#  DCPT/21/14, DESY 21-107, IPPP/21/07, MCnet-21-13, SAGEX-21-16 Logarithmic
corrections to the QCD component of
same-sign W-pair production for VBS studies

Jeppe R. Andersena, Bertrand Duclouéb, Conor Elrickb,
Andreas Maierc, Graeme Nailb, Jennifer M. Smillieb

(a Institute for Particle Physics Phenomenology,
University of Durham, Durham, DH1 3LE, UK
b Higgs Centre for Theoretical Physics, University of Edinburgh,
Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK
c Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany)

###### Abstract

We present the results of the first calculation of the logarithmic corrections
to the QCD contribution to same-sign $W$-pair production, $pp\to
e^{\pm}\nu_{e}\mu^{\pm}\nu_{\mu}jj$, for same-sign charged leptons. This
includes all leading logarithmic contributions which scale as
$\alpha_{W}^{4}\alpha_{s}^{2+k}\log^{k}(\hat{s}/p_{\perp}^{2})$. This process
is important for the study of electroweak couplings and hence the QCD
contributions are usually suppressed through a choice of Vector Boson
Scattering (VBS) cuts. These select regions of phase space where logarithms in
$\hat{s}/p_{\perp}^{2}$ are enhanced. While the logarithmic corrections lead
to a small change for the cross sections, several distributions relevant for
experimental studies are affected more significantly.

###### Contents

  1. 1 Introduction
  2. 2 Construction of Leading-Logarithm Amplitudes
    1. 2.1 Logarithmic Accuracy
    2. 2.2 Amplitudes for $pp\to W^{\pm}W^{\pm}+\geq 2j$ Within HEJ
    3. 2.3 Matching to Fixed Order
  3. 3 Impact of Leading-Logarithm Corrections
  4. 4 Conclusions
  5. A Momentum Configurations for Phase Space Slices
  6. B Simulation Parameters
  7. C Exclusive Jet Rates Without Lepton Isolation Requirement

##  1 Introduction

Vector boson scattering (VBS) describes the process $pp\to VV+2j$ where each
$V$ may be a $W$ or $Z$ boson. All possible $V^{*}V^{*}\to VV$ processes can
be inserted between two quark lines to give this final state, e.g. figure
1(a), and it provides a key opportunity to study the mechanism of electroweak
symmetry breaking, and in particular the means by which the Higgs boson
unitarises $VV$-scattering. Other topologies may also contribute at this
order, $\mathcal{O}(\alpha_{W}^{4})$ in the squared matrix element _and_ the
same final state may also be generated by gluon-exchange between the quark
lines which contributes at $\mathcal{O}(\alpha_{W}^{2}\alpha_{s}^{2})$. There
is also an interference channel between the two at
$\mathcal{O}(\alpha_{W}^{3}\alpha_{s})$. Given the experimental importance of
this process, next-to-leading order (NLO) QCD corrections to the different
channels have been extensively studied [1, 2, 3, 4, 5, 6, 7, 8, 9] and matched
with parton shower corrections [10, 11, 12, 13]. Calculations of NLO
electroweak corrections [14] and combined NLO QCD and electroweak corrections
are now also available [15, 16, 17, 18, 19]. See e.g. [20] for a recent
review.

(a) (b) a

Figure 1: Possible tree-level diagrams which contribute to $pp\to VVjj$ at (a)
$\mathcal{O}(\alpha_{W}^{4})$ and (b)
$\mathcal{O}(\alpha_{W}^{2}\alpha_{s}^{2})$ to the squared matrix element.
Interference between the two contributes at
$\mathcal{O}(\alpha_{W}^{3}\alpha_{s})$.

In order to cleanly study the electroweak (EW) processes of interest, analyses
typically apply cuts which require a large invariant mass between two jets
and/or a large rapidity separation between two jets [21, 22, 23]. These are
very effective at enhancing the EW contributions over the QCD channels, but it
is important to accurately predict the purity of the sample which remains. In
particular, the requirement of large invariant mass or large rapidity
separation enhances the importance of logarithmic corrections of the form
$\log(\hat{s}/p_{t}^{2})$ which arise at all orders of $\alpha_{s}$ (the
leading logarithmic (LL) corrections scale as
$\alpha_{s}^{k+2}\log^{k}(\hat{s}/p_{t}^{2})$). The combination
$\alpha_{s}\log(\hat{s}/p_{t}^{2})$ is not necessarily small in these regions
of phase space so that the logs can damage the convergence of the perturbative
expansion. This means that the very cuts which suppress the QCD contribution
make it harder to calculate a reliable estimate of that contribution.

Similar cuts are applied to separate the electroweak vector boson fusion (VBF)
and QCD gluon fusion (GF) components in vector boson fusion analyses of $pp\to
H+2j$. In this case, predictions including all LL corrections and matched to
fixed order are available within the HEJ framework [24, 25, 26, 27]. Further,
that framework also allows the inclusion of finite quark-mass effects for
arbitrary jet multiplicities [28] which is currently only available at LO up
to 3-jets. Here it was seen that while the inclusive cross sections for GF
were in good agreement between the LL predictions of HEJ and fixed-order
calculations at NLO, significant differences in the invariant-mass and
rapidity distributions lead to a large difference in the predicted cross
sections after cuts on invariant-mass and rapidity-separation were applied. At
NLO, the impact of requiring $|\Delta y_{j_{1}j_{2}}|>2.8$ and
$m_{j_{1}j_{2}}>400$ GeV reduced the cross section to 9% of the inclusive
value while at LL with finite quark masses, the impact of the same cuts
reduced the cross section to 4% of the corresponding inclusive value,
indicating the cuts were roughly a factor of two _more_ effective. This
behaviour is largely due to the dominance of the $gg$-component within the
inclusive cuts, and the difference in the $m_{jj}$-spectrum of the $gg$, $qg$
and $qq$ components[29, 30]. The same behaviour has also been seen in other
comparisons between combinations of fixed-order, parton shower and HEJ
predictions, e.g. [31, 32, 33].

