\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + W^\pm}
\nonumber \\
&\quad
=  \left[ \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H W^\pm}(\Lambda^2, m_V^2; s) \int_{m_V^2}^{s} \!  \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2} \Delta_{V}(m_V^2, k_T^2; k_T^2) \right]
\nonumber\\
&\quad\quad \times\left( \hat \sigma^{B}_{{q'}_1^H q_2^H \to \ell_1^H \ell_2^H} A^{W^\pm}_{q_1^H} + \hat \sigma^{B}_{q_1^H {q'}_2^H \to \ell_1^H \ell_2^H} A^{W^\pm}_{q_2^H} + \hat \sigma^{B}_{q_1^H q_2^H \to {\ell'}_1^H \ell_2^H} A^{W^\pm}_{\ell_1^H}+ \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H {\ell'}_2^H} A^{W^\pm}_{\ell_2^H} \right)
 \nonumber\\
&\quad
=  \left[ \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H W^\pm}(\Lambda^2, m_V^2; s) \, I_1 (m_V^2, s) \right]
\nonumber\\
&\quad\quad \times\left( \hat \sigma^{B}_{{q'}_1^H q_2^H \to \ell_1^H \ell_2^H} A^{W^\pm}_{q_1^H} + \hat \sigma^{B}_{q_1^H {q'}_2^H \to \ell_1^H \ell_2^H} A^{W^\pm}_{q_2^H} + \hat \sigma^{B}_{q_1^H q_2^H \to {\ell'}_1^H \ell_2^H} A^{W^\pm}_{\ell_1^H}+ \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H {\ell'}_2^H} A^{W^\pm}_{\ell_2^H} \right)
\,,
 \nonumber\\
 \label{eq:RealWFinal}
 \end{align}
where $f'$ is the fermion $f$ becomes after having radiated a $W^\pm$ that is $u'=d$, $d'=u$, $l'=\nu$ and $\nu'=l$ and for any flavor set which allows a $W^\pm$ emission there is one of the Born cross section which is zero because its electromagnetic charge is not conserved. Also, we broke $A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H}$ in its component related to the emission of a $W^\pm$, that is
\begin{align}
A^{\rm SU(2)}_{q_1^H q_1^H\ell_1^H\ell_2^H} = A^{W^3}_{q_1^H q_1^H\ell_1^H\ell_2^H}+\sum_i A^{W^\pm}_i
 \,,
\end{align}
with
\begin{align}
A^{W^\pm}_{f^L}=\frac{\alpha_2}{4\pi}
\,,\qquad
A^{W^\pm}_{f^R}=0
 \,,
\qquad
A^{W^3}_{q_1^H q_1^H\ell_1^H\ell_2^H}=\frac{\alpha_2}{2\pi} \sum_i (T_i^3)^2
 \,.
\end{align}

For the emissions of $Z$ bosons and photons, one needs to take into account the mixing between the third component of the SU(2) gauge symmetry and the U(1) gauge symmetry.
The emission of a $W^3$ boson is given by
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + W^3}
&=  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
A^{\rm W^3}_{q_1^H q_1^H\ell_1^H\ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)
\nonumber\\
& \qquad  \times
\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s) \,
\int \! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2} \Delta_{V}(m_V^2, k_T^2; k_T^2) \, ,
\end{align}
while for the emission of a U(1) boson $B$
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + B}
&=  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H}
A^{\rm U(1)}_{q_1^H q_1^H\ell_1^H\ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)
\nonumber\\
& \qquad  \times
\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s) \,
\int \! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2}
\,.\end{align}
To combine these two expressions into the emission of a $Z$ boson, we have to mix the amplitude of the emission of a $W^3$ with the amplitude of the emission of a $B$. Each individual emission is given by breaking the coefficient $A^{W^3}_{q_1^H q_1^H\ell_1^H\ell_2^H}$ and $A^{\rm U(1)}_{q_1^H q_1^H\ell_1^H\ell_2^H}$ in their components for an emission from one fermion, and the amplitude is given by the square root of this emission. We mix the amplitudes using that
\begin{align}
Z = s_W B - c_W W^3
 \,,
\end{align}
where $c_W = \cos(\theta_W)$ and  $s_W = \sin(\theta_W)$. This gives
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + Z}
&=  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)  \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)
\nonumber\\
& \qquad \times \int_{m_V^2}^{s} \! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2}
 R_{q_1^H q_2^H \ell_1^H \ell_2^H}(m_V^2, k_T^2)
 \,,
\end{align}
where we have defined
\begin{align}
R_{q_1^H q_2^H \ell_1^H \ell_2^H}(m_V^2, k_T^2) = \alpha_W \sum_i \left(s_W \sqrt{\frac{A^{U(1)}_{i}}{\alpha_W}} - c_W \sqrt{\frac{A^{W^3}_{i}}{\alpha_W} \Delta_{W}(m_V^2, k_T^2; k_T^2)} \right)^2
 \,,
\end{align}
with
\begin{align}
A^{\rm U(1)}_{f^H} = \frac{\alpha_1 }{2\pi}Y_{f^H}^2 \, , \qquad A^{W^3}_{f^H} = \frac{\alpha_1 }{2\pi}(T^3_{f^H})^2
 \,.
\end{align}
By using the simple relations
\begin{align}
\sqrt{\frac{A^{\rm U(1)}_{f^H}}{\alpha_W}} = \sqrt{\frac{1}{2\pi}} \frac{s_W}{c_W} Y_{f^H}
\,,
\qquad
\sqrt{\frac{A^{W^3}_{f^H}}{\alpha_W}} = \sqrt{\frac{1}{2\pi}} T^3_{f^H}
\,,\end{align}
we obtain the final result for the $Z$ boson emission cross section
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + Z}
&=  \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)  \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)
\nonumber \\ \\
& \qquad
\times \int_{m_V^2}^{s} \! \frac{\mathrm{d} k_T^2}{k_T^2} \ln \frac{s}{k_T^2}
\Bigg{(}
s_W^2 A^{\rm U(1)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} - A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \sqrt{\Delta_{W}(m_V^2, k_T^2; k_T^2)}
 \nonumber\\
& \qquad\qquad\qquad
 +c_W^2 A^{W^3}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \Delta_{W}(m_V^2, k_T^2; k_T^2) \Bigg{)}  \nonumber\\
&= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)  \, \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)\, \nonumber\\ & \qquad\qquad
\Bigg{(} s_W^2 A^{\rm U(1)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \, \frac{1}{2} \ln^2 \frac{m_V^2}{s}
\nonumber\\
& \qquad\qquad
  - A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H}
\, I_{\frac{1}{2}}(m_V^2,s)
+
c_W^2 \, A^{W^3}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \, I_{1}(m_V^2,s)
\Bigg{)} \,,
 \label{eq:RealZFinal}
\end{align}
where
\begin{align}
A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H} = \frac{\alpha_{\rm em}}{\pi} \sum_{i}  T^3_{i} Y_{i}
\,.
\end{align}

