Probes of the Standard Model effective field theory extended with a right-handed neutrino

If neutrinos are Dirac particles and, as suggested by the so far null LHC results, any new physics lies at energies well above the electroweak scale, the Standard Model effective field theory has to be extended with operators involving the right-handed neutrinos. In this paper, we study this effective field theory and set constraints on the different dimension-six interactions. To that aim, we use LHC searches for associated production of light (and tau) leptons with missing energy, monojet searches, as well as pion and tau decays. Our bounds are generally above the TeV for order one couplings. One particular exception is given by operators involving top quarks. These provide new signals in top decays not yet studied at colliders. Thus, we also design an LHC analysis to explore these signatures in the $t\overline{t}$ production. Our results are also valid if the right-handed neutrinos are Majorana and long-lived.


Introduction
Despite its impressive success in describing particle physics phenomena across a wide range of energies, the Standard Model (SM) is known to be incomplete. The main experimental indication of new physics is the non-vanishing neutrino masses. On the theory side, the hierarchy problem, the flavour puzzle and the unification of couplings among others, also suggest the existence of particles beyond the SM. In light of the null results at the Large Hadron Collider (LHC), new particles might lie at energies, Λ, well above the electroweak (EW) scale, established by the Higgs vacuum expectation value (VEV), v ∼ 246 GeV.
For such scenarios, new physics can be described by the SM effective field theory (SMEFT) [1]. This theory extends the renormalisable SM Lagrangian with a tower of effective operators respecting the SM gauge symmetries, but not necessarily the global (accidental) ones. Within this framework, neutrinos are predicted to be Majorana, their mass being m ν ∼ α 5 v 2 /Λ with α 5 being the Wilson coefficient of the Weinberg operator [2]. Up to flavour indices, the latter can be considered to be the only independent effective operator of dimension five. At dimension six, this number goes up to 59 [3], assuming baryon number conservation. However, if neutrinos are Dirac particles, the SMEFT has to be extended with effective operators containing the right-handed (RH) neutrino, N .
This extended EFT, known as νSMEFT, was first considered in Ref. [4], where also a complete set of operators in the νSMEFT to dimension six was worked out. (See also Ref. [5] for dimension five and Ref. [6] for dimension seven.) This set was shown to be redundant in Ref. [7], where an actual basis was provided. It is shown in Tab. 1 for completeness. In this paper, we set constraints on these operators upon using searches for one lepton (e, µ, τ ) and missing energy at the LHC, monojet searches at the LHC as well as pion and tau decays. For those operators not yet constrained by these observables, most of them involving top quarks, we suggest a novel search strategy based on a new rare top decay at the LHC.
A comment about cosmological constraints is in order. It is well known that in the absence of any other interactions, sterile neutrinos do not contribute to the number of relativistic species, N eff . This is no longer true in the νSMEFT with a low cutoff Λ. Indeed, the interaction rate for producing RH neutrinos out of the thermal bath behaves as Γ ∼ T 5 /Λ 4 . The latter are in thermal equilibrium at the time of the Big Bang Nucleosynthesis (BBN) provided Γ T 2 BBN /m P , with m P being the Planck mass and T BBN ∼ MeV; namely if Λ 200 GeV ∼ v. (There is no experimental evidence of the pre BBN era -some models even require reheating right before T ∼ MeV [8,9] -so there are no actual cosmological constraints on the νSMEFT if N decouples before BBN.) Finally, N could be also a very light Majorana neutrino with mass m N . Even in the standard cosmological history, the contribution of such particle to N eff is below the current limit from Planck, ∆N eff 0.3 [10], provided m N 10 MeV [11]. Our analysis still applies to this case if the lifetime of N , τ N ∼ 256 π 3 Λ 4 /m 5 N is larger than the scales relevant for the experiments considered in this work. Namely, for m N 10 −3 Λ 4/5 . Thus, for Λ v, N behaves also as a stable particle if m N 0.1 GeV. In summary, our study works within the νSMEFT if the cutoff is above the EW scale and provided neutrinos are Dirac particles, or Majorana particles with 0.01 GeV m N 0.1 GeV; the upper limit being significantly larger if the EFT starts being valid only above the TeV.
This article is organised as follows. In section 2 we introduce the formalism of the νSMEFT. We discuss which operators contribute to each of the observables considered in this work and compute the relevant equations. In section 3 we compare the obtained results with the experiments, and set global constraints on the (four-fermion) operators of the νSMEFT. We emphasise that some of them are actually not bounded by current data, and thus develop a new collider search designed for the HL-LHC in section 4. We conclude in section 5.

