CP3-20-63, DESY 20-230, MCNet-20-23

# Probing the Weinberg Operator at Colliders

Benjamin Fuks<EMAIL_ADDRESS>Sorbonne Université, CNRS, Laboratoire de
Physique Théorique et Hautes Énergies, LPTHE, F-75005 Paris, France  Institut
Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris, France
Jonas Neundorf<EMAIL_ADDRESS>Krisztian Peters
<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron, Notkestraße 85,
22607 Hamburg, Germany  Richard Ruiz<EMAIL_ADDRESS>Centre for
Cosmology, Particle Physics and Phenomenology (CP3),
Université Catholique de Louvain, Chemin du Cyclotron, Louvain la Neuve,
B-1348, Belgium  Matthias Saimpert<EMAIL_ADDRESS>CERN, Geneva,
Switzerland

###### Abstract

Motivated by searches for $0\nu\beta\beta$ decay in nuclear experiments and
collider probes of lepton number violation at dimension $d\geq 7$, we
investigate the sensitivity to the $d=5$ Weinberg operator using the non-
resonant signature $pp\to\ell^{\pm}\ell^{\prime\pm}jj$ at the LHC. We develop
a prescription for the operator that is applicable in collisions and decays,
and focus on the $\ell\ell^{\prime}=\mu\mu$ channel, which is beyond the reach
of nuclear decays. For a Wilson coefficient $C^{\mu\mu}_{5}=1$, scales as
heavy as $\Lambda\sim 8.3~{}(13)$ TeV can be probed with
$\mathcal{L}=300~{}{\rm fb}^{-1}~{}(3~{}{\rm ab}^{-1})$. This translates to
effective $\mu\mu$ Majorana masses of $|m_{\mu\mu}|\sim 7.3~{}(5.4)$ GeV, and
establishes a road map for testing the Weinberg operator at accelerators.

Introduction – Among the most pressing mysteries shared in cosmology, nuclear,
and high-energy physics is whether neutrinos are their own antiparticles Ellis
_et al._ (2019); Eur (2020). This importance follows from Majorana neutrinos
being necessary ingredients for standard leptogenesis, grand unification, as
well as new gauge symmetries. Discovering that neutrinos are Majorana
particles would indicate that lepton number (LN) symmetries are not conserved
below the electroweak (EW) scale, and demonstrate the existence of a mass-
generating mechanism beyond those responsible for chiral and EW symmetry
breaking (EWSB).

Motivated by this, broad, complementary approaches are taken to explore the
nature of neutrinos Atre  _et al._ (2009); Deppisch  _et al._ (2015); Cai  _et
al._ (2017, 2018); Cirigliano  _et al._ (2018); Dolinski  _et al._ (2019). A
foremost probe is the search for the neutrinoless $\beta\beta$ process
$({0\nu\beta\beta})$ in decays of heavy nuclei. This is characterized by the
transition $(A,Z)\to(A,Z+2)$ and the appearance of two same-sign electrons but
an absence of neutrinos in the final state. While no discovery has been
confirmed, and assuming that the decay is mediated solely by three neutrinos,
on-going searches place stringent upper limits of $79-180$ meV at 90%
confidence level (CL) Agostini  _et al._ (2020) on the so-called effective
$\beta\beta$ Majorana mass, given by Klapdor-Kleingrothaus  _et al._ (2001)

$|m_{ee}|=\left|\sum_{k=1}^{3}U_{ek}m_{\nu_{k}}U_{ek}\right|.$ (1)

In this definition, $m_{\nu_{k}}$ are the mass eigenvalues of the light
neutrinos and $U_{\ell k}$ are the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
mixing matrix elements.

From the perspective that the Standard Model (SM) of particle physics is a
low-energy effective field theory (EFT), Majorana neutrino masses, and hence
$|m_{ee}|$, can be generated most minimally Ma (1998); Bonnet  _et al._ (2012)
at dimension $d=5$ from the LN-violating Weinberg operator Weinberg (1979):

${\cal L}_{5}=\
\frac{C_{5}^{\ell\ell^{\prime}}}{\Lambda}\big{[}\Phi\\!\cdot\\!\overline{L}^{c}_{\ell}\big{]}\big{[}L_{\ell^{\prime}}\\!\\!\cdot\\!\Phi\big{]}.$
(2)

Here, $\Lambda$ is the scale at which the particles responsible for neutrino
masses become relevant degrees of freedom; $C_{5}^{\ell\ell^{\prime}}$ is a
flavor-dependent Wilson coefficient; $L_{\ell}^{T}=(\nu_{\ell},\ell)$ is the
left-handed (LH) lepton doublet; and $\Phi$ is the SM Higgs doublet, whose
vacuum expectation value (vev) $v=\sqrt{2}\langle\Phi\rangle\approx
246{\rm~{}GeV}$ generates the quantity

${\color[rgb]{0,0,0}m_{\ell\ell^{\prime}}=C_{5}^{\ell\ell^{\prime}}v^{2}/\Lambda}.$
(3)

As the Weinberg operator can be realized by the Type I-III Seesaw models Ma
(1998), limits on $|m_{ee}|$ translate into lower bounds on the Seesaw scale
of about $(\Lambda/C_{5}^{ee})\gtrsim(3.3-7.6)\cdot 10^{14}{\rm~{}GeV}$. While
stringent, a caveat of this constraint is its flavor dependency. $C_{5}^{ee}$
can, for instance, be zero due to a flavor symmetry Jenkins (2009). More
generally, the production of same-sign leptons involving muons or taus in
$(A,Z)\to(A,Z+2)$ decays is kinematically forbidden. Their production requires
higher energies, implying a lack of sensitivity for non-electron flavors of
$C_{5}^{\ell\ell^{\prime}}$.

