Probing the Weinberg Operator at Colliders

Motivated by searches for $0\nu\beta\beta$ decay in nuclear experiments and collider probes of lepton number violation at dimension $d\geq7$, we investigate the sensitivity to the $d=5$ Weinberg operator using the non-resonant signature $pp\to \ell^\pm \ell'^{\pm} j j$ at the LHC. We develop a prescription for the operator that is applicable in collisions and decays, and focus on the $\ell\ell'=\mu\mu$ channel, which is beyond the reach of nuclear decays. For a Wilson coefficient $C^{\mu\mu}_5=1$, scales as heavy as $\Lambda\sim 8.3~(11)$~TeV can be probed with $\mathcal{L}=300~{\rm fb}^{-1}~(3~{\rm ab}^{-1})$. This translates to an effective $\mu\mu$ Majorana mass of $\vert m_{\mu\mu}\vert\sim7.3~(5.4)$~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 [1,2]. 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 [3][4][5][6][7][8][9][10]. A foremost probe is the search for the neutrinoless ββ process (0νββ) in decays of nuclei. This is characterized by the transition (A, Z) → (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 the light neutrinos observed in nature, searches place upper limits of 79 − 180 meV at 90% confidence level (CL) [11] on the so-called effective ββ Majorana mass, given by [12] |m ee | = 3 k=1 U ek m ν k U ek . (1) In this definition, m ν k are the mass eigenvalues of the three light neutrinos and U 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 [13,14] at dimension d = 5 from the LN-violating Weinberg operator [15]: Here, Λ is the scale at which the particles responsible for LN violation become relevant degrees of freedom; C 5 is a flavor-dependent Wilson coefficient; L T = (ν , ) is the left-handed (LH) lepton doublet; and Φ is the SM Higgs doublet, whose vacuum expectation value (vev) v = √ 2 Φ ≈ 246 GeV generates the quantity m = C 5 v 2 /Λ.
As the Weinberg operator can be realized by treeand loop-level Seesaw models [7,13,14], limits on |m ee | translate into lower bounds on the Seesaw scale of about (Λ/C ee 5 ) (3.3 − 7.6) · 10 14 GeV. While stringent, a caveat of this constraint is its flavor dependence. For instance, C ee 5 can be zero due to a flavor symmetry [16], or be immeasurably small due to accidental cancellations [17][18][19]. More generally, the production of samesign leptons involving muons or taus in (A, Z) → (A, Z + 2) decays is kinematically forbidden. Their production requires higher energies, implying a lack of sensitivity for non-electron flavors of C 5 at 0νββ decay experiments.
Motivated by these limitations and by sensitivity projections for interactions at d ≥ 7 in same-sign W ± W ± scattering [20][21][22][23], we report an investigation into the realization of the 0νββ process at d = 5 in high-energy proton collisions. As shown in Fig. 1, the transition proceeds from W ± W ± scattering into same-sign charged lepton pairs ± ± of arbitrary flavor and bridged via the coupling m ∝ C 5 /Λ. While related, this work differs from studies on the "inverse" 0νββ process [22][23][24][25], which focus on d ≥ 7 operators or their realizations. Moreover, this work relies on a new method for modeling 1. Diagrammatic representation of same-sign charged lepton production through same-sign W W scattering in proton collisions when mediated at dimension d = 5.
the Weinberg operator that is applicable to meson and lepton decays, and establishes a road map to the Weinberg operator at accelerators. Finally, we release an implementation of this method in new and publicly available software 1 for Monte Carlo (MC) simulations.
The Standard Model at Dimension Five -To describe Majorana neutrino masses and the 0νββ process from d = 5 interactions, we work in the SM effective field theory [26] and extend the SM Lagrangian (L SM ) by gauge-invariant operators of d > 4. In the canonical representation [27], the Lagrangian is given by [15] where L 5 is defined in Eq. (2). By the power counting of Ref. [27], the Weinberg operator is the only gaugeinvariant operator at d = 5 in the SM [15,28]. After EWSB, the Higgs field can be expanded about its vev, which in the unitary gauge reads √ 2Φ ≈ v + h, where h is the Higgs boson. The resulting Lagrangian is Here, C 5 is defined in the flavor basis. The minus signs above originate from the SU(2) L -invariant product Φ · L c = Φ i ε ij L cj , with ε 12 = 1. While the first two terms in Eq. (5) signify double-and single-Higgs couplings to neutrinos of flavors , the third term generates the 3 × 3 LH Majorana mass matrix m , as defined in Eq. (3). After rotating m into the mass basis, the resulting eigenvalues parametrize the three neutrino mass eigenstates m ν k that describe neutrino oscillation data.
