PHYSICAL REVIEW D 100, 075033 (2019)

Accessible lepton-number-violating models and negligible neutrino masses

† ‡ Andr´e de Gouvêa ,1,* Wei-Chih Huang,2, Johannes König,2, and Manibrata Sen 1,3,§ 1Northwestern University, Department of Physics and Astronomy, 2145 Sheridan Road, Evanston, Illinois 60208, USA 2CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark 3Department of Physics, University of California Berkeley, Berkeley, California 94720, USA

(Received 18 July 2019; published 25 October 2019)

Lepton-number violation (LNV), in general, implies nonzero Majorana masses for the Standard Model neutrinos. Since neutrino masses are very small, for generic candidate models of the physics responsible for LNV, the rates for almost all experimentally accessible LNV observables—except for neutrinoless double- beta decay—are expected to be exceedingly small. Guided by effective-operator considerations of LNV phenomena, we identify a complete family of models where lepton number is violated but the generated Majorana neutrino masses are tiny, even if the new-physics scale is below 1 TeV. We explore the phenomenology of these models, including charged-lepton flavor-violating phenomena and baryon- number-violating phenomena, identifying scenarios where the allowed rates for μ− → eþ-conversion in nuclei are potentially accessible to next-generation experiments.

DOI: 10.1103/PhysRevD.100.075033

I. INTRODUCTION Experimentally, in spite of ambitious ongoing experi- Lepton number and baryon number are, at the classical mental efforts, there is no evidence for the violation of level, accidental global symmetries of the renormalizable lepton-number or baryon-number conservation [5]. There Standard Model (SM) Lagrangian.1 If one allows for are a few different potential explanations for these (neg- generic nonrenormalizable operators consistent with the ative) experimental results, assuming degrees-of-freedom SM gauge symmetries and particle content, lepton number beyond those of the SM exist. Perhaps the new particles are and baryon number will no longer be conserved. Indeed, either very heavy or very weakly coupled in such a way that lepton-number conservation is violated by effective oper- phenomena that violate lepton-number or baryon-number ators of dimension five or higher while baryon-number conservation are highly suppressed. Another possibility is conservation (sometimes together with lepton number) is that the new interactions are not generic and that lepton- violated by effective operators of dimension six or higher. number or baryon-number conservation are global sym- In other words, generically, the addition of new degrees-of- metries of the beyond-the-Standard-Model Lagrangian. freedom to the SM particle content violates baryon-number Finally, it is possible that even though baryon number or and lepton-number conservation. lepton number are not conserved and the new degrees-of- freedom are neither weakly coupled nor very heavy, only a subset of baryon-number-violating or lepton-number- violating phenomena are within reach of particle physics *[email protected] † experiments. This manuscript concentrates on this third [email protected] ‡ option, which we hope to elucidate below. [email protected] §[email protected] The discovery of nonzero yet tiny neutrino masses is 1At the quantum level these symmetries are anomalous, i.e., often interpreted as enticing—but certainly not definitive— they are violated by nonperturbative effects [1,2]. These are only indirect evidence for lepton-number-violating new physics. relevant in extraordinary circumstances (e.g., very high temper- In this case, neutrinos are massive Majorana fermions and atures) much beyond the reach of particle physics experiments one can naturally “explain” why the masses of neutrinos [3,4]. Nonperturbative effects still preserve baryon-number- minus-lepton number, the nonanomalous possible global sym- are much smaller than those of all other known massive metry of the renormalizable SM Lagrangian. particles (see, e.g., [6–8] for discussions of this point). Searches for the nature of the neutrino—Majorana fermion Published by the American Physical Society under the terms of versus Dirac fermion—are most often searches for lepton the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to number violation (LNV). The observation of LNV implies, the author(s) and the published article’s title, journal citation, generically, that neutrinos are Majorana fermions [9], and DOI. Funded by SCOAP3. while Majorana neutrino masses imply nonzero rates for

2470-0010=2019=100(7)=075033(21) 075033-1 Published by the American Physical Society DE GOUVÊA, HUANG, KÖNIG, and SEN PHYS. REV. D 100, 075033 (2019)