Figure 1:  Diagrammatic representation of same-sign charged lepton production
through same-sign $WW$ scattering in proton collisions when mediated at
dimension $d=5$.

Motivated by this limitation and sensitivity projections for interactions at
$d\geq 7$ in same-sign $W^{\pm}W^{\pm}$ scattering Dicus  _et al._ (1991);
Datta  _et al._ (1994); Aoki  _et al._ (2020); Costantini  _et al._ (2020);
Fuks  _et al._ (2020), we report an investigation into the realization of the
${0\nu\beta\beta}$ process at $d=5$ in high-energy proton collisions. As shown
in Fig. 1, the transition proceeds from same-sign $W^{\pm}W^{\pm}$ scattering
into same-sign charged lepton pairs $\ell^{\pm}\ell^{{}^{\prime}\pm}$ that are
of arbitrary flavor and bridged via the coupling
$m_{\ell\ell^{\prime}}\\!\propto\\!C_{5}^{\ell\ell}/\Lambda$. While related,
this work differs from studies on the “inverse” ${0\nu\beta\beta}$ process
Rizzo (1982); Belanger  _et al._ (1996); Aoki  _et al._ (2020); Fuks  _et al._
(2020), which focus on $d\geq 7$ operators or their realizations. Moreover,
this work relies on a new method for modeling the Weinberg operator that is
applicable to meson and lepton decays, and establishes a road map to the
Weinberg operator at accelerators.

The Standard Model at Dimension Five – To describe Majorana neutrino masses
and the ${0\nu\beta\beta}$ process from $d=5$ interactions, we work in the SM
effective field theory Weinberg (1980) and extend the SM Lagrangian
$(\mathcal{L}_{\rm SM})$ by gauge-invariant operators of $d>4$. In the
canonical representation Grzadkowski  _et al._ (2010), the Lagrangian is given
by Weinberg (1979)

$\displaystyle\mathcal{L}_{\rm SM~{}EFT}=\mathcal{L}_{\rm
SM}+\mathcal{L}_{5}+\mathcal{O}\left(\Lambda^{-2}\right),$ (4)

where $\mathcal{L}_{5}$ is defined in Eq. (2). By the power counting of Ref.
Grzadkowski  _et al._ (2010), the Weinberg operator is the only gauge-
invariant operator at $d=5$ in the SM Weinberg (1979); Kobach (2016).

After EWSB, the Higgs field can be expanded around its vev, which in unitary
gauge reads $\sqrt{2}\Phi\approx v+h$, with $h$ being the SM Higgs boson. The
result is the Lagrangian

$\displaystyle\mathcal{L}_{5}$
$\displaystyle=-\frac{C_{5}^{\ell\ell^{\prime}}}{2\Lambda}hh\overline{\nu^{c}_{\ell}}\nu_{\ell^{\prime}}-\frac{C_{5}^{\ell\ell^{\prime}}v}{\Lambda}h\overline{\nu^{c}_{\ell}}\nu_{\ell^{\prime}}-\frac{C_{5}^{\ell\ell^{\prime}}v^{2}}{2\Lambda}\overline{\nu^{c}_{\ell}}\nu_{\ell^{\prime}},$
(5)

where $C^{\ell\ell^{\prime}}_{5}$ is defined in the flavor basis. The minus
signs above originate from the SU$(2)_{L}$-invariant product
$\Phi\cdot\overline{L^{c}}=\Phi^{i}\varepsilon_{ij}\overline{L^{cj}}$, with
$\varepsilon_{12}=1$. While the first two terms in Eq. (5) signify double- and
single-Higgs couplings to neutrinos of flavors $\ell\ell^{\prime}$, the third
term generates the $3\times 3$ LH Majorana mass matrix
$m_{\ell\ell^{\prime}}$, as defined in Eq. (3). After rotating
$m_{\ell\ell^{\prime}}$ into the mass basis, the resulting eigenvalues
parametrize the three neutrino mass eigenstates $m_{\nu_{k}}$ that describe
neutrino oscillation data.

We make no assumption on the structure of $C_{5}^{\ell\ell^{\prime}}$. It is
therefore possible under this framework that one neutrino is massless, as
allowed by data Wyler and Wolfenstein (1983); that all masses scale as
$m_{\nu_{k}}\sim\mathcal{O}(m_{\ell\ell^{\prime}})$, indicating minor fine
tuning; or that $m_{\nu_{k}}\ll m_{\ell\ell^{\prime}}$, indicating strong
cancellations among the $m_{\ell\ell^{\prime}}$ elements. As nuclear searches
are only sensitive to $|m_{ee}|$, the latter possibilities remain
underexplored.

The ${0\nu\beta\beta}$ Process at Dimension Five – The goal of this work is to
estimate the sensitivity of the LHC to the Weinberg operator using the
${0\nu\beta\beta}$ process. When simulating the Weinberg operator at LHC
energies, difficulties arise if working in the neutrinos’ mass eigenbasis. In
this basis, $d=5$ vertices are proportional to $m_{\nu_{k}}$, which are
unknown and small on LHC scales, and to $U_{\ell k}$, which possess unknown
phases. So while the transition in Fig. 1 may proceed through a non-trivial
incoherent sum of intermediate mass states, individual contributions may be
too small for practical computations.