Given the identical type of cuts usually applied to the jets when studying VBS
to those in VBF, one would expect the impact of adding LL corrections to also
be significant in this process. In this paper, we calculate the
$\alpha_{s}^{k}\log^{k}(\hat{s}/p_{t}^{2})$ corrections to the
$\mathcal{O}(\alpha_{W}^{2}\alpha_{s}^{2})$ component of VBS. We do this
within the HEJ framework. These could then be straight-forwardly combined with
predictions for electroweak channels to give a complete description of $pp\to
W^{\pm}W^{\pm}+\geq 2j$111We use this notation from now on to represent _same-
sign_ $WW$ production in association with jets, i.e. it is shorthand for
$pp\to VV+\geq 2j$, $V=W^{+}$ or $V=W^{-}$., for example using the method
described in [17]. This was done previously for $pp\to W+2j$ in [34] where LL
predictions of the QCD component from HEJ were combined with electroweak
predictions from Powheg+Pythia [35, 36, 37, 38] and were found to describe
data very well, especially at large invariant mass.

In the following section, we outline how LL accuracy is obtained in the
predictions from HEJ and then focus on how to construct the necessary
amplitudes to describe $pp\to W^{\pm}W^{\pm}+\geq 2j$ at this order. In
section 3, we illustrate the impact of the LL corrections on distributions
which are typically studied at the LHC, both before and after VBS cuts are
applied. We conclude in section 4.

##  2 Construction of Leading-Logarithm Amplitudes

###  2.1 Logarithmic Accuracy

The Born-level scattering amplitudes for the same-sign $W$-pair production
process ${pp\to W^{\pm}W^{\pm}+\geq 2j}$ depicted in figure 1(b) can be
expanded in powers of $\hat{s}/p_{t}^{2}$ considered in the limit of large
$\hat{s}\to\infty$ with fixed $p_{\perp j_{1}}\sim p_{\perp j_{2}}\sim
p_{\perp}$, where $p_{\perp}$ is an arbitrary (but fixed) transverse scale. In
this limit, large $\hat{s}$ with fixed $p_{\perp}$ is reached by increasing
rapidity differences between the jets. For processes permitting a gluon
(colour-octet) exchange between the two jets (which the production of a same-
sign $W$-pair does), this expansion of $|\mathcal{M}|^{2}$ starts at
$\alpha_{s}^{2}\hat{s}^{2}/p_{\perp}^{4}$.

The leading logarithmic corrections to the cross section are controlled by the
scattering amplitudes in this so-called Multi-Regge Kinematic limit, where the
invariant mass between each pair of coloured particles is large but the
transverse scales are fixed. The higher-order corrections to the cross section
for these processes will contain terms of
$\alpha_{s}\log(\hat{s}/p_{t}^{2})$[39, 40, 41], and such corrections have
traditionally been resummed through the use of the formalism developed by
Balitskii, Fadin, Kuraev and Lipatov[42].

The all-order, leading logarithmic accuracy in $\log(\hat{s}/p_{t}^{2})$ is
ensured by a systematic control of the power-expansion to
$\hat{s}/(p_{t}^{2})^{n}$ of $|\mathcal{M}|^{2}$ of each real correction with
$n$ legs, and the expansion into terms of
$(\alpha_{s}\log(\hat{s}/p_{t}))^{n}$ of the virtual corrections, with
subtraction terms organising the cancellation of divergences between real and
virtual corrections. In order to control just the leading logarithmic
corrections to the Born-level process, one needs to consider just the leading
expansion in $\hat{s}/p_{t}^{2}$ of each multiplicity, and its virtual
corrections. This leading term in the expansion in $\hat{s}/p_{t}^{2}$ has
contributions only from processes which allow a colour octet exchange between
each particle considered in order of increasing rapidity. So processes with
the rapidity ordering (or equivalently light-cone momentum ordering in the MRK
limit) $qQ\to Qq$ (incoming and outgoing states ordered in light-cone
momentum, and dropping the $W$s from the listing, since their ordering is
irrelevant for the discussion) will not contribute at leading order in the
expansion, whereas the leading order term _will_ receive contributions from
the same partonic process with the states ordered as $qQ\to qQ$.

For states of higher multiplicity, the leading contribution is from orderings
such as $qQ\to qgQ$, which permit a colour octet (gluon, spin-1) exchange
between each parton considered in the ordering of rapidity. The orderings
$qQ\to gqQ$ require a colour _singlet_ (spin-$\frac{1}{2}$) exchange between
the $gq$. The contribution to the square of the scattering amplitudes in the
MRK limit is then suppressed by one power of $\hat{s}$ compared to the
contribution from the ordering $qQ\to qgQ$. This power suppression of the
amplitude leads to a logarithmic suppression of the contribution to the cross
section (see e.g. [43, 26, 44]).

The factorisation of the scattering amplitudes in the MRK limit[40, 43]
implies that the MRK limit of these amplitudes can be constructed to the
required accuracy in $s/(p_{t}^{2})^{n}$ using building blocks which depend on
a reduced set of the momenta. The cross section can be calculated to leading
logarithmic accuracy using an amplitude consisting of so-called _impact
factors_ , describing the forward and backward particle production, and a
_Lipatov vertex_ describing the production of a gluon of central rapidity. The
Lipatov vertex depends on the momenta of the incoming and outgoing quarks, and
the emitted gluon. The impact factor222In the formalism of HEJ this impact
factor is a current. depends on the momenta of the incoming quark $p_{a}$, the
outgoing quark $p_{1}$ and the $W$ (or its decay products) only. These impact
factors are in fact the same as those used for single $W$ production in
association with dijets[24, 45]. The cross sections discussed in the current
study will be matched to full fixed-order accuracy of high multiplicity (up to
6 jets), see section 2.3. Earlier studies have shown that with such high-
multiplicity matching, the overall impact of sub-leading logarithmic
contributions on two and three-jet observables is very minor indeed. Therefore
we will in this study calculate to just leading logarithmic accuracy, matched
to high multiplicity fixed order.

###  2.2 Amplitudes for $pp\to W^{\pm}W^{\pm}+\geq 2j$ Within HEJ

In this subsection we present the new amplitudes required to apply the HEJ
method to the process $pp\to W^{\pm}W^{\pm}+\geq 2j$. At this point we include
the leptonic decay of the $W$ bosons as we will need this to compute
predictions for typical experimental cuts and analyses. As described in the
previous subsection, for a given multiplicity, a HEJ amplitude is built out of
impact factors and Lipatov vertices. These multiplicities are then combined
with the Lipatov ansatz for virtual corrections in order to achieve leading
logarithmic accuracy in $\hat{s}/p_{t}^{2}$ at all orders in $\alpha_{s}$.