Al −14 lepton-number-violating phenomena. The most powerful ∶ − þ ≳ 10 ð Þ COMET Phase-I Rμ e : 1:3 probes of LNV are searches for neutrinoless double-beta decay (0νββ, see [10] for a review); several of these are For a recent, more detailed discussion, see [20]. ongoing, e.g., [11–13]. The growing excitement behind There are several recent phenomenological attempts at searches for 0νββ is the fact that these are sensitive enough understanding whether there are models consistent with to detect LNV mediated by light Majorana neutrino current experimental constraints where the rate for exchange if the neutrino masses are above a fraction on μ− → eþ-conversion in nuclei is sizable [19,21,22]. The an electronvolt. In many models that lead to Majorana main challenges are two-fold. On the one hand, the light- neutrino masses, including, arguably, the simplest, most Majorana-neutrino exchange contribution to μ− → eþ- elegant, and best motivated ones, LNV phenomena are conversion in nuclei is tiny. On the other hand, while it predominantly mediated by light Majorana neutrino is possible to consider other LNV effects that are not exchange. Hence, we are approaching sensitivities to captured by light-Majorana-neutrino exchange, most of 0νββ capable of providing nontrivial, robust information these scenarios lead, once the new degrees of freedom are on the nature of the neutrino. integrated out, to Majorana neutrino masses that are way Other searches for LNV are, in general, not as sensitive too large and safely excluded by existing neutrino data. 0νββ as those for . Here, we will highlight searches for In [19], an effective operator approach, introduced and μ− → eþ-conversion in nuclei, for a couple of reasons. One – − þ exploited in, e.g,, [23 26], was employed to both diagnose is that, except for searches for 0νββ, searches for μ → e - the problem and identify potentially interesting directions conversion in nuclei are, arguably, the most sensitive to for model building. generic LNV new physics.2 Second, several different − − New-physics scenarios that violate lepton-number con- experiments aimed at searching for μ → e -conversion servation at the tree level in a way that LNV low-energy in nuclei are under construction, including the COMET phenomena are captured by the “all-singlets” dimension- [15] and DeeMe experiments [16] in J-PARC, and the nine operator: Mu2e experiment [17] in Fermilab. These efforts are expected to increase the sensitivity to μ− → e−-conversion 1 L ⊃ O ; where O ¼ ecμcucucdc dc; ð1:4Þ by, ultimately, four orders of magnitude and may also be Λ5 s s able to extend the sensitivity to μ− → eþ-conversion in nuclei by at least a few orders of magnitude. and Λ is the effective scale of the operator, were flagged as − þ The best bounds on the μ → e -conversion rate relative very “inefficient” when it comes to generating neutrino − to the capture rate of a μ on titanium were obtained by the Majorana masses. According to [19], the contribution SINDRUM II experiment [18] over twenty years ago: to Majorana neutrino masses from the physics that leads to Eq. (1.4) at the tree level saturates the upper bound on Γðμ− þ Ti → eþ þ CaÞ neutrino masses for Λ ∼ 1 GeV. This means that, for Ti ≡ Rμ−eþ − Λ ≫ 1 Γðμ þ Ti → νμ þ ScÞ GeV, the physics responsible for Eq. (1.4) will lead to neutrino masses that are too small to be significant 1.7 × 10−12 ðGS; 90% CLÞ < ; ð1:1Þ while the rates of other LNV phenomena, including 3.6 × 10−11 ðGDR; 90% CLÞ μ− → eþ-conversion in nuclei, may be within reach of next-generation experiments. According to [19], this where GS considers scattering off titanium to the ground happens for Λ ≲ 100 GeV. state of calcium, whereas GDR considers the transition to The effective operator approach from [23–26] is mostly a giant dipole resonance (GDR) state. Next-generation powerless when it comes to addressing lepton-number- experiments like Mu2e, DeeMe, and COMET have the conserving, low-energy effects of the same physics that potential to be much more sensitive to μ− → eþ-conversion. leads to Eq. (1.4). One way to understand this is to The authors of [19] naively estimated the future sensitivities appreciate that lepton-number-conserving phenomena are of these experiments to be captured by qualitatively different effective operators and, in general, it is not possible to relate different “types” of Al −16 operators in a model-independent way. Concrete results can Mu2e∶ R − þ ≳ 10 ; ð1:2Þ μ e only be obtained for ultraviolet (UV)-complete scenarios. In this manuscript, we systematically identify all possible UV-completemodels that are predominantly captured, when O 2 it comes to LNV phenomena, by s at the tree level. All It was recently pointed out that searches for nonstandard these models are expected to have one thing in common: neutrino interactions from long-baseline neutrino experiments are also sensitive to certain LNV new physics and involve all lepton potentially large contributions to LNV processes combined flavors [14]. In some cases, the resulting limits are stronger than with insignificant contributions to the light neutrino masses. those from μ− → eþ-conversion. Such models are expected to manifest themselves most