We propose a solution to this complication by working in the neutrino flavor
basis and treating the mass term in Eq. (5) as a “two-point vertex”. From this
perspective, the Weinberg operator in Fig. 1 couples one massless, LH neutrino
of momentum $p$ and flavor $\ell$ with the conjugate of a second neutrino of
momentum $p$ and flavor $\ell^{\prime}$. After contracting Dirac matrices, the
LN-violating $(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$ current in Fig. 1 reduces
to the ratio of $m_{\ell\ell^{\prime}}$ and the virtuality $p^{2}$.
Explicitly, its graph simplifies to:

Up to corrections of $\mathcal{O}(|m_{\ell\ell^{\prime}}^{2}/p^{2}|)$, which
are assumed small, one can identify the rightmost ratio as the right-handed
(RH) helicity state of an intermediate fermion with mass
$m_{\ell\ell^{\prime}}$ and momentum $p$. That is, one can write

$\displaystyle\gamma^{\alpha}P_{L}\frac{i\left(\not\\!p+m_{\ell\ell^{\prime}}\right)}{p^{2}-m_{\ell\ell^{\prime}}^{2}}\gamma^{\beta}P_{R}$
$\displaystyle=\gamma^{\alpha}P_{L}\frac{im_{\ell\ell^{\prime}}}{p^{2}-m_{\ell\ell^{\prime}}^{2}}P_{L}\gamma^{\beta}$
(6)
$\displaystyle=\gamma^{\alpha}P_{L}\frac{im_{\ell\ell^{\prime}}}{p^{2}}P_{L}\gamma^{\beta}$
$\displaystyle\times\left[1+\mathcal{O}\left(\left|\frac{m_{\ell\ell^{\prime}}^{2}}{p^{2}}\right|\right)\right],$
(7)

where $P_{R/L}=\frac{1}{2}(1\pm\gamma^{5})$ are the usual chiral projection
operators in four-component notation, and recover the same ratio at leading
power of the expansion.

Following Refs. Kayser (1982); Mohapatra and Pal (1998), fermions
participating in the LN-violating
$(\ell^{+}\nu_{\ell}\nu_{\ell^{\prime}}^{c}\ell^{{}^{\prime}+})$ current in
Fig. 1 experience a parity inversion beyond the standard SU$(2)_{L}$ chiral
couplings. In terms of Feynman rules Denner  _et al._ (1992a, b), this
manifests as a chiral inversion of the
$(W\ell^{\prime}\nu_{\ell^{\prime}}^{c})$ vertex, i.e.,
$\gamma^{\beta}P_{L}\to\gamma^{\beta}P_{R}$. In the absence of additional new
physics, this ensures Han  _et al._ (2013); Ruiz (2020) the presence of the
projection operators enveloping the $(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$
current in Eq. (6), and hence that $(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$
propagates in the RH helicity state.

As a result, up to corrections of
$\mathcal{O}(|m_{\ell\ell^{\prime}}^{2}/p^{2}|)$, the
$(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$ current itself can be modeled as an
unphysical Majorana neutrino $N$ with mass $m_{\ell\ell^{\prime}}$ that
couples to the $W$ boson and all charged leptons $\ell$ via the Lagrangian

$\displaystyle\Delta\mathcal{L}=-\frac{g_{W}}{\sqrt{2}}W^{+}_{\mu}\sum_{\ell=e}^{\tau}\overline{N}\gamma^{\mu}P_{L}\ell^{-}+{\rm
H.c.}$ (8)

Here, $g_{W}\approx 0.65$ is the SU$(2)_{L}$ weak coupling constant. Up to
factors of active-sterile mixing, Eq. (8) is identical to the interaction
Lagrangian in the Phenomenological Type I Seesaw model del Aguila and Aguilar-
Saavedra (2009); Atre  _et al._ (2009), and therefore can also be applied to
LN-decays of unstable hadrons and leptons.

Signal and Background Simulation – To simulate the ${0\nu\beta\beta}$ process
in LHC collisions using mainstream Monte Carlo (MC) tools, we exploit the
above observation that the intermediate $(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$
current in Fig. 1 can be modeled as an unphysical Majorana neutrino with mass
$m_{\ell\ell^{\prime}}=C^{\ell\ell^{\prime}}_{5}v^{2}/\Lambda$. We implement
the Lagrangian of Eq. (4) into the FeynRules software package (version 2.3.36)
Christensen and Duhr (2009); Christensen  _et al._ (2011); Degrande  _et al._
(2012); Alloul  _et al._ (2014) by extending the FeynRules implementation of
the SM (version 1.4.7) by a single Majorana neutrino $N$ with mass $m_{N}$ and
EW boson couplings governed by Eq. (8). We ensure that conventional factors
are kept according to Ref. Alloul  _et al._ (2014). To account for all
$\ell^{\pm}\ell^{{}^{\prime}\pm}$ flavor permutations accessible at LHC
energies, we make $m_{N}$ an internally calculated quantity that is set by

$m_{N}=\left|C^{ee}_{5}+C^{e\mu}_{5}+C^{e\tau}_{5}+C^{\mu\mu}_{5}+C^{\mu\tau}_{5}+C^{\tau\tau}_{5}\right|\frac{v^{2}}{\Lambda}.$
(9)

Using Refs. Degrande (2015); Hahn (2001), we extract renormalization and
$R_{2}$ counterterms up to the first order in the quantum chromodynamic (QCD)
coupling $\alpha_{s}$. Feynman rules are collected into a set of
public111Available from feynrules.irmp.ucl.ac.be/wiki/SMWeinberg universal
FeynRules output (UFO) libraries that we call the SMWeinberg libraries.

