# Dijet Production in Neutral Current and Charged Current Polarized Deep
Inelastic Scattering

Ignacio Borsa<EMAIL_ADDRESS>Departamento de Física and IFIBA, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad
Universitaria, Pabellón 1 (1428) Buenos Aires, Argentina  Daniel de Florian
<EMAIL_ADDRESS>Iván Pedron<EMAIL_ADDRESS>International Center for
Advanced Studies (ICAS), ICIFI and ECyT-UNSAM, 25 de Mayo y Francia, (1650)
Buenos Aires, Argentina

###### Abstract

We calculate the fully differential Next-to-Leading Order (NLO) corrections to
(longitudinally) polarized dijet production in deep inelastic scattering,
featuring both neutral current and charged current processes. We also analyze
their phenomenological impact in the Electron-Ion Collider (EIC) kinematics.

##  I Introduction

The way in which the proton’s spin emerges from the share of its constituents
is still an open question and a key area of research in particle physics. The
amount of spin carried by quarks and gluons is codified in terms of polarized
parton distributions functions (pPDFs), which can be probed in high energy
collision processes with longitudinally polarized nucleons. In spite of the
significant progress achieved in the determination of pPDFs in the last twenty
years, our knowledge on these distributions remains limited, specially
compared to that of their unpolarized counterparts. This difference is mainly
due to the lack of a large amount of complementary measurements of observables
probing a wide kinematical range, associated to the technical challenges
involved in the production of polarized beams.

Although the amount of spin carried by quarks and antiquarks has been
confirmed to be lower than the naive-parton-model value of $\hbar/2$ by fixed-
target polarized deep inelastic scattering measurements at CERN, SLAC, DESY
and JLAB Aidala et al. (2013), it is still unknown how much of the remaining
spin is carried by gluons or is associated the orbital angular momentum of
partons. Stronger constraints on the gluon distributions were obtained from
measurements carried out by the RHIC spin program Aschenauer et al. (2013a),
albeit for a reduced range in the proton momentum fraction $x$. In addition to
the question regarding the separation of the quarks and gluon spin, the way in
which each of the different flavors of quarks contributes to the proton spin
is also work in progress. The determination of flavor-discriminated parton
distribution is primarily driven by semi-inclusive deep inelastic scattering
(SIDIS) data, which require previous knowledge on the parton-to hadron
fragmentation functions, as well as charged weak vector boson production in
proton-proton collisions data, again covering only a rather limited
kinematical range.

In that sense, the construction of the future Electron-Ion-Collider (EIC),
with a wider coverage in both the proton momentum fraction $x$ and the boson
virtuality $Q^{2}$, and reaching an unprecedented precision for polarized
measurements Accardi et al. (2016), is likely to shed some light on these
questions and provide new insights on the spin structure of the proton
Aschenauer et al. (2012, 2015, 2020); Boughezal et al. (2018). Besides
extending our knowledge on pPDFs towards lower values of $x$ and providing
additional constraints on the gluon distribution through scaling violations,
measurements of polarized DIS mediated by electroweak bosons at the EIC will
be particularly important to discriminate the helicity associated to each of
the different quark flavors. Electron-proton scattering processes receives
contributions from the exchange of virtual electroweak bosons $Z$ or
$W^{\pm}$, which become significantly relevant at higher values of the virtual
boson virtuality $Q^{2}$. Both charged current (CC) and neutral current (NC)
DIS offer crucial complementary information on the nucleon spin structure,
since they probe different partonic combinations than their purely photonic
counterpart. While the use of CC DIS data has become a standard tool to
improve flavor separation in modern unpolarized PDFs extractions, there is
currently no data on CC DIS taken on longitudinally polarized nucleons. Thus,
new measurements of this process at the EIC would provide a valuable addition
to the corpus of existing data, providing new independent constraints to
polarized PDFs Aschenauer et al. (2013b). Additionally, $Z$ exchange in
neutral current (NC) processes has the advantage of being accessible even at
lower values of $Q^{2}$ due to the $\gamma Z$-interference with the photon in
DIS. This means that the NC measurements at the EIC could provide new
electroweak precision tests, with accurate determinations of the electroweak
vector and axial-vector couplings, as well as be a probe for beyond-standard-
model physics Zhao et al. (2017); Abdul Khalek et al. (2021).

New high precision measurements should be accordingly accompanied by high
precision theoretical calculations of the corresponding observables. Even
though significant advances were made during the past 30 years in the
computation of higher order corrections for unpolarized processes, setting
next-to-next-to-leading order (NNLO) as the standard for Large-Hadron-Collider
(LHC) calculations and even reaching the following order in some cases, the
picture for polarized calculations is not yet as developed. Until recently,
NNLO QCD corrections for polarized processes were only known for completely
inclusive Drell-Yan Ravindran et al. (2004) and DIS Zijlstra and van Neerven
(1994), in addition to the helicity splitting functions Vogt et al. (2008);
Moch et al. (2014, 2015). Within the past year, the first polarized NNLO
exclusive cross section were obtained both for jet production in DIS Borsa et
al. (2020, 2021) and $W$ boson production in proton-proton collisions
Boughezal et al. (2021).

Following our previous work, the first step to reach NNLO accuracy for jet
production in electroweak DIS lies in the calculation of the NLO dijet cross
section. In this paper we present NLO calculation for dijet production in
polarized lepton-nucleon scattering, based on our extension of the Catani-
Seymour dipole subtraction method Catani and Seymour (1997) to account for
polarized initial-state particles. The computation is fully differential in
the jet and lepton momenta. We analyze the impact of higher order corrections
in the EIC kinematics, its perturbative stability and phenomenological
implications, for both neutral and charged current processes.

The rest of the paper is organized as follow: in section II we begin by
defining the kinematics for dijet production in DIS. Then, in section III we
go into some of the intricacies of the calculation of polarized electroweak
processes within dimensional regularization. In section IV we present the
phenomenological results for NLO inclusive dijet production at the EIC in the
Breit-frame for both NC and CC processes. Finally, in section V we summarize
our work and present our conclusions.