For the emission of a photon, we use that, for a scale higher than the electroweak bosons masses, the photon is a mixing of the $B$ and $W^3$ bosons.
\begin{align}
\gamma = c_W B + s_W W^3
 \,,
\end{align}
while, for a scale lower than the electroweak bosons masses, the photon can still be produced proportionally to the derivative of its no-branching probability $\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(k_T^2, m_V^2; s)$.
\begin{align}
\hat \sigma^{\rm LL}_{q_1^H q_2^H \to \ell_1^H \ell_2^H + \gamma}
&= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s)
\nonumber\\
& \qquad \times \Bigg{[}
\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)
\bigg{(}
c_W^2 A^{\rm U(1)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \frac{1}{2}\log^2(\frac{m_V^2}{s})
\nonumber\\
& \qquad  \qquad + A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H}
I_{\frac{1}{2}}(m_V^2,s) +
s_W^2 A^{W^3}_{q_1^Hq_2^H\ell_1^H\ell_2^H}
I_{1}(m_V^2,s)
\bigg{)} +
\nonumber\\
& \qquad \qquad \int_{\Lambda^2}^{m_V^2} \! \mathrm{d} k_T^2 \! \frac{\mathrm{d}}{\mathrm{d} k_T^2} \left[\Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(k_T^2, m_V^2; s) \right] \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, k_T^2; s)
\Bigg{]}
\nonumber\\
&= \hat \sigma^{B}_{q_1^H q_2^H \to \ell_1^H \ell_2^H} \, \Delta_{q_1^Hq_2^H\ell_1^H\ell_2^H}(m_V^2, s; s) \Delta^{\rm em}_{q_1^Hq_2^H\ell_1^H\ell_2^H}(\Lambda^2, m_V^2; s)
\nonumber\\
& \qquad \times \Bigg{[}
c_W^2 A^{\rm U(1)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \frac{1}{2}\log^2(\frac{m_V^2}{s})+ A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H}
I_{\frac{1}{2}}(m_V^2,s) +
\nonumber\\
& \qquad  \qquad  s_W^2 A^{W^3}_{q_1^Hq_2^H\ell_1^H\ell_2^H}
I_{1}(m_V^2,s) + \frac{\alpha Q^2_{\rm tot}}{4 \pi} \left( \ln^2 \frac{\Lambda^2}{s} - \ln^2 \frac{m_V^2}{s} \right)
\Bigg{]}
\,.
\label{eq:RealPhotonFinal}
\end{align}

\section{Results}\label{sec:Results}
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{plotincl13TeV}
\caption{The cross-section summed over all lepton flavors for the 13 TeV LHC. On the top we show the individual corrections relative to the Born cross-section as defined in~\eq{upperplot}, in the middle the relative size of the resummation as defined in \eq{middleplot}, while on the bottom we show the total perturbartive correction relative to the Born as defined in \eq{lowerplot}.
Virtual corrections are shown in black, while real corrections with a $Z$, photon, $W^+$, $W^-$ are shown in green, orange, red and blue. Resummed corrections are shown in solid lines, while fixed order results are dashed. The x-axis denotes the fraction of the partonic center of mass energy relative to the collider center of mass energy. }
\label{fig:Inclusive13}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{plotincl100TeV}
\caption{The cross-section summed over all lepton flavors for a 100 TeV $pp$ collider on the right. All colors are the same as in \fig{Inclusive13}.}
\label{fig:Inclusive100}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat[13 TeV LHC]{\includegraphics[width=0.5\textwidth]{plotee13TeV}}
\hfill%
\subfloat[100 TeV collider]{\includegraphics[width=0.5\textwidth]{plotee100TeV}}
\caption{The cross-section for $e^+ e^-$ for the 13 TeV LHC on the left, and 100 TeV $pp$ collider on the right.
All colors are the same as in \fig{Inclusive13}. Note that the scaling of the $y$ axis is different for the LHC and the 100 TeV collider.}
\label{fig:ee}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat[13 TeV LHC]{\includegraphics[width=0.5\textwidth]{plotnunu13TeV}}
\hfill%
\subfloat[100 TeV collider]{\includegraphics[width=0.5\textwidth]{plotnunu100TeV}}
\caption{The cross-section for $\bar \nu\nu$ for the 13 TeV LHC on the left, and 100 TeV $pp$ collider on the right. All colors are the same as in \fig{Inclusive13}.}
\hspace{-2ex}
\label{fig:nunu}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat[13 TeV LHC]{\includegraphics[width=0.5\textwidth]{plotenu13TeV}}
\hfill%
\subfloat[100 TeV collider]{\includegraphics[width=0.5\textwidth]{plotenu100TeV}}
\caption{The cross-section summed for $e^+ \nu$ for the 13 TeV LHC on the left, and 100 TeV $pp$ collider on the right. All colors are the same as in \fig{Inclusive13}.}
\hspace{-2ex}
\label{fig:enu}
\end{figure}
\begin{figure}[h!]
\centering
\subfloat[13 TeV LHC]{\includegraphics[width=0.5\textwidth]{plotnue13TeV}}
\hfill%
\subfloat[100 TeV collider]{\includegraphics[width=0.5\textwidth]{plotnue100TeV}}
\caption{The cross-section for $\bar\nu e^-$ for the 13 TeV LHC on the left, and 100 TeV $pp$ collider on the right.  All colors are the same as in \fig{Inclusive13}.}
\hspace{-2ex}
\label{fig:nue}
\end{figure}

In this section we analyze the results presented in the last two sections
for both the the 13 TeV LHC and a 100 TeV proton-proton collider. The main reason to present the results for a 100 TeV collider is that it is the main candidate to succeed to the LHC and the importance of our results increase with the energy. Our results show that resumming both the real and virtual logarithms is essential for a 100 TeV collider.

We begin by explaining the format used in all of our plots: solid lines represent the resummed LL corrections, while dashed lines are the fixed order double logarithmic correction. Black lines correspond to the virtual corrections while the real corrections with $Z$, photon, $W^+$ and $W^-$ are shown in red, orange, blue and green, respectively. For each plot, we show on the top the perturbative corrections relative to the Born, in the middle the relative size of the resummation and on the bottom the total perturbative correction after summing virtual and real relative to the Born. To be more precise, on the top we plot for virtual and real
\begin{align}
\label{eq:upperplot}
{\rm Virt:} \quad \frac{\sigma_{pp \to \ell_1 \ell_2}-\sigma^{\rm B} _{pp \to \ell_1 \ell_2}}{\sigma^{\rm B} _{pp \to \ell_1 \ell_2}}
\, , \qquad\qquad
{\rm Real\, (V):} \quad \frac{\sigma_{pp \to \ell_1 \ell_2V}}{\sigma^{\rm B} _{pp \to \ell_1 \ell_2}}
\,,\end{align}
using either the resummed or fixed order expression.  In the middle we show for the virtual contribution
\begin{align}
\label{eq:middleplot}
{\rm Virt:} \quad \frac{\sigma^{\rm FO}_{pp \to \ell_1 \ell_2} - \sigma^{\rm Res}_{pp \to \ell_1 \ell_2}}{\sigma^{\rm Res}_{pp \to \ell_1 \ell_2}}
 \,, \qquad \qquad
{\rm Real \, (V):}\quad
\frac{\sigma^{\rm FO}_{pp \to \ell_1 \ell_2 V} - \sigma^{\rm Res}_{pp \to \ell_1 \ell_2 V}}{\sigma^{\rm Res}_{pp \to \ell_1 \ell_2 V}}
\,,\end{align}
while for the lower plot we show
\begin{align}
\label{eq:lowerplot}
\frac{\sigma_{pp \to \ell_1 \ell_2} + \sigma_{pp \to \ell_1 \ell_2V}-\sigma^{\rm B} _{pp \to \ell_1 \ell_2}}{\sigma^{\rm B} _{pp \to \ell_1 \ell_2}}
\,,\end{align}
for both fixed order and resummed. All effects are shown as a percentage.