With this UFO, proton collisions are simulated at next-to-leading (NLO) in QCD
with the event generator MadGraph5_aMC@NLO (version 2.7.1.2) Stelzer and Long
(1994); Alwall  _et al._ (2014); Frixione and Webber (2002); Frederix  _et
al._ (2009); Hirschi  _et al._ (2011); Hirschi and Mattelaer (2015). Parton
showering (PS) and modeling of non-perturbative phenomena are handled by
Pythia8 (version 243) Sjöstrand  _et al._ (2015). Hadron-level events are
passed through Delphes (version 3.4.2) de Favereau  _et al._ (2014) for the
simulation of an ATLAS-like detector. Hadron clustering is handled according
to the anti-$k_{T}$ algorithm at $R=0.4$ Catani  _et al._ (1993); Ellis and
Soper (1993); Cacciari  _et al._ (2008) as implemented in FastJet Cacciari and
Salam (2006); Cacciari  _et al._ (2012). We tune our simulation tool chain as
in the study on $W^{\pm}W^{\pm}$ scattering by Ref. Fuks  _et al._ (2020),
whose methodology we also follow to model SM backgrounds.

Table 1:  The total cross section at NLO in QCD for the process in Eq. (10)
for various choices of EFT scale $\Lambda$ and collider energy $\sqrt{s}$ at a
Wilson coefficient
$C_{5}^{\ell\ell^{\prime}}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$, and the
corresponding mass $m_{N}$, uncertainties, and NLO $K$-factor.

$\Lambda$ [TeV] |  $m_{N}$ [GeV] |  $\sqrt{s}$ [TeV] |  $\sigma^{\rm NLO}$ [ab] | $\delta_{\rm Scale}$ | $\delta_{\rm PDF}$ | $K^{\rm NLO}$
---|---|---|---|---|---|---
10 | 6 | 13 | $133$ | ${}^{+0.8\%}_{-0.8\%}$ | ${}^{+0.9\%}_{-0.9\%}$ | $0.968$
100 | 0.6 | 13 | $1.42$ | ${}^{+1.0\%}_{-0.6\%}$ | ${}^{+0.9\%}_{-0.9\%}$ | $0.978$
200 | 0.3 | 13 | $0.361$ | ${}^{+0.7\%}_{-0.6\%}$ | ${}^{+1.0\%}_{-1.0\%}$ | $0.952$
400 | 0.15 | 13 | $0.0904$ | ${}^{+0.6\%}_{-0.8\%}$ | ${}^{+0.9\%}_{-0.9\%}$ | $0.988$
200 | 0.3 | 27 | $1.21$ | ${}^{+0.9\%}_{-0.8\%}$ | ${}^{+0.9\%}_{-0.9\%}$ | $1.04$
200 | 0.3 | 100 | $6.56$ | ${}^{+1.4\%}_{-1.2\%}$ | ${}^{+0.9\%}_{-0.9\%}$ | $1.03$

The $d=5$, ${0\nu\beta\beta}$ Process at the LHC – In LHC collisions the LN-
violating ${0\nu\beta\beta}$ process occurs through the scattering of two
same-sign $W$ bosons that are sourced from quarks and antiquarks, and exit as
two high-$p_{T}$ jets. At the hadronic level, the collider signature is given
by

$p~{}p~{}\to~{}j~{}j~{}\ell^{\pm}~{}\ell^{{}^{\prime}\pm}+X,$ (10)

where $X$ represents the additional hadronic and electromagnetic activity that
may exist in the inclusive process.

To identify the dependence of Eq. (10) on the Weinberg operator, we consider
the Effective $W$ Approximation Dawson (1985); Kane  _et al._ (1984); Kunszt
and Soper (1988) and treat the incoming $W^{\pm}W^{\pm}$ pair as perturbative
constituents of the proton. In this limit, we find that the
$W^{\pm}W^{\pm}\to\ell^{\pm}\ell^{{}^{\prime}\pm}$ sub-process is dominated by
the scattering of longitudinal $W$ bosons. After summing over all external
helicities, the spin-averaged, parton-level cross section for the $2\to 2$
process is given by

$\displaystyle\hat{\sigma}(W^{+}W^{+}$
$\displaystyle\to\ell^{+}\ell^{{}^{\prime}+})$
$\displaystyle=\frac{(2-\delta_{\ell\ell^{\prime}})}{2\pi~{}3^{2}}\left|\frac{C_{5}^{\ell\ell^{\prime}}}{\Lambda}\right|^{2}+\mathcal{O}\left(\frac{m_{W}^{2}}{M_{WW}^{2}}\right),$
(11)

which shows that, like in nuclear experiments, the ${0\nu\beta\beta}$ rate at
the LHC scales as
$\sigma\sim|m_{\ell\ell^{\prime}}|^{2}\propto|C^{\ell\ell^{\prime}}_{5}/\Lambda|^{2}$.

Figure 2:  Total cross sections at NLO in QCD (upper) and the corresponding
NLO $K$-factors (lower) for the process in Eq. (10), as a function of EFT
scale $\Lambda$ with
$C^{\ell\ell^{\prime}}_{5}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$, and
$\sqrt{s}=13$, 27, 100 TeV. Bands represent scale uncertainties.