We make no assumption on the structure of C 5 . It is therefore possible under this framework that one neutrino is massless, as allowed by data [29]; that all masses scale as m ν k ∼ O(m ), indicating minor fine tuning; or that m ν k m , indicating strong cancellations among the m elements. As nuclear searches are only sensitive to |m ee |, the latter possibilities remain under-explored.
The 0νββ Process at Dimension Five -A goal of this work is to estimate the sensitivity of the Large Hadron Collider (LHC) to the 0νββ process, and hence the Weinberg operator. When simulating the Weinberg operator at the LHC, difficulties arise if working in the neutrinos' mass eigenbasis. There, d = 5 vertices are proportional to m ν k , which are unknown and small on LHC scales, and to U k , which carry unknown phases. So while the transition in Fig. 1 may proceed through a non-trivial incoherent sum of intermediate 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 with the conjugate of a second neutrino of momentum p and flavor . After contracting Dirac matrices, the LN-violating (ν ν c ) current in Fig. 1 reduces to the ratio of m and the squared virtuality p 2 . Explicitly, its graph simplifies to: p Up to corrections of O(|m 2 /p 2 |), which are assumed small, one can identify the rightmost ratio as the righthanded (RH) helicity state of an intermediate fermion with mass m and momentum p. That is, one can write where P R/L = 1 2 (1 ± γ 5 ) are the usual chiral projection operators in four-component notation, and recover the same ratio at leading power of the expansion. Intuitively, this identification follows from the inversion of helicity in LN-violating currents as discussed in Refs. [23,[30][31][32][33][34][35].
As a result, up to corrections of O(|m 2 /p 2 |), the (ν ν c ) current itself can be modeled as an unphysical Majorana neutrino N with mass m that couples to the W boson and all charged leptons via the Lagrangian Here, g W ≈ 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 [3,36], and therefore can also be employed in LN-violating decays of hadrons and leptons.
Signal and Background Simulation -To simulate the 0νββ process in LHC collisions using mainstream MC tools, we exploit the above observation that the intermediate (ν ν c ) current in Fig. 1 can be modeled as an unphysical Majorana neutrino with mass m = C 5 v 2 /Λ. We implement the Lagrangian of Eq. (4) into the Feyn-Rules software package (version 2.3.36) [37][38][39][40] 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. [40]. To account for all ± ± flavor permutations accessible at LHC energies, we make m N an internally calculated quantity that is set by Using Refs. [41,42], we extract renormalization and R 2 counterterms up to the first order in the quantum chromodynamic (QCD) coupling α s . Feynman rules are collected into a set of public universal FeynRules output (UFO) libraries that we call the SMWeinberg libraries.