We start with the result for the fully inclusive cross-section, where we sum over the flavors of all final state particles. In other words, we are summing over the virtual corrections for any lepton flavor, and the real corrections for any lepton flavor and gauge boson type.
In terms of equations, for the fixed order result, the virtual are obtained by summing over all terms in Eqs. (\ref{eq:FOVirtW}), (\ref{eq:FOVirtZ}), and  (\ref{eq:FOVirtPhoton}); while the real are given by Eqs. (\ref{eq:FORealZ}), (\ref{eq:RealWp}), (\ref{eq:RealWm}), and (\ref{eq:RealPhoton}). The resummed result are given by Eqs. (\ref{eq:virtualResummed}), (\ref{eq:RealZFinal}), (\ref{eq:RealWFinal}), and (\ref{eq:RealPhotonFinal}). All those results are for $\Lambda=m_Z$, that is we are resolving the photon only down to the mass of the $Z$. A lower $\Lambda$ would reduce all the exclusive cross section but the real cross section for photon as calculated in eq. (\ref{eq:RealPhotonFinal}) which will increase up to a value of $\Lambda$ low enough than
$\frac{\alpha Q^2_{\rm tot}}{4 \pi} \left( \ln^2 \frac{\Lambda^2}{s} - \ln^2 \frac{m_V^2}{s} \right) = 1 - c_W^2 A^{\rm U(1)}_{q_1^Hq_2^H\ell_1^H\ell_2^H} \frac{1}{2}\log^2(\frac{m_V^2}{s})- A^{\text{mixing}}_{q_1^Hq_2^H\ell_1^H\ell_2^H} I_{\frac{1}{2}}(m_V^2,s) - s_W^2 A^{W^3}_{q_1^Hq_2^H\ell_1^H\ell_2^H} I_{1}(m_V^2,s)$,
and then decrease if $\Lambda$ continues to be lowered.

The results for the virtual and real contributions both at fixed and resummed order are shown in \fig{Inclusive13} for the LHC and in \fig{Inclusive100} for the 100 TeV collider. For the LHC, the corrections from virtual contributions range from ${\cal O}(15\%)$ at $\sqrt{s} \sim 1$ TeV to ${\cal O}(30\%)$ at $\sqrt{s} \sim 3$ TeV, while the real corrections for a given gauge boson are about a factor of 3 smaller individually. However, as can be seen from the ratio plot in the middle, the relative effect of the resummation is more than twice bigger for the real correction compared to the virtual correction. This clearly shows that the size of the resummation effect cannot be inferred from the size of the fixed order correction alone. The relative effect of the resummation for the virtual reaches from ${\cal O}(10\%)$ at $\sqrt{s} \sim 1$ TeV to ${\cal O}(20\%)$ at $\sqrt{s} \sim 3$ TeV while  the relative effect of the resummation for the real reaches from ${\cal O}(20\%)$ at $\sqrt{s} \sim 1$ TeV to ${\cal O}(50\%)$ at $\sqrt{s} \sim 3$ TeV.

In the lower part of the plot for the fixed order, one can see that after summing the virtual and real, the perturbative corrections largely cancel, but a small effect at the  ${\cal O}(1\%)$ level persists. For a fully inclusive cross section the logarithmically enhanced virtual and real corrections cancel against each other, up to the fact that the $pp$ initial state is not an iso-singlet. The large cancellation can be understood from \eq{InclusivePartonic}, which shows that switching the flavor of initial state anti-quark changes the sign of the partonic cross-section. Since pdf's for sea quarks are similar in magnitude, one expects $f_{\bar u / p} \sim f_{\bar d / p}$, explaining the cancellation. For the resummed result on the other hand, there is no cancellation. That is because even if the initial state is an SU(2) singlet, the cancellation would occur only for an inclusive result, that is summing the virtual to the real for any number of gauge boson. The remaining correction for the resummed result is thus mostly due to the production of more than one gauge boson with also an order 1\% correction due to the initial state not being an SU(2) singlet. This  remaining correction ranges from $2\%$ at $\sqrt{s} \sim 5$ TeV to $7\%$ at $\sqrt{s} \sim 25$ TeV.

For the 100 TeV collider, the results are qualitatively the same, but given the much larger reach in energy, the overall size of the effects are much larger. The virtual contributions range from ${\cal O}(30\%)$ at $\sqrt{s} \sim 5$ TeV to ${\cal O}(60\%)$ at $\sqrt{s} \sim 25$ TeV, with the real corrections again roughly a factor of 3 smaller. The relative size of the resummation, is also much larger, and at $\sqrt{s} \sim 25$ TeV changes the result by ${\cal O}(50\%)$ for the virtual and by ${\cal O}(200\%)$ for the real corrections. Thus, at such large energies, resummation has to be included to get a reliable estimate of the effects, not only for virtual corrections but also for the real emissions. Once the virtual and real corrections are added at fixed order, the total corrections again are very small, at the percent level; while for the resummed result, the total correction ranges from ${\cal O}(10\%)$ at $\sqrt{s} \sim 5$ TeV to ${\cal O}(30\%)$ at $\sqrt{s} \sim 25$ TeV.

Next, we consider the results for final states with specific leptons flavors. From the Figures \ref{fig:ee} to \ref{fig:nue}, one can see that at the LHC, the virtual corrections range from ${\cal O}(15\%)$ at $\sqrt{s} \sim 1$ TeV to ${\cal O}(30\%)$ at $\sqrt{s} \sim 3$ TeV, with the exact numbers depending  on the leptonic final state chosen, while at a 100 TeV collider they can exceed $50\%$ at $\sqrt{s} \sim 25$ TeV. Resummation at the LHC changes the virtual corrections by ${\cal O}(10\%)$ at  $\sqrt{s} \sim 1$ TeV to ${\cal O}(20\%)$ at  $\sqrt{s} \sim 3$ TeV, while at a 100 TeV collider the effect can become as large as $50\%$. Resummation at the LHC changes the real corrections by ${\cal O}(20\%)$ at  $\sqrt{s} \sim 1$ TeV to ${\cal O}(60\%)$ at  $\sqrt{s} \sim 3$ TeV, while at a 100 TeV collider the effect can become as large as $200\%$.
After summing over virtual and real corrections, the remaining perturbative corrections grow with energy are much larger than in the fully inclusive case. This is of course expected, since by specifying the leptonic final state, we are not considering an inclusive final state any longer.

\section{Conclusions}\label{sec:Conclusions}

In this paper, we considered Drell-Yan production of leptons at a hadron collider, and presented result for the logarithmic resummation at LL accuracy for electroweak corrections to the total cross-section in simple analytic form.