Using Eq. (11) as a check of the SMWeinberg UFO, we list in Table 1 the total
$2\to 4$, hadronic cross section $\sigma$ for Eq. (10) at NLO in QCD for
representative cutoff scales $\Lambda$, assuming a Wilson coefficient of
$C_{5}^{\ell\ell^{\prime}}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$, for the
$\sqrt{s}=13{\rm~{}TeV}$ LHC and proposed experiments at
$\sqrt{s}=27{\rm~{}TeV}$ and 100 TeV. Also listed are the corresponding
(unphysical) Majorana neutrino mass $m_{N}$, as defined by Eq. (9), the nine-
point scale uncertainty $\delta_{\rm Scale}$, the parton distribution function
(PDF) uncertainty $\delta_{\rm PDF}$, and the QCD $K$-factor, which is defined
as the ratio $K=\sigma/\sigma^{LO}$, where $\sigma^{\rm LO}$ is the cross
section at lowest order.

For $\Lambda=10{\rm~{}TeV}$ and $100{\rm~{}TeV}$ at $\sqrt{s}=13{\rm~{}TeV}$,
we observe a cross section scaling of
$\sigma(\Lambda=10{\rm~{}TeV})/\sigma(\Lambda=100{\rm~{}TeV})\sim 93$, which
undershoots the $100\times$ scaling expected from Eq. (11). We attribute this
to a breakdown of Eq. (7), which requires the mass $m_{N}\sim v^{2}/\Lambda$
to be small compared to the virtuality of
$(\nu_{\ell}\nu_{\ell^{\prime}}^{c})$. At larger $\Lambda$ we find, for
example, that
$\sigma(\Lambda=100{\rm~{}TeV})/\sigma(\Lambda=200{\rm~{}TeV})\sim 3.93$ and
$\sigma(\Lambda=200{\rm~{}TeV})/\sigma(\Lambda=400{\rm~{}TeV})\sim 3.99$,
indicating behavior more inline with Eq. (11). We conclude that choices of
$\Lambda\gtrsim 200{\rm~{}TeV}$ generate sufficiently small $m_{N}$ so that
Eq. (7) remains valid.

Given this guidance and by re-scaling from baseline rates for
$\Lambda=200{\rm~{}TeV}$ at $\sqrt{s}=13,~{}27$, and 100 TeV, we show in the
upper panel of Fig. 2 the hadronic cross section $\sigma$ at NLO in QCD for
the full $2\to 4$, ${0\nu\beta\beta}$ process as a function of $\Lambda$,
again assuming
$C_{5}^{\ell\ell^{\prime}}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$. The
band thickness for each curve, which reaches $\mathcal{O}(0.5\%-1.5\%)$,
denotes the nine-point scale uncertainty. While not shown, PDF uncertainties
reach about $\mathcal{O}(1\%)$. At $\sqrt{s}=13{\rm~{}TeV}$ and for
$\Lambda=10{\rm~{}TeV}$, or $m_{\mu\mu}\sim 6{\rm~{}GeV}$, we find $\sigma\sim
0.14{\rm~{}fb}$. Conversely, we find the rate at
$\sqrt{s}=13~{}(27)~{}[100]{\rm~{}TeV}$ reaches the $\sigma\sim 1{\rm~{}ab}$
threshold at about $\Lambda\sim 120~{}(220)~{}[510]{\rm~{}TeV}$, which
correspond to $m_{\mu\mu}\sim 500~{}(275)~{}[120]{\rm~{}MeV}$. As a measure of
the QCD corrections to the total cross section, we show in the lower panel of
Fig. 2 the QCD $K$-factor. We report that $\mathcal{O}(\alpha_{s})$
corrections are mild, reaching $K\sim 0.95-1.05$ across $\sqrt{s}$.

Table 2: Particle identification and signal region definitions

Particle Identification Cuts
---
$p_{T}^{e~{}(\mu)~{}[j]}>10~{}(10)~{}[25]{\rm~{}GeV}$, Anti-$k_{T}$($R$=0.4)
$|\eta^{e~{}(\mu)~{}[j]}|<2.5~{}(2.7)~{}[4.5]$
Signal Region Cuts
$p_{T}^{\mu_{1}~{}(\mu_{2})}>27~{}(10){\rm~{}GeV}$, $n_{\mu}=2$, $n_{j}\geq
2$,
$n_{e}=n_{\tau^{\mathrm{had}}}=0$, $Q_{\mu_{1}}\times Q_{\mu_{2}}=1$,
$M(j_{1},j_{2})>700{\rm~{}GeV}$
${E_{\rm T}^{\rm miss}}<30{\rm~{}GeV}$,
$(H_{\mathrm{T}}/p_{\mathrm{T}}^{\mu_{1}})<1.6$

To estimate the LHC’s discovery potential of the Weinberg operator, we focus
on the $\ell\ell^{\prime}=\mu\mu$ channel with benchmark inputs
$C_{5}^{\ell\ell^{\prime}}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$ and
$\Lambda=200{\rm~{}TeV}$ (or $m_{\mu\mu}\approx 300{\rm~{}MeV}$), and design
an analysis inspired by Run 2 performance of the ATLAS detector ATLAS
Collaboration (2008, 2010). We employ particle identification requirements on
electrons, muons, and jets that are summarized in the top of Table 2. For
simplicity, we ignore particles originating from pileup interactions as they
would mostly be subtracted with dedicated algorithms in real experiments.

