Phenomenology of TeV-scale scalar Leptoquarks in the EFT

We examine new aspects of leptoquark (LQ) phenomenology using effective field theory (EFT). We construct a complete set of leading effective operators involving SU(2) singlets scalar LQ and the SM fields up to dimension six. We show that, while the renormalizable LQ-lepton-quark interaction Lagrangian can address the persistent hints for physics beyond the Standard Model in the B-decays $\bar B \to D^{(*)} \tau \bar\nu$, $\bar B \to \bar K \ell^+ \ell^-$ and in the measured anomalous magnetic moment of the muon, the LQ higher dimensional effective operators may lead to new interesting effects associated with lepton number violation. These include the generation of one-loop sub-eV Majorana neutrino masses, mediation of neutrinoless double-$\beta$ decay and novel LQ collider signals. For the latter, we focus on 3rd generation LQ ($\phi_3$) in a framework with an approximate $Z_3$ generation symmetry, and show that one class of the dimension five LQ operators may give rise to a striking asymmetric same-charge $\phi_3 \phi_3$ pair-production signal, which leads to low background same-sign leptons signals at the LHC. For example, with $M_{\phi_3} \sim 1$ TeV and a new physics scale of $\Lambda \sim 5$ TeV, we expect about $5000$ positively charged $\tau^+ \tau^+$ events via $pp \to \phi_3 \phi_3 \to \tau^+ \tau^+ + 2 \cdot j_b$ ($j_b$=b-jet) at the 13 TeV LHC with an integrated luminosity of 300 fb$^{-1}$. It is interesting to note that, in the LQ EFT framework, the expected same-sign lepton signals have a rate which is several times larger than the QCD LQ-mediated opposite-sign leptons signals, $gg,q \bar q \to \phi_3 \phi_3^* \to \ell^+ \ell^- +X$. We also consider the same-sign charged lepton signals in the LQ EFT framework at higher energy hadron colliders such as a 27 TeV HE-LHC and a 100 TeV FCC-hh.


I. INTRODUCTION
The electroweak (EW) and strong interactions of the SM have been very successfully tested at the low-energy (GeV-scale) and high-energy (EW-scale) frontiers as well as in precision measurements [1]. However, despite the impressive success of the SM at sub-TeV energies, it is widely believed that it is an effective low-energy framework of a more complete UV theory that should address the experimental and theoretical indications for new physics beyond the SM (BSM), such as the indirect detection of dark matter and dark energy, the measurements of neutrino masses, the flavor and hierarchy problems residing in the SM's scalar sector and the long sought higher symmetry which unifies the fundamental forces.
While the scale of the new physics (NP) that may shed light on these fundamental questions in particle physics and address the deficiencies of the SM might be beyond the reach of present and future high-energy colliders, the mass scale of the particle content of the desired UV theory may span over many orders of magnitudes, similar to the hierarchical mass pattern observed in nature and embedded in the SM. Indeed, although direct searches at high-energy colliders have not yet led to a discovery of new heavy particles, in the past several years there have been intriguing and persistent hints of new TeVscale degrees of freedom from anomalies associated with possible violations of lepton universality in B-decays: B → D ( * ) τν [2][3][4] andB →K + − [5], as well as in the anomalous magnetic moment of the muon [6].
Out of these three anomalies, the most striking and least expected is the anomalous enhancedB → D ( * ) τν rate measured by BaBar [2], Belle [3] and LHCb [4] (a ∼ 4σ effect). In the SM this decay occurs at tree-level and is mediated by the W cb charged current coupling, so that the measured deviation requires a relatively large tree-level NP contribution near the TeV scale to compete with the "classic" SM tree-level diagram. Promising candidates that address this large effect inB → D ( * ) τν are TeV-scale leptoquarks (LQ's); in addition to this phenomenological role, these particles also appear naturally in theories that address some of the most fundamental questions in particle physics (see [7] and references therein) such as grand unification [8] and compositeness [9], where they can also arise as pseudo-Nambu Goldstone bosons [10] and lead to interesting collider signals [11,12]. They are also involved in models for neutrino masses [13]. In some cases, the effects of scalar LQ are similar to that of the scalar partners of the quarks in R-parity violating supersymmetry models [14,15], which can have similar couplings to quark-lepton pairs.
Given their theoretical appeal, and their potential role in addressing the B anomalies, it is of interest to study LQ phenomenology within the context of BSM physics. That is, allowing for the presence of excitations heavier than the LQs. This we shall do using an effective theory, which will include the LQs as (relatively) low-energy excitations, and the effective interactions generated by heavier physics of scale Λ. We will see that LQ effective interactions produce unique collider signatures that are observable at the LHC, and, in some cases, at rates that are higher than for the usual channels. We will also see that the physics at scale Λ, responsible for the effective LQ interactions, is also intimately connected with various possible mechanism of neutrino mass generation, so that a study of LQ phenomenology at the LHC can provide also information about the neutrino sector. In this work we will concentrate on the study of the interactions and phenomenology of TeV-scale scalar LQs, which are SU(2) singlets and transform either as a righthanded down-type quark, 1 φ(3, 1, − 1 3 ), or as a righthanded up-type quark, φ(3, 1, 2 3 ), under the SM gauge group; since the BSM effects of both types of LQ have similar characteristics, in the bulk of the paper we will explore the effects and underlying physics of the down-type LQ, and towards the end of the paper we will shortly address the underlying physics and effects that are expected for an up-type LQ.