Formalism
The basis of dimension-six operators containing N and respecting lepton and baryon numbers is shown in Tab. 1. In the following subsections, we compute different cross sections and decay widths involving N in the final state. Therefore, up to corrections proportional to the neutrino masses, the νSMEFT operators mediating the corresponding processes do not interfere with their SMEFT counterparts. As a result, any bounds computed on the νSMEFT Wilson coefficients by using solely our equations are conservative. Hereafter,

Operator Notation
Operator Notation we assume all these coefficients to be real. For simplicity, we also restrict to only one N family; expressions for the case of several N fields can be trivially obtained from our results. Moreover, we explicitly do not consider the operators involving flavour violation between the first and second fermion families. We focus mostly on observables sensitive to the four-fermion operators in the classes RRRR, LLRR and LRRL, where L (R) denotes left (right) handed fermions. These are typically more relevant because they can be generated at tree level in UV completions of the SM. Moreover, operators such as O N B and O N W are very much constrained by measurements of neutrino dipole moments when the latter are Dirac [12,13]. In the Majorana case they must be suppressed in order for N to be long-lived; otherwise it would decay rather promptly into two body final states [14,15]. In such case, the collider signals are very different from the ones considered in this article and related works [16][17][18]; they will be presented elsewhere. We only include the contribution of the SF operators (see Tab. 1) in the relevant equations hereafter for the sake of completeness.

Searches for one lepton and missing energy at the LHC
where i = 1 (2) for electron (muon). The imprint of these operators in searches for + / E T can be better observed in the tail of the distribution of the transverse mass of the lepton because, contrary to the SM background, the cross section of the signal grows with the energy due to the absence of propagator suppression.
A number of searches have been carried out at the LHC in this regard. Here, we focus on the ATLAS study of Ref. [24], based on 36 fb −1 of data collected at √ s = 13 TeV.
The main selection criteria are the requirement of exactly one light lepton with p T > 55 GeV (65 GeV) in the muon (electron) channel; and likewise for / E T . The events are subsequently categorised according to the variable m T , with m 2 . The numbers of predicted SM events and of observed events in four m T regions as provided in Ref. [24] are shown in Tab. 2. For convenience, we also show the maximum number of signal events, s max , in each bin separately. We have obtained them using the CL s method [25] including the quoted uncertainties.
Following Eq. 2.1, the number of events in the signal side in any of the bins can be expressed as: where A 1 , A 2 , · · · , A 5 are bin-dependent coefficients to be determined by simulation after recasting the experimental analysis. Our findings are also reported in Tab   bins; see Tab. 3 which, as in the previous case, also reports the number of predicted SM events, the observed number of events and s max , as well as the values of the different coefficients (this time dubbed B) as obtained from simulation.
In principle, the same operators but involving second generation quarks also contribute (without interfering with the previous ones) to this process. However, the corresponding A and B coefficients are expected to be a factor of ∼ 10 smaller due to the smaller parton distribution functions (PDFs).  Table 4: Coefficients in TeV 4 and rounded to two significant figures for pp → N N g(q) obtained upon recasting the experimental analysis of Ref. [28] for L = 35.9 fb −1 .