Using these analytical results, we have analyzed the size of the corrections numerically for the 13 TeV LHC and a possible future 100 TeV $pp$ collider. Our results show than the real resummation is as important as the virtual resummation and its importance increases fast with the partonic center of mass energy: at the LHC, the resummation effect over the fixed order goes from around 10\% at 1 TeV to around 20\% at 3 TeV for the virtual with any leptonic final state and goes from around 20\% at 1 TeV to around 50\% at 3 TeV for all kinds of real emissions with any leptonic final state. The resummed virtual over the Born cross-section goes from around 10\% at 1 TeV to 30\% at 3 TeV, while each resummed reals are around the third of the resummed virtual. At a future 100 TeV $pp$ collider, the resummation effects over the fixed order can be much larger, and are around 50\% for the virtual and 200\% for the real at 25 TeV in most cases. The resummed virtual over the Born cross-section is around 80\% at 25 TeV.

For final states with fixed lepton flavors, summing the virtual and real corrections does not reduce the magnitude of the double logarithmic terms, as expected for a result that is not inclusive. After summing over all lepton flavors, a large cancellation is observed at fixed order, but a small logarithmic sensitivity remains, due to the fact that protons are not SU(2) singlets. On the other hand, for the resummed results, a large effect remains because we are including only the first emission for the real. The cancellation will happen only for an inclusive result, that is summing the virtual and the real for any number of emitted bosons.

In this paper, we have only considered the resummation of the real corrections to the total cross-section, and not given results that are differential in the emitted vector boson. That implies that we were only able to present results that were differential in $s$, and not $m_{\ell\ell}$, since these differ for real emissions. However, one can use the same approach used in this work, namely using the analogy to a parton shower, to obtain results that are fully differential. This will be the subject of a future paper.

\acknowledgments
{We would like the thank Aneesh Manohar for several stimulating discussions, and Michelangelo Mangano for helpful comments. This work was supported by by the Office of High Energy Physics of the U.S. Department of Energy under contract DE-AC02-05CH11231.
}

\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


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text:
Home Food & Home Cooking Tips & Nutrition The Truth Behind the Label
From the June 2019 Metro Parent magazine
The Truth Behind the Label
The best way to buy healthy food is to read the label, but what if it's trying to mislead you?
Carly Adams
BROUGHT TO YOU BY United Dairy Industry of Michigan
Picture this: you're walking through the grocery store trying to find food that's healthy, nutritious and safe for your family. You see some strawberries that scream "GMO-free" across the packaging, so you toss them into your cart because they must be healthier. But all strawberries in the U.S. are GMO-free, even though the packaging might not say so. So how can you tell which labels to trust and which are just marketing tactics? Registered dietitian Michael DeAngelis, vice president of Integrated Communications for New England Dairy & Food Council, says it's time to think critically about food labels.
"Some labels are tied to specific rules and regulations," DeAngelis says. However, many are used to give one product a marketing edge over another. "It's unfortunate, because we have people steering clear of products that are perfectly healthy, perfectly good and perfectly delicious because they're fearful of something that's just not factual."
What is 'clean'?
"Clean," for example, is a term often used on food packaging in hopes of attracting the health-conscious consumer. But, according to DeAngelis, the term has no definition according to any regulatory body. "It's completely a marketing term that people throw around."
Generally, consumers looking for "clean" options are seeking food with labels they can easily read or foods that are minimally processed. DeAngelis believes dairy fits into this perspective. "Milk has three ingredients: milk, vitamin A and vitamin D. I'd consider that a simple, clean food. Although it is pasteurized for food safety, it's a minimal form of processing."
Processed doesn't mean bad
The processing that milk goes through doesn't impact how healthy it is, but helps to ensure it's safe for consumption and meets the consumers' demands.
"If you were to have milk in its natural form, it would separate. That's what it used to do, and people didn't like having to always shake their milk. So as the years went by, people pointed out that they prefer their milk to be homogenized, which keeps the cream from separating. Now that's pretty much the standard, but it's technically a process," he says. "Processed doesn't necessarily mean it's bad."
Don't necessarily fear the ingredients
For those who think an easy-to-read ingredient list is the key to good health, think again.
"I was at a meeting and asked, 'Who here had dihydrogen monoxide today?' Everyone looked worried," DeAngelis says. "But that's just a chemical name for water."
When you see the chemical name for something, it might not be as scary as it looks.
Citric acid, for example, is naturally occurring in citrus fruits and many people have enjoyed citrus for their entire life! "You might add citric acid to something to help preserve it, or to add flavor or acidity to it. Those are safe and healthy things, but they look bad when it's a chemical name on the label," he notes.
The GMO mystery
What about the "non-GMO" labels? GMOs are bad, right? Wrong.
DeAngelis points out that GMOs have been around for a long time yet there is no evidence showing any negative health effects for anyone as a result of consuming genetically modified or bioengineered food. "It's just the unknown that people fear. They don't understand what it is, so they are concerned."
GMO is another way to say something is bioengineered, says DeAngelis, comparing it to an enhanced breeding process.
"Typically, you would take two crops with desirable traits and crossbreed them to have those desirable characteristics," he says. But with that process, you can have mutations. "What bioengineering does is looks at those desired characteristics on the genetic level and then inserts that gene into the seed. That way we don't end up with uncontrolled consequences."
Still, many insist that GMOs add things to our food unnecessarily. "It's just not true," he emphasizes, and many GMO advances have helped our crops become disease resistant. "Papayas were going to be extinct unless they found a solution for a viral disease. Bioengineering solved that."
nine crops in the United States are bioengineered: alfalfa, canola, cotton,
maize, papaya, potato, soybean, squash and sugar beet. So "GMO strawberries,"
for example, just don't exist. "Strawberries aren't bioengineered."
"Organic," however, is a legitimate term with real regulations. Organic specifically speaks to the production practices used by the farm.
"A lot of times it has to do with what kind of fertilizer and pesticides are used to raise a crop. Organic farms still use pesticides," DeAngelis adds, "just non-synthetic ones."
Milk and milk products that are labeled "USDA organic" are made using milk from cows never treated with antibiotics. But DeAngelis notes that all milk sold in the U.S is actually antibiotic-free.
When a dairy cow falls ill and needs antibiotics, it is permanently moved to a non-organic producing herd. However, that dairy cow's milk must test clean of antibiotics before the cow is allowed to produce milk for consumers again. "Whether it's labeled or not, it doesn't matter. The milk is antibiotic-free," he says.
Although DeAngelis says while there's no scientific evidence finding that organically-produced foods are safer or healthier than conventionally-produced product, "it does meet some people's values – that they just want it to be produced in a particular way – and that's perfectly fine."
A matter of values
When it comes to buying food with your values in mind rather than dietary needs, many consumers look for labels like "responsibly sourced," "grass fed" or "cage-free."
"Nowadays, consumers want to know farmers are raising food or animals in a way that's not negatively impacting the environment," he says. "Be assured that all farmers are looking to leave the land better than when they received it."
on family-operated farms, though, farmers are always looking for ways to
produce crops and livestock responsibly. "Be assured that all farmers are
looking to leave the land better than when they received it."
Navigating the labels
So what labels should you be on the lookout for? DeAngelis says that if you have any dietary restrictions, sensitivities or allergies – or if your doctor has advised you to stay clear of something – then those should be avoided. Also keep in mind that anything labeled "raw" should be eaten with caution. Raw can potentially come with pathogens normally killed during cooking or 'processing.'
Beyond that, "the best thing you can do is eat a variety of foods from a variety of sources. Eat a balanced diet and enjoy."
Brought to you by Michigan Dairy Farm Families. Approved by the Michigan Dairy Market Program Committee. For more information, visit milkmeansmore.org.
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redpajama_set_name: RedPajamaCommonCrawl