We begin with the impact factors which are independent of the number of gluons
in the amplitude and hence can be derived from the lowest order process where
they occur. The starting point is therefore the leading order (LO) process:

$\displaystyle
q(p_{a})Q(p_{b})\to(W_{1}^{\pm}\to)\ell(p_{\ell_{1}})\bar{\ell}(p_{\bar{\ell}_{1}})(W_{2}^{\pm}\to)\ell^{\prime}(p_{\ell_{2}})\bar{\ell}^{\prime}(p_{\bar{\ell}_{2}})q^{\prime}(p_{1})Q^{\prime}(p_{2}),$
(1)

where $q$ and $Q$ represent different quark or anti-quark flavours. There are
eight Feynman diagrams which contribute at LO, each similar to figure 1(b),
which arise from the $qq^{\prime}W$ vertices for each boson being assigned to
different points on different quark lines.

We define the following current to describe the production of a $W$ boson from
a quark line with an off-shell gluon,
$q(p_{i})\to(W\to\ell\bar{\ell})q^{\prime}(p_{o})g^{*}$:

$\displaystyle\begin{split}j^{W}_{\mu}(p_{i},p_{o},p_{\ell},p_{\bar{\ell}})=&\
\frac{g_{W}^{2}}{2}\
\frac{1}{(p_{\ell}+p_{\bar{\ell}})^{2}-m_{W}^{2}+i\Gamma_{W}m_{W}}\
\left[\bar{u}^{-}(p_{\ell})\gamma_{\alpha}v^{-}(p_{\bar{\ell}})\right]\\\
&\times\left(\frac{\bar{u}^{-}(p_{o})\gamma^{\alpha}(\not{p}_{o}+\not{p}_{\ell}+\not{p}_{\bar{\ell}})\gamma_{\mu}u^{-}(p_{i})}{(p_{o}+p_{\ell}+p_{\bar{\ell}})^{2}}+\frac{\bar{u}^{-}(p_{o})\gamma_{\mu}(\not{p}_{i}-\not{p}_{\ell}-\not{p}_{\bar{\ell}})\gamma^{\alpha}u^{-}(p_{i})}{(p_{i}-p_{\ell}-p_{\bar{\ell}})^{2}}\right),\end{split}$
(2)

as illustrated in figure 2.

Figure 2: Schematic illustration of the current,
$j^{W}_{\mu}(p_{i},p_{o},p_{\ell},p_{\bar{\ell}})$, defined in Eq. (2) to
describe the production of a $W$ boson from a quark line with an off-shell
gluon (shown as a zigzag line).

The exact tree-level result can then be compactly expressed as the following
two contractions of two such currents:

$\displaystyle\begin{split}i\mathcal{M}^{\rm
HEJ,tree}=g_{s}^{2}&\left(\frac{j_{\mu}^{W}(p_{a},p_{1},p_{\ell_{1}},p_{\bar{\ell}_{1}})g^{\mu\nu}j_{\nu}^{W}(p_{b},p_{2},p_{\ell_{2}},p_{\bar{\ell}_{2}})}{q^{2}}\right.\\\
&\left.\qquad\qquad+\
\frac{j_{\mu}^{W}(p_{a},p_{1},p_{\ell_{2}},p_{\bar{\ell}_{2}})g^{\mu\nu}j_{\nu}^{W}(p_{b},p_{2},p_{\ell_{1}},p_{\bar{\ell}_{1}})}{\tilde{q}^{2}}\right),\end{split}$
(3)

where $q=p_{a}-p_{1}-p_{\ell_{1}}-p_{\bar{\ell}_{1}}$ and
$\tilde{q}=p_{a}-p_{1}-p_{\ell_{2}}-p_{\bar{\ell}_{2}}$. This amplitude
remains exact at $\mathcal{O}(\alpha_{W}^{2}\alpha_{s})$ within the HEJ
framework. In particular, our full amplitudes already achieve LO accuracy
without the need for further matching with the exception of channels with
identical leptons or quarks. The extra contributions arising in these special
cases are suppressed in the MRK limit and do not affect the logarithmic
accuracy. They are nonetheless included through the fixed-order matching
described in the next subsection.

In order to achieve LL accuracy in the inclusive predictions, we supplement
the tree-level amplitude with corrections arising from real and virtual
corrections at $\alpha_{s}^{3}$ and above. Although separately divergent in
$4d$, we follow the procedure outlined in [24, 46] to arrive at a finite
amplitude for $qQ\to W^{\pm}W^{\pm}q^{\prime}g...gQ^{\prime}$ for any number
of intermediate gluons. It is built from [24]:

  * •

a skeleton function for the process (often the LO matrix-element), given by a
contraction of impact factors

  * •

a Lipatov vertex, $V^{\mu}$, for each real gluon emission with transverse
momentum above scale $\lambda_{cut}$,

  * •

finite exponential factors for each $t$-channel propagator which arise from
the sum of the virtual corrections at LL (given by the Lipatov Ansatz) and
integration over all unresolved real emissions with $p_{t}<\lambda_{cut}$.

We typically choose values of $\lambda_{cut}$ to be around 0.2 GeV, and have
checked that our final results (cross sections and distributions) are not
strongly dependent on this value. We have compared results for the total cross
section at values of $\lambda_{cut}$ between 0.2 GeV and 2 GeV and observe
discrepancies of a few percent at most, with no clear trend. This is a non-
trivial check that the implementation is correct.

The construction above is built around the concept of effective $t$-channel
momentum (although far more than contributions from $t$-channel diagrams are
included). These are the momenta which would correspond to a planar
$t$-channel diagram when the outgoing coloured particles are ordered in
rapidity. For $qQ\to W^{\pm}W^{\pm}q^{\prime}g...gQ^{\prime}$, it is already
clear at lowest order that there is not a unique definition of a $t$-channel
momentum, as it depends on the pairing of $W$ bosons with quark lines (see Eq.
(3)). However, the addition of extra gluons does not make this problem any
worse than at leading order, and hence for any number of coloured particles in
the final state there will be two sets of planar $t$-channels. We will keep
both (and the interference between them) which significantly complicates the
expressions compared to the HEJ description of single-$W$ production [45]. It
is similar to the treatment of $pp\to Z+2j$ in HEJ [46]. Specifically, we
write the amplitude using two skeleton functions, each with its own tower of
real and virtual corrections. Interference between these is immediately
included upon squaring the amplitude.