To define our signal-enriched region we demand events to have at least two
jets, with the leading pair carrying a large invariant mass, and exactly two
muons with the same charge. Events with additional leptons, including
hadronically decaying $\tau$ leptons, are vetoed. To further reduce
backgrounds we take into account two qualitative differences between our
signal and background processes: (i) Unlike SM processes with the same final
state, our signal does not contain outgoing neutrinos. As neutrinos go
undetected in LHC experiments, their presence gives rise to missing transverse
momentum ${E_{\rm T}^{\rm miss}}$, which is defined as the $p_{T}$ recoil
against all visible objects. We therefore require that events have a small
${E_{\rm T}^{\rm miss}}$ in accordance with the detector resolution. (ii) Due
to the lack of QCD color flowing between the two hadrons in Fig. 1, the
hadronic activity is much milder than the QCD and $W^{\pm}V$ backgrounds
Pascoli  _et al._ (2018, 2019); Fuks  _et al._ (2019). We therefore impose an
upper limit on the ratio $(H_{T}/p_{T}^{\mu_{1}})$, where $H_{T}$ is the
scalar sum of jet $p_{T}$. To guide our precise cut choice, we plot in Fig. 3
the $(H_{T}/p_{T}^{\mu_{1}})$ distribution for our signal and leading
backgrounds after applying all other selection cuts.

At this stage, the leading backgrounds consist of mixed EW-QCD production of
$W^{\pm}W^{\pm}jj$, pure EW production of $W^{\pm}W^{\pm}jj$, and the
inclusive diboson+jets spectrum $W^{\pm}V+nj$ with
$V\in\\{\gamma^{*}/Z/Z^{*}\\}$. We checked that other processes, e.g.,
$t\overline{t}W^{\pm}$, do not appreciably survive our signal region
selection. We neglect processes that are especially difficult to simulate from
MC methods alone. This includes, for example, when muons are assigned the
wrong charge during event reconstruction. While sub-dominant for dimuon final
states, such backgrounds are relevant for the electron and tau channels Aad
_et al._ (2016); Alvarez  _et al._ (2017); CMS (2017); Pascoli  _et al._
(2019). We account for these backgrounds with a more conservative uncertainty
in our background estimate.

Figure 3: At $\sqrt{s}=13{\rm~{}TeV}$ with $\mathcal{L}=300{\rm~{}fb^{-1}}$,
the $(H_{T}/p_{T}^{\mu_{1}})$ distribution at NLO+PS for signal and
backgrounds in the signal region, for
$C_{5}^{\ell\ell^{\prime}}=\delta_{\ell\mu}\delta_{\ell^{\prime}\mu}$ and
$\Lambda=200{\rm~{}TeV}$.

We summarize our signal region definition in the bottom of Table 2. About
$\varepsilon\sim 12\%$ of generated signal events pass all identification and
selection cuts. For our inputs the signal (total background) rate reaches
$\sigma\sim 42$ zb $(\sigma\sim 25{\rm~{}ab})$. In Table 3 we summarize the
expected number of signal and background events after all cuts for the LHC
(HL-LHC) with $\mathcal{L}=300{\rm~{}fb^{-1}}~{}(3{\rm~{}ab^{-1}})$.

Sensitivity to the Weinberg Operator – To quantify any excess of events, we
apply a Poisson-counting likelihood with a background rate uncertainty that is
constrained by an auxiliary Poisson measurement ATL (2020); Cousins  _et al._
(2008). Assuming a $\delta_{b}=20\%$ uncertainty on the background, we
estimate the sensitivity at 95% CL to $|C_{5}^{\mu\mu}|/\Lambda\propto
m_{\mu\mu}$ by fixing our signal significance to $Z\approx 2$ and the number
of signal events $n_{s}$ to

$n_{s}=n_{s}^{0}\times|C_{5}^{\mu\mu}|^{2}\left(\frac{200{\rm~{}TeV}}{\Lambda}\right)^{2},$
(12)

where $n_{s}^{0}$ is the number of signal events for our benchmark inputs. We
then solve this equality for $|C_{5}^{\mu\mu}|$. With
$\mathcal{L}=300{\rm~{}fb^{-1}}~{}(3{\rm~{}ab^{-1}})$, we report that the LHC
(HL-LHC) is sensitive at 95% CL to scales below

${\color[rgb]{0,0,0}\Lambda/|C_{5}^{\mu\mu}|\lesssim 8.3~{}(13){\rm~{}TeV},}$
(13)

which translate into an effective $\mu\mu$ Majorana masses of

${\color[rgb]{0,0,0}|m_{\mu\mu}|\gtrsim 7.3~{}(5.4){\rm~{}GeV}.}$ (14)

With an outlook to potential successors of the LHC Ellis  _et al._ (2019); Eur
(2020), we estimate the sensitivity of a $\sqrt{s}=100{\rm~{}TeV}$ proton
collider by noting the signal rate increase of about $18\times$ in Table 1.
For $\mathcal{L}=30{\rm~{}ab^{-1}}$ of data, this translates to about
$180\times$ larger statistics. Assuming an analysis comparable to what we have
built, we find that the reach reported in Eq. (13) can improve by roughly to a
factor $\sqrt{180}\sim 14$.