•	Recently a new rainfed agriculture atlas was released highlighting the policy biases that are making farming unviable for many in these areas.
•	There has been negligence toward rainfed areas which is leading to lower incomes for farmers in these areas.
•	Further farmers in rainfed areas are receiving 40% less of their income from agriculture in comparison to those in irrigated areas.
•	Even though rainfed agriculture contributes to 60 percent of the value of agriculture GDP of India, there is a clear-cut bias towards irrigated areas when it comes to public investment in agriculture in the country.
•	This neglect, together with unsuitable programme design, has ensured that potential of rainfed areas remains unrealized.
•	In terms of procurement over the decade between 2001-02 and 2011-12, the government procured ₹5.4 lakh crore of wheat and rice. On the other hand, coarse cereals, which are grown in rainfed areas, only had ₹3,200 crore worth of procurement in the same period.
•	Flagship government schemes, such as seed and fertiliser subsidies and soil health cards, are designed for irrigated areas and simply extended to rainfed farmers without taking their needs into consideration.
•	More balanced approach is needed, to give rainfed farmers the same research and technology focus, and production support that their counterparts in irrigation areas have received over the last few decades.
•	There is an urgent need to do more R&D in rainfed agriculture and bring in more policy perspective like tweaking of government schemes by considering the needs of rainfed agricultural areas.
•	In the long run, cash incentives and income support like the PM-KISAN scheme announced in the interim budget 2019 are better than extensive procurement as they are inclusive in character, and doesn't distinguish between farmers in one area or another, growing one crop or another.
•	Along with income support to help farmers through the current crisis, it is now the time to design better structured interventions for the future.
•	Like the ease of doing business, ease of doing farming should be done on the parameters of seeds, soil, water in rainfed areas to make agriculture attractive in the long term.


The best things in life are free. Yes, free! I am giving away "Free Entrance and Free Ride Tickets in Paradizoo and Zoomanity Parks." I will be choosing two winners of 4 free entrance tickets and 4 free ride tickets. Each ticket is good for 2 persons. Enjoy!
By joining you agree to the "Terms and Conditions" of The Lifestyle Avenue. Winners will be contacted through email and contact details you provided in Rafflecopter.
sana makapunta kami ng pamilya ko dito..sigurado mageenjoy kami..
Great for relaxation, to reward yourself from the daily stresses in life. raffymichel from 8share philippines!
An interesting place to unwind yourself. raffymichel from 8share Philippines!!!
Yes I agree. Thanks Raffymichel!
Thank you! It's really a nice place!
Perfect! Jlnnrmss from 8share Philippines.
Hi Mai! This will be a great family bonding!
Hi French! Thanks for the comment!
I think this will lead to no.1 touristpots in tagytay .!more power to you paradizoo!
Cool place and nice give away, sana makapunta din kami dyan..
sana po mai from 8share ako po makauha ng libreng ticket, ipapasyal ko po ang mga kapatid ko saparadizoo.
Thank you 8share for making this website to everyone.
What a perfect treat for the whole family this summer!
thank u 4 sharing this free passes ticket good 4 person in paradizoo i wish to go there with my family,..
i hope to have vacation in this place soon. it's very nice.
Nakakawala siguro ng stress dito….
Hello Iam Clyde Anthony Capricho 5080 Malolos st. Brgy.tejeros makati City Good Evening Po !!!
Its a best place to go with your bestfriends!
It would be nice to have a vacation there.
a great place to have bonding!
if i will be given a chance to win, i will share it with my bf cause i know he needs to relax because of his very busyat work..
a simple gift i can say and all the effort he is giving to our relationship.
Thanks 8share for introducing this perfecr summer getaway. I'd love to visit this place the soonest.
Thank you for giving us opportunity like this. I wish I can enjoy this privilege to visit this wonderful place.


The new Fridja F-1400 is Fridja's most advanced steamer yet! Transform tired looking, creased garments and fabrics into presentable pieces quickly and easily.
Traditional ironing is time consuming and laborious, particularly in a commercial or retail setting. With a Fridja steamer you can avoid taking garments from their hanger by ironing them directly whilst still hung up. The Fridja steamer allows you to simply get professional results with the minimum of fuss.
The Fridja F-1400 steamer has a large 3.2 litre water tank that produces continuous steam for up to 100 minutes. Simply switch it on using the control panel or remote control in the handle and in just 45 seconds the steamer is ready to use.
One of the greatest advantages of using a steamer rather than an iron is that they are safe to use on nearly any fabric, even delicate fabrics like silk, satin, chiffon and clothes with decorative details like beads and sequins.
They are also safe to use on dry clean only clothes, like suits and winter coats.
A folding hanger with adjustable shoulders is also supplied with the steamer that allows you to hang garments for speedy crease and crinkle removal. Steam is also effective in refreshing and sanitising fabrics, so you could use it on sofas, bedding, baby's clothes and toys. The telescopic pole can be adjusted from 80cm to 132cm, making it easy for anyone to use and suiting different environments.
The Fridja F-1400 professional garment steamer also includes a host of accessories including a trouser clamp, mini ironing mat, upholstery brush and a hand-protecting glove. The fantastic Fridja F-1400 professional garment steamer can benefit both domestic and commercial environments.


* Fully functional date chronograph subdial at 6.
* Independently adjustable GMT hand.
* Rolex crown engraved on the stem.
* All correct markings and engravings.
* Exactly like the Authentics Green Hologram sticker in the back.
* Perfect weight and feel.
* Exactly like the Authentics Strong construction.
* Full automatic movement, with sweeping second hand.
* Threaded screws in links, as per the original, and not pins.


Homeowners can also enjoy the versatility and energy efficient warmth of propane. Comfort, cost savings and transportability make it an ideal fuel source for homes in rural areas throughout Alberta. Find out more about residential/home heating solutions. Don't forget to ask us about our membership option.
Propane burns clean, and it is both efficient and economical for all agriculture-related applications. We provide a dependable service and matched versatility delivered right to your farm or ranch, whether you are grain drying or heating calving barns. No matter what your agricultural/farm supply needs, Mutual Propane has got you covered.
For more information regarding this service and our membership program, call us toll-free at 1-800-688-8258 or call the Mutual Propane office.