These skeleton functions are defined as

$\displaystyle\begin{split}B&=j_{\mu}^{W}(p_{a},p_{1},p_{\ell_{1}},p_{\bar{\ell}_{1}})g^{\mu\nu}j_{\nu}^{W}(p_{b},p_{n},p_{\ell_{2}},p_{\bar{\ell}_{2}}),\\\
\tilde{B}&=j_{\mu}^{W}(p_{a},p_{1},p_{\ell_{2}},p_{\bar{\ell}_{2}})g^{\mu\nu}j_{\nu}^{W}(p_{b},p_{n},p_{\ell_{1}},p_{\bar{\ell}_{1}}),\end{split}$
(4)

which relate to the two possible combinations of leptons and quark lines.
These have corresponding planar $t$-channel momenta:

$\displaystyle\begin{split}q_{1}&=p_{a}-p_{1}-p_{\ell_{1}}-p_{\bar{\ell}_{1}},\qquad
q_{i}=q_{i-1}-p_{i}\quad i=2,...,n-1,\\\
\tilde{q}_{1}&=p_{a}-p_{1}-p_{\ell_{2}}-p_{\bar{\ell}_{2}},\qquad\tilde{q}_{i}=\tilde{q}_{i-1}-p_{i}\quad
i=2,...,n-1.\end{split}$ (5)

We refer to the corresponding momenta-squared as $t_{i}=q_{i}^{2}$ and
$\tilde{t}_{i}=\tilde{q}_{i}^{2}$ for $i=1,...,n-1$. We also define the
rapidity differences of consecutive quarks/gluons to be $\Delta
y_{i}=y_{i+1}-y_{i}$. The LL accurate matrix element is then given by

$\displaystyle\begin{split}&|\mathcal{M}^{\rm HEJ,reg}_{qQ\to
W_{1}^{\pm}W_{2}^{\pm}q^{\prime}(n-2)gQ^{\prime}}|^{2}\\\ =&\
g_{s}^{4}\frac{C_{F}}{8N_{c}}\ (g_{s}^{2}C_{\\!A})^{n-2}\ \\\
&\times\Bigg{(}\frac{|B|^{2}}{t_{1}t_{(n-1)}}\exp(\omega^{0}(q_{(n-1)\perp})\Delta
y_{n-1})\prod^{n-2}_{i=1}\frac{-V^{2}(q_{i},q_{(i+1)})}{t_{i}t_{(i+1)}}\exp(\omega^{0}(q_{i\perp})\Delta
y_{i})\\\ &\quad+\
\frac{|\tilde{B}|^{2}}{\tilde{t}_{1}\tilde{t}_{(n-1)}}\exp(\omega^{0}(\tilde{q}_{(n-1)\perp})\Delta
y_{n-1})\prod^{n-2}_{i=1}\frac{-V^{2}(\tilde{q}_{i},\tilde{q}_{(i+1)})}{\tilde{t}_{i}\tilde{t}_{(i+1)}}\exp(\omega^{0}(\tilde{q}_{i\perp})\Delta
y_{i})\\\ &\quad+\
\frac{2\Re\\{B\tilde{B}\\}}{\sqrt{t_{1}\tilde{t}_{1}}\sqrt{t_{(n-1)}\tilde{t}_{b(n-1)}}}\exp(\omega^{0}(\sqrt{q_{(n-1)\perp}\tilde{q}_{(n-1)\perp}})\Delta
y_{n-1})\\\ &\hskip
85.35826pt\;\times\prod^{n-2}_{i=1}\frac{-V(q_{i},q_{(i+1)})\cdot
V(\tilde{q}_{i},\tilde{q}_{(i+1)})}{\sqrt{t_{i}\tilde{t}_{i}}\sqrt{t_{(i+1)}\tilde{t}_{(i+1)}}}\exp(\omega^{0}(\sqrt{q_{i\perp}\tilde{q}_{i\perp}})\Delta
y_{i})\Bigg{)},\end{split}$ (6)

where

$\displaystyle\begin{split}V^{\mu}(q_{1},q_{2})&=-(q_{1}+q_{2})^{\mu}+\frac{p_{a}^{\mu}}{2}\left(\frac{q_{1}^{2}}{p\cdot
p_{a}}+\frac{p\cdot p_{b}}{p_{a}\cdot p_{b}}+\frac{p\cdot p_{n}}{p_{a}\cdot
p_{n}}\right)+p_{a}\leftrightarrow p_{1},\\\
&\qquad-\frac{p_{b}^{\mu}}{2}\left(\frac{q_{2}^{2}}{p\cdot p_{b}}+\frac{p\cdot
p_{a}}{p_{a}\cdot p_{b}}+\frac{p\cdot p_{1}}{p_{b}\cdot
p_{1}}\right)-p_{b}\leftrightarrow p_{n}\qquad{\rm{for}}\ p=q_{1}-q_{2},\\\
\omega^{0}(q_{\perp})&=-\frac{g_{s}^{2}C_{\\!A}}{4\pi^{2}}\log\left(\frac{q_{\perp}^{2}}{\lambda_{cut}^{2}}\right).\end{split}$
(7)

One can immediately check that at $\mathcal{O}(\alpha_{W}^{4}\alpha_{s}^{2})$
this exactly agrees with the summed, averaged and squared amplitude of Eq.
(3). One can extend this test to higher orders by comparing the result of Eq.
(6) with the virtual corrections removed (i.e. setting
$\omega^{0}(q_{\perp}^{2})=0$) at a fixed order in $\alpha_{s}$ to the
corresponding LO result. In the MRK limit, these should match. We illustrate
this in figure 3 for squared matrix elements for sample channels at
$\alpha_{W}^{4}\alpha_{s}^{3}$ and $\alpha_{W}^{4}\alpha_{s}^{4}$.

(a) (b)

Figure 3: Phase space explorer plots for $uc\to W^{+}W^{+}+n\textrm{ jets}$
for (a) 3 jets and (b) 4 jets.

The rapidities of the outgoing particles in these slices are:

(a) |  $y_{d}=y_{e}=y_{\nu_{e}}=\Delta,\ y_{s}=y_{\mu}=y_{\nu_{\mu}}=-\Delta,\ y_{g}=0$,
---|---
(b) |  $y_{d}=y_{e}=y_{\nu_{e}}=\Delta,\ y_{s}=y_{\mu}=y_{\nu_{\mu}}=-\Delta,\ y_{g_{1}}=\Delta/3,\ y_{g_{2}}=-\Delta/3$.