As described above, our modeling of the Weinberg operator can be applied to
LN-violating decays of mesons. Therefore, updating the limits and projections
on $|m_{\mu\mu}|$ from $B^{\pm}\to\pi^{\pm}\mu^{\mp}\mu^{\mp}$ and
$K^{\pm}\to\pi^{\mp}\mu^{\pm}\mu^{\pm}$ decays using Refs. Atre  _et al._
(2005); Aaij  _et al._ (2016); Cortina Gil  _et al._ (2019), we find that LHCb
with $\mathcal{L}=300{\rm~{}fb^{-1}}$ can only probe
$\Lambda/|C_{5}^{\mu\mu}|\lesssim 9{\rm~{}MeV}$ and that NA-62 has excluded
with its 2017 data set $\Lambda/|C_{5}^{\mu\mu}|\lesssim 1.1{\rm~{}TeV}$.

Assuming that neutrino masses are described completely at $d=5$, we summarize
in Fig. 4 our sensitivities to $|m_{\mu\mu}|$ in comparison to the values
Esteban  _et al._ (2020) allowed by Eq. (1) (as generalized for arbitrary
$\ell\ell^{\prime}$ Frigerio and Smirnov (2002)) for normal (NO) and inverse
ordering (IO) of neutrino masses. While the reach of $W^{\pm}W^{\pm}$
scattering greatly exceeds our LHCb and NA-62 benchmarks, improvements at
these and similar experiments are anticipated.

Table 3: Expected number of background and ${0\nu\beta\beta}$ signal events in
the signal region with $\mathcal{L}=300~{}{\rm~{}fb^{-1}}$ (3 ab-1).

Collider | QCD $W^{\pm}W^{\pm}jj$ | EW $W^{\pm}W^{\pm}jj$ | $W^{\pm}V$ | Total | Signal
---|---|---|---|---|---
LHC | $<0.01$ | 6.40 | 1.16 | 7.56 | 0.013
HL-LHC | $<0.01$ | 64.0 | 11.6 | 75.5 | 0.13

Figure 4:  Projected sensitivity to $|m_{\mu\mu}|$ at the
$\sqrt{s}=13{\rm~{}TeV}$ LHC and a successor proton collider at
$\sqrt{s}=100{\rm~{}TeV}$, observed limits set by NA-62 with its 2017 data set
Cortina Gil  _et al._ (2019), and allowed values by best-fits to neutrino
oscillation data Esteban  _et al._ (2020).

Conclusions – If the Weinberg operator is present in nature and is accessible
at LHC energies, then a key prediction are processes that violate LN, such as
the ${0\nu\beta\beta}$ transition, and possibly also charged lepton flavor.
Motivated by the flavor limitations of nuclear decay experiments, we report an
investigation into the sensitivity to the Weinberg operator using the
${0\nu\beta\beta}$ process at the LHC, which permits muon- and tau-flavored
final states. To conduct this study, we have developed a new modeling
prescription for the Weinberg operator in high-energy scattering and resonance
decays, and which has been implemented into a set of publicly available UFO
libraries.

As a case study, we have focused on the LN-violating
$pp\to\mu^{\pm}\mu^{\pm}jj$ channel at $\sqrt{s}=13{\rm~{}TeV}$. After
employing a robust analysis that exploits the ${0\nu\beta\beta}$ process’s
weak vector boson scattering topology, we find sensitivity that exceeds
representative searches at current $B$\- and $K$-meson factories. In all cases
search strategies can be improved, such as through detector-specific selection
cuts and optimization. Whatever their final results entail, our work
establishes a complementarity across accelerator facilities in searching for
the Weinberg operator.

Acknowledgements – JN and KP acknowledge support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. RR is supported
under the UCLouvain fund “MOVE-IN Louvain.” This work has received funding
from the European Union’s Horizon 2020 research and innovation programme as
part of the Marie Skłodowska-Curie Innovative Training Network MCnetITN3
(grant agreement No. 722104), FNRS “Excellence of Science” EOS be.h Project
No. 30820817. Computational resources have been provided by the supercomputing
facilities of the Université catholique de Louvain (CISM/UCL) and the
Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles
(CÉCI) funded by the Fond de la Recherche Scientifique de Belgique
(F.R.S.-FNRS) under convention 2.5020.11 and by the Walloon Region.

Appendix – In this appendix we provide additional details of our methodology
and simulation setup.

In the SMWeinberg UFO, the full interaction Lagrangian by which we extend the
SM Lagrangian is

$\displaystyle\Delta\mathcal{L}=$
$\displaystyle-\frac{g_{W}}{\sqrt{2}}W^{+}_{\mu}\sum_{\ell=e}^{\tau}\overline{N}\gamma^{\mu}P_{L}\ell^{-}+{\rm
H.c.}$ (15)
$\displaystyle-\frac{g_{W}}{2\cos\theta_{W}}Z_{\mu}\sum_{\ell=e}^{\tau}\overline{N}\gamma^{\mu}P_{L}\nu_{\ell}+{\rm
H.c.}$ (16)
$\displaystyle-\frac{g_{W}m_{N}}{\sqrt{2}m_{W}}h\sum_{\ell=e}^{\tau}\overline{N}P_{L}\nu_{\ell}+{\rm
H.c.}$ (17)

The model’s input parameters along with their FeynRules and Les Houches Skands
_et al._ (2004) information are summarized in Table 4. Using this model file,
the syntax used to import the UFO into MadGraph5_aMC@NLO and simulate at NLO
the process

$\displaystyle
q_{1}~{}q_{2}~{}\to~{}q^{\prime}_{1}~{}q^{\prime}_{2}~{}\mu^{\pm}~{}\mu^{\pm},$
(18)

where $q$ is any light quark or antiquark, is

    import model SMWeinbergNLO
    generate     p p > mu+ mu+ j j QED=4 QCD=0
        $$ w+ w- [QCD]
    add process  p p > mu- mu- j j QED=4 QCD=0
        $$ w+ w- [QCD]

Table 4: Parameters for the SMWeinberg UFO.