##  II Dijet production kinematics

We start by briefly reviewing the case of leading dijet production in DIS. We
study the process

$l(k)+P(p)\rightarrow
l^{\prime}(k^{\prime})+\mathrm{jet}(p_{T,1},\eta_{1})+\mathrm{jet}(p_{T,2},\eta_{2})+X,$

where $k$ and $p$ are the momenta of the incoming lepton and proton,
respectively, and $k^{\prime}$ is the momentum of the outgoing lepton. We
consider both neutral and charged current processes, mediated by the exchange
of a virtual boson with momentum $q=k-k^{\prime}$ and virtuality
$Q^{2}=-q^{2}$ fully determined by the lepton kinematics. The Bjorken variable
$x$ and the inelasticity $y$ are then defined as usual by

$x=\frac{Q^{2}}{2p\cdot q},\qquad y=\frac{q\cdot p}{k\cdot p}.$ (1)

For CC electron-proton scattering it should be noted that, while the
kinematics of the outgoing neutrino are not experimentally accessible, the
values of $x$ and $Q^{2}$ can be reconstructed from the hadronic final state
using the Jacquet-Blondel method.

In addition to these variables, used in the analysis of fully inclusive DIS, a
deeper insight on the underlying partonic kinematics can be obtained through
the study of the final-state jets, which can be characterized in terms of
their transverse momentum $p_{T,i}$ with respect to the beam, and its
pseudorapidity $\eta_{i}$. The availability of two jet allows for an even more
in-depth study of the partonic kinematics. As in the H1 Andreev et al. (2015,
2017) and ZEUS Abramowicz et al. (2010) experiments, and in addition to the
jets transverse momentum and pseudorapidities, the dijet production cross
section can be studied in term of variables such as the invariant mass
$M_{12}$ of the two-jet system, the dijet momentum fraction $\xi_{2}$, as well
as the pseudorapidity difference $\eta^{*}$ in the Breit-Frame, which are
defined by

$\begin{split}M_{12}&=\sqrt{(p_{1}+p_{2})^{2}},\\\
\xi_{2}&=x\big{(}1+\frac{M^{2}_{12}}{Q^{2}}\big{)},\\\
\eta^{*}&=\frac{1}{2}|\eta_{1}^{B}-\eta_{2}^{B}|.\end{split}$ (2)

It is worth noticing that, at the LO of dijet production, $\xi_{2}$ is the
momentum fraction carried by the incoming parton.

Dijet production can be better studied in the Breit frame (B), which is
defined as the one that satisfies $2x\vec{p}+\vec{q}=0$. At the
$\mathcal{O}(\alpha_{s}^{0})$ DIS process, this implies that the virtual boson
and incoming parton collide head-on, completely reversing the momentum of the
parton (hence the commonly used nickname brick-wall frame), as its represented
schematically in Fig. 1. The first non-vanishing contribution to dijet
production is then obtained at $\mathcal{O}(\alpha_{s})$, with two final-state
partons with completely opposite transverse momentum, and receives
contributions from both initial-state quarks and gluons.

Figure 1: $\mathcal{O}(\alpha_{S}^{0})$ Breit frame kinematics for the process
$p_{1}(xp)+\gamma^{*}(q)\rightarrow p_{2}(xp+q)$.

##  III Calculation of higher order corrections

The calculation of cross sections beyond lowest order in QCD necessarily
involves cancellations between individually divergent pieces stemming from
infrared real emission and virtual diagrams, in addition to the factorization
contributions, which can be archived via various subtraction or phase space
slicing methods Catani and Grazzini (2007); Gehrmann-De Ridder et al. (2005);
Boughezal et al. (2012); Cacciari et al. (2015); Czakon (2010); Binoth and
Heinrich (2004); Anastasiou et al. (2004); Somogyi et al. (2007); Stewart et
al. (2010). In our code POLDIS, the NLO dijet cross section is obtained by an
implementation of the Catani-Seymour dipole subtraction Catani and Seymour
(1997), extended to account for initial-state polarized particles Borsa et al.
(2020, 2021).

In addition to the counterterms for the real emission parts of the
calculation, the handling of the divergences associated to the virtual
diagrams and the integrated dipoles requires the use of a regularization
method. In the dimensional regularization scheme the number of dimensions is
set to $d=4-2\epsilon$, and then those divergences appear as poles in
$\epsilon$. Problems arise with the $\gamma^{5}$ matrix and the
$\epsilon^{\mu\nu\rho\sigma}$ tensor, since they are only properly defined in
the four-dimensional spacetime. A consistent way to treat $\gamma^{5}$ and
Levi-Civita tensor in $d$-dimensions is the ’t Hooft, Veltman, Breitenlohner,
and Maison (HVBM) scheme ’t Hooft and Veltman (1972); Breitenlohner and Maison
(1977), which splits the $d$-dimensional Minkowski space into the usual four-
dimensional one and a $(d-4)$-dimensional subspace where, for instance, the
$(d-4)$-dimensional part of the $\gamma^{\mu}$ matrices, represented as
$\hat{\gamma}^{\mu}$, commutes with the strictly four-dimensional
$\gamma^{5}$. Dealing consistently with $\gamma^{5}$ in $d$-dimensional is
necessary in both polarized and electroweak DIS, since the matrix appears both
in the polarization projectors and axial couplings.

###  III.1 Electroweak couplings and treatment of $\gamma^{5}$

Due to the fact that the electroweak vertices involve the appearence of terms
$\gamma^{\mu}\gamma^{5}$, the utilization of the HVBM scheme brings additional
issues to the calculations of electroweak processes. Since the
$(d-4)$-dimensional part of the $\gamma^{\mu}$ in the vertex commutes with
$\gamma^{5}$, the location of the projector affects the $(d-4)$-dimensional
part of the vertex. Thus, in order to avoid some of the issues arising in
$d$-dimensions, the definition of the vertex requires the symmetrization Larin
(1993); Korner et al. (1990); Buras and Weisz (1990)

$\begin{split}\gamma^{\mu}\gamma^{5}\ \rightarrow\
\frac{1}{2}(\gamma^{\mu}\gamma^{5}-\gamma^{5}\gamma^{\mu})=\tilde{\gamma}^{\mu}\gamma^{5},\end{split}$
(3)

where $\tilde{\gamma}^{\mu}$ is the four-dimensional part of the
$\gamma^{\mu}$. With this prescription, the electroweak vertex between two
flavors of quarks or leptons $f_{1}$ and $f_{2}$ can be consistently expressed
in terms of the vector and axial parts as