The rapidity separation of the quarks and gluons is controlled by the
parameter $\Delta$ and hence the MRK limit is approached at the right-hand
side of the plots. The other parameters used are given in Appendix A, but the
behaviour seen is not sensitive to the exact values of the transverse momentum
or the azimuthal angles. In both plots, we show the squared matrix element
divided by $\hat{s}^{2}$, to achieve a finite non-zero value in the MRK limit.
The exact LO result (black, dotted line) and the approximation within HEJ
(solid, red line) are very similar throughout the range and converge to the
same limiting value at large $\Delta$. The plots also show the rich dynamics
of the matrix elements which would be missed at small values of $\Delta$ if
the limiting value was used throughout phase space.

The LL accurate cross section is then given by the following sum over
multiplicities and integration over all phase space (where $\ell_{l}$ numbers
the four leptons from the $W$ decays):

$\displaystyle\begin{split}\sigma^{\rm LL}_{pp\to
W^{\pm}W^{\pm}+2j}=&\sum_{f_{i1},f_{i2}}\ \sum_{n=2}^{\infty}\
\prod_{i=1}^{n}\left(\int\frac{\mathrm{d}^{2}\mathbf{p}_{i\perp}}{(2\pi)^{3}}\
\int\frac{\mathrm{d}y_{i}}{2}\right)\
\prod_{l=1}^{4}\left(\int\frac{\mathrm{d}^{2}\mathbf{p}_{\ell_{l}\perp}}{(2\pi)^{3}}\
\int\frac{\mathrm{d}y_{\ell_{l}}}{2}\right)\\\
&\times\frac{\overline{\left|\mathcal{M}^{\rm HEJ,reg}_{qQ\to
W_{1}^{\pm}W_{2}^{\pm}q^{\prime}(n-2)gQ^{\prime}}\right|}^{2}}{\hat{s}^{2}}\\\
&\times\ \ x_{a}f_{A,f_{i1}}(x_{a},Q_{a})\ x_{2}f_{B,f_{i2}}(x_{b},Q_{b})\
(2\pi)^{4}\
\delta^{2}\\!\\!\left(\sum_{k=1}^{n}\mathbf{p}_{k\perp}+\sum_{m=1}^{4}\mathbf{p}_{\ell_{m}\perp}\right)\
\mathcal{O}_{2j}(\\{p_{j}\\}).\end{split}$ (8)

We emphasise here that no approximation is made to the phase space being
integrated over, only within the matrix element itself. This integral can be
efficiently implemented in an exclusive Monte Carlo event generator giving
full flexibility to implement experimental cuts and distributions. Before this
integration, we first multiply the squared matrix element by reweighting
factors to implement fixed-order accuracy, as discussed in the next section.

###  2.3 Matching to Fixed Order

In the previous subsection, we have described how to construct the cross
section for $pp\to W^{\pm}W^{\pm}+\geq 2j$ at LL accuracy in
$\hat{s}/p_{t}^{2}$. In order to increase the validity of the approach, we
will supplement this with subleading terms which will provide leading-order
accuracy at each order in $\alpha_{s}$, up to the point where this is
computationally feasible. For this process, in this study, that is samples
with 2, 3, 4, 5 and 6 jets at LO. We observe that the impact of adding higher
multiplicity fixed-order samples decreases with each multiplicity, and in
particular that the $6$-jet sample has at most a few percent effect in any
distribution so we are confident that our results have converged.

The matching is then implemented using the methods of HEJ2 [27] which
reorganises the integral over phase space to supplement fixed-order samples at
each order with real and virtual corrections such that leading-logarithmic
accuracy is maintained at all orders in $\alpha_{s}$ and additionally leading-
order accuracy is achieved for the $n$-jet components for $n=2$–$6$. This
method is implemented in the exclusive event generator, HEJ2 [47], which is
publicly available at https://hej.hepforge.org. The process $pp\to
W^{\pm}W^{\pm}+\geq 2j$ will be included in a future public release of HEJ2.
The fixed-order input is given as Les Houches events and can be taken from any
generator. In this study we have used Sherpa [48] to generate the fixed-order
input.

Finally, we rescale all the final HEJ2 predictions to match the total cross
section to the inclusive NLO cross section for each scale choice. However, for
the setup described in section 3, and as discussed there, this turns out to
have a negligible impact in this case.

##  3 Impact of Leading-Logarithm Corrections

We will now show the predictions constructed and matched as described in the
previous section for observables commonly studied at the LHC for the process
$pp\to W^{\pm}W^{\pm}+\geq 2j$, where one $W^{\pm}$ decays in the electron
channel and one in the muon channel. We will compare these with an NLO
calculation of the same process (here taken from Sherpa [48] using COMIX [49]
with the extension of OpenLoops [50]) to assess the impact of the new LL
corrections. Also included for comparison is a calculation including MC@NLO
matching [51] with the shower generator CS Shower [52] packaged with Sherpa.
We use the NNPDF3.0 NLO PDF set [53] as provided by LHAPDF6 [54]. We choose
the central factorisation and renormalisation scale as the geometric mean of
the transverse momenta of the two leading jets,
$\mu_{F}=\mu_{R}=\sqrt{p_{\perp;j_{1}}p_{\perp;j_{2}}}$. These scales are
varied independently by a factor of two around this central value, with the
constraint that their ratio is kept between 0.5 and 2. The uncertainty bands
shown in the plots are obtained from the envelope of these variations.

In figures 5–9 we show distributions measured in a recent CMS analysis [22].
It is not meaningful to compare to the data points in that study as these
include the large $\mathcal{O}(\alpha_{W}^{4})$ contributions. The cuts
applied to the predictions are listed in Table 1. Those in the first group
form the inclusive cuts which are applied to all plots. The additional three
criteria (below the second horizontal line) give the extra cuts on leading
dijet invariant mass, leading jet pseudorapidity separation and the Zeppenfeld
variable [55]

$\displaystyle
z_{l}=\frac{\eta_{l}-\frac{1}{2}(\eta_{j_{1}}+\eta_{j_{2}})}{|\eta_{j_{1}}-\eta_{j_{2}}|},$
(9)

where $j_{1}$ and $j_{2}$ are the two hardest jets in the event. These cuts
are used to try to suppress the QCD contribution to this process. We will
refer to them as “VBS cuts” and will show results before and after these extra
criteria. The output of HEJ is exclusive in the momenta of all outgoing
particles. Here we have used the functionality of linking HEJ directly with
Rivet[56] to apply the cuts and fill the histograms.