The d = 5, 0νββ Process at the LHC -In LHC collisions the LN-violating 0νββ 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 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 [56][57][58] and treat the incoming W ± W ± pair as perturbative constituents of the proton. In this limit, we find that the W ± W ± → ± ± 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 → 2 process is given bŷ This shows that like in nuclear experiments the 0νββ rate at the LHC scales as σ ∼ |m | 2 ∝ |C 5 /Λ| 2 . Using this scaling behavior, we have checked that setting Λ 200 TeV in simulations with the SMWeinberg UFO will generate unphysical cross sections. This is due to a breakdown of the expansion in Eq. (7), which requires v 2 /Λ to be small compared to the virtuality of the internal (ν ν c ) current. For the LHC and beyond, physical rates can be obtained by choosing, for example, Λ = 200 TeV and using the relationship Given this guidance, we show in the top of Fig. 2    To estimate the LHC's discovery potential of the Weinberg operator, we focus on the = µµ channel with benchmark inputs C 5 = δ µ δ µ and Λ = 200 TeV (or m µµ ≈ 300 MeV), and design an analysis inspired by the Run 2 performance of the ATLAS detector [59,60]. We employ particle identification requirements on electrons, muons, and jets that are summarized in the top of Table I. 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 τ 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 give rise to missing transverse momentum E miss T , which is defined as the p T recoil against all visible objects. We therefore require that events have a small E miss T , 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 ± V backgrounds. Following past studies [61][62][63], we impose an upper limit on the ratio (H T /p µ1 T ), 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 µ1 T ) 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 ± W ± jj, pure EW production of W ± W ± jj, and the inclusive diboson+jets spectrum W ± V + nj, with V ∈ {γ * /Z/Z * }. We checked that other processes, e.g., ttW ± , do not appreciably survive our event selection. We neglect processes that are especially difficult to simulate from MC methods alone. This includes 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 [62,[64][65][66]. We account for such backgrounds with a more conservative uncertainty in our background estimate. We summarize our signal region definition in Table I. About ε ∼ 12% of generated signal events with Λ = 200 TeV pass all identification and signal region cuts. For our inputs the signal (total background) rate reaches σ ∼ 42 zb (σ ∼ 25 ab).
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 [67,68]. Assuming a δ b = 20% systematic uncertainty in the background, we estimate the sensitivity at 95% CL to |C µµ 5 |/Λ ∝ m µµ by fixing our signal significance to Z ≈ 2 and the number of signal events n s to where n 0 s is the number of signal events for our benchmark inputs. We then solve this equality for |C µµ 5 |. With L = 300 fb −1 (3 ab −1 ), we report that the LHC (HL-LHC) is sensitive at 95% CL to scales below Λ/|C µµ 5 | 8.3 (11) TeV.
With an outlook to potential successors of the HL-LHC [1, 2], we estimate the sensitivity of a √ s = 100 TeV proton collider. We employ our LHC analysis but with changes listed at the bottom of Table I. We set δ b = 5% to account for improved detector resolution and control region modeling. For L = 30 ab −1 of data, we find sensitivity to Λ/|C µµ 5 | 48 TeV at 95% CL. Our precise choice of cuts are for illustration and optimization should be investigated. This is especially relevant as we neglect an O(30%) statistical uncertainty on our W V ± simulation despite starting from 10 7 NLO+PS events.
As described above, treating the Weinberg operator as an unphysical Majorana fermion is applicable to LNviolating decays of mesons, so long as the expansion in Eq. (7) is satisfied. Using Refs. [69][70][71], we update the limits and projections on |m µµ | from B ± → π ± µ ∓ µ ∓ and K ± → π ∓ µ ± µ ± decays. We find that LHCb with L = 300 fb −1 can only probe Λ/|C µµ 5 | 9 MeV while NA-62 has excluded with its 2017 data set Λ/|C µµ 5 | 1.1 TeV. Assuming that neutrino masses are described completely at d = 5, we summarize in Fig. 4 our sensitivities to |m µµ | in comparison to the values [72] allowed by Eq. (1) (generalized for arbitrary [73]) for normal (NO) and inverse ordering (IO) of neutrino masses. The reach of W ± W ± scattering greatly exceeds our LHCb and NA-62 benchmarks. Nevertheless, improvements at these and similar experiments are anticipated.
Conclusions -If the Weinberg operator is present in nature and is accessible at collider energies, then a key prediction are processes that violate LN, such as the 0νββ transition, and possibly charged lepton flavor. Motivated by the flavor limitations of nuclear decay experiments, we have investigated the LHC's sensitivity to the Weinberg operator using the W ± W ± → ± i ± j process, which permits muon-and tau-flavored final states. We find sensitivity that exceeds representative searches at B-and K-meson factories, and establishes a complementarity across accelerator facilities in the search for the Weinberg operator. Appendix: Technical details on methodology -In this appendix we provide additional discussions and details of our methodology.