Monojet searches at the LHC
Four-fermion operators containing two light quarks and two RH neutrinos, i.e. O uN , O dN , O qN , can lead to monojet searches at the LHC if for example a gluon is emitted from one of the initial quarks in pp collisions. They also modify the π 0 width but chirality suppressed. Furthermore, this width is experimentally known to be sensibly large [27]. Different monojet searches have been performed by ATLAS and CMS. In this work, we consider the CMS analysis of Ref. [28], based on 35.9 fb −1 of data collected at √ s = 13 TeV. The main cuts of this study are missing transverse energy above 250 GeV, at least one hard jet with p T > 250 GeV and no isolated leptons. The analysis defines different signal regions depending on the value of / E T ; see Tab. 4. The interference between the three aforementioned operators is chirality suppressed. Therefore, in very good approximation, one can estimate the number of signal events in any region of the analysis as with C 1 , C 2 , C 3 depending again on the signal region. Note that in this case we are neglecting the contribution of the operator O HN , which in principle interferes with the previous ones, because it does not grow at large energies. At any rate, as commented previously, we focus on scenarios with only four-fermion operators when setting bounds in section 3. Likewise, other four-fermion operators contribute to the monojet channel via νN . They of course do not interfere with the ones in Eq. 2.3. Besides, they are constrained by other observables, so we conservatively neglect them.

Pion decays
The operators modifying the tail of + / E T in pp collisions can also enhance the pion decays, provided N is the actual RH component of a Dirac neutrino. (Also if N is Majorana with low enough mass.) For the computation of the corresponding matrix elements, we neglect O i11 ldqN , whose tensor form factor is hard to estimate. We obtain where p = p i + p N . A similar expression holds for π + → + i N . Following Ref. [29], we assume the expectation values of the quark currents to be determined by current algebra: 0|V µ |π ± = f π p µ and 0|S|π ± = f π m 2 π /(m u + m d ) with f π ∼ 131 MeV. Then, the corresponding decay width reads (2.5) The measured values of the pion decay widths into electrons and muons read Γ(π → e + inv) = (310 ± 1) × 10 −23 GeV and Γ(π → µ + inv) = (25279 ± 5) × 10 −21 GeV, respectively [27]. We bound the coefficients entering this equation by requiring that the corresponding contribution is smaller than twice the experimental error.

Tau decays
The following operators give contributions to τ → + inv: When there are more than one term, only those in boldface are relevant. These operators lead to the following decay widths of the tau lepton: The measured values of the tau decay widths into electrons and muons are Γ(τ → e + inv) = (4.03 ± 0.02) × 10 −13 GeV and Γ(τ → µ + inv) = (3.93 ± 0.02) × 10 −13 GeV, respectively [27]. Following the same procedure as in the pion case, we bound these operators by requiring that the corresponding theoretical decay width is not larger than twice the experimental error.

Top decays
Finally, the following four-fermion operators contribute to flavour-conserving top decays: When there are more than one term, only those in boldface are relevant. The corresponding decay width reads and similarly for t → b − i N . In this expression, we have neglected the contribution of O HN e and O N W . The reason is that in those cases the top decay is approximately two body, followed by the leptonic decay of an on-shell W . This does not only allow to disentangle the two contributions, but the interference is also very small. Similar results have been pointed out in the SMEFT; see e.g. Ref. [30].
These operators can be mostly probed only in top decays. In particular, O duN e does not provide any other interaction. This is in contrast with analogous operators in the SMEFT, such as for example ∼ (t R γ µ t R )(l L γ µ l L ), because the left-handed (LH) neutrinos always come along with charged leptons. These operators are therefore better tested at lepton facilities; see e.g. Ref. [31].
This discussion applies also to the flavour-violating top operators: The terms in boldface lead to the rare top decays t → j + inv (where j stands for jet); again without charged lepton counterpart unlike the SMEFT [32,33]. The corresponding decay widths read: Similar expressions hold of course for second generation quarks. The sensitivity of measurements at colliders is expected to be similar in both cases, just as in other flavourviolating top decays [33][34][35][36][37][38].
Current bounds on the top width are not constraining enough in any of these cases, though. Instead, dedicated analyses at colliders are to be performed to bound these operators. In section 4 we develop one such analysis for the flavour-conserving top decay at the LHC. The flavour-violating processes involve a light quark and two sources of missing energy, making these much appropriate search channels at future lepton colliders, in which all the three components of the missing momentum are measured.