075033-2 ACCESSIBLE LEPTON-NUMBER-VIOLATING MODELS AND … PHYS. REV. D 100, 075033 (2019) efficiently in LNV phenomena like μ− → eþ-conversion in These different contractions, however, lead to estimates nuclei, lepton-number-conserving phenomena, including for the rates of the processes of interest which are roughly charged-lepton flavor-violating (CLFV) observables, or the same. baryon-number-violating phenomena, including neutron– An effective operator of mass dimension d is suppressed antineutron oscillations. Furthermore, if the rates for by (d − 4) powers of the effective mass-scale Λ of the new μ− → eþ-conversion in nuclei are indeed close to being physics, i.e., accessible, we find that all tree-level realizations of Os g require the existence of new degrees-of-freedom with L ⊃ Od þ H:c:; ð2:1Þ masses that are within reach of TeV-scale colliders like Λd−4 the LHC. The following sections are organized as follows. where gs are dimensionless coupling constants. Note that g Λ In Sec. II, we study the all-singlets effective operator and and are not independently defined; one can resolve this Λ illustrate its contributions to neutrino masses, 0νββ, and issue, e.g., by defining such that the largest g is one. The − þ Λ μ → e -conversion in nuclei. In Sec. III, we list the effective scale indicates the maximum laboratory energy different UV-complete models that are associated to the beyond which the effective-operator description breaks all-singlets effective operator at tree level. We discuss down, i.e., the effective-theory description is valid at energy Λ various bounds arising from searches for baryon-number- scales which are at most of order . violating and CLFV processes. In Sec. IV, we comment on With this arsenal, the dimension-nine all-singlets oper- some salient collider signatures of the different new particles ators are proposed in this work. Finally, in Sec. VI, we briefly αβ c c c c c c comment on possible extensions of these scenarios which Os ¼ lαlβu u d d ð2:2Þ can account for the observed neutrino masses, summarize c c c c our results, and conclude. where lα ≡ e , μ or τ . Os are formed from all the ð2Þ SU L-singlet fields. If all quarks are of the same II. THE EFFECTIVE ALL-SINGLETS generation, there is only one independent Lorentz con- αβ c σ c c ρ c c c σ_ OPERATOR Os traction: ðlαÞ ðlβÞσðu Þ ðu Þρðd Þσ_ ðd Þ , where σ, ρ and σ_ are the Lorentz indices; all other possible contractions are In the context of particle physics phenomenology, differ- related to this via Fierz transformations.3 ent notations are prevalent in the literature. Before pro- Oαβ ceeding, we outline the notation used in this paper, which At different loop-orders, s will contribute to Majorana follows that in [24,25]. The SM is constructed using only neutrino masses as well as different LNV processes. In what follows, we estimate in some detail the contributions left-chiral Weyl fields: Q ≡ ðuL;dLÞ, L ≡ ðνL;lLÞ are the ð2Þ c c lc of these operators to Majorana neutrino masses, 0νββ left-chiral SU L doublets, while u , d and are the left- ð2Þ and μ− → eþ-conversion in nuclei. The idea [24] is to chiral SU L singlet fields. The corresponding Hermitian- conjugated fields are identified with a bar (e.g., L;¯ ec). start with the effective operator, and add SM interactions to generate the relevant processes. The results presented in Thus, unbarred fields Lσ correspond to the ð1=2; 0Þ representation of the Lorentz algebra, while barred fields this section agree with those in [19]. ¯ † Neutrino Majorana masses are generated by the LNV Lσ_ ≡ Lσ transform under the ð0; 1=2Þ representation of the Weinberg operator [27], algebra. In this terminology, the familiar four-component Dirac spinor consisting of the electron and the positron can fαβ ¼ð cÞT L ⊃ ð α Þð β Þ ð Þ be written as e eL; e . Throughout, color indices are Λ L H L H : 2:3 implicit and hence omitted. Also, the SM Higgs doublet is W taken to be H ≡ ðHþ;H0ÞT, where H0 acquires a vacuum ð2Þ ð1Þ These are dimension-five operators, violate lepton number expectation value (vev) v to break the SU L ×U Y ð1Þ by two units, and, after electroweak symmetry breaking, gauge-symmetry spontaneously to U EM. α β L ⊃ mαβν ν Gauge singlets can be formed by either contracting the lead to neutrino Majorana mass terms, , m ¼ fv2=Λ . Λ is the effective scale of the Weinberg SUð2Þ indices using the antisymmetric tensor ϵij or the W W L operator, related to but not the same as Λ, the effective scale Kronecker δij (for conjugated fields). Additionally, flavor couplings are, unless explicitly shown, implicitly con- tracted. The flavor structure of the effective operators 3If we consider up-type and down-type quarks of different gene- can be used to infer contributions to different new-physics rations, there are two independent contractions. One can choose c c c c c c c c c μ c c c those to be ðe μ Þðu t Þðd b Þ and ðe u Þðμ σ d Þðt σμb Þ, processes, as we shall see. We also do not explicitly c where, for convenience, we fix the two different lα to be the show the Lorentz structure of the different operators. electron and the muon and the two different generations of quarks Note that, for the same operator, there can be different to be the first and third generations. All other contractions can be contractions associated with the gauge and Lorentz indices. expressed as combinations of these two.

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Neutrino oscillation data constrain only the neutrino mass-squared differences. Nonetheless, one can use the atmospheric and the solar mass-squared differences to set lower bounds on the masses of the heaviest and the next-to- heaviest neutrinos. The atmospheric mass-squared differ- ence, for example,p dictatesffiffiffiffiffiffiffiffiffiffiffiffiffiffi that at least one neutrino has 2 to be heavier than jΔm32j ≃ 0.05 eV [28]. On the other FIG. 1. Four-loop contribution to Majorana neutrino masses hand, cosmic surveys limit the sum of masses of the Oee from the dimension-nine all-singlets operator s . Given the neutrinos to be ≲0.12 eV [29–31]. For concreteness, we large loop suppression, the contribution to neutrino masses is assume that the largest element of the neutrino mass matrix only relevant for very low effective scales. The blob represents lies between mν ∈ ð0.05 − 0.5Þ eV. In this case, Eq. (2.4) the effective operator while the × represents the Higgs-boson Oαβ vacuum expectation value. implies that the effective scale of s [19] is