Variable | Selection Cut
---|---
Lepton pseudorapidity | $|\eta_{l}|<2.5$
Jet pseudorapidity | $|\eta_{j}|<4.7$
Leading/subleading lepton $p_{\textrm{T}}$ |  $p_{T}>25/20$ GeV
Missing transverse momentum |  $E^{\textrm{miss}}_{\textrm{T}}>30$ GeV
Jet $p_{\textrm{T}}$ |  $p_{\textrm{T}}>50$ GeV
Lepton isolation |  $\Delta R(l,\textrm{jet})>0.4$ o/w jet is removed
Di-lepton mass |  $m_{ll}>20$ GeV
Di-lepton mass restriction |  $|m_{ll}-m_{Z}|>15$ GeV
Di-jet mass |  $m_{j_{1}j_{2}}>500$ GeV
Jet rapidity separation | $|\Delta\eta_{j_{1}j_{2}}|>2.5$
Max lepton Zeppenfeld variable (Eq. 9) | $\max(z_{l})<0.75$
Table 1: The selection cuts used in the analysis where the lepton cuts apply
only to the charged leptons. The last three rows define the additional VBS
cuts.

Before discussing the distributions, we give the cross sections obtained at
NLO, with MC@NLO and with HEJ2 before and after the application of VBS cuts in
Table 2 for the central scale choice above. Both before and after VBS cuts the
values are remarkably similar for the central value of the renormalisation and
factorisation scales. This is a marked difference to other processes where
similar cuts have been applied. For example in $pp\to H+\geq 2j$ despite
relative agreement at the inclusive level, the HEJ2 predictions were
significantly more suppressed by VBF cuts than those at NLO (by about a factor
of 2) [28]. This result is sensitive to the scale choice; in the window of
variations we studied, the ratio between the HEJ2 and NLO cross sections
varies by as much as 22% in either direction. It is also clear from the
distributions that follow that this agreement is not flat in phase space (even
for the central scale choice), but arises from different regions where the
HEJ2 result is greater and less than NLO.

Cross Section (fb) |  _without_ VBS cuts, $\sigma_{\rm incl}$ |  _with_ VBS cuts, $\sigma_{\rm VBS}$ |  $\sigma_{\rm VBS}$/$\sigma_{\rm incl}$
---|---|---|---
HEJ2 $W^{+}W^{+}$ | $1.428\pm 0.002$ | $0.1219\pm 0.0004$ | $0.0854\pm 0.0003$
NLO $W^{+}W^{+}$ | $1.41\pm 0.05$ | $0.12\pm 0.07$ | $0.08\pm 0.02$
MC@NLO $W^{+}W^{+}$ | $1.285\pm 0.003$ | $0.1033\pm 0.0006$ | $0.0804\pm 0.0005$
HEJ2 $W^{-}W^{-}$ | $0.6586\pm 0.0003$ | $0.0402\pm 0.0001$ | $0.0610\pm 0.0002$
NLO $W^{-}W^{-}$ | $0.68\pm 0.02$ | $0.04\pm 0.01$ | $0.06\pm 0.02$
MC@NLO $W^{-}W^{-}$ | $0.6186\pm 0.0004$ | $0.0371\pm 0.0002$ | $0.0600\pm 0.0002$
Table 2: This table gives the total cross section calculated with the new HEJ2
LO+LL predictions in this paper compared to the result at NLO accuracy, both
before and after the VBS cuts given in the text.

In figure 4, we show the exclusive jet rates. For only the inclusive selection
criteria, we see a steady decrease at each multiplicity, but in the HEJ2
predictions the $4$-, $5$\- and $6$-jet rates remain at 21%, 6% and 2% of the
exclusive $2$-jet rate respectively. After VBS cuts, the relative importance
of the higher multiplicity rates in the HEJ2 predictions is enhanced with the
$2$-jet and $3$-jet rates being very similar and the $4$-, $5$\- and $6$-jet
rates now increasing to 40%, 13% and 3% of the exclusive $2$-jet rate
respectively. In the NLO sample after VBS cuts, the $3$-jet rate is a third
larger than the $2$-jet rate for the central scale choice. The scale variation
bands here are very large; however, for any one choice the $3$-jet was always
comparable to or greater than the $2$-jet rate. This is already one measure of
the importance of the higher-order corrections in $\alpha_{s}$ (i.e.
$\alpha_{s}^{4}$ and above). The MC@NLO predictions shows that the effect of
adding a parton shower to the NLO predictions is to distribute the $3$-jet
component among the higher jet-rate bins. The large contribution to the cross
section from events with three or more jets suggests that additional jet
vetoes could be used to further suppress the relative QCD contribution to
$pp\to VV+2j$.

The jet rate plots are affected by the lepton isolation cut (see table 1). Any
jet which satisfies $\Delta R(l,\textrm{jet})<0.4$ for any charged lepton is
removed from the event, but the event is still kept provided there are at
least two further jets. This means that events which arise from a theoretical
calculation with e.g. 4 jets can appear in the plot in the 2-jet or 3-jet bin.
For comparison, we show the equivalent plots from HEJ2 without lepton
isolation applied in appendix C.

(a)

(b)

Figure 4: Exclusive jet rates for $pp\to W^{\pm}W^{\pm}+\geq 2j$, (a) without
and (b) with additional VBS cuts.

Figure 5 shows the comparison for the difference in pseudorapidity between the
two leading jets333We use pseudorapidity to match the convention in the Ref.
[22]. In practice, there is little difference in NLO and HEJ2 predictions when
one chooses to use rapidity or pseudorapidity as the parton multiplicity
within a jet is relatively low in each case.. Here we see only modest
differences in shape between the two descriptions which are slightly enhanced
once VBS cuts are applied in the right-hand plot. However, any differences lie
mostly within the scale variation bands. Here, and in the remaining
distributions, we observe only a small impact of adding a parton shower to the
pure NLO calculation, with no significant changes in shapes of distributions.