We construct the complete set of effective operators up to dimension six that involve the LQs and SM fields, and use this LQ EFT framework to demonstrate the impact of heavy physics on φ collider phenomenology, and on low-energy lepton number violating phenomena such as Majorana neutrino masses and neutrino-less double beta decay. This model-independent formalism pro-vides a broader and a more reliable view of the expected physics associated with TeV-scale LQs, and lays the ground for further investigations of φ-related phenomenology at high-energy colliders. For example, we find that the higher dimensional LQ interactions in the EFT framework may lead to very interesting, essentially background free, same-sign lepton signals at the LHC and/or at future colliders.
The paper is organized as follows: in the following section we summarize the effects of the renormalizable LQ interaction Lagrangian L φSM ; in section III we review the LHC phenomenology of the scalar LQ in the φSM framework and in section IV we construct the effective theory beyond L φSM , listing all the higher-dimensional effective operators involving the down-type LQ φ(3, 1, − 1 3 ) up to dimension six. In section V we study the ∆L = 2 lowenergy effects associated with the dimension five operators and in section VI we explore the leading signals of the down-type and up-type LQ, φ(3, 1, − 1 3 ) and φ(3, 1, 2 3 ), in the EFT framework at the 13 TeV LHC as well as at higher energy (27 and 100 TeV) hadron colliders. In Section VII we summarize and in the appendix we list all dimension six operators for the down-type LQ.

II. RENORMALIZABLE LQ INTERACTIONS
We define the renormalizable extension of the SM which contains the LQ as: where, for the down-type LQ φ(3, 1, − 1 3 ), the Yukawa-like and scalar interaction pieces are: with q and the SU(2) left-handed quark and lepton doublets, respectively, while u, d, e are the right-handed SU(2) singlets; also, ψ c = Cψ T . A few comments are in order regarding the φSM Lagrangian defined in Eqs. 1-3: • The last two Yukawa-like φ-quark-quark terms of L Y,φ in Eq. 2 violate Baryon number and can potentially mediate proton decay (see e.g., [16]). The Yukawa-like LQ couplings involving the 1st and 2nd 1 In our notation X(c, w, y), indicates that particle X transforms under SU(3) representation c, SU(2) dimension w and carries hypercharge y.
generations are then either vanishingly small (i.e., (y L qq ) ij , (y R ud ) ij → 0 for i, j = 3) or are forbidden, e.g., by means of a symmetry.
• The LQ -Higgs interaction term ∝ λ φH in Eq. 3 may play an important role in stabilizing the EW vacuum [29].
• As will be discussed below, within the renormalizable φSM framework, L φSM , LQ phenomenology and leading signals at the LHC are completely determined by the two Yukawa-like parameters y L q , y R ue and the LQ mass M φ (ignoring the baryon number violating couplings).
The search for LQ is then performed assuming two distinct LQ decay channels that correspond to its two Yukawa-like interactions in the φSM : φ → e i j and φ → νj, with Γ(φ → e i j/νj) ∼ |y| 2 m φ /16π, where y is the corresponding φ-lepton-quark coupling, and the quark and lepton masses are neglected. Thus, the overall LQ signatures at the LHC contain either two leptons and two jets with large transverse momentum, e + i e − j jj and/or e i jj + missing E T , when the LQ are pair-produced [39][40][41][42][43][44][45], or two leptons and a jet with large transverse momentum, e + i e − j j and e i j + missing E T , when the LQ is singly produced.
Indeed, searches for 1st and 2nd generations LQ pairproduction (i.e. for LQ with couplings only to quarklepton pairs of the 1st and 2nd generations) yield stronger bounds than the ones for 3rd generation LQ, since the detector sensitivity to the different flavors of high-p T leptons and quarks varies. In addition, these bounds strongly depend on the LQ decay pattern, i.e., branching ratios to the different quark-lepton pairs. For example, the current bounds on the mass of a 1st(2nd) generation LQ assuming pp → φφ * → e + e − /µ + µ − + jj and BR(φ → e/µ + j) ∼ 1 is M φ > ∼ 1.5 TeV [41,45]. Third generation LQ are particularly motivated, due to their potential role in explaining the observed anomalies in B-physics discussed above, but also on more general aspects concerning the underlying UV physics, e.g., the dynamical generation of fermion masses in composite scenarios [46]. Recent searches for a pair-produced 3rd generation scalar LQ, decaying via φ → tτ, bν τ and/or φ → bτ , have yielded weaker bounds: M φ > ∼ 1 TeV [39,40,[42][43][44]47]. On the other hand, the bound on the mass of a φ(3, 1, − 1 3 ) that couples exclusively to a top-muon pair (and can, therefore, address the anomalous muon magnetic moment and the anomaly measured inB →K + − ), obtained in the search for pp → φφ * → ttµ + µ − , is M φ > ∼ 1.4 TeV [43], i.e., comparable to the lower limit on the mass of a 1st and 2nd generation LQ.
Finally, a search for a singly produced 3rd generation scalar LQ which decays exclusively via φ → bτ has also been performed recently by CMS; they exclude such a LQ up to a mass of 740 GeV [48].