Acknowledgements
In a generic gauge, the Higgs field in terms of the EW Goldstone bosons G ±,0 is √ 2Φ = (−i √ 2G + , v + h + iG 0 ) T . Explicit contraction of SU(2) L indices then gives This allows us to express the full Weinberg operator as Here and below, the Hermitian conjugate is understood to apply to the full expression, not simply the final line. With this, the interaction Lagrangian by which we ex- tend the SM Lagrangian in the SMWeinberg UFO is To further understand the identification in Eq. (7), we recall that the fermions in the LN-violating ( + ν ν c + ) current in Fig. 1 experience an additional parity inversion beyond the standard SU(2) L chiral couplings [30,31]. In terms of Feynman rules [32,33], this manifests as a chiral inversion of the (W ν c ) vertex, i.e., γ β P L → γ β P R . In the absence of additional new physics, this ensures [34,35] the presence of the P R/L projection operators that envelope the (ν ν c ) current in Eq. (6), and hence that the (ν ν c ) current propagates in the RH helicity state.
The model's input parameters along with their Feyn-Rules and Les Houches [74] information are summarized in Table II. The syntax used to import the UFO into MadGraph5 aMC@NLO and simulate the process where q is any light quark or antiquark, at NLO 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] 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 [75]: We employ the NNPDF3.1 NLO+LUXqed parton distribution function set (lhaid=324900) [76][77][78], with scale evolution driven by LHAPDF [79], and PDF uncertainties are extracted using the replica method [78,79]. We fix the collinear factorization (µ f ), QCD renormalization (µ r ), and shower matching (µ s ) scales to the default values in Ref. [44]. The uncertainty in choosing µ f and µ r is quantified by scaling their baseline values by factors of 0.5, 1 and 2 to obtain a nine-point uncertainty band.
As a check of the SMWeinberg UFO, we consider the amplitudes for the W ± (p W 1 , λ W 1 )W ± (p W 2 , λ W 2 ) → ± (p 1 , λ 1 ) ± (p 2 , λ 1 ) process. Explicit calculation reveals that in the high-energy limit, i.e., when M 2 W W = (p W 1 + p W 2 ) 2 m 2 W , the 2 → 2 process is dominated by the scattering of two longitudinally polarized W ± bosons. For the (λ W 1 , λ W 2 ) = (0, 0) helicity configuration, the exact helicty amplitude is where r W = m 2 W /M 2 W W ; θ 1 and φ 1 are respectively the polar and azimuthal angles of (p 1 ) in the (W W )-frame, and the kinematic invariants are defined by t = (p W 1 − p 1 ) 2 and u = (p W 1 − p 2 ) 2 . Further evaluation of t and u results in the somewhat simple expression The J = 0 partial wave is subsequently given by Since s-wave perturbative unitarity requires that |a J | < 1, one obtains the constraint that After evaluating the exact helicity amplitude for each (λ W 1 , λ W 2 ) permutation, taking their sum, and then taking the high-energy limit, we obtain The Kronecker δ accounts for the 1/2! symmetry factor needed for amplitudes with identical final-state particles. This implies a totally differential cross section of Integration over the full solid angle recovers Eq. (11). Using Eq. (11) as a check of the SMWeinberg UFO, we list in Table III the total 2 → 4, hadronic cross section σ for Eq. (10) at NLO in QCD for representative cutoff scales Λ, assuming a Wilson coefficient of C 5 = δ µ δ µ , for the √ s = 13 TeV LHC and proposed experiments at √ s = 27 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 δ Scale , the parton distribution function (PDF) uncertainty δ PDF , and the QCD K-factor, which is defined as the ratio For Λ = 10 TeV and 100 TeV at √ s = 13 TeV, we observe a cross section scaling of σ(Λ = 10 TeV)/σ(Λ = 100 TeV) ∼ 93, undershooting the 100× scaling expected from Eq. (11). We attribute this to a breakdown of Eq. (7), which requires the mass m N ∼ v 2 /Λ to be small compared to the virtuality of (ν ν c ). At larger Λ we find, for example, that σ(Λ = 100 TeV)/σ(Λ = 200 TeV) ∼ 3.93 and σ(Λ = 200 TeV)/σ(Λ = 400 TeV) ∼ 3.99, indicating behavior more inline with Eq. (11). We conclude that choices of Λ 200 TeV generate sufficiently small m N so that Eq. (7) remains valid for √ s 13 TeV. For benchmark signal inputs of Λ = 200 TeV and C 5 = δ µ δ µ , the expected number of background and 0νββ signal events in the signal region with L = 300 fb −1 (3 ab −1 ).