Global constraints
Taking into account the observables computed in Eqs. 2.2 (for light leptons and taus; see Tabs. 2 and 3, respectively), 2.3, 2.5 and 2.18-2.22, we can set constraints on the different operators of Tab. 1. The bounds are given by the non-boldfaced numbers in Tabs. 5-7.
When relying on high energy searches at colliders, we derive the bounds on the operator coefficients from the several bins for each of the analyses and keep the bounds yielding the strongest result. However, we also ensure that we do not go very high in energies in order to abide by EFT validities. Specifically, we remain within energy bins of less than 1 TeV.
Deriving the bounds on α 11 uN , α 11 dN and α 11 qN is particularly simple as they enter the analytical expression for the number of events, for the monojet analysis, without any interference. In this case, the / E T bin of [740 − 790] GeV yields the strongest limits. For the + / E T analysis, it is straightforward to constrain α 11i quN l and α 11i duN e as they do not interfere with any operator in the expression for the number of events. For the operators that do interfere, viz. O i11 lN qd and O i11 ldqN , we marginalise over each coefficient to set a bound on the second one. Thus, in order to constrain α i11 ldqN , we fix α i11 lN qd to the value that minimises the number of events for each value of α i11 ldqN , from where the procedure of setting bounds reduces to the one dimension as in the previous cases. The reverse process can be followed to constrain α i11 lN qd . The bounds obtained this way must be taken if N is a Majorana neutrino with mass above m π . For Dirac neutrinos, we bound α i11 lN qd instead by using the pion decay width in Eq. 2.5 (neglecting the m i piece). The corresponding limit is a factor of 1.9 (1.8) more stringent for electrons (muons) than the LHC counterpart. This is the constraint we show in tables. Operators involving electrons are more constrained than those involving muons due to the larger sensitivity of the analysis of Ref. [24] to electrons. Also, the pion decay width to µ + inv is significantly larger than that for e + inv, therefore leaving more space for new physics.
The τ + / E T follows a similar technique and helps us bound α 113 quN l , α 113 duN e , α 311 ldqN and α 311 lN qd . We use the respective bins of [600 − 1000] GeV in m T for the + / E T analysis and [500 − 1000] GeV for the τ + / E T case. When the aforementioned operators are considered with second generation quarks instead of the first family, the bounds get weaker by a factor of ∼ 3. Thus, without relying on flavour observables, one can estimate the bounds on operators involving transitions between the first and second quark families to be those  In this case, muonic and electronic operators get more or less equally constrained. Similar operators but involving transitions between the first and second lepton families could be constrained using muon decays. Naively, the bounds would be a factor of 10 − 100 stronger. However, the Fermi constant would be also redefined on account of the Wilson coefficients, making the process of bounding these operators more subtle.
Among those operators not constrained by the observables considered in this paper, we find O N N , which only involves RH neutrinos. We also have O ii eN and O ii lN ; these could be tested in monophoton searches at lepton colliders. Third generation flavour conserving O 33 dN , O 33 uN and O 33 qN could be tested in searches for bb + / E T and tt + / E T . Most of the rest of operators induce new top decays, either into b + inv, bτ + inv or j + inv. The latter one is hard to probe at hadron colliders, because it involves two sources of missing energy and only light jets. (Moreover, is flavour-violating.) The second one involves a tau lepton, making this signal less promising than the first one. The first signal gives rise to a final state which is identical to the SM leptonic top decay (which complicates its study), but interestingly in this case the lepton and the missing energy do not reconstruct a W boson. This provides a completely new signal not yet explored experimentally, for which we design a completely novel search strategy. For completeness, we advance the prospective bounds on the corresponding operators by the boldfaced numbers in Tabs. 5 and 7.  Table 6: Maximum (minimum) value of α (Λ) for Λ = 1 TeV (α = 1) allowed by the observables quoted in the last column for LLRR operators. The numbers outside (inside) the parentheses refer to = e (µ).