Big chunks of meat don't always need to be the only source of protein in your diet. Cutting back on meat occasionally and replacing meat with another source of protein can be a good change for the body. And by other sources of protein we don't mean protein shakes and protein powder.
You've probably heard many times before that vegetables do not have much protein and it would be very hard to satisfy your daily protein needs. Yes, vegetables are not complete protein sources, but combining them with other veggies or combining them with eggs and dairy products can be a great way to boost your protein intake.
In this article, we present you with a list of 20 vegetables that will help you boost your protein intake and provide you with fiber at the same time.
Edamame is one of the tastiest and healthiest appetizers. One cup of it or cooked soybeans has an enormous amount of protein. Always opt for an organic type, since the majority of soybeans in the US are actually genetically modified and grown with lots of pesticides. You can try frying it along with scallions, eggs, and asparagus.
Tempeh is made from fermented soybeans and mixing them in a dense cake that can later be fried in a pan like tofu or it can be sliced. It's easily chewable, feels very nutty and has a lot higher amount of fiber and protein in comparison to tofu and since it is fermented, it can be more digestible for some people.
Tofu is the standard vegetarian food known even to non-vegetarians and what vegetarians are most known for to eat. It's made from curdled soy milk that can be fried in a pan, sautéed and stirred or sometimes scrambled. It might not be as full of protein as the tempeh is, but the taste can be more tolerable to some. Always go for the organic types to avoid GMO soy and nasty pesticides.
These beans have a high content of fiber, folate, potassium, vitamin B6 and various types of phytonutrients. They can also be made into brownies.
Steel cut oats are not only a rich source of protein they have a lot smaller glycemic index in comparison to rolled oats. That means that they don't cause big blood sugar spikes, and chances are you'll be much more satiated and will have fewer cravings after you've finished eating them. You can add all sorts of fruit like peach, blueberries, cranberries or various veggies.
Besides having a moderate amount of protein, cashews provide around twenty percent of the daily recommended magnesium intake, as well as twelve percent of the daily recommended vitamin K intake which are two crucial nutrients when it comes to bone health and growth.
Four grams may not seem a big amount to you but considering it is a green vegetable, it sure is. And spinach salad is not the only thing you can make with it. Cooking it with cheese is one way you can increase its content of protein and make it a lot more delicious.
Broccoli offers an amazing amount of fiber and has an amazing amount of protein considering it's a vegetable. Additionally, it's been shown to have amazing anti-cancer compounds including sulforaphane. You can increase your intake by preparing a salad with broccoli and peanuts which combines two plants rich in protein in one easy-to-cook meal.
Tiny Brussels sprouts are needlessly scorned for having a not-so-great taste, especially once frozen, but when it comes to having lots of nutrients they pack a powerful punch. Beside protein, they also have large amounts of vitamin K and potassium.


text:
2. History 2.5 Roman History MS 118
MS Short Title	 SUETONIUS
Text	 SUETONIUS: VITAE 12 CAESARUM
Description	 MS in Latin on paper, Austria, ca. 1470, 169 ff. (complete), 24x16 cm, single column, (16x10 cm), 30 lines in a square, slightly sloping humanistic book script, headings and sidenotes in pale red, 12 5- to 6-lines illuminated initials in highly burnished and punched gold, entwined with coloured leafy plantstems on blue, red or green grounds, heightened with tracery and clusters of dots, the first one with 3/4 scrolling border of coloured and gold plantstems including a bird.
Binding	 France, before 1878, blue morocco gilt, sewn on 5 cords, gilt inner borders, marbled endleaves, green silk markers, by Lortic.
Provenance	 1. Ambroise Firmin-Didot (1790-1876); 2. Adolphe Labitte (Librairie Firmin-Didot & Cie), Paris, 6.6.1878:59; 3. Maggs Bros., London (1959); 4. Major J.R. Abbey (1959-1969), J.A.6963; 5. Sotheby's 1.12.1970:2885; 6. Sotheby's 21.6.1988:89.
Commentary	 This classical text covers the period from Caesar (100-44 BC) to Domitian (81-96 AD). Gaius Suetonius Tranquillus (ca. 70 – ca. 135) wrote the biographies of Caesar (100-44 BC) and Augustus (27 BC-14 AD) while he was secretary at the imperial palace, where he had access to the imperial archives. After he was dismissed by Hadrian in 122, he wrote the biographies of the 10 following emperors up to Domitian (81-96). Suetonius is the source of the well-known stories of Caesar crossing Rubicon, his assassination in the Senate, Tiberius' atrocities at his villa in Capri, Nero singing while Roma was burning, Vespasian's statement that money do not smell, etc. He also wrote the biographies of Terence, Vergil, Horace and Lucan. There are many mss surviving of this classical text, but this is the only ms written in Austria, and it has some text variants.
Image featured in
Place of origin	 Austria
Dates	 ca 1470 AD


text:
Start your path to creative Immortality today...
The Immortal Awards is a global, free-to-enter advertising award for all Little Black Book's members.
Launched in 2018 as a modern award for the new world of advertising, The Immortal Awards is simply about celebrating the best creative work in the world – with no confusing categories or chunky entrance fees.
Inspired by Sir John Hegarty's belief that creative companies rarely create more than five genuinely award-worthy pieces per office a year, The Immortal Awards only allows entrants a maximum of five entries.
The limit ensures only the very best creative work is entered, creating a level playing field for all entrants and recognising the creative ideas – and the creative talent behind them – that will live forever.
Entry comes at no additional cost to our members because we understand the importance of awards in furthering the careers of creatives but recognise just how tough the industry is now, particularly financially. As long as your company or office is a member of Little Black Book, you'll be able to enter the award show free of charge.
Whilst entering the awards is easy, winning an award is difficult - because that's the way it should be. The world's best advertising should be judged by the world's best creatives, which is why we put together the strongest jury that we possibly can.
Our world-class jury is under no obligation to Immortalise entrants or award Commendation and Finalist status, so only the very best will be taking home a prize. We have no quotas to fill and there are no numbers games from our side, with just four projects taking home iconic Immortal awards in our debut year.
With every entry receiving a score from our jury, every entry really does count. The results power our League Table of Creativity, a transparent ranking of the world's creative companies. And the winning work is taken on a global roadshow for The Immortal Awards Showcase tour, bringing creative immortality to the world – from London and LA to New York and the Netherlands.
If you're not creating, you're dying.