IV. EFT BEYOND THE φSM FRAMEWORK
In this section we focus on the EFT extension of the renormalizable Lagrangian in Eqs. 1-3, for the down-type LQ φ(3, 1, − 1 3 ). The effects of the NP which underlies the φSM framework in Eqs. 1-3 can be parameterized by a series of effective operators O i , which are constructed using the φSM fields and whose coefficients are suppressed by inverse powers of the NP scale Λ, where n is the mass dimension of O (n) i and we assume decoupling and weakly-coupled heavy NP, so that n equals the canonical dimension. The dominating NP effects are then expected to be generated by contributing operators with the lowest dimension (n value) that can be generated at tree-level in the underlying theory.
Before listing the specific form of the higher dimension operators, O (n) i , it is useful to denote their generic structure in the form where a, b, c, d are integers representing the multiplicity of the corresponding factors: O fields -both violating lepton number by two units. To see that, note that the dimension-five operators with c = 0 in Eq. 5 are all absent because of gauge invariance. Furthermore, operators of the form φ 2 ψ 2 must contain the fermion bilinearψ L ψ R , so that only a single gauge invariant dimension five operator of this form survives (with two possible SU(3) color contractions which are not specified): which violates lepton number by two units. The diagrams that can generate the dimension five operatordd c φ 2 at tree-level in the underlying heavy theory are depicted in Fig. 1; the corresponding heavy NP must contain a heavy scalar Φ(6, 1, − 2 3 ) and/or the heavy fermions Ψ(1, 1, 0), Ψ(8, 1, 0).
Dimension five operators of the class φψ 2 D can be shown to be equivalent to operators without a derivative using integration by parts and, therefore, can be ignored. Thus, the remaining class of dimension five operators is of the form φψ 2 H and, therefore, must also contain the fermion bilinearψ L ψ R . The only gauge invariant operator of this form, which also violates lepton number by two units is: The heavy physics generating this operator at treelevel must contain a heavy scalar Φ(3, 2, 1 6 ) and/or the heavy fermions Ψ(1, 1, 0), ψ(1, 3, 0) or Ψ(3, 2, − 5 6 ), see Fig. 2.
We recall that there is also a unique dimension five operator that can be constructed using the SM fields only; the so called Weinberg operator [49]: that can be generated in the underlying theory at treelevel by an exchange of a heavy scalar Φ(1, 3, 0) and/or the heavy fermions Ψ(1, 1, 0), Ψ(1, 3, 0). Therefore, the overall dimension five effective operator extension of L φSM is: where we have kept a general notation assigning each of these operators their own effective scale. Note, for example, that the heavy fermionic state Ψ(1, 1, 0) can generate all three dimension five operators in Eq. 9, in which case they will have a common scale. On the other hand, as we will see below, the Weinberg operator¯ cH H † and the operator¯ dHφ * generate Majorana masses for the SM neutrinos, so their effective scale, ∼ f /Λ, must be considerably suppressed in order to obtain sub-eV masses (we ignore the possibility of cancellations between these two contributions because of the extreme fine tuning this would require). This leaves us with a single viable dimension five operator,dd c φ 2 , whose scale can be low enough for it to be relevant for collider LQ phenomenology.
In the appendix we construct the complete set of the dimension six operators involving the down-type scalar LQ φ(3, 1, − 1 3 ) and the SM fields. 2

V. THE DIMENSION FIVE OPERATORS AND LOW ENERGY ∆L = 2 EFFECTS
As mentioned earlier, while the φSM renormalizable interaction Lagrangian, L φSM , can address the BSM effects associated with the current B-physics anomalies, other aspects of NP associated with lepton number violation require new higher-dimensional effective interactions of the LQ with the SM fields. In particular, the dimension five operators in Eq. 9 violate lepton number by two units and can, therefore, generate Majorana neutrino masses, mediate neutrinoless double beta decay and also give rise to interesting same-sign lepton signals at the LHC.
In this section we investigate in more detail the low energy ∆L = 2 effects associated with these operators, while in the next section we discuss the potential ∆L = 2 collider signals.

A. Majorana Neutrino masses
As is well known, the dimension five Weinberg oper-ator¯ cH H † can generate a tree-level Majorana neutrino mass through the type I (if it is generated by the exchange of the heavy fermion Ψ(1, 1, 0)) and/or type III (if it is generated by Ψ(1, 3, 0)) seesaw mechanisms. In either case, the resulting Majorana neutrino mass is: where v is the Higgs Vacuum Expectation Value (VEV) and f W and Λ W are the Wilson coefficient and NP scale of the Weinberg operator (see Eq. 9). Therefore, there are two extreme cases for generating . In both cases the effect of the Weinberg operator at TeV-scale energies is negligible.
The operator¯ dHφ * can also generate a Majorana neutrino mass term at 1-loop via the diagram depicted in Fig. 3, which involves both the dimension five coupling strength f dφH and the Yukawa-like LQ-quark-lepton where Λ = Λ dφH and f = f dφH are the NP scale and Wilson coefficient of the dimension five operator¯ dHφ * ; m d is the mass of the down-quark in the loop and M φ is the leptoquark mass. Thus, setting e.g., Λ = 5 TeV and M φ = 1 TeV, we obtain: so that, for f ·y L q ∼ O(1), the resulting Majorana mass is TeV, in order to obtain e.g., m ντ < ∼ 1 eV (ignoring off-diagonal generation couplings). We note that other interesting mechanisms for generating light Majorana neutrino masses from 1-loop LQ exchanges that are intimately related to the down-quark mass matrix have been discussed in [51][52][53][54][55][56][57]. These studies, however, were based on renormalizable LQ extensions of the SM.