(a)

(b)

Figure 5: The differential distribution in the pseudorapidity separation of
the two leading jets in $pp\to W^{\pm}W^{\pm}+\geq 2j$, (a) without and (b)
with additional VBS cuts.

Figure 6 exhibits greater differences in shape in the distribution of the
transverse momentum of the leading jet. For the inclusive cuts in (a), the
HEJ2 prediction starts much lower than the NLO prediction but increases with
respect to it until the predictions cross around 200 GeV. Above this value the
prediction from HEJ2 falls more slowly leading to a prediction of a harder
spectrum in $p_{j_{1};\perp}$. A very similar behaviour is seen in (b) after
the application of VBS cuts. This distribution clearly emphasises that the
close agreement of the total cross section values is a coincidence of the
experimental setup used. If the transverse momentum requirement of the jets
had been larger, the HEJ2 cross section would have also been correspondingly
larger than that from NLO.

(a)

(b)

Figure 6: The differential distribution in the transverse momentum of the
hardest jet in $pp\to W^{\pm}W^{\pm}+\geq 2j$, (a) without and (b) with
additional VBS cuts.

Similarly, in figure 7 we see that the HEJ2 and NLO predictions for the
invariant mass distribution of the two leading jets have a different shape
with a ratio which increases steadily from 0.5 at $m_{j_{1}j_{2}}=0$ GeV to
1.4 by $m_{j_{1}j_{2}}=2$ TeV where it roughly plateaus. A similar effect is
seen after VBS cuts are imposed, although of course here the lower region has
been removed. The point where the predictions cross has moved to a slightly
higher value of $m_{j_{1}j_{2}}$ as a result of the cuts on the other
variables which form part of the VBS cuts.

(a)

(b)

Figure 7: The differential distribution in the invariant mass of the two
leading jets in $pp\to W^{\pm}W^{\pm}+\geq 2j$, (a) without and (b) with
additional VBS cuts.

In figure 8, we show the distributions in the invariant mass of the two
charged leptons from the decays of the $W$ bosons. This is related to the
invariant mass of the jets if one considers event topologies where the $W$
bosons follow the direction of the associated quark lines. For modest
transverse momenta, the invariant mass between particles is driven by their
rapidity difference. The leptons, though, are required to be more central than
the jets and we see more modest differences between the NLO and HEJ2
predictions.

(a)

(b)

Figure 8: The differential distribution in the invariant mass of the two
charged leptons in $pp\to W^{\pm}W^{\pm}+\geq 2j$, (a) without and (b) with
additional VBS cuts.

The final distribution we show in this section is the Zeppenfeld variable
$z_{e}$ of the electron, defined in Eq. (9). This measures the relative
position of the electron in these events with respect to the jet system. Both
before and after VBS cuts, the predictions from NLO and from HEJ2 are in very
close agreement, and the ratio between the two remains largely flat throughout
the region showing that this variable is largely insensitive to the
logarithmic corrections at higher orders in $\alpha_{s}$.

(a)

(b)

Figure 9: The differential distribution in the Zeppenfeld variable for the
electron in $pp\to W^{\pm}W^{\pm}+\geq 2j$, without and with additional VBS
cuts.

In this section, we have compared the new predictions for $pp\to
e^{\pm}\nu_{e}\mu^{\pm}\nu_{\mu}+\geq 2j$ available in HEJ2 (which include the
leading logarithmic corrections in $\hat{s}/p_{\perp}^{2}$ at all orders in
$\alpha_{s}$) with those obtained at next-to-leading order in QCD. We have
seen close agreement in the total cross sections obtained in the two
approaches at the central scale choice, but a study of distributions in
$p_{j_{1};\perp}$, $m_{j_{1}j_{2}}$ and $m_{ll}$ show large differences in
shape which make this agreement appear to be a coincidence of the specific
values chosen in the experimental cuts.

##  4 Conclusions

We have presented the calculation of all leading logarithmic contributions
which scale as
$\alpha_{W}^{4}\alpha_{s}^{k+2}\log^{k}(\hat{s}/p_{t}^{2})$ to the production
of a same-sign $W$-pair which decays leptonically, i.e. the QCD contribution
to the process $pp\to e^{\pm}\nu_{e}\mu^{\pm}\nu_{\mu}+\geq 2j$. In order to
separate the electroweak and QCD contributions to this process, so-called VBS
cuts are usually applied to require large invariant mass and rapidity
separation of the tagging jets. These cuts exactly select regions of phase
space where the logarithms above become important. The equivalent corrections
have been seen to be significant in the QCD component of $pp\to H+\geq 2j$
through vector boson fusion, where similar cuts are used. To assess their
impact in $pp\to e^{\pm}\nu_{e}\mu^{\pm}\nu_{\mu}+\geq 2j$, we have compared
our new predictions to those obtained at NLO within the experimental setup of
a recent 13 TeV CMS analysis.

We have found that the HEJ2 cross section is very close to the NLO prediction
both for inclusive cuts, and after VBS cuts have been applied. However, it is
clear from the distributions that this agreement arises from cancellations
across phase space rather than being true throughout. The distributions in
transverse momentum of the leading jet in figure 6, in invariant mass of the
leading jets in figure 7 and in invariant mass of the charged leptons in
figure 8 show clear differences in shape with differences of up to 50% between
HEJ2 and NLO. There are other distributions, $\Delta\eta_{j_{1}j_{2}}$ and
$z_{e}$ where the two sets of predictions show close agreement, indicating
that these distributions are more stable with respect to higher-order
logarithmic corrections.

Previous studies of this process have seen that the $3$-jet component is
significant in typical experimental analyses, enhanced within VBS cuts. We
also find this, and that it extends beyond $3$-jets. The exclusive jet
components within HEJ2 are matched to leading-order accuracy for $2$–$6$ jets.
We showed in figure 4 that the VBS cuts do indeed increase the significance of
the $3$–$6$-jet components relative to the $2$-jet component. The contribution
from $3$-jets is similar to $2$-jet in HEJ2 and greater than the $2$-jet at
NLO. The $4$–$6$-jet components steadily decrease but such that the $6$-jet
components still contributed at the order of a few percent in some
distributions.

We therefore conclude that logarithmic corrections of the form
$\alpha_{W}^{4}\alpha_{s}^{k+2}\log^{k}(\hat{s}/p_{t}^{2})$ are numerically
significant at the 13 TeV LHC, and should be included in accurate modelling of
the QCD background to vector boson scattering.