$-ie\,\gamma^{\mu}\left(C_{V}^{f_{1}f_{2}}+C_{A}^{f_{1}f_{2}}\gamma^{5}\right)\
\rightarrow\
-ie\,\left(C_{V}^{f_{1}f_{2}}\gamma^{\mu}+C_{A}^{f_{1}f_{2}}\,\tilde{\gamma}^{\mu}\gamma^{5}\right).$
(4)

The values of the vector and axial couplings $C_{V}^{f_{1}f_{2}}$ and
$C_{A}^{f_{1}f_{2}}$ depend on the exchanged boson and are given by the
following expressions:

$W^{-}$: |  $C_{V}^{f_{1}f_{2}}=\frac{1}{2\sqrt{2}\sin\theta_{W}}(\tau_{+})_{f1f2}\ V_{f1f2}$, |  $C_{A}^{f_{1}f_{2}}=C_{V}^{f_{1}f_{2}}$,
---|---|---
$W^{+}$: |  $C_{V}^{f_{1}f_{2}}=\frac{1}{2\sqrt{2}\sin\theta_{W}}(\tau_{-})_{f1f2}\ V^{\dagger}_{f1f2}$, |  $C_{A}^{f_{1}f_{2}}=C_{V}^{f_{1}f_{2}}$,
$Z$: |  $C_{V}^{f_{1}f_{2}}=\frac{1}{2\sin(2\theta_{W})}(\tau_{3})_{f1f2}-\delta_{f1f2}\,e_{f1}\,\tan\theta_{W}$, |  $C_{A}^{f_{1}f_{2}}=\frac{1}{2\sin(2\theta_{W})}(\tau_{3})_{f1f2}$,
$\gamma$: |  $C_{V}^{f_{1}f_{2}}=\delta_{f_{1}f_{2}}\ e_{f1}$, |  $C_{A}^{f_{1}f_{2}}=0$.
(5)

Here $\theta_{W}$ is the electroweak mixing angle, $\tau_{\pm}=(\tau_{1}\pm
i\tau_{2})/2$ and $\tau_{3}$ are the weak isospin Pauli matrices, $V$ is the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, and $e_{f}$ is the electric
charge of the corresponding quark/lepton ($e_{f}=2/3$ for $u$, $c$, $t$,
$e_{f}=-1/3$ for $d$, $s$, $b$, and $e_{f}=-1$ for $e$, $\mu$ and $\tau$).

When working in the HVBM scheme, the presence of $\gamma^{5}$ in both the
symmetrized vertex and the quirality projectors over initial-state polarized
particles leads to so called “evanescent” terms of order $\mathcal{O}(d-4)$,
which in combination with divergent terms can result in spurious finite
contributions. Infrared anomalous terms cancel out between virtual and real
contributions Korner et al. (1985), but in the case of the ultraviolet
divergencies appropriate additional finite renormalization terms may be
required to maintain the correct form of the chiral Ward identities Larin
(1993); Ahmed et al. (2021). In this case (up to order $\alpha_{s}^{2}$), the
appropriate counterterm to be subtracted is simply given by

$\left(\Delta\right)C_{T}=\frac{\alpha_{s}}{2\pi}2C_{F}\left(\Delta\right)\hat{\sigma}^{A}_{LO},$
(6)

where $(\Delta)\hat{\sigma}^{A}_{LO}$ are the contributions to the (polarized)
unpolarized leading order partonic cross section with at least one axial
coupling between the weak boson and the quarks.

For the polarized case, additional finite subtraction terms are usually added
within the mass factorization (in addition to those associated to the
$\overline{\mathrm{MS}}$ factorization scheme) in order to enforce helicity
conservation in quark lines (due to the use, by convention, of four-
dimensional kernels instead of $d$-dimensional ones), as pointed out in our
previous paper Borsa et al. (2021).

We note that there are multiple ways of symmetrizing the vertex in Eq. (4).
For example, when working with the left and right pieces of the coupling, by
performing the modification in the projectors as
$\gamma^{\mu}\frac{1}{2}(1\pm\gamma^{5})\rightarrow\frac{1}{4}(1\mp\gamma^{5})\gamma^{\mu}(1\pm\gamma^{5})$.
We have checked that those choices lead to the same results. The key point is
to remove the $(d-4)$-dimensional part of the $\gamma^{\mu}$ when accompanied
by a $\gamma^{5}$ in order to avoid the appearance of spurious finite terms
originating from UV-divergent pieces.

###  III.2 New terms in electroweak boson exchange DIS

Compared to purely photonic DIS, the inclusion of processes mediated by axial-
vector bosons require, in principle, the calculation of additional terms
proportional to the axial coupling of those bosons. It should be noted,
however, that since the calculation of the matrix elements for quark-initiated
processes, both for the polarized cross sections and for the axial-vector
interchange cross sections, involves the presence of $\gamma^{5}$ in the
fermionic traces, many of the new contributions associated to the axial part
of the electroweak coupling can be inferred from the polarized and unpolarized
matrix elements already calculated in Borsa et al. (2021) for photon exchange.
For that purpose is useful to separate the partonic cross sections in a
parity-violating (PV) and a non-parity-violating (NPV) piece, as

$\hat{\sigma}_{q}=\hat{\sigma}^{PV}_{q}+\hat{\sigma}^{NPV}_{q},$ (7)

where the subscript $q$ is used to indicate the contribution from quark-
initiated processes. For unpolarized (polarized) DIS, the processes
$q+W/Z\rightarrow q$, $q+W/Z\rightarrow q+g$ and $q+W/Z\rightarrow q+g+g$
contribute to the parity-violating part when an odd (even) number of
$\gamma^{5}$ matrices are present in the partonic trace, while the
contribution to the non-parity violating part comes from traces with an even
(odd) number of $\gamma^{5}$’s. It is then possible to see that, up to order
$\alpha_{s}^{2}$

$\begin{split}\hat{\sigma}^{PV}_{q}&=\Delta\hat{\sigma}^{NPV}_{q},\\\
\Delta\hat{\sigma}^{PV}_{q}&=\hat{\sigma}^{NPV}_{q}.\end{split}$ (8)

These relations are trivial for the contributions from real diagrams, since
within the dipole subtraction formalism they are dealt in four-dimensions and
one can anticommute $\gamma^{5}$ freely in the trace. Care should be taken,
however, when dealing with virtual diagrams, where dimensional regularization
is needed. The anticommutation of the $\gamma^{5}$ is only valid after
performing the additional finite renormalization Larin and Vermaseren (1991);
Larin (1993). We wish to stress that although we did calculate the new parity
violating pieces from scratch, the relations from Eq. (8) were useful to ease
the implementation in the Monte Carlo code.