αβ Λ ∈ ð100 MeV − 1 GeVÞ: ð2:5Þ of Os . Experimental information on neutrino masses point 14 to ΛW ∼ 10 GeV. Starting from the all-singlets operator, Fig. 1 illustrates how the Weinberg operator is obtained Figure 2 depicts the tree-level, two-loop and four-loop 0νββ Oee at the four-loop level. In Fig. 1, the blob represents the contributions to from s . The half-life for such a ee ee decay is estimated as [19] effective operator Os , for concreteness. Clearly, since Os involves only SUð2Þ -singlet fields, neutrino masses       L ð2Þ Λ2 4 1 2 2 2 2 2 2 require six Yukawa insertions so one can “reach” the ln GF yt ybyev T0νββ ¼ pffiffiffi L jg j2 Q11 2 q2 ð16π2Þ4 corresponding lepton-doublets and the Higgs-doublet ee     H. The contribution to the neutrino mass matrix can be 2 1 2 1 −1 þ pGFffiffiffi ytybyev þ ð Þ estimated as 2 2 2 2 8 : 2:6 2 q ð16π Þ Λ Λ 2 gαβ yαyβðy y vÞ ¼ t b ð Þ The effective Q-value of the decay process can be extracted mαβ 2 4 ; 2:4 Λ ð16π Þ from analyses of the data from the KamLAND-Zen experi- ment [32] and turns out to be Oð10 MeVÞ. The factor of where y are the different charged-lepton and quark Yukawa ð1=q2Þ comes from the neutrino propagator and is typically of couplings, Λ and g are the effective scale and couplings of order 100 MeV,the inverse distance-scale between nucleons. αβ Os , respectively, and we assumed third-generation quarks, Combining these, our estimate for the half-life as a function of as these are associated to the largest Yukawa couplings. Λ is depicted in the left panel of Fig. 3.ForOð1Þ couplings, Note that the α, β indices in Eq. (2.4) are not summed over. the current lifetime lower-bound—Λ ≳ 5 TeV—and the

ee FIG. 2. Feynman diagrams contributing to 0νββ from the dimension-nine all-singlets operator Os at the tree level (left), two-loop level (middle), and four-loop level (right). The blob represents the effective operator while the × represents the Higgs-boson vacuum expectation value.

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FIG. 3. Left: The lifetime associated to 0νββ, T0νββ, as a function of the cutoff scale Λ, from the dimension-nine all-singlets operator ee 4 10 Os . For values Λ ≲ 10 TeV the lifetime is dominated by the tree-level contribution and scales like ∝ Λ , whereas for larger values of Λ, the lifetime is dominated by the four-loop contribution and scales ∝ Λ2. The current experimental bound from KamLAND-Zen is depicted as a horizontal black line. Right: The normalized rate Rμ−eþ of muon to positron conversion as a function of the cutoff scale Λ, μe 2 from the dimension-nine all-singlets operator Os . For scales Λ ≲ 10 TeV, the tree-level contribution dominates and the rate scales like ∝ Λ−10. For scales Λ ≳ 104 TeV the four-loop contribution is most relevant and the rate scales like ∝ Λ−2. Between those regions, the two-loop contribution is most important and the rate scale like ∝ Λ−6. The current experimental bound from SINDRUM II and the sensitivity of Mu2e are depicted as a horizontal black and purple lines, respectively.

    — — 2 3 neutrino mass requirements Eq. (2.5) are incompatible. GF Z 2 Γ − ∝ pffiffiffi eff ð Þ μ 3 Q ; 2:7 This strongly suggests that if there is new physics that 2 πða0m =mμÞ ee e manifests itself via Os at the tree level, this new physics is not responsible for generating the observed nonzero neutrino masses. where Zeff is the effective atomic number, a0 the Bohr Figure 4 depicts the tree-level, two-loop and four-loop radius, and Q the estimated typical energy of the process, of μ− → þ Oeμ contributions to e -conversion from s . In order to order the muon mass mμ. While estimating Rμ−eþ , the term estimate Rμ−eþ , as defined in Eq. (1.1), we estimate the in the second parentheses in Eq. (2.7) cancels out in the muon capture rate, as outlined in [19],tobe ratio, yielding

− þ μe FIG. 4. Feynman diagrams contributing to μ → e -conversion from the dimension-nine all-singlets operator Os at the tree level (left), two-loop level (middle), and four-loop level (right). The blob represents the effective operator while the × represents the Higgs- boson vacuum expectation value.