As noted in the previous section, the heavy fermionic states Ψ(1, 1, 0) and Ψ(1, 3, 0) can generate at tree-level both the Weinberg operator¯ cH H † and the operator dHφ * . Therefore, in this setup there are two scenarios that do not require small coupling constants: 1. The heavy fermionic states Ψ(1, 1, 0) and Ψ (1,3,0) are responsible for generating both operators cH H † and¯ dHφ * , with a typical mass scale of M Ψ ∼ O(10 14 ) GeV. In this case, the Majorana neutrino mass term will be generated at tree-level through the type I or type III seesaw mechanisms by the Weinberg operator¯ cH H † and the 1-loop contribution from the operator¯ dHφ * will be subdominant. This holds also in the case that the Weinberg operator is generated by the heavy scalar GeV and a corresponding O(1) Wilson coefficient.
2. The Weinberg operator is not relevant to neutrino masses, i.e., there are no heavy Φ(1, 3, 0), Ψ(1, 1, 0) and Ψ(1, 3, 0) states in the underlying theory. In this case, neutrino masses are not generated through the seesaw mechanism, but they may be still generated at 1-loop by the dimension five operator¯ dHφ * as described above, if this operator is generated at tree-level in the underlying theory by the heavy states Φ(3, 2, 1 6 ) and/or Ψ(3, 2, − 5 6 (see previous section).

B. Neutrinoless double beta decay
The dimension five operatordd c φ 2 can mediate neutrinoless double beta decay (0νββ) via the diagram depicted in Fig. 4. This requires both the dimension five operator dd c φ 2 and the Yukawa-like renormalizable coupling of φ to the right-handed 1st generation u-quark and electron, i.e., the term ∝ y R ue in L Y,φ (see Eq. 2). If φ is a 3rd generation leptoquark, we expect y R ue 1 (see discussion in the next section) in which case the 0νββ decay rate will be significantly suppressed.
The limit on 0νββ decay is usually expressed in terms of the electron-electron element of the neutrino mass matrix. The current bound is |(m ν ) ee | < 0.1 − 0.5 eV, depending on the 0νββ experiment, see e.g., [58]. This translates into a bound on the corresponding parton-level amplitude for 0νββ [59]: where p eff ∼ 100 MeV is the neutrino effective momentum obtained by averaging the corresponding nuclear matrix element contribution. In our case, the 0νββ amplitude corresponding to the diagram in Fig. 4 can be estimated as: where f = f d 2 φ 2 and Λ = Λ d 2 φ 2 . Therefore, using Eq. 13 we obtain: In particular, we find that no useful bound can be imposed on the scale of the dimension five operator dd c φ 2 , assuming f d 2 φ 2 ∼ O(1) and a TeV-scale LQ mass, M φ ∼ O(1 TeV), if the LQ φ is a 3rd generation LQ (as assumed below), i.e., having a suppressed Yukawalike coupling to the 1st generation right-handed fermions: y R ue < 0.1.
All cross-sections presented in this section were calculated using MadGraph5 [64] at LO parton-level, for which a dedicated universal FeynRules output (UFO) model for the LQ-SM EFT framework defined in Eq. 4 was produced for the MadGraph5 sessions using FeynRules [65]. The LO nnpdf3 PDF set (NNPDF30-lo-as-0130 [66]) was used in all the calculations presented below. Also, all cross-sections were calculated with a dynamical scale choice for the central value of the factorization (µ F ) and renormalization (µ R ) scales corresponding to the sum of the transverse mass in the hard-process, and, for consistency with the EFT framework, a cut on the center of mass energy of √ŝ < Λ was placed using Mad-Analysis5 [67], where several values of Λ (the scale of NP) were used for the processes considered below. 4 Furthermore, we will assume throughout the rest of the paper that φ(3, 1, − 1 3 ) and φ(3, 1, 2 3 ), under consideration in this section, are 3rd generation leptoquarks and denote them generically by φ 3 . In particular, we assume that the LQ-lepton-quark Yukawa-like couplings of φ 3 to the 1st and 2nd generations SM fermions in the corresponding renormalizable φSM Lagrangian are much smaller than its couplings to the 3rd generation quark-lepton pair, e.g., to a tτ and/or bν τ pairs in the case of the down-type LQ φ(3, 1, − 1 3 ) (see Eqs. 2). This scenario can be realized by imposing an approximate Z 3 generation symmetry under which the physical states of the SM fermions (i.e., mass eigenstates) transform as: where k is the generation index and α(ψ k ) are the Z 3 charges of ψ k . Consider for example the down-type LQ φ(3, 1, − 1 3 ): if the Z 3 charges equal the generation index, i.e., α(ψ k ) = k, and α(φ) = 3, then only terms in L φSM involving the 3rd generation are allowed. In particular, assuming Baryon number conservation and thus ignoring the Z 3allowed LQ interactions with the 3rd generation quarks (i.e., φt c R b R and φt c L b L ) that would in general allow for proton decay, we have: where we will assume that the above Yukawa-like LQquark-lepton 3rd generation couplings are O(1). The Z 3 generation symmetry is exact in the limit where the quark mixing CKM matrix V is diagonal, so that Z 3 -breaking effects will in general be proportional to the square of the small off-diagonal CKM elements |V cb | 2 , |V ub | 2 , |V ts | 2 , |V td | 2 , and will, therefore, be suppressed (see also [28,61,62]). In particular, the Z 3 generation symmetry is assumed to be broken in the underlying heavy theory and can, therefore, be traced to the higher dimensional operators. For example, the off-diagonal SM Yukawa couplings may be generated by the dimension six operators: (18) where, if e.g., Λ ∼ 1.5, 3 or 5 TeV and f uH , f dH ∼ O(1), then the resulting effective Yukawa couplings, , respectively, where y SM q are the corresponding Yukawa couplings in the SM (see [63]).