One of the biggest common student worries is what we're going to do when we graduate. As a Music student, I am more than used to people telling me that they are worried for my career and future; it's not perceived as the most "employable" subject to take at degree level. Most people think that it's a bit of fun for three years and then we leave University, only to have to settle for mundane jobs in a totally non-musical setting. This certainly isn't the case for the majority of Music graduates, especially from Surrey – they have gone on to do amazing things! And personally, my passion is teaching; I'd love to be a Secondary School music teacher when I finish University.
I recently attended the University of Surrey's Autumn Careers and Placement Fair, on Wednesday 12th October 2016 in the marquee on the PATS field. This was an amazing opportunity for students to explore the career possibilities available for them not only after they graduate, but also during their degrees.
I had never actually attended the Autumn Careers Fair before – I either didn't get round to registering, was busy with my course or felt that it "wasn't for me". How wrong could I be! Going this time round completely changed my perspective on the potential I have in the future.
At the Careers Fair, I was drawn to the various employers looking for teachers to join their schemes – this included TeachFirst, Teach Surrey and many others. I found that the exhibitors were incredibly friendly and approachable, which encouraged me to communicate clearly about what I want to do in my career and the impact that I want to make.
There were 150 different companies and employers there, and the event was free and open to all students; what an amazing opportunity! So after I'd chatted with some of the exhibitors in the realm of work that I'm interested in, I went round the fair and met various employers from companies I'd never heard of. I learnt about companies looking for engineering and computer science students, such as BAE Systems, AECOM and TRL, and also companies who employ Languages/Arts/Social Sciences students such as Jacobs Douwe Egberts.
There was a company handing out loads of free chocolate (Express Vending Ltd) which was one of my personal favourites! But from going round the fair for a couple of hours and having the opportunity to chat with all of the exhibitors, I got the impression that many roles in these companies can suit the individual, meaning that you don't have to fit a really strict mould. Individuality is still valued, and there are so many exciting work opportunities out there.
For a full list of the companies that were at the fair, and further information about what kind of student they are looking for as part of their various schemes and placements, follow this link.
The companies all had varied and attractive stands, piled with information and freebies. Plus, it was an excellent opportunity to stock up on all of those pens you miraculously lost after Freshers' Fair… and to get your hands on all sorts of quirky merchandise too! There was food, stationery, an array of water bottles, keyrings, bags and all sorts of other fun freebies to take away and remember the companies by. The variety of companies present at the fair really gave a sense of community and the atmosphere was buzzing throughout the event.
I think that the Careers Fair is really important for students to attend, as well as all of the other exciting opportunities for personal development that the Careers Service provide. Particularly for second years thinking about placement and final years thinking about postgraduate opportunities, this is a unique chance to really explore what's out there and ultimately make the decision about what you want to do when you leave University. I know it feels like it's a while away, but the time flies at a scary rate and it really does pay to be prepared!
If you feel you want some help at any stage during your degree, The University of Surrey has an active and approachable Careers Service. They run tons of events, workshops, CV writing skills sessions and opportunities to meet employers, as well as one-to-one sessions with careers advisors. The Careers Service website is also really useful, highlighting the numerous sessions that students are able to go to. They occur almost daily – check out the event's calendar right here!
Careers Fair: 3 Questions with Enterprise Rent-A-Car, Expedia & Vodafone Group!


text:
Q: Build gradle system app as part of AOSP build I have a custom rom based on AOSP and I am working on a system app that is packaged during the rom build just like any other system app.
Is it possible to switch this app to a gradle style app and build that specific app with gradle during the AOSP build? i.e. - Start the gradle build from a makefile?

A: I didn't find a perfect solution for this. But you can try:
Android.mk:
LOCAL_PATH := $(call my-dir)
$(info $(shell ($(LOCAL_PATH)/gradlew build -p $(LOCAL_PATH)/)) )
$(info $(shell ($(LOCAL_PATH)/finalize.sh $(PRODUCT_OUT))))

finallize.sh:
BASEDIR=$(dirname $0)
PRODUCT_PATH=$1
echo $PRODUCT_PATH
java -Xmx1024m -jar $BASEDIR/../../../out/host/linux-x86/framework/signapk.jar $BASEDIR/../../../build/target/product/security/platform.x509.pem $BASEDIR/../../../build/target/product/security/platform.pk8 $BASEDIR/main/build/outputs/apk/main-normal-release-unsigned.apk $PRODUCT_PATH/system/app/MYAPP.apk

It worked for me

redpajama_set_name: RedPajamaStackExchange


D4DOGZ stocks only the best Dog products, all tried and tested by the crew.
All our dog products are subjected to rigorous assessment by our 'consultants': Buster, a Jack Russell x mini foxy; Shadow, a black Labrador; and Charli, a Jack Russell x Aussie terrier.
We sell gorgeous outfits and dog clothing for a range of climates, including Rogz Jackets that are Harness compatible that will keep your precious pup toasty during the colder months. we also stock Puppia, and Hamish McBeth. We have dog socks and boots to protect their paws (and your floors).
Our fabulous range of dog toys includes puzzles and Games to help keep your dog entertained, Plush Dog Toys, Water Toys, Treat dispensers, Noisy Toys and Lots more.. Brands we stock also include Petstages, Rogz, Hartz, Outward Hound, Kong, Kyjen, Nina Ottosson, and many more.
We have a fantastic range of Harnesses Dog Collars and Leads, including leather, restraint and harness styles. Brands include he famous Puppia Brand, Rogz, Dogue, Hamish McBeth, Kurgo, and Sporn halters, just to name a few.
We also stock car restraints, car seats, Hammocks and Door Guards to protect your car and keep your dog safe.
We also cater for all your dog's treat needs. D4DOGZ ensure that our range of dog treats are the best you will find, we pride ourselves on supplying only the finest of rewards for your dogs, and if we haven't got the stock we are not happy with what is available, we alos do not sell yogurt/carob drops due to the Palm oil content.
And what better way to serve dinner to your pooch than in a specially chosen Dog Bowl or travel bowl, We also stock treat bags and accessories.
Our dog grooming products include luxurious shampoos and fragrances, combs and brushes. Look no further for products to keep your dog's breath fresh and teeth clean with Petzlife.
Other accessories for the pampered pooch and owner include, Gift items, xmas toys, Dog Tags, Dog waste dispensers, and much much more .
If you don't see what you're looking for, just let us know and we'll do our best to track it down for you. After all, our aim at D4DOGZ is to provide anything and everything for everyone and their dog.
We ran the idea past Buster, Shadow and Charli, whose tails started wagging at the thought of all the fun dog toys, special treats, cool outfits, and other fascinating and useful items they would have ready access to. They have never really been interested in our computer business, which we continue to run; so our idea got a big 'paws up' from the family.
We began researching dog products and putting together a plan to open a retail store in Beechworth, a popular tourist destination in north-east Victoria. We wanted the place to look stunning - like a boutique in London or Paris - with quality fittings and fixtures to match the quality products.
Finally, after purchasing a beautiful building and spending four months renovating and restoring it, we opened D4DOGZ in June 2008. The store was exactly what we had set out to achieve: a luxury dog boutique. We were extremely pleased with the outcome.
Then, on 21 April 2009, a fire broke out in our building and destroyed our computer business, our office, and all D4DOGZ' stock and fittings.
We finally re-opened our even better store at the end of January 2011, we decided in mid 2012 to concentrate on online only.
We thank YOU for taking the time to learn a little about us.