Analogously, the parity-violating contributions to unpolarized DIS arising
from the gluon-initiated processes $g+W/Z\rightarrow q+\bar{q}$ and
$g+W/Z\rightarrow q+\bar{q}+g$, can be obtained as

$\hat{\sigma}^{PV}_{g}=-\Delta\hat{\sigma}^{NPV}_{q},$ (9)

where the minus sign arises from the crossing of the initial quark into the
final state. Note that a similar relation does not hold for polarized DIS,
since in this case there is no helicity projector in the quark trace. That
being said, parity-violating terms in processes with an initial gluon actually
vanish after integration due to charge conjugation arguments, since this
contribution is antisymmetric under the exchange of a parton $a$ with an
antiparton $\bar{a}$ ($\hat{\sigma}_{a}=-\hat{\sigma}_{\bar{a}}$). These terms
are only relevant if, e.g., the charge of the final jets is observed.

Special attention should be given to the contributions arising the processes
$q+W/Z\rightarrow q+q^{\prime}+\bar{q^{\prime}}$, since they can involve more
than one partonic trace. Depending on trace structure of the matrix element
considered, relations similar to those of equations 8 and 9 may be used. The
other non-trivial case are the new anomalous diagrams involving quarks
triangles, which only contribute to $Z$-boson exchange. These cancel out if
the two members of each weak isospin doublet are considered.

##  IV Results of Polarized NLO Dijet Production

###  IV.1 Dijet production in neutral current DIS

In this section we present our results for polarized inclusive dijet
production at NLO in NC DIS in the Breit frame. Since the structure of higher
order corrections and the scale dependence of the cross section was already
analyzed in Borsa et al. (2021), for the case of virtual photon interchange,
in the present work we will focus primarily on the effects of the inclusion of
$Z$ boson contribution at NLO. As in our previous publication, we concentrate
on the EIC kinematics, considering electron-proton collisions with beam
energies of $E_{e}=18$ GeV and $E_{p}=275$ GeV, and reconstruct the jets with
the anti-$k_{T}$ algorithm and $E$-scheme recombination ($R=1$). The
renormalization and factorization scales are fixed at central values of
$\mu_{F}^{2}=\mu_{R}^{2}=\frac{1}{2}(Q^{2}+\langle
p^{B}_{T}\rangle^{2})\equiv\mu_{0}^{2}$, with $\alpha_{s}$ evaluated at NLO
accuracy with $\alpha_{s}(M_{z})=0.118$, and we require that the pair of
leading jets satisfy the following kinematical cuts:

$p^{B}_{T,1}>5\ \mathrm{GeV}$,
---
$p^{B}_{T,2}>4\ \mathrm{GeV}$,
$|\eta^{L}|<3.5$,
(10)

where the $p_{T}$ and $\eta$ cuts are imposed in the Breit and laboratory
frame, respectively. The lepton kinematics is restricted by

$0.2<y<0.6$,
---
$25\,\mathrm{GeV}^{2}<Q^{2}<2500\,\mathrm{GeV}^{2}$.
(11)

Figure 2: Inclusive dijet production distributions as a function of the
variables $Q^{2}$, $\log_{10}(\xi_{2})$, $M_{12}$ and $\eta^{*}$, for DIS
mediated by pure photon (red) or full NC $Z/\gamma$ (blue). The lower boxes
show the ratio between the two cross sections. Figure 3: Same as Fig. 2, but
for the polarized case.

For the $Z$ boson we use a mass $M_{z}=91.1876$ GeV and a decay-width of
$\Gamma_{z}=2.4952$ GeV, with an electromagnetic coupling constant
$\alpha=1/137$ and the Weinberg angle given by $\sin^{2}\theta_{W}=0.23122$.
The parton distributions sets used were the NLOPDF4LHC15 Butterworth et al.
(2016) and DSSV de Florian et al. (2014, 2019) for the unpolarized and
polarized case, respectively.

We start by presenting the NLO results for the dijet production cross section
in unpolarized DIS in Fig. 2, both for processes mediated exclusively by a
virtual photon as well as for the full neutral current interchange
($Z/\gamma$). The cross sections are presented as distributions in the boson
virtuality $Q^{2}$, the logarithm of the dijet momentum fraction
$\log_{10}(\xi_{2})$, the invariant mass of the dijet system $M_{12}$, and the
pseudorapidity difference in the Breit frame $\eta^{*}$. The lower inlets in
the figure show the ratio between the $Z/\gamma$ mediated cross section to the
one with pure photon exchange. As has been already noted by Mirkes and
Zeppenfeld (1997); Mirkes (1997), due to the $Z$ boson propagator suppression,
the contribution to the cross section from processes with a $Z$ are small for
the values of $Q^{2}$ to be covered by the future EIC (surpassing 5% only for
$Q^{2}>1000$), and comes fundamentally from the $Z\gamma$ interference terms.

Fig. 3 presents the same distributions as Fig. 2, but for longitudinally
polarized scattering. While the polarized cross section has the same
propagator suppression as the unpolarized case, the corrections associated to
the $Z$ boson exchange are far more pronounced for the former. As a function
of $Q^{2}$, the increase of the cross section range between 5% (low $Q^{2}$)
and 50% (high $Q^{2}$), while for the remaining distributions is typically of
order 15%. The larger EW effects in the polarized case are associated to the
fact that the contributions coming from different partonic channels can have
relative signs, leading to cancellations between channels. Since those
cancellations result in a suppression of the cross section at low $Q^{2}$, if
the $Z$ boson corrections are not the same for each of the channels canceling
out, the relative corrections can be enhanced. This is exactly the case for
dijet production: for polarized scattering, the contribution from gluon-
initiated processes is negative and becomes comparable to that of the quarks
for low $Q^{2}$, as the emission of soft gluons becomes more prominent. Since
the gluon contribution to parity violating piece of the cross section (related
to the $g_{4}$ and $g_{5}$ structure functions) cancels after the integration
over the phase space, while the quark net contribution is non-zero, the
inclusion of $Z$ boson interchange leads to more sizable corrections. This
difference is significant over the whole $Q^{2}$ range, since at high $Q^{2}$
the cancellation between the $u$ and $d$ flavors also becomes relevant. As
mentioned, a detailed discussion of the different partonic contributions to
the dijet cross section, both in the polarized and unpolarized case, can be
found in Borsa et al. (2021).