Parameter | FR name | Type | LH block | LH counter
---|---|---|---|---
$\Lambda$ | Lambda | External (Real) | NUPHYSICS | 1
$C_{5}^{\ell\ell^{\prime}}$ | Cll | External (Real) | NUPHYSICS | 2-7
$m_{N}$ | mN | Internal (Real) | MASS | 9900012
$\Gamma_{N}$ | wN | Internal (Real) | WIDTH | 9900012

For SM inputs, we approximate the quark sector by $n_{f}=5$ massless quarks
that do not mix. Values of couplings and masses are set to global averages
reported in the 2020 Particle Data Group review Zyla  _et al._ (2020):

$\displaystyle m_{t}(m_{t})$ $\displaystyle=172.76{\rm~{}GeV},$ $\displaystyle
m_{h}=125.1{\rm~{}GeV},$ (19) $\displaystyle M_{Z}$
$\displaystyle=91.1876{\rm~{}GeV},$ $\displaystyle\alpha_{\rm
QED}^{-1}(M_{Z})=127.952,$ $\displaystyle G_{F}$ $\displaystyle=1.1663787\cdot
10^{-5}$ $\displaystyle{\rm~{}GeV}^{-2},~{}\alpha_{s}(M_{Z})=0.118.$

We employ the NNPDF3.1 NLO+LUXqed parton distribution function set
(lhaid=324900) Manohar  _et al._ (2016, 2017); Bertone  _et al._ (2018), with
scale evolution driven by LHAPDF Buckley  _et al._ (2015), and PDF
uncertainties are extracted using the replica method Buckley  _et al._ (2015);
Bertone  _et al._ (2018). We fix the collinear factorization $(\mu_{f})$, QCD
renormalization $(\mu_{r})$, and shower matching $(\mu_{s})$ scales to the
default values in Ref. Alwall  _et al._ (2014). The uncertainty in choosing
$\mu_{f}$ and $\mu_{r}$ is quantified by scaling their baseline values by
factors of 0.5, 1 and 2 to obtain a nine-point uncertainty band.

To quantify the LHC’s sensitivity to the Weinberg operator, we define our
signal significance $Z$ as ATL (2020); Cousins  _et al._ (2008)

$\displaystyle Z$ $\displaystyle=\frac{(n-n_{b})}{|n-n_{b}|}\sqrt{2\left[n\log
x-\frac{n_{b}^{2}}{\delta_{b}^{2}}\log y\right]},\quad\mathrm{with}$ (20)
$\displaystyle x$
$\displaystyle=\frac{n(n_{b}+\delta_{b}^{2})}{n_{b}^{2}+n\delta_{b}^{2}},\quad\mathrm{and}\quad
y=1+\frac{\delta_{b}^{2}(n-n_{b})}{n_{b}(n_{b}+\delta_{b}^{2})}.$ (21)

Here, $n=n_{s}+n_{b}$ is the total number of observed events,
$n_{s}~{}(n_{b})$ is the predicted number of signal (background) events, and
$\delta_{b}$ is the uncertainty on $n_{b}$

Under the parametrization of the PMNS matrix

$\displaystyle U_{\rm PMNS}=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix}$
$\displaystyle\cdot\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta_{\rm CP}}\\\
0&1&0\\\ -s_{13}e^{i\delta_{\rm CP}}&0&c_{13}\\\ \end{pmatrix}$
$\displaystyle\cdot\begin{pmatrix}c_{12}&s_{12}&0\\\ -s_{12}&c_{12}&0\\\
0&0&1\\\ \end{pmatrix}$ $\displaystyle\cdot\begin{pmatrix}e^{i\eta_{1}}&0&0\\\
0&e^{i\eta_{2}}&0\\\ 0&0&1\\\ \end{pmatrix},$ (22)

where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, and $\delta_{\rm
CP}$, $\eta_{1}$, and $\eta_{2}$ are imaginary phases, the regions for the
allowed effective Majorana massses $|m_{\beta\beta}|$ and $|m_{\mu\mu}|$ are
obtained using

$\displaystyle U_{e1}$ $\displaystyle=c_{12}c_{13}e^{i\eta_{1}},$ (23)
$\displaystyle U_{e2}$ $\displaystyle=s_{12}c_{13}e^{i\eta_{2}},$ (24)
$\displaystyle U_{e3}$ $\displaystyle=s_{13}e^{-i\delta_{\rm CP}},$ (25)
$\displaystyle U_{\mu 1}$
$\displaystyle=-s_{12}c_{23}e^{i\eta_{1}}-c_{12}s_{13}s_{23}e^{i(\delta_{\rm
CP}+\eta_{1})},$ (26) $\displaystyle U_{\mu 2}$
$\displaystyle=c_{12}c_{23}e^{i\eta_{2}}-s_{12}s_{13}s_{23}e^{i(\delta_{\rm
CP}+\eta_{2})},$ (27) $\displaystyle U_{\mu 3}$ $\displaystyle=c_{13}s_{23}.$
(28)

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