You will become a local business owner and also get the 2 tier commission program for your affiliate marketing efforts by aiding other individuals. Share the ultimate digital marketing platform that will conserve you big time loan. The very best value as well as most cost effective platform on the net.
3 web site builders, an e-mail marketing platform, animated video developer, sales funnels, lead capture devices and Builderall has over a 1000 prebuilt templates for all type of niches.
Its great for affiliate marketing, neighborhood marketing and comes with its very own style workshop. What a lot more do you need? O incidentally, they have over 26 devices and also more are coming soon.
You can even develop membership websites as well as e-learning centers. Market your own items as well as have other affiliates sell your products. It does not obtain any kind of much better than that.
This product is a suit for small business owners, business owners, affiliate marketing, local firms and also also new folks that want to find out how to earn money online. You can start in any particular niche you such as. Utilize their prebuilt sales funnels to get started.
Set up your Boss Email Marketing system and also your ready to get the word out. It's cost effective for everybody. Builderall is the best value that you will locate on the internet for all the devices that you reach make use of.
Meet Rodney V referred to as the Mobile Search Engine Optimization Person. Why did we begin promoting Builderall? It is the utmost all in one marketing system at a budget friendly rate. They have around 26 devices that can get your digital marketing rolling. Builderall has one of the very best commission programs.
We have actually been involved in online marketing considering that 2014. Regional marketing and also search engine optimization has constantly been our favored. You reach deal with local companies right in your area.
There has actually been a significant change when it involves building web sites. A great deal of folks are starting to get involved in do it yourself (DIY) marketing. Some individuals like others to do for you (DFY).
We like Builderall because you obtain the best of both globes. You can use a Do It Yourself or a DFY service. In either case you still earn money. If folks want to do it yourself, after that just supply them the digital marketing package.
They obtain all the tools they require to market their business. Offer them the very same bundle plus fee for anything you will need to develop for them.
In addition, you can enter the Builderall Business Pro package. This way we can offer other individuals a business opportunity with affiliate marketing. It appears to be the simplest way of marketing the product due to the fact that they provide you step by step instruction using their viral sales funnels.
So you obtain numerous options to gain an easy income. It's fantastic for us since we do a great deal of taking a trip as well as similar to delighting in life. When you get come down all you reached do is work a number of hours advertising on social media sites.
At first your mosting likely to need to construct your foundation and also work wise. Your hard work will settle as it has for us. It's a fantastic opportunity and we'll be surprise how much business you get traveling around.
Look it, I'm an old-fashioned individual. I'm in my 60's as well as if we can do it most any individual need to have the ability to make an easy earnings with this system. It's enabled me to not benefit any individual yet myself.
I don't even draw social safety and security. LOL. So get in where you fit in and also begin constructing on your own a regular monthly income. Required some concepts or a little coaching. We'll provide you an aiding hand and guide you in the right direction.
All we do nowadays is a little angling, traveling, enjoy all the sports we can take care of as well as take some time for my lady. We still reached connect in concerning an hour or more a day with a little digital marketing initiatives so that revenue keeps boosting.
It's great just doing what you such as to do. Here is my guidance. Anybody that can adhere to directions can do well. Just don't quit and also keep connecting along. All your effort will certainly pay off.
Do not discharge your boss yet. Learn your abilities that you require as well as begin working in your spare time to develop on your own an income that you will take pride in. It will certainly start some clever and hard work when you first start.
You need to be willing to hone your skills and also discover as high as you can about digital marketing. Whatever you do, ensure that you are passionate concerning it. Choose a niche you enjoy.
An additional area that you will need to get proficient at is social media. You'll need to understand social media to aid you construct your brand as well as additionally to drive traffic to your internet site.
You can work at your own pace. Anywhere as well as whenever you such as. All you require is an internet connection. You need to use a laptop or a computer to perform your job. People take a trip all over the globe constructing their online business.
You'll wish to maintain reinvesting into your business so it will grow much faster. Paid advertising and marketing is the fastest method to shed loan or make your business grow. The factor is to make sure you build a strong business structure as well as find out the abilities you require before you tackle paid marketing.
You can get involved in the fast lane by joining other individuals similar to you. Gain from people that are already generating income online. You can enter the Affiliate Marketing Insiders group.
It's an excellent means to launch prior to you even sign up with Builderall. You'll be able to ask inquiry, see just how other individuals are earning money and get some free training to get you rolling. Just call us for all the details for this personal training.
The fast track training covers a lot of ground. It easy to follow and it permits you to construct your business quick so you can begin to make money. You'll have the ability to also call me directly anytime you have a question or need guidance.
Learn specifically what you need to do to begin. When you bring somebody aboard you can welcome them to this free training as well as spend more time marketing so you can make more loan or have even more time for yourself.
You can also pass your business on down to your kids if you would certainly like. We make use of the program for several income streams. This is an evergreen business. It's everything about the hosting.
Every internet site needs organizing and also you get it with Builderall. You can construct a business so your household will be dealt with. You can even market your business back to Builderall. Who does that?


Classic fried zucchini with garlic and mayonnaise and recipes in batter, without flour, with vegetables. How to cook in a frying pan with carrots, cheese, tomatoes, onions, chicken. What is combined with, the options for cold and hot snacks.
Stewed zucchini with vegetables: cooking in a frying pan, in a multi-store, oven and in pots. Recipes with carrots, onions, potatoes, tomatoes, meat, sour cream, mushrooms. Dietary options and dishes for children.
Recipes of stuffed eggplants with meat, vegetables, mushrooms, nuts and cottage cheese. How to prepare "blue" for baking, and why should they soak. Options for harvesting sauerkrained aubergines with a filling.


text:
The Montelbaanstoren, Amsterdam, Netherlands, 1895. Creator: Unknown.
The Montelbaanstoren, Amsterdam, Netherlands, 1895. Tower built in 1512 as a part of the city's fortifications. The steeple was added in 1606. From "Round the World in Pictures and Photographs: From London Bridge to Charing Cross via Yokohama and Chicago". [George Newnes Ltd, London, 1895]
People Related
George Newnes: British: Publisher
------ Netherlands
--------- North Holland
------------ Amsterdam
Amsterdam The Netherlands
george newnes
george newnes ltd
montelbaanstoren
Newnes George


text:
/* eslint-disable react/no-multi-comp */
import React from 'react';
import classNames from 'classnames';
import BootstrapMixin from './BootstrapMixin';
import CustomPropTypes from './utils/CustomPropTypes';

const Tooltip = React.createClass({
  mixins: [BootstrapMixin],

  propTypes: {
    /**
     * An html id attribute, necessary for accessibility
     * @type {string}
     * @required
     */
    id: CustomPropTypes.isRequiredForA11y(React.PropTypes.string),

    /**
     * Sets the direction the Tooltip is positioned towards.
     */
    placement: React.PropTypes.oneOf(['top', 'right', 'bottom', 'left']),

    /**
     * The "left" position value for the Tooltip.
     */
    positionLeft: React.PropTypes.number,
    /**
     * The "top" position value for the Tooltip.
     */
    positionTop: React.PropTypes.number,
    /**
     * The "left" position value for the Tooltip arrow.
     */
    arrowOffsetLeft: React.PropTypes.oneOfType([
      React.PropTypes.number, React.PropTypes.string
    ]),
    /**
     * The "top" position value for the Tooltip arrow.
     */
    arrowOffsetTop: React.PropTypes.oneOfType([
      React.PropTypes.number, React.PropTypes.string
    ]),
    /**
     * Title text
     */
    title: React.PropTypes.node
  },

  getDefaultProps() {
    return {
      placement: 'right'
    };
  },

  render() {
    const classes = {
      'tooltip': true,
      [this.props.placement]: true
    };

    const style = {
      'left': this.props.positionLeft,
      'top': this.props.positionTop
    };

    const arrowStyle = {
      'left': this.props.arrowOffsetLeft,
      'top': this.props.arrowOffsetTop
    };

    return (

            {this.props.children}

      );
  }
});

export default Tooltip;

redpajama_set_name: RedPajamaGithub