The Z 3 breaking terms in the LQ sector will also be generated in the effective theory through higher dimensional operators. To demonstrate that consider for example the dimension five operatordd c φ 2 in Eq. 6. As was shown in section IV, this operator can be generated at tree-level in the UV theory by exchanging e.g., a heavy scalar Φ(6, 1, − 2 3 ) (see diagram (a) in Fig. 1). Thus, if Φ(6, 1, − 2 3 ) couples to the 1st and/or 2nd generation down-quarks, then the Z 3 generation symmetry is broken and the scale of generation breaking is the mass of Φ(6, 1, − 2 3 ), M Φ . In particular, the Z 3 generation breaking effects in this case will be proportional to g Φdd · g Φφφ /M Φ , where g Φdd and g Φφφ are the couplings of the heavy Φ(6, 1, − 2 3 ) to a dd-pair and a φφpair, respectively. The matching to the EFT framework of Eq. 9 can be done by replacing M Φ → Λ d 2 φ 2 and g Φdd · g Φφφ → f d 2 φ 2 .
As mentioned above, LQ phenomenology changes in the presence of the higher dimensional effective operators. In particular, additional potentially interesting φ 3 production channels are opened at the LHC. However, most of them will have a too small cross-section at the 13 TeV LHC, due to the 1/Λ n suppression in the EFT expansion, so that the leading effects are produced by the dimension five operators involving φ 3 in Eq. 9. Recall, however, that the operator¯ dHφ * is expected to have suppressed effects because of a large effective scale, as required for consistency with sub-eV neutrino masses (cf. the previous section).
We are therefore left with only one dimension five operator,dd c φ 2 , that can potentially mediate interesting φ 3 pair-production signals at the LHC. In particular, we find that this operator may yield a strikingly large asymmetric same-sign(charge) φ 3 φ 3 signal at the LHC via dd → φ 3 φ 3 , which is more than an order of magnitude larger than the charged conjugate channel dd → φ * 3 φ * 3 , due to the different fractions of d andd in the incoming protons, see Fig. 5. The hard cross-section for dd → φ 3 φ 3 (which equals that of the charged conjugate onedd → φ * 3 φ * 3 ) is: where (cf. Eq. 9) √ŝ is the center of mass energy of the hard process. For example, if Λ d 2 φ 2 = 5 TeV (and with a cut on the φ 3 φ 3 invariant mass, M φ3φ3 < 5 TeV), we find: 6 This can be compared to the gluon-fusion crosssection of the opposite-charge φ 3 φ * 3 pair-production signal, pp (gg) → φ 3 φ * 3 , for which the hard cross-section (see e.g., [30,31]): drops with the energy as 1/ŝ and yields a cross-section of (again with M φ3φ * 3 < 5 TeV): 6 There are no SM contributions to the processes studied here and also none of the tree-level generated dimension six operators that we list in the appendix contribute to them. Thus, the dimension five operators that we consider generate the leading contributions to these processes. In particular, potential corrections to the leading-order cross-sections presented in this section can be generated either by loop-generated dimension six operators and/or by dimension seven operators. The former are suppressed by a factor of E/(16π 2 Λ) (E is the typical energy of the process) and can, therefore, be neglected here, while the latter are suppressed typically by (E/Λ) 2 and, therefore, their size depend on the relevant energy scale of the process. In particular, for the s-channel process (see Fig. 1a) the corrections can reach 50%, while for t or u channel processes (see Fig. 1b) the relevant energy scale is much smaller and the corrections are again negligible.
We thus see that the same-sign φ 3 φ 3 rate is expected to be larger than the opposite-sign φ 3 φ * 3 rate at the 13 TeV LHC, in particular, Taking into account the leading φ 3 decays φ 3 → tτ − and φ 3 → bν τ , this signal will in turn give rise to the new asymmetric signatures (j b = b-jet): with a cross-section which is more than an order of magnitude larger than the charged conjugate channels.
While pp → 2 · j b + E T may not be unique to φ 3 pairproduction, and may be more challenging due to the larger background expected in this channel, the signal of same-sign top-quark pair in association with a pair of same-sign negatively charged τ -leptons, pp → ttτ − τ − , and the single top -single τ signature, pp → tτ − +j b + E T , may give striking new asymmetric φ 3 φ 3 signals.