Figure 4: Double spin asymmetries for dijet production, as a function of
$Q^{2}$, $\log_{10}(\xi_{2})$, $M_{12}$ and $\eta^{*}$, for DIS mediated by
pure photon (red) or full NC $Z/\gamma$ (blue). The lower boxes show the ratio
between the asymmetry for $Z/\gamma$ exchange, and that of purely photonic
exchange.

The difference in the magnitude of between the $Z$ boson contribution to the
polarized and unpolarized distributions is reflected in the double spin
asymmetries, presented in Fig. 4, which are defined as the ratio of the
polarized to the unpolarized cross sections $A_{LL}=\Delta\sigma/\sigma$. The
asymmetry is enhanced at the high $Q^{2}$ region, where it increases up to a
30% with respect to the pure photon exchange. As high $Q^{2}$ correlates with
higher values of $\xi_{2}$, the asymmetry also increases significantly at
dijet momentum fractions closer to $1$. Conversely, in both the asymmetries in
$M_{12}$ and $\eta^{*}$, this enhancement is more evenly distributed, with
ratios of $\sim 1.13$ over most of the available range. It should be noted
that, due to the origin of this behavior, the enhancement of the asymmetries
is already present at the LO. Overall, due to the combination of the channel
cancellations in the polarized process and the nature of the NC contributions,
the double spin asymmetries are noticeably increased even at the relatively
small values of $Q$ (compared to $M_{Z}$) to be reached at the EIC, making it
an extremely useful tool to disentangle the contribution from the different
partonic channels, particularly the less constrained sea-quark distributions.

###  IV.2 Dijet production in charged current DIS

In this section we present our results for polarized inclusive dijet
production at NLO in CC DIS. The cut imposed on the jets and the reconstructed
lepton kinematics are the same as the ones in the previous section. For the
$W$ boson we use a mass $M_{W}=80.379$ GeV and a decay-width of
$\Gamma_{W}=2.085$ GeV. The values used for the CKM matrix are
$|V_{ud}|=0.9737$, $|V_{us}|=0.2245$, $|V_{ub}|=0.00382$, $|V_{cd}|=0.2210$,
$|V_{cs}|=0.987$ and $|V_{cb}|=0.041$.

We start by analyzing unpolarized CC dijet production in the EIC kinematics.
In Fig. 5 we present the unpolarized LO and NLO cross sections as a function
of the $W$ boson virtuality $Q^{2}$, $\log_{10}(\xi_{2})$, the invariant mass
$M_{12}$ and the pseudorapidity difference $\eta^{*}$. The bands shown
correspond to the estimation of the theoretical uncertainty, obtained
performing the seven-point variation of the factorization and renormalization
scales as $\mu_{R},\mu_{F}=[1/2,2]\,\mu_{0}$ (with the additional constrain
$1/2\leq\mu_{F}/\mu_{R}\leq 2$). The lower box in each distribution shows the
corresponding K-factor, defined as the ratio to the LO cross section
$\sigma_{LO}$, in order to quantify the effect of the higher order
corrections. To that end, the same NLO sets of PDF were also used for the
calculation of the LO distributions.

As expected, the distributions shown in Fig. 5 are highly suppressed compared
to the NC ones, especially at low $Q^{2}$, due to the massive boson
propagator. The propagator suppression also accounts for the shift in the
$\xi_{2}$ distributions to higher momentum fractions, since $Q^{2}$ and
$\xi_{2}$ are correlated. This means that in this case the kinematics limit
the probing of small momentum fractions of the initial parton. In terms of the
structure of higher order corrections, while the NLO distributions show a
general reduction of the scale-dependence going to NLO, the corrections are
still sizable. As mentioned in Borsa et al. (2021), the low-$p_{T}$ cuts
imposed to the dijets systems implies that the region $M_{12}>10$ GeV is
forbidden at LO and becomes available only from NLO, making the calculation
for that region effectively LO, and leading to perturbative instabilities.

Figure 5: Cross section for dijet production in unpolarized, charged current
electron-proton DIS, as a function of $Q^{2}$, $\log_{10}(\xi_{2})$, $M_{12}$
and $\eta^{*}$, at Leading (pink) and Next-to-Leading Order (green). The bands
correspond to the theoretical uncertainty obtained performing the seven-point
variation of the factorization and renormalization scales. The lower boxes
show the K-factor, i.e, the ratio of the NLO cross section to the LO one.
Figure 6: Same as in Fig. 5, but for polarized electron-proton scattering.

In Fig. 6 we present the polarized CC dijet distributions. It is interesting
to notice that, as opposed to the NC case, the distributions for polarized DIS
are not as suppressed compared with those for unpolarized DIS (differing only
up to an order of magnitude at low-$Q^{2}$). Due to the nature of the $W$
boson, having equal vector and axial couplings, the parity violating part of
the cross section is not suppressed as in the NC case. Since this contribution
stems only from initial-quark processes, the initial-gluon channel becomes
overall less relevant, thus reducing the cancellations between channels in the
polarized cross section. When comparing to the unpolarized case, this effect
is further amplified by a suppression of the gluon helicity distribution at
lower momentum fractions. The different low-$Q^{2}$ behavior of the polarized
cross section and K-factors is then a consequence of the smaller relevance of
the gluonic contribution.

To highlight this point, Figs. 7 and 8 presents the same distributions as
Figs. 5 and 6, but distinguishing the quark and gluon contributions to the
cross section. The lower boxes show the ratios between the gluon-initiated
pieces and the quark-initiated ones. While for the unpolarized distribution,
the low-$Q^{2}$ cross section is mainly driven by the gluon channel (reaching
ratios of up to 2), the aforementioned suppression in the polarized case
results in a significantly smaller cross section in this region. This general
reduction of the relevance of the gluon-initiated process can also be observed
in the ratios for the other distributions.