075033-5 DE GOUVÊA, HUANG, KÖNIG, and SEN PHYS. REV. D 100, 075033 (2019)       6 2 1 2 2 2 2 2 2 Q GF yt ybyμyev Rμ− þ ¼jg μj pffiffiffi e e Λ2 2 q2 ð16π2Þ4   pffiffiffi  2 2 1 y y yμv 2 1 þ t b þ ð Þ 2 2 2 2 8 : 2:8 q ð16π Þ Λ GF Λ

The normalized conversion rate for this process as a function of Λ is depicted in Fig. 3 along with the current bounds on the process from the SINDRUM II collaboration [33], and the expected Mu2e sensitivity, Eq. (1.2). The current bound from SINDRUM II implies that Λ ≳ 10 GeV FIG. 5. Topologies that realize the all-singlets dimension-nine operator O at tree level. Topology 1 (left) involves only new for Oð1Þ couplings. Again, the neutrino mass requirements s μ− → þ bosons while Topology 2 (right) requires both new fermions and are inconsistent with the existing e -conversion bosons. In both topologies, the bosons can be scalars or vectors. bounds. If all gαβ are of the same magnitude, current constraints new-physics boson or couple to a new-physics on Λ from 0νββ—Λ ≳ 1 TeV for gee of order one—would translate into unobservable rates for μ− → eþ-conversion in boson and a new-physics fermion. This is depicted nuclei. However, there are no model-independent reasons in the right panel of Fig. 5. Again, the new-physics bosons can be scalars or vectors. to directly relate, e.g., gμe to gee, hence the bounds from 0νββ need not apply directly to searches for μ− → eþ- In order to systematically analyze the different internal conversion. Model-dependent considerations are required particles that can appear in Fig. 5, we determine the quantum numbers of pairs and triplets of the external in order to explore possible relations between gee and gμe. − þ SM fermions of interest. The different combinations of On the other hand, observable rates for μ → e -conver- pairs of fermions determine the possible quantum numbers sion require Λ ≲ 100 GeV and hence new particles with of the bosons in the internal lines in Fig. 5. Similarly, masses around (or below) the weak scale. It is natural to different combinations of triplets of fermions determine the suspect that models associated to such small effective quantum numbers of the potential new fermions in the scales are also vulnerable to lepton-number conserving, internal fermion line in Fig. 5 (right). low-energy observables, especially searches for CLFV. Table I lists all possible ways of pairing up any two As argued in the introduction, these phenomena can only SUð2Þ -singlet SM fermions.4 Generation indices, for both be addressed within UV-complete models, which we L leptons and quarks, have been omitted. Topology 1 can be introduce and discuss in the next section. realized by choosing three bosons with the same quantum numbers as these pairs, keeping in mind that there are two III. ULTRAVIOLET COMPLETIONS c c c αβ αβ fermions of each type—u , d , l —in Os . For bilinear OF THE EFFECTIVE OPERATOR Os combinations of the same generation of quarks, only Here we discuss tree-level UV-completions of the all- products symmetric in the color indices, i.e., forming a Oαβ 6¯ ð3Þ ð cÞαið cÞj ¼ singlets dimension-nine operator s , introduced in the 6or of SU c, exist since, e.g., u u α c i c αj c αj c i previous section, Eq. (2.2). As all fields in the effective −ðu Þαðu Þ ¼ðu Þ ðu Þα, where α is the dummy ð Þ ð3Þ operator are fermions, all new interactions involving SM Lorentz index and i; j are the SU c indices. Here, fields are either Yukawa or gauge interactions, i.e., they we will be concentrating on new-physics involving first- are all 3-point vertices. Furthermore, relevant interactions generation quarks, as we are interested in models that involving only new-physics fields are at most also 3-point mediate μ− → eþ-conversion at the tree level (left panel of vertices. This is due to the fact that the operator in question Fig. 4). Unless otherwise noted, we will not consider has six fermions in the final state and we are only interested models that “mix” different generations of the same quark- αβ in tree-level realizations of Os . Since only 3-point vertices flavor. αβ There are five different “minimal” realizations of are possible, there are only two topologies that lead to Os at the tree level [34,35]: Topology 1. Two of them involve heavy scalar bosons (1) All new particles are bosons. Each boson couples to only, while the remaining three require new-physics vector “ a pair of SM fermions, and three new-physics bosons and scalar bosons. Of course, one can consider less- ” define a new interaction vertex. This is depicted in minimal scenarios where one includes bosons with differ- the left panel of Fig. 5. The new-physics bosons can ent quantum numbers associated to the same fermion-pair, be scalars or vectors. (2) All new interactions involve one boson and two fermions. The new particles are bosons and fermi- 4Excluding left-handed antineutrinos νc. We will comment on ons and SM fields either couple pair-wise with a those later in this section.