t → b N at the LHC
We can constrain the operators O duN e , O dqN , O N qd and O qu N in tt production with one of the tops decaying exactly as in the SM in the hadronic mode and the other decaying leptonically through the modified vertex. We focus on the high luminosity run of the LHC at 14 TeV with an integrated luminosity of 3 ab −1 . The final state consists of bb / E T + jets. (We perform our analysis only in the muonic final state. We assume the results not to be significantly different for the case of electrons.) The dominant background to this process comes from the SM semi-leptonic tt channel. It is a very challenging task to reduce the background, as the topology is essentially the same in both the signal and the background. In particular, the hadronic top decays in the exact same manner in both cases.
Our analysis strategy starts by generating the signal and background events at √ s = 14 TeV using the NNPDF23LO PDF set [39]. We generate our samples at the leading order and then multiply by a flat next-to-next-to leading order (NNLO) + next-to-next-to leading logarithm (NNLL) k-factor of ∼ 1.63 corresponding to m t = 172.5 GeV and for the central value of the cross section. The central value of the 14 TeV NNLO+NNLL cross section is 984.50 pb [40]. At the generation level, we demand p T (j, b, ) > 15, 15, 10 GeV, |η(j, b, )| < 5, 3, 3 and ∆R(jj, bb, bj, j , b ) > 0.3 and ∆R( ) > 0.2. We shower the events using Pythia v8 [20,21]. We construct jets upon using the anti-k t algorithm [41] with a jet parameter R = 0.4 in the FastJet v3 framework [22]. We require the jets to have p T > 30 GeV and the b-tagged jets to have |η| < 2.5. The leptons are required to have p T > 10 GeV and |η| < 2.5. The b-tagged jets are constructed by requiring the B-meson tracks to be within ∆R = 0.2 of a jet. Moreover, we implement a flat b-tagging efficiency of 70%. In our simplified analysis, we also consider c quarks and light quarks faking a b-jet with probabilities of 10% and 1%, respectively. Finally, isolated leptons are defined by requiring that the hadronic activity around ∆R = 0.2 of the corresponding lepton is smaller than 10% of its p T . Subsequently, we demand exactly two b-tagged jets, one isolated lepton and at least two light jets. In order to ascertain proper top-mass reconstruction, we build two hadronic top mass variables (m t had,1 and m t had,2 ). We include the two hardest light jets (j 1,2 ) when reconstructing both m t had,1 and m t had,2 . However, we include the harder (softer) b-tagged jet to reconstruct m t had,1 (m t had,2 ). For each event we check which of these reconstructed hadronic top masses is closest to the actual top mass, that we take to be m t = 172.5 GeV. The b-jet giving the poorer reconstruction is assigned to the leptonic top. Next, we demand that the best reconstructed hadronic top (m had t ) and the hadronic W (m had W ), reconstructed out of the two hardest light jets, lie respectively within 40 GeV and 30 GeV of the top and W masses (with m W taken as 80.385 GeV).
We reconstruct the p z of the neutrino by solving for the leptonic top mass. There is a two-fold ambiguity in this process, which results into two values for the momentum of the ν system, corresponding to the "+" and "−" solutions of the quadratic equation. We denote these two solutions by W 1 and W 2 , respectively. (Note, however, that in the signal they do not need to be close to the W mass.) We only select those events where the solution to the quadratic equation for the top mass has a positive discriminant. The aforementioned trigger cuts, isolation cuts and analysis cuts have the respective efficiencies of 4.4% and 4.7% for the signal and the SM background.
The distribution of m  and-count analysis. We therefore perform a simple boosted decision tree (BDT) analysis in the TMVA framework [42]. We include the following observables: the four-momenta of the lepton and the b-tagged jet best reconstructing the leptonic top, the two components of the transverse missing momentum, m had W , m ,1 W , m ,2 W , m had t , / E T , ∆R b,b , ∆R b ,j 1/2 , ∆R ,b h and ∆R ,j 1/2 , where b (b h ) is the b-jet associated with the leptonic (hadronic) top. We ensure that there is no overtraining by checking that the Kolmogorov-Smirnov statistic TeV and L = 10 ab −1 (right).
for neither the signal nor the background falls below the critical value of 0.01 [43]. As expected, m ,1 W and m ,2 W serve as the best discriminating variables. The other variables do not show significant prowess in the discriminating procedure.
We get a statistical significance of ∼ 13 for BR(t → b N ) ∼ 2×10 −4 . However, it drops dramatically as soon as systematic uncertainties are included, due to the very small S/B ∼ 0.41%. The asymmetry between the signal and the background in the BDT variable (see Fig. 2) can however be used to better control the systematic uncertainties. To this aim, we slide the bins from left to right in the corresponding distribution, and construct the variable A = (N right − N lef t )/(N right + N lef t ), where N lef t (N right ) denotes the number of events to the left (right) of a chosen bin including the bin itself. Systematic uncertainties cancel in this ratio. The number of standard deviations between the asymmetry in the signal + background and the SM alone is shown in Fig. 3 as a function of BR(t → b N ). It can be seen that branching ratios as low as ∼ 2 × 10 −4 can be tested at the LHC in the long run. Scaling with the larger tt cross section, we also estimate the reach at √ s = 27 TeV with L = 10 ab −1 , that improves by a factor of ∼ 4. The projected 95% C.L. bounds on the various couplings entering this rare decay, as described in Eq. 2.27, are shown in Tabs. 5 and 7 for the HL-LHC analysis.
In summary, we find that the BDT combined with the asymmetry variable helps in constraining these EFT couplings. More studies are required in this direction and a control over systematics is warranted.