Figure 7: Same distributions as in Fig. 5, but separating the contributions of
quark and gluon-initiated processes. The lower inlets show the ratio between
the two partonic channels Figure 8: Same as in Fig. 7, but for polarized
electron-proton scattering.

The reduction of cancellations between channel in the polarized cross section
leads, in turn, to higher values of the double spin asymmetries with CC, which
are presented in Fig. 9. This effect also results in milder K-factors of the
asymmetry (compared to the photon-exchange case in Borsa et al. (2021)), with
values of $\sim 0.7$ at low-$Q^{2}$, that then approach unity at higher boson
invariant mass. Note that, in spite of the small value of the cross section,
the asymmetries for charged current DIS are typically of order $\sim 0.8$ for
the relevant high-$Q^{2}$ region, providing an additional valuable constraint
to the helicity parton distribution functions.

Figure 9: Double spin asymmetries as distributions in $Q^{2}$,
$\log_{10}(\xi_{2})$, $M_{12}$ and $\eta^{*}$ for CC electron-proton
scattering.

##  V Conclusions

In this paper we presented the fully exclusive NLO calculation of dijet
production in polarized DIS, considering the exchange of both neutral and
charged currents. The calculation is done with the dipole subtraction method,
extended to allow for longitudinal polarization of initial partons. In
particular, we analyzed the production of dijets in the Breit-frame with EIC
kinematics. The cross sections were studied as distributions in the virtuality
$Q^{2}$, invariant jet mass $M_{12}$, the pseudorapidity difference
$\eta^{*}$, and the dijet momentum fraction $\xi_{2}$. For NC DIS, the
contribution of the $Z$ boson to the unpolarized cross section is, as
expected, very small overall, with the exception of the highest $Q^{2}$ bins
for which the enhancement is of order 10%. In the polarized case, however, due
to the cancellations between partonic channels, the effect is not only non-
negligible at low $Q^{2}$, but can reach 50% for the high-$Q^{2}$ region,
leading to an increase of the double spin asymmetry. In the case of the
distributions for CC DIS, while an overall reduction in the scale dependence
is observed for the higher order terms, the corrections are still sizable. In
terms of the experimental observables, higher asymmetries are obtained
compared with the NC case, mainly due to a strong suppression of the gluon
channel (which is negative for polarized DIS), reaching values of $\sim 0.8$
at higher $Q^{2}$.

The results presented on this paper highlight the potential of the electroweak
boson exchange processes to further improve our knowledge on polarized parton
distributions, providing complementary constraints to determine the flavor
decomposition of the proton spin, as well as the relevance that higher order
QCD corrections will have in the precise description of the jet observables to
be measured in the future EIC. Our dijet results will also be a fundamental
step towards the obtention of the fully exclusive NNLO calculation for jet
production in DIS with electroweak boson exchange.

###### Acknowledgements.

I.B. wishes to thanks the University of Tübingen for its hospitality during
the last stages of this project. This work was partially supported by CONICET
and ANPCyT.

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modulus. As anticipated, this condition identifies the generalized average
phase estimation problem, with an arbitrary number of modes between $1$ and
$d$. Evaluated at its minimum point and for $\bm{v}=\bm{v}_{\rm ave}$, Eq.
(105) becomes

$\min_{\bm{U},\,|\alpha_{1}|^{2},\dots,|\alpha_{d}|^{2}}\Delta^{2}(\bm{v}_{\rm
ave}\cdot\bm{\theta})_{\rm
emom}=\frac{e^{-2r}\bar{n}_{c}+\bar{n}_{s}/d}{d\bar{n}_{c}^{2}}.$ (109)

Finally, we prove Eq. (21) for $\bar{n}_{c}\gg\bar{n}_{s}$ [equivalent to
$\bar{n}_{T}\gg(d+1)\bar{n}_{s}$]. Setting $|\alpha_{j}|^{2}=\bar{n}_{c}$ for
all $j$ in Eq. (VII.3) and taking the limit $\bar{n}_{c}\gg\bar{n}_{s}$, one
gets

$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
eQCR}=\frac{e^{-2r}-1}{\bar{n}_{c}}\left(\sum_{i=1}^{d}\tilde{u}_{i}v_{i}\right)^{2}+\frac{1}{d\bar{n}_{c}}.$
(110)

This is just Eq. (105) lacking the last term. As already discussed above, the
function in Eq. (110) is minimized by choosing
$\bm{\tilde{u}}=\sqrt{d}\bm{v}$, which corresponds to the condition used to
get the upper bound. The saturation of the bound is only realized for
$\bm{v}=\bm{v}_{\rm ave}$ since only in that case an even distribution of
photons between the coherent modes, $|\alpha_{j}|^{2}=\bar{n}_{c}$ for all
$j$, can be used to attain the minimum of
$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm eQCR}$. The right hand side of Eq.
(21) is obtained, for any value of $\bar{n}_{T}$, assuming that conditions
$\bm{\tilde{u}}=\sqrt{d}\bm{v}_{\rm ave}$ and $|\alpha_{j}|^{2}$ for all $j$
identify the optimal configuration of the sensor network, for any
$\bar{n}_{T}$. These assumptions are confirmed by the results of numerical
simulations shown in Fig. 2.

###  VII.5 Demonstration of Eq. (26)

To derive Eq. (26), we refer to Eq. (25) which gives the definition of
$\mathcal{G}_{2}(\bm{v})$. We work in the regime $\bar{n}_{c}\gg
e^{2r}\bar{n}_{s}$, which implies also $\bar{n}_{c}\gg
e^{2r^{\prime}_{i}}(\bar{n}^{\prime}_{s})_{i}$ for $i=1,\dots,d$ since, under
the constraint $\mathcal{C}_{2}$, we have
$\sum_{i=1}^{d}(\bar{n}^{\prime}_{s})_{i}=\bar{n}_{s}$. The sensitivity of the
entangled strategy, $\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm emom}$, can be
obtained from Eq. (105) by neglecting the last term, which needs not to be
considered in such regime. The expression of
$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm smom}$, the sensitivity of the
separable strategy, is derived from Eq. (7). In the limit considered, we only
keep the first term both in the numerator and in the denominator of the
inverse moment matrix, getting