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ð2Þ TABLE I. Quantum numbers of all possible pairs of the SU L TABLE III. Pairs of Standard Model fermions that share the gauge singlet Standard Model fermions lc, uc, dc. same gauge quantum numbers. The pairs of interest here are in bold. The pair lclc does not transform like any other pair of Pairs (Lorentz) Representation under ðSUð3Þ ; SUð2Þ Þ ð1Þ c c C L U Y SM fields; the same is true of the color-symmetric pairs of u u c c c c and d d . l l (scalar) ð1; 1Þ1 × ð1; 1Þ1 ¼ð1; 1Þ2 lc c ð1 1Þ ð3¯ 1Þ ¼ð3¯ 1Þ u (scalar) ; 1 × ; −2=3 ; 1=3 ð ð3Þ ð2Þ Þ Fermion pairs transforming as SU C; SU L Uð1Þ c c ð1 1Þ ð3 1Þ ¼ð3 1Þ Y l d (vector) ; 1 × ; −1=3 ; 2=3 c c ¯ ¯ ¯ lc νc ð1; 1Þ−1 scalar u u (scalar) ð3; 1Þ−2 3 × ð3; 1Þ−2 3 ¼ð3 ; 1Þ−4 3 þð6 ; 1Þ−4 3 LL, = = a = s = c c c c c c ¯ d u , l ν ð1; 1Þ−1 vector u d (vector) ð3; 1Þ−2 3 × ð3; 1Þ−1 3 ¼ð1; 1Þ−1 þð8; 1Þ−1 = = lcuc, uc dc, Q2, L¯ Q¯ , dcνc ð3¯ 1Þ c c ¯ ; 1=3 scalar d d ð3; 1Þ−1 3 × ð3; 1Þ−1 3 ¼ð3 ; 1Þ−2 3 þð6 ; 1Þ−2 3 (scalar) = = a = s = c c ð6 1Þ u d , QQ ; 1=3 scalar dcdc c νc ð3 1Þ , u ; 2=3 scalar dclc ¯ cνc ð3 1Þ , LQ, u ; 2=3 vector c c ucuc c lc ð3 1Þ e.g., the combination u d can connect to vector bosons in , d ; −4=3 scalar ð3Þ c c c c ð1 1Þ two different SU c representations. ν ν , ν ν ; 0 scalar ¯ ¯ Similarly, Table II lists all possible combinations of three LL, QQ, lclc, dcdc, ucuc, νcνc ð1; 1Þ0 vector ð2Þ ¯ c c c c ð8 1Þ SU L-singlet SM fermions. The different new-physics QQ, d d , u u ; 0 vector fermions that can make up Topology 2 must have the same quantum numbers as the combinations listed in the table. This list is exhaustive, and to get all possible diagrams, complete list, see Tables I, II, and III in [19]). Unlike one needs to consider all allowed, distinct permutations of the all-singlets operator, all other dimension-nine operators the triplets. In order to realize Topology 2, for each such saturate the constraints associated to nonzero neutrino combination, one needs to consider the possible ways of masses for Λ values that translate into tiny rates for arranging fermion pairs, listed in Table I. It can be shown μ− → eþ-conversion, see Fig. 7 in [19]. that this yields eighteen different “minimal” realizations of In order to systematically address this issue, we list all Topology 2, not considering the different representations the relevant SM fermion pairs that transform in the same for the same combination of SM fermions. αβ way in Table III. The pairs relevant for Os are shown in Next, we want to ensure that, at the tree level, the bold. From the table, one can see that a new particle that different new-physics scenarios lead to the all-singlets couples to, e.g., lc with uc or d¯c can also couple to L¯ Q¯ , operator but not to other dimension-nine (or lower dimen- and so on. The table reveals that there are two avenues for sional) LNV operators. New particles with the same avoiding unwanted couplings. One is to have one of the quantum numbers as some of the combinations in new bosons couple to the pair lclc, which is not degen- Table I can also couple to pairs of SM fermions that erate, quantum-number-wise, with any other pair of SM ð2Þ contain the SU L-doublets L, Q. For example, the pair fermions. The other is to add a new fermion and a new lc c ð3¯ 1Þ u transforms like a ; 1=3. A scalar that couples to this boson such that lc couples to them in Topology 2. The ¯ ¯ pair of SM fermions can also couple to L Q, since the latter reason for this is that all other pairings involving lc have an has identical quantum numbers. These new bosons would unwanted “match,” see Table III. This extra requirement lead to, along with the all-singlets operator, other six- drastically reduces the total number of minimal models for fermion operators, including ðL¯ Q¯ ÞðL¯ Q¯ Þðdc dcÞ (for a the two topologies, and allows us to write down all possible

ð2Þ lc c TABLE II. Quantum numbers of all the possible triplets of SU L gauge singlet Standard Model fermions , u , dc (with at most two identical fields).

Triplets Representation under ðSUð3Þ ; SUð2Þ Þ ð1Þ C L U Y lclc c ð1 1Þ ð1 1Þ ð3¯ 1Þ ¼ð3¯ 1Þ u ; 1 × ; 1 × ; −2=3 ; 4=3 c c c ð1 1Þ ð1 1Þ ð3 1Þ ¼ð3 1Þ l l d ; 1 × ; 1 × ; −1=3 ; 5=3 lc c c ð1 1Þ ð3¯ 1Þ ð3¯ 1Þ ¼ð3 1Þ þð6¯ 1Þ u u ; 1 × ; −2=3 × ; −2=3 a; −1=3 s; −1=3 c c c ð1 1Þ ð3 1Þ ð3 1Þ ¼ð3¯ 1Þ þð6 1Þ l d d ; 1 × ; −1=3 × ; −1=3 a; 1=3 s; 1=3 c c c ð1 1Þ ð3¯ 1Þ ð3 1Þ ¼ð1 1Þ þð8 1Þ l u d ; 1 × ; −2=3 × ; −1=3 ; 0 ; 0 c c c ð3¯ 1Þ ð3¯ 1Þ ð3 1Þ ¼½ð3 1Þ þð6¯ 1Þ ð3 1Þ u u d ; −2=3 × ; −2=3 × ; −1=3 a; −4=3 s; −4=3 × ; −1=3 ¼ð3¯ 1Þ þð6 1Þ þð3¯ 1Þ þð15¯ 1Þ ; −5=3 ; −5=3 ; −5=3 ; −5=3 c c c ð3¯ 1Þ ð3 1Þ ð3 1Þ ¼ð3¯ 1Þ ½ð3¯ 1Þ þð6 1Þ u d d ; −2=3 × ; −1=3 × ; −1=3 ; −2=3 × a; −2=3 s; −2=3 ¼ð3 1Þ þð6¯ 1Þ þð3 1Þ þð15 1Þ ; −4=3 ; −4=3 ; −4=3 ; −4=3