For example, if the scale of the NP underlying L φSM is Λ = 5 TeV, the LQ mass is M φ3 ∼ 1 TeV and its leading branching ratios are BR(φ 3 → tτ − ) = BR(φ 3 → bν τ ) = 0.5, then we expect σ(pp → ttτ − τ − ) ∼ 3.4 fb; while σ(pp →ttτ + τ + ) ∼ 0.07 fb, see Fig. 5. The former is about five times larger than the rate for the gluonfusion φ 3 φ * 3 signal pp → ttτ + τ − , for which a dedicated search has already been performed by CMS [44] with null results. With an integrated luminosity of ∼300 fb −1 , Λ = 5 TeV and M φ3 ∼ 1 TeV, about 1000 ttτ − τ − events with an invariant mass smaller than 5 TeV are expected. After the top-quarks decay hadronically via t → W + b → 2·j+b (j =light jet) with a BR(t → W + b → 2 · j + b) ∼ 2/3, we expect about 450 same-sign τ − τ − events with a high jet-multiplicity signature: pp → τ − τ − + 4 · j + 2 · j b and with a statistical error of ∼ √ 450 ∼ 20 events and no irreducible background. 6 Note also that roughly the same number of events are expected for the tτ − production signal pp → tτ − + j b + E T , which leads to pp → τ − + 2 · j +2·j b + E T , when the top-quark decays hadronically via t → W + b → 2 · j + b. This single-τ signal lack a unique characterization akin the same-sign lepton signature in pair LQ production and might, therefore, be harder to trace.
It is also useful to define the inclusive same-charge τ τ asymmetry: where we have assumed again that the top-quark decays hadronically via t → W + b → 2 · j + b and X j stands for any accompanying jets in the final state. When the φ 3 mass is in the range 1 TeV < ∼ M φ3 < ∼ 2 TeV, we expect A τ τ → 1 since this asymmetry receives its most significant contribution from the φ 3 φ 3 and φ * 3 φ * 3 channels (see Fig. 5). The same-sign τ − τ − events have no irreducible background, so we expect this asymmetry to remain close to a 100% even after the reducible background is taken into account.
The statistical significance, N SD , with which this asymmetry can be detected at the LHC is: where σ τ τ is the inclusive cross-section σ(pp → τ − τ − + X j ) and is the corresponding combined efficiency for the simultaneous measurement of this final state. Thus, with an integrated luminosity of 300 inverse fb (recall that σ(pp → τ − τ − + X j ) ∼ 1.5 fb for Λ = 5 TeV and M φ3 = 1 TeV), and a combined efficiency of ∼ 0.01, this asymmetry can be detected with about a ∼ 2σ significance. At the high-luminosity LHC with 3000 inverse fb this asymmetry should be accessible with a statistical significance of N SD ∼ 7.
B. The up-type scalar LQ φ(3, 1, 2 3 ) We wish to briefly comment here on the phenomenology and LHC signals expected for an up-type LQ φ (3, 1, 2 3 ) in the EFT framework. The renormalizable Yukawa-like interactions of this LQ contain only the term y R is anti-symmetric due to SU(3) (color) gauge invariance. Note, however, that this di-quark LQ coupling violates baryon number and, in the presence of the higher dimensional LQ couplings to quark-lepton pairs (see below), may mediate proton decay. We therefore, assume that it is either negligibly small or forbidden due to a symmetry.
In the up-type φ(3, 1, 2 3 ) case, we find that there are four dimension five operators (in addition to the Weinberg operator of Eq. 8): 7 The fourth operator in Eq. 25,ūu c φ 2 , will give rise to a similar same-sign asymmetric φ 3 φ 3 signals via uu → φ 3 φ 3 (and the much smaller charged conjugate oneūū → φ * 3 φ * 3 ), with a considerably larger cross-section than the same-sign down-type LQ pair-production one, due to the larger u-quark content/PDF in the protons. For example, with Λ u 2 φ 2 = 5 TeV and the invariant mass cut M φ3φ3 < 7 This estimate does not include the τ -decay branching ratio into a specific final state. 8 We note that if both the down-type and up-type LQ are included as light degrees of freedom in the low-energy framework, then four more dimension five operators can be constructed in the EFT extension: where we have used here the subscripts d and u to distinguish between them. 5 TeV, we find for the up-type LQ case: which is about 25(600) times larger than the expected opposite-charged φ 3 φ * 3 signal for M φ3 = 1(2) TeV, see Eq. 22.
In contrast to the case of the down-type LQ (which decays via its renormalizable couplings to quark-lepton pairs), the decay pattern of the up-type LQ considered here will be controlled by its dimension five interactions with the SM fields in Eq. 25. In particular, it will decay via either φ → de + and/or φ → uν, where d, u, e, ν stand here for a down-quark, up-quark, charged lepton and neutrino of any generation, with a corresponding coupling which is suppressed by ∼ v/Λ, e.g., for the decay φ → uν the coupling is f luφH · (v/Λ luφH ). Thus, assuming as an example that its dominant dimension five couplings are to the 3rd generation SM fermions, then here also, when it decays via either φ 3 → bτ + and/or φ 3 → tν τ , we expect the new asymmetric signals: each having a cross-section which is several orders of magnitude larger than the charged conjugate channels.
Despite obvious parallels, there are important differences between the above signals and the ones expected for the down-type LQ : j b for the up-type LQ has opposite lepton charges than the corresponding signal for the down-type LQ. Therefore, the asymmetry A τ τ flips signs in the up-type LQ case.