$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
smom}=\bm{v}^{T}\bm{\mathcal{M}^{-1}}\bm{v}=\sum_{i=1}^{d}\frac{e^{-2r^{\prime}_{i}}}{\bar{n}_{c}}v_{i}^{2}.$
(111)

We then optimize the sensitivities of both strategies. In the previous section
we proved that, when the rightmost term in Eq. (105) is negligible, the
entangled strategy is optimized by the choice $\bm{\tilde{u}}=\sqrt{d}\bm{v}$,
with

$\min_{\bm{U}}\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
emom}=\frac{e^{-2r}}{d\bar{n}_{c}}.$ (112)

In order to make the minimization of Eq. (111) as simple to perform, we need
an extra condition that allows us to express $e^{2r^{\prime}_{i}}$ as a simple
function of $(\bar{n}^{\prime}_{s})_{i}=\sinh^{2}r^{\prime}_{i}$. As a first
case, we assume $r^{\prime}_{i}\gg 1$, so that $e^{2r^{\prime}_{i}}\approx
4(\bar{n}^{\prime}_{s})_{i}$. The method of Lagrange multipliers is very
easily applied to this case, predicting the minimum

$\min_{r^{\prime}_{1},...,r^{\prime}_{d}}\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
smom}=\frac{e^{-2r}}{\bar{n}_{c}}\left(\sum_{i=1}^{d}|v_{i}|\right)^{2},$
(113)

which is achieved for
$(\bar{n}^{\prime}_{s})_{i}=\bar{n}_{s}|v_{i}|/\sum_{i=1}^{d}|v_{i}|$. We also
find $\lambda=-(\sum_{i=1}^{d}|v_{i}|)^{2}/4\bar{n}_{s}^{2}$, $\lambda$
denoting the Lagrange multiplier associated with this optimization problem. It
is easy to see that the ratio between the two sensitivities corresponds to Eq.
(26). Notice that, for $v_{i}\to 0$, one has $(\bar{n}^{\prime}_{s})_{i}\to 0$
too, so that the approximation $r^{\prime}_{i}\gg 1$ cannot be valid in such
limit. We thus consider a second case, the limit $r^{\prime}_{i}\ll 1$, which
justifies the truncated series expansion $e^{-2r^{\prime}_{i}}\approx
1-2r^{\prime}_{i}\approx 1-2\sqrt{(\bar{n}^{\prime}_{s})_{i}}$. Applying again
the method of Lagrange multipliers, this time we find

$\min_{r^{\prime}_{1},...,r^{\prime}_{d}}\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
smom}=\frac{1}{d\bar{n}_{c}}-2\frac{\sqrt{\bar{n}_{s}}}{\bar{n}_{c}}\sqrt{\sum_{i=1}^{d}v_{i}^{4}}$
(114)

and $(\bar{n}^{\prime}_{s})_{i}=\bar{n}_{s}v_{i}^{4}/\sum_{i=1}^{d}v_{i}^{4}$,
with $\lambda=\sqrt{\sum_{i=1}^{d}v_{i}^{4}}/\sqrt{\bar{n}_{s}}$. The
expression of the gain is easily verified to be

$\mathcal{G}_{2}(\bm{v})=\left(1-2d\sqrt{\bar{n}_{s}}\sqrt{\sum_{i=1}^{d}v_{i}^{4}}\right)e^{2r}.$
(115)

In principle, both of the formulas for $\mathcal{G}_{2}(\bm{v})$, Eqs. (26)
and (115), are not expected to provide accurate predictions for non-uniform
vectors $\bm{v}$, when $r^{\prime}_{i}\gg 1$ and $r^{\prime}_{i}\ll 1$ occur
simultaneously, for different components $v_{i}$ of the same $\bm{v}$. So, it
is interesting to note that Eqs. (26) and (115) still give the correct value
of the gain, $\mathcal{G}_{2}(\bm{v})=1$, when applied to the estimation of a
single phase. Equation (26) is found in good agreement with numerical results
for all values of $\bm{v}$.

###  VII.6 Demonstration of Eqs. (28) and (29)

We demonstrate here Eq. (29), which refers to the gain
$\mathcal{G}_{2}(\bm{v}_{\rm ave})$ achieved in the estimation of the
generalized average phase. Starting again from Eq. (7), with
$|\alpha_{j}|^{2}=\bar{n}_{c}$, this time we can impose the less strict
condition $\bar{n}_{c}\gg\bar{n}_{s},(\bar{n}_{s}^{\prime})_{i}$. As a
consequence, we are only allowed to neglect the second term in the denominator
of the inverse moment matrix, ending up with

$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
smom}=\bm{v}^{T}\bm{\mathcal{M}^{-1}}\bm{v}=\sum_{i=1}^{d}\frac{\bar{n}_{c}e^{-2r^{\prime}_{i}}+(\bar{n}_{s}^{\prime})_{i}}{\bar{n}_{c}^{2}}v_{i}^{2}.$
(116)

Making use of the general relation
$e^{2r_{i}^{\prime}}=1+2(\bar{n}_{s}^{\prime})_{i}+\sqrt{[1+2(\bar{n}_{s}^{\prime})_{i}]^{2}-1}$,
with $(\bar{n}_{s}^{\prime})_{i}=\sinh^{2}r_{i}^{\prime}$, which holds for any
value of $r^{\prime}_{i}$, and setting $\bm{v}=\bm{v}_{\rm ave}$, or
$v_{i}^{2}=1/d^{2}$, we get

$\Delta^{2}(\bm{v}_{\rm ave}\cdot\bm{\theta})_{\rm
smom}=\frac{1}{d^{2}\bar{n}_{c}^{2}}\sum_{i=1}^{d}\frac{\bar{n}_{c}}{X_{i}+\sqrt{X^{2}_{i}-1}}+\frac{X_{i}-1}{2}.$
(117)