Conclusions
By using LHC searches for light leptons or taus with missing energy, monojet analyses, and measurements of different pion and tau decay widths, we have singled out the most constrained directions in the νSMEFT and, as a consequence, those others in which new physics can be more likely hiding. (Our results are valid for a cutoff above the EW scale provided N is the RH component of the SM Dirac neutrino or a new Majorana particle with 0.01 GeV m N 0.1 GeV.) In the first category, our limits range from ∼ 380 (1400) GeV to ∼ 3 (11) TeV for couplings of order ∼ 1 (4π); corresponding to leptonic operators with taus and first generation quark-lepton operators, respectively. In the category of unconstrained operators, we find interactions involving heavier quarks, as well as operators that could be better probed at lepton facilities. However, the operators giving genuinely new signals are those providing new top decays. In particular, the operators O duN e , O ldqN , O lN qd and O quN l trigger the process t → b +inv, with the missing energy and the lepton not reconstructing a W boson. This not yet explored process is the only sensible signature of these operators. We have worked out a BDT analysis sensitive to this decay channel in semileptonic tt production relying mostly on the invariant mass of the lepton and the missing energy; obtained upon requiring the top decay products to reconstruct the top mass.
The BDT variable turns out to be sensibly shifted in the signal with respect to the background. In order to minimise systematic uncertainties, we have based the statistical analysis on an asymmetry built out of the BDT variable. Considering one single lepton family, we have shown that, at the LHC with √ s = 14 (27) TeV and 3 (10) ab −1 of collected luminosity, branching ratios of the top quark into this exotic decay mode as small as ∼ 2 × 10 −4 (∼ 5 × 10 −5 ) can be probed at the 95 % C.L. This translates into a prospective lower bound on Λ ∼ 330 (460) GeV for α ∼ 1; Λ ∼ 1.2 (1.6) TeV for α ∼ 4π. These numbers can rise up to Λ ∼ 1.8 (2.5) TeV in the strongly interacting limit if both electrons and muons as well as three RH neutrinos are present. Finally, let us emphasise that the aforementioned results are also useful to constrain certain operators in the usual SMEFT. Indeed, although SMEFT operators contributing to t → b + inv typically also induce quark-charged lepton interactions (because LH neutrinos share SU (2) L doublet with the charged leptons; and in this respect the νSMEFT is qualitatively different), some directions can be mostly constrained upon exploiting the new top decay we have just studied. For example, let us expand the operator O = (l iL γ µ l iL )(q 3L γ µ q 3L ) + (l iL γ µ σ I l iL )(q 3L γ µ σ I q 3L ) = (ν iL γ µ ν iL )(t L γ µ t L ) + ( iL γ µ iL )(b L γ µ b L ) + ( iL γ µ ν iL )(t L γ µ b L ) + h.c. . (5.1) The non-boldface interactions are very hard to probe at hadron colliders. At lepton facilities, one could study the e + e − → bb, but any departure from the SM could be attributed to e.g. RRRR operators, unless the rare top decay is also tagged.
Note added. During the final stages of this work, Ref. [44] appeared, in which the bounds on some operators coming from neutrino experiments are derived.