2. Similarly, in single LQ production, the final τ lepton is positive for the up-type LQ and negative for the down-type.
3. The same-sign τ τ signal has a lower jet multiplicity than in the case of the down-type LQ.
Thus, the most promising signals in up-type φ 3 φ 3 pairproduction are pp → τ + τ + + 2 · j b and pp → tt + E T → + + +2·j b + E T , containing two positive charged leptons (for which the background is low) and two high-p T tagged b-jets. For Λ = 5 TeV, M φ3 = 1 TeV and assuming BR(φ 3 → bτ + ) = BR(φ 3 → tν τ ) = 0.5, the overall cross-sections for these signals (with an invariant mass smaller than 5 TeV) are expected to be: where, as mentioned above, for the same-charged topquark pair signal, pp → tt + E T → + + + 2 · j b + E T , this cross-section applies to any one of the same-charged leptons, i.e., + + = e + e + , µ + µ + or τ + τ + , when the top-quarks decay leptonically with Considering the same-sign dilepton asymmetry defined in Eq. 23, in the up-type LQ case we find that A τ τ may be detected with a statistical significance of N SD ∼ 8, with an integrated luminosity of 300 inverse fb and a combined efficiency of ∼ 0.01 (see Eq. 24). On the other hand, a statistically significant signal of the asymmetries A ee/µµ will require the 13 TeV HL-LHC with an integrated luminosity of 3000 inverse fb. Pair-production cross-sections of the down-type and up-type LQ, as a function of the LQ mass, for a NP scale Λ = 5 TeV, at a 27 TeV HE-LHC (upper plot) and a 100 TeV FCC-hh (lower plot): the QCD cross-section via gg, qq → φ3φ * 3 (solid line), the same-charge up-type LQ pairproduction cross-section via uu → φ3φ3 (dashed line) and the same-charge down-type LQ pair-production cross-section via dd → φ3φ3 (dashed-dotted line). See also text.
As can be seen from Fig. 5, the LQ production crosssections sharply drop with the LQ mass at the 13 TeV LHC for LQ masses M φ > 1 TeV. This is due to the limited phase space at the 13 TeV LHC for producing TeV-scale heavy particles and, hence, the currently relatively poor discovery potential for such new heavy particles. In particular, the detection of NP scales Λ > 5 TeV and/or heavy new particles with masses of several TeV, will require in general higher energy colliders with higher luminosities. For example, for a LQ mass of M φ3 ∼ 4 TeV, the opposite-charge φ 3 φ * 3 pair-production cross-section (via gg, qq → φ 3 φ * 3 ) at the 13 TeV LHC is σ(pp → φ 3 φ * 3 ) ∼ 10 −6 fb. The new same-charge φ 3 φ 3 signal discussed above is also too small at the 13 TeV LHC for M φ3 ∼ 4 TeV; σ(pp → φ 3 φ 3 ) ∼ 10 −4 fb, if the NP scale is Λ ∼ 10 TeV. Therefore, heavy LQ with masses of several TeV are not accessible at the 13 TeV LHC with or without the new EFT interactions from the higher dimensional effective operators.
A better sensitivity to multi-TeV LQ and, in particular, to the LQ EFT dynamics presented in this work, can be obtained at future higher energy hadron colliders such as the HE-LHC and the FCC-hh mentioned above. In Figs. 6-8 we plot the same-charge LQ pairproduction cross-sections pp → φ 3 φ 3 for both the downtype and up-type LQ (i.e., the underlying hard-processes being dd → φ 3 φ 3 and uu → φ 3 φ 3 , respectively), as well as the opposite-charge LQ pair-production (QCD) cross- , for a NP scale of Λ = 5, 10 and 15 TeV. Here also, for consistency with the EFT framework, all cross-sections are calculated with an invariant mass cut on the LQ pair M φ3φ3 < Λ, i.e., M φ3φ3 < 5, 10, 15 TeV for Λ = 5, 10, 15 TeV, respectively. We note that the cross sections in Figs. 6-8 for a 3rd generation LQ are insensitive to the the Yukawa couplings in Eq. 2, so the results for 1st and 2nd generation LQ are expected to be comparable.
We see that the production rate of positively-charged up-type LQ pair (in the EFT framework) can reach σ(pp → φ 3 φ 3 ) ∼ O(1) fb at the 100 TeV FCC-hh, for a rather heavy LQ with M φ3 ∼ 7 TeV and a NP scale of Λ ∼ 15 TeV, whereas the corresponding opposite-charged φ 3 φ * 3 signal (i.e., for M φ3 ∼ 7 TeV) is expected to be about two orders of magnitudes smaller. A 27 TeV HE-LHC is also sensitive to a several TeV LQ and a NP scale of O(10) TeV, e.g., expecting an O(1) fb cross-section for pair production of positively-charged up-type LQ pair when M φ3 ∼ 4 TeV and a NP scale of Λ ∼ 10 TeV.

VII. SUMMARY
We have explored the phenomenology of the EFT expansion of a low-energy TeV-scale framework, where the "light" degrees of freedom contain the SM fields and a down-type scalar LQ φ(3, 1, − 1 3 ) or an up-type LQ φ (3, 1, 2 3 ). We found that there are only two dimension five operators that can be assigned to the down-type LQ φ(3, 1, − 1 3 ) and four dimension five operators for the up-type LQ φ (3, 1, 2 3 ); all these dimension five operators violate lepton number by two units. We have also identified the distinct underlying heavy physics that can generate these operators at tree-level.
We have shown that for each LQ type one of these operators can generate sub-eV Majorana neutrino masses at 1-loop, provided its effective scale is sufficiently high f /[Λ/TeV] ∼ 10 −6 , where f is the corresponding Wilson coefficient derived from the underlying heavy theory. We also found that another dimension five operator involving the down-type LQ, which is relevant to current collider phenomenology, may mediate neutrinoless double beta decay.