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TABLE IV. All new particles required for all different tree-level Table IV. Explicitly, they are (1) ζΦΣ, (2) χΔΣ, (3) ψΔΦ, Oαβ αβ realizations of the all-singlets dimension-nine operator s , and (4) ΦΣΔ. The first three realize Os via topology 2 αβ according to the restrictions discussed in the text. All particles [Fig. 5(right)] while the last one realizes O via topology 1 are SUð2Þ singlets. The fermions ψ, ζ, and χ come with a partner s L [Fig. 5(left)]. (ψ c, ζc, and χc respectively), not listed. We do not consider fields that would couple to the antisymmetric combination of same- flavor quarks since these cannot couple quarks of the same 1. Model ζΦΣ generation. Here, the SM particle content is augmented by a couple ¯ New particles ðSUð3Þ ; SUð2Þ Þ ð1Þ Spin ζ ≡ ð3 1Þ ζc ≡ ð3 1Þ C L U Y of vectorlike fermions ; −5=3 and ; 5=3, Φ ∼ ð1 1Þ Φ ≡ ðlc lcÞ ð1; 1Þ−2 Scalar the color-singlet scalar ; −2, and the colored c c ð6 1Þ scalar Σ ∼ ð6; 1Þ4 3. The most general renormalizable Σ ≡ ðu u Þ ; 4=3 Scalar = c c ð6 1Þ Δ ≡ ðd d Þ ; −2=3 Scalar Lagrangian is c c C ≡ ðu d Þ ð1; 1Þ1; ð8; 1Þ1 Vector c c c c ψ ≡ ð c c cÞ ð3¯ 1Þ LζΦΣ ¼ L þ L þ yΦαβΦlαl þ yΣ Σu u u l l ; 4=3 Fermion SM kin β u ¯ ζ ≡ ð c c cÞ ð3; 1Þ Fermion c c c c d l l −5=3 þ yΦζc Φζ d þ yΣζΣζd þ mζζζ c c c ¯ χ ≡ ðl u u Þ ð6; 1Þ−1 3 Fermion = þ VðΦ; Σ; 0ÞþH:c:; ð3:1Þ N ≡ ðlcdcucÞ ð1; 1Þ0, ð8; 1Þ0 Fermion L L where SM is the SM Lagrangian, kin contains the kinetic- energy terms for the new particles, and VðΦ; Σ; 0Þ is the UV completions with no more than three new particles. The Φ Σ final allowed combinations and the corresponding new most general scalar potential involving the scalars , , particles are listed in Table IV. The list is exhaustive, and all written out explicitly in Appendix A. By design, lepton Oαβ number is violated by two units but it is conserved in the possible UV completions of s at the tree level can be limit where any of the new Yukawa couplings vanishes. On implemented with a subset of less than or equal to three of the other hand, baryon number is conserved. In units where these particles. the quarks have baryon-number one, Σ can be assigned It is also important to consider whether new interactions þ2 ζ ζc −1 þ1 ð2Þ νc baryon-number , ; baryon-number ; , respec- would materialize if neutrino SU L-singlet fields, , Φ νc tively, and baryon-number zero. were also present. Pairings that include are also included Oαβ in Table III. Given all constraints discussed above, there are It is easy to check that this model realizes s via no new couplings involving νc other than the neutrino topology 2 [Fig. 5(right)] and c Yukawa coupling and ν Majorana masses for new-physics μ gαβ yΦαβyΦζc yΣζyΣu models that do not contain the vector C ∼ ð1; 1Þ1 field. ≡ : ð3:2Þ μ 5 2 2 In models that contain C , one need also consider the Λ MΦMΣmζ lcσ¯ μνc interaction term i Cμ. We return to the left-handed antineutrinos and the mechanism behind neutrino masses Here, yΦαβ controls the lepton-flavor structure of the model. μ− → þ j j2 in Sec. VI. e -conversion rates are proportional to yΦeμ , while 2 In the following subsections we list all the different those for 0νββ are proportional to jyΦeej . models. We divide them into different categories. Some The new-physics states will also mediate CLFV phe- models contain new vector bosons, others contain only nomena, sometimes at the tree level. In what follows, new-physics scalars or fermions. Since all new particles we write down the effective operators that give rise to need to be heavy, including potential new vector bosons, different CLFV processes, and estimate bounds on the no-vectors models are easier to analyze since, as is well- effective scales of these operators. The CLFV observables known, consistent quantum field theories with massive of interest are: ∓ vector bosons require extra care. There are, altogether, eight (1) μ → e e e decay: The effective Lagrangian models: four with and four without new massive vector giving rise to this decay, generated at the tree fields. We discuss the no-vectors models first. We will also level, is broadly distinguish models based on whether they also lead yΦ μy to the violation of baryon-number conservation and L ¼ e Φee ðμc cÞð c cÞ ð Þ μ→3e 2 e e e ; 3:3 whether any flavor-structure naturally arises. MΦ