## Acknowledgements

We are grateful to our collaborators within HEJ for useful discussions
throughout this project. The predictions presented in section 3 were produced
using resources from PhenoGrid which is part of the GridPP Collaboration[57,
58]. We are pleased to acknowledge funding from the UK Science and Technology
Facilities Council, the Royal Society, the ERC Starting Grant 715049
“QCDforfuture” and the Marie Skłodowska-Curie Innovative Training Network
MCnetITN3 (grant agreement no. 722104).

##  Appendix A Momentum Configurations for Phase Space Slices

We give here the momentum configurations used for the plots of the matrix
elements in figure 3. For the 3 jet final state in figure 3(a), we use:

$\begin{split}p_{i}&=p_{i;\perp}(\cos(\phi_{i}),\sin(\phi_{i}),\sinh(y_{i}),\cosh(y_{i}))\\\
p_{d;\perp}&=p_{\ell;\perp}=p_{g;\perp}=40\textrm{ GeV}\\\
p_{\nu_{\ell};\perp}&=\frac{m_{W}^{2}}{2p_{\ell;\perp}\left(\cosh\left(y_{\ell}-y_{\nu_{\ell}}\right)-\cos\left(\phi_{\ell}-\phi_{\nu_{\ell}}\right)\right)}\\\
p_{s;\perp}&=-(p_{d;\perp}+p_{e;\perp}+p_{\nu_{e};\perp}+p_{\mu;\perp}+p_{\nu_{\mu};\perp}+p_{g;\perp})\\\
\phi_{d}&=2\pi/3,\quad\phi_{e}=\pi/4,\quad\phi_{\mu}=-\pi/2,\quad\phi_{\nu_{e}}=-\pi/4,\quad\phi_{\nu_{\mu}}=+\pi/2,\quad\phi_{g}=0.4\\\
y_{d}&=y_{e}=y_{\nu_{e}}=\Delta,\quad
y_{s}=y_{\mu}=y_{\nu_{\mu}}=-\Delta,\quad y_{g}=0\end{split}$ (10)

For the 4 jet final state in figure 3(b) we use:

$\begin{split}p_{i}&=p_{i;\perp}(\cos(\phi_{i}),\sin(\phi_{i}),\sinh(y_{i}),\cosh(y_{i}))\\\
p_{d;\perp}&=p_{\ell;\perp}=p_{g_{1};\perp}=p_{g_{2};\perp}=40\textrm{ GeV}\\\
p_{\nu_{\ell};\perp}&=\frac{m_{W}^{2}}{2p_{\ell;\perp}\left(\cosh\left(y_{\ell}-y_{\nu_{\ell}}\right)-\cos\left(\phi_{\ell}-\phi_{\nu_{\ell}}\right)\right)}\\\
p_{s;\perp}&=-(p_{d;\perp}+p_{e;\perp}+p_{\nu_{e};\perp}+p_{\mu;\perp}+p_{\nu_{\mu};\perp}+p_{g_{1};\perp}+p_{g_{2};\perp})\\\
\phi_{d}&=\pi,\quad\phi_{e}=\pi/4,\quad\phi_{\mu}=-\pi/2,\quad\phi_{\nu_{e}}=-\pi/4,\quad\phi_{\nu_{\mu}}=+\pi/2,\quad\phi_{g_{1}}=\pi/2,\quad\phi_{g_{2}}=-\pi/3\\\
y_{d}&=y_{e}=y_{\nu_{e}}=\Delta,\quad
y_{s}=y_{\mu}=y_{\nu_{\mu}}=-\Delta,\quad y_{g_{1}}=\Delta/3,\quad
y_{g_{2}}=-\Delta/3\end{split}$ (11)

The qualitative effects seen in figure 3 are only sensitive to the rapidity
values (and not to the exact choices of transverse momenta or azimuthal
angle).

##  Appendix B Simulation Parameters

In Table 3 we summarise the parameters used to generate the predictions in
section 3.

Variable | Value
---|---
PDF | NNPDF30_nlo_as_0118 (lhapdf number 260000)
$m_{Z}$ | 91.1876 GeV
$\Gamma_{Z}$ | 2.4952 GeV
$m_{W}$ | 80.385 GeV
$\Gamma_{W}$ | 2.085 GeV
$\mu_{F}$ | $\sqrt{p_{\perp;j_{1}}p_{\perp;j_{2}}}$
$\mu_{R}$ | $\sqrt{p_{\perp;j_{1}}p_{\perp;j_{2}}}$
$\alpha_{s}(m_{Z})$ | 0.118 (set to match PDF)
$1/\alpha_{w}$ | 132.232
$G_{F}$ | $1.16639\times 10^{-5}\ \textrm{GeV}^{-2}$
Sherpa Version | 2.2.2
Openloops Version | 1.3.1
Parton Shower | CS Shower (packaged with Sherpa)
Table 3: The input parameters used for the plots in section 3.

Sherpa uses the complex mass scheme for vector boson masses and $\alpha_{w}$
is calculated in the $G_{\mu}$ scheme from these.

##  Appendix C Exclusive Jet Rates Without Lepton Isolation Requirement

In figure 4 we showed the exclusive jet rates obtained at NLO and with HEJ2
for the experimental setup described in Table 1. This includes a lepton
isolation requirement where any jet which satisfies $\Delta R(l,{\rm
jet})<0.4$ is removed from the event. This means that many events appear in
bins with lower numbers of jets than that present in the calculation of the
weight of that event. To more closely reflect the impact of the higher orders
in the calculation, in figure 10 we show the jet rates from HEJ2 with the
lepton isolation requirement removed (all other cuts remain the same).

(a)

(b)

Figure 10: Exclusive jet rates for $pp\to W^{\pm}W^{\pm}+\geq 2j$ as in figure
4 except now the lepton isolation requirement has been removed, (a) without
and (b) with additional VBS cuts.

The differences with only inclusive cuts are modest, with a slight decrease in
the first bin and slight increases in the higher bins. After VBS cuts, the
effect is more pronounced. The $3$-jet rate is now slightly above the $2$-jet
rate and there is then a bigger step down to the $4$, $5$ and $6$ jet rates
which have each risen slightly from the values after lepton isolation cuts.
They are now 47%, 16% and 4% respectively of the exclusive $2$-jet rate
compared to 40%, 13% and 3% after the lepton isolation cut is applied.

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