We have then focused on collider phenomenology of both the down and up-type scalar LQ in the EFT framework. In particular, motivated by the current anomalies in B-decays, we have suggested an approximate Z 3 generation symmetry and studied the signals of 3rd generation down-type and up-type LQs (φ 3 ) at the LHC. We found that the dimension five operators may give rise to striking asymmetric, same-charge dilepton final states in the reactions pp → φ 3 φ 3 for both the down and up-type scalar LQs, that have low background.
For example, for the 3rd generation down-type LQ with a mass M φ3 ∼ 1 TeV and a NP scale Λ ∼ 5 TeV, the resulting same-sign lepton signature is pp → φ 3 φ 3 → τ − τ − + 4 · j + 2 · j b (j=light jet and j b =b-jet), which is expected to yield about 500 such τ − τ − events at the 13 TeV LHC with a luminosity of 300 fb −1 . For the 3rd generation up-type LQ, we expect about 6000 events of same-sign positively charged τ + τ + from the process pp → φ 3 φ 3 → τ + τ + + 2 · j b , if Λ ∼ 5 TeV. Moreover, for similar parameters, the same-charge up-type φ 3 φ 3 pair production process can also generate events with pairs of same-charge top quarks pp → tt + E T (when each LQ decays via φ 3 → tν), leading to about 50 same-sign dilepton events pp → + + + 2 · j b + E T (when each top-quark decays leptonically via t → W + b → + ν b), for any of the three charged leptons, = e, µ, τ .
We have also defined a double lepton-charge asymmetry that may be useful for detection and disentangling these same-sign lepton signals.
Finally, since the LQ production cross-sections sharply drop with the LQ mass at the 13 TeV LHC, due to its limited phase-space for producing multi-TeV heavy particles, we have also calculated the projected samecharge LQ pair production cross-sections, σ(pp → φ 3 φ 3 ), at 27 and 100 TeV hadron colliders; the future planned HE-LHC and FCC-hh, respectively. As expected, we find that these future higher energy hadron colliders can extend the sensitivity to the LQ EFT dynamics up to masses of M φ > ∼ 5 TeV and a NP scale of Λ ∼ 15 TeV.
Appendix: Dimension six operators for the down-type scalar LQ φ(3, 1, − 1 3 ) There are several classes of dimension six operators which correspond to the generic form of Eq. 5, which will be listed here.
The only φ 6 operator is: There are no operators of the form φ 5 H due to gauge invariance and out of the φ 4 H b D 2−b type operators there are only two non-redundant gauge invariant operators corresponding to the b = 0 and b = 2 cases: φ 4 D 2 = |φ| 2 |Dφ| 2 . (29) Out of the operators that contain a φ 3 factor, the ones of the form φ 3 H b D 3−b are absent since they violate either gauge (b odd) or Lorentz (b even) invariance. On the other hand, in the class φ 3 ψ 2 operators there are four gauge invariant combinations which can be constructed, all of the form |φ| 2 φψ L ψ R : φ 3 ψ 2 ∈ |φ| 2 (ql c φ) , |φ| 2 (ue c φ) , |φ| 2 (q c qφ) , |φ| 2 (u c dφ) , where the last two φ 3 ψ 2 operators above violate both baryon and lepton number. The operators that contain a φ 2 factor can be divided into two categories: the ones proportional to gauge invariant factor |φ| 2 and the ones that contain φ 2 or (φ * ) 2 . The former case is straight forward, since it includes all operators involving an SU(3) singlet φ † φ of the form: where O 4 SM includes all the dimension 4 renormalizable terms of the SM Lagrangian. In addition, there are operators involving the SU(3) octet φ † φ states of the form: where λ a are the SU(3) Gell-Mann matrices and B µν is the SM SU(1) field strength. The latter case (i.e., operators which contain a φ 2 factor) is more elaborate, but it can be shown that there are only two non-redundant gauge invariant operators of this class, both in the form φ 2 ψ 2 D, where ψ 2 is composed out of one quark and one lepton: where here a, b, c are color indices. Finally, the dimension six operator which contain only one LQ field have to be of the form φψ 2 H b D 2−b , where 0 ≤ b ≤ 2 and ψ 2 is either a quark-lepton or quark-quark pair. For the b = 2(b = 1) case we find six(five) gauge invariant operators: φψ 2 H 2 ∈ |H| 2 φ † qq c , |H| 2 φql c , |H| 2 φd c u , |H| 2 φue c , φ † (qH)(H † q c ) , φ(qH)(H † l c ) , O (6) φψ 2 HD ∈ (qH)γ µ u c D µ φ † , (qH)γ µ d c D µ φ † , (qH)γ µ e c D µ φ , (lH)γ µ u c D µ φ , (lH)γ µ d c D µ φ , where we have omitted the color indices and the antisymmetric tensor abc in the above operators containing 3 ⊗ 3 ⊗ 3 and 3 ⊗ 3 ⊗ 3 states.
The case of b = 0, i.e., operators of the type φψ 2 D 2 , contain four possible combinations of ψ 2 fields of the form: where the notation above indicates that the two derivatives are to act on any of the fields; note though that D µ D µ acting on a field gives a redundant operator, but [D µ , D ν ] does not. Thus, for example, D 2 × ql c φ † corresponds to: D 2 × ql c φ → (qD µ l c )D µ φ , (qσ µν l c )B µν φ , (qσ µν σ I l c )W µν I φ , (qσ µν λ A l c )G µν A φ .