Here, we have set $X_{i}\equiv 1+2(\bar{n}_{s}^{\prime})_{i}$ for convenience.
The function above has to be minimized with respect to the set of variables
$\\{X_{i}\\}_{i=1,\dots,d}$ under the constraint $\sum_{i=1}^{d}X_{i}=X$,
which comes from the original constraint
$\sum_{i=1}^{d}(\bar{n}^{\prime}_{s})_{i}=\bar{n}_{s}$. Notice that
$X=d+2\bar{n}_{s}$. Relying again on the method of Lagrange multipliers, we
are able to show that the minimum point is $X_{i}=X/d$, which corresponds, as
expected, to splitting $\bar{n}_{s}$ evenly between the $d$ MZIs used in the
separable strategy: $(\bar{n}_{s}^{\prime})_{i}=\bar{n}_{s}/d$. In this case,
we find the following value for the Lagrange multiplier:
$\lambda=1/(d^{2}\bar{n}_{c}^{2})[\bar{n}_{c}(1-X/\sqrt{X^{2}-d^{2}})+1/2]$.
Evaluated at its minimum point and for $\bm{v}=\bm{v}_{\rm ave}$, Eq. (116)
becomes

$\min_{r^{\prime}_{1},...,r^{\prime}_{d}}\Delta^{2}(\bm{v}_{\rm
ave}\cdot\bm{\theta})_{\rm
smom}=\frac{\bar{n}_{c}e^{-2r^{\prime}}+\bar{n}_{s}/d}{d\bar{n}_{c}^{2}},$
(118)

with $r^{\prime}={\rm arcsinh}\sqrt{\bar{n}_{s}/d}$. Taking the ratio with Eq.
(27), which expresses the optimal sensitivity of the entangled strategy, we
find Eq. (29).

###  VII.7 Further discussion on Eq. (36)

In Fig. 8 we further clarify the behaviour of $\mathcal{G}_{4}(\bm{v}_{\rm
ave})$ in a broad parameter regime. The dashed red line is the analytical Eq.
(36), which is expected to be accurate for $\bar{n}_{T}\gg(d+1)\bar{n}_{s}$.
The analytical formula predicts $\mathcal{G}_{4}(\bm{v}_{\rm ave})=1$ for
$\bar{n}_{T}\gg de^{2r}\bar{n}_{s}$ and a gain up to
$\mathcal{G}_{4}(\bm{v}_{\rm ave})=d$ when $\bar{n}_{T}\ll(d+1)\bar{n}_{s}$.
The solid line is $\mathcal{G}_{4}(\bm{v}_{\rm ave})$ where minimization over
$\bm{U}$ is performed numerically. The quantity diverges at
$\bar{n}_{T}=2d\bar{n}_{s}$ due to the divergence of Eq. (34), while
$\min_{\bm{U}}\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm emom}$ remains finite.
The dot dashed line shows the gain of the entangled strategy (with sensitivity
calculate within the multimode moment-matrix approach) with respect to the
quantum Cramer-Rao bound of the separable strategy:

$\mathcal{\tilde{G}}_{4}(\bm{v})=\frac{\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
sQCR}}{\min_{\bm{U}}\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm emom}},$ (119)

where

$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
sQCR}=\frac{1}{d[\bar{n}_{c}^{\prime}e^{2r}+\bar{n}_{s}]},$ (120)

under the constraints considered here. As we see from the figure, we obtain
$\mathcal{\tilde{G}}_{4}(\bm{v}_{\rm ave})\leq 1$, even where
$\mathcal{G}_{4}(\bm{v}_{\rm ave})$ diverges. In particular,
$\mathcal{G}_{4}(\bm{v}_{\rm ave})=\mathcal{\tilde{G}}_{4}(\bm{v}_{\rm
ave})=1$ for $\bar{n}_{T}\gg de^{2r}\bar{n}_{s}$.

Figure 8: $\mathcal{G}_{4}(\bm{v}_{\rm ave})$ as a function of $\bar{n}_{T}$.
The solid blue line is the result of a numerical optimization. The red dashed
line is the analytical Eq. (36). The dot-dashed black line is Eq. (119).
Vertical dotted lines mark interesting values of $\bar{n}_{T}$ (see text).
Here $d=10$ and $\bar{n}_{s}=100$.

###  VII.8 Demonstration of Eqs. (38) and (39)

Let us demonstrate Eq. (38): the demonstration of Eq. (39) is analogous.
Equation (38) is a direct consequence of

$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm
emom}=\bm{v}^{T}\bm{\mathcal{M}}^{-1}\bm{v}\geq|\bm{v}|^{4}/\bm{v}^{T}\bm{\mathcal{M}}\bm{v},$
(121)

$\bm{v}^{T}\bm{\mathcal{M}}\bm{v}\leq\bm{v}_{\mu_{\rm
max}}^{T}\bm{\mathcal{M}}\bm{v}_{\mu_{\rm max}}=\mu_{\rm max}$ and
$|\bm{v}|^{2}=1/d$. These imply the bound
$\Delta^{2}(\bm{v}\cdot\bm{\theta})_{\rm emom}\geq 1/(\mu_{\rm max}d)$, which
is saturable for $\bm{v}/|\bm{v}|=\bm{v}_{\mu_{\rm max}}$. The inequality
(121) follows from the Cauchy-Schwarz inequality
$(\bm{f}^{T}\bm{f})(\bm{g}^{T}\bm{g})\geq(\bm{f}^{T}\bm{g})^{2}$ with
$\bm{f}=\bm{\mathcal{M}}^{1/2}\bm{v}$ and
$\bm{g}=\bm{\mathcal{M}}^{-1/2}\bm{v}$ and is saturated if and only if
$\bm{f}=\lambda\bm{g}$ for some real number $\lambda$, namely if and only if
$\bm{v}$ is an eigenvector of $\bm{\mathcal{M}}$. Note that
$\bm{\mathcal{M}}^{1/2}$ is well definite since $\bm{\mathcal{M}}\geq 0$: this
follows from $\bm{\mathcal{M}}=\bm{C}^{T}\bm{\Gamma}^{-1}\bm{C}$,
$\bm{\Gamma}\geq 0$ being a covariance matrix, and $\bm{\Gamma}^{-1}\geq 0$
being the inverse of a positive semidefinite matrix.

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  * (90) To be more precise, the lower bound in Eq. (15) holds for $\bar{n}_{s}\ll\bar{n}_{T}\ll\bar{n}_{s}e^{2r}$, where the first term in Eq. (16) can be neglected with respect to first one. The upper bound in Eq. (15) instead requires $(d+1)\bar{n}_{s}\ll\bar{n}_{T}\ll\bar{n}_{s}e^{2r}\mathcal{W}$, under which condition the first term in Eq. (17) can be neglected. We conclude by noticing that $\mathcal{W}\geq 1$.
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