and the relevant Feynman diagram is depicted in A. No-vectors models the left panel of Fig. 6. The strongest bounds on Here, all no-vectors models are discussed in turn. Models μþ → eþe−eþ come from the SINDRUM spectrom- are named according to the new-physics field content, see eter experiment [36]:

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FIG. 6. Feynman diagrams contributing to the CLFV processes μ → 3e (left), μþ → eþγ (middle), and μ− → e−-conversion (right) in model ζΦΣ.

Brðμþ → eþe−eþÞ < 1.0 × 10−12: ð3:4Þ as depicted in the right panel of Fig. 6. The effective Lagrangian can be estimated as Assuming the phase-space distributions are similar μ ðμ → ν¯ ν Þ to those of ordinary -decay e e μ , this y yΦ βyΦζc y c L ¼ Φμβ e Φζ ðμc cÞð c cÞ ð Þ translates into [37,38] μ→e 16π2Λ2 e d d ; 3:9 j j2 yΦeμyΦee −22 −4 ≤ 1.4 × 10 GeV ; ð3:5Þ where Λ is the effective scale, a function of MΦ and M4 Φ Mζ, and the lepton-index β is summed over. Note that this operator is also sensitive to the yΦeτ and or MΦ ≥ 290 TeV for Oð1Þ couplings. The Mu3e yΦμτ couplings. An extra contribution comes from experiment, under construction at PSI, aims to reach the middle panel of Fig. 6, where the photon is put sensitivities better than 10−15 on this channel [39] and offshell, and radiates a qq¯ pair. hence sensitivity to Φ-masses around 1000 TeV [7]. þ þ The SINDRUM II experiment at PSI constrains (2) μ → e γ decay: At the one-loop level, Φ-exchange − − μ → e -conversion in gold [33]: also mediates, as depicted in the middle panel of Fig. 6, μþ → eþγ. The effective operator governing Γðμ− þ → − þ Þ μ → eγ is Au Au e Au Rμ−e− ≡ − Γðμ þ Au → νμ þ PtÞ ð2 Þ yΦμμyΦμe e yμ αβ −13 L þ þ ¼ ð ¯ Þσ c ð Þ 7 10 ð90% Þ ð Þ μ →e γ 2 2 LH e Fαβ; 3:6 < × CL : 3:10 16π MΦ

where yμ is the muon Yukawa coupling. Experi- Using Eq. (2.7), we estimate [37] mentally, the most stringent constraints come from pffiffiffi 2 the MEG experiment at PSI [40] 2 yΦμβyΦeβyΦζc yΦζc − − ¼ ð Þ Rμ e 2 2 : 3:11 GF 16π Λ Brðμþ → eþγÞ¼4.2 × 10−13: ð3:7Þ Oð1Þ Λ ≥ 30 Using results from [41], we get For couplings, this yields TeV. Stronger sensitivity is expected from the next-gen- 2 eration experiments COMET [15], DeeMe [16], jyΦμμyΦμej ðμþ → eþγÞ ≈ 5 3 10−6 and Mu2e [17], as discussed in the introduction. Br . × 4 ð Þ MΦ TeV Ultimately, one would be sensitive to Λ scales up to < 4.2 × 10−13; ð3:8Þ a few 100 TeV. (4) Muonium-antimuonium oscillations ðμþe− → μ−eþÞ: − þ which leads to MΦ ≳ 60 TeV, given Oð1Þ cou- Muonium ðMuÞ is the bound state of an e and a μ , plings. As expected, the μþ → eþγ bound is weaker whereas its anti-partner, the antimuonium ðMuÞ is than that of μ → 3e as the former is loop-suppressed. the bound state of an eþ and a μ−. Muonium- The upgraded MEG-II experiment plans to reach antimuonium oscillation is a process where muonium a sensitivity of 10−14 with three years of data converts to antimuonium, thereby changing both taking [42]. electron-number and muon-number by two units (3) μ− → e−-conversion in nuclei: In this model, [37]. Here, the effective Lagrangian governing this μ− → e−-conversion occurs at the one-loop level, process at the tree level is

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yΦμμy ≳ 734 L ¼ Φee ðμcμcÞð c cÞ ð Þ the discrepancy, which leads to MΦ GeV, Mu−Mu 2 e e : 3:12 Oð1Þ MΦ given the couplings. This bound is weaker than most of the previous ones discussed here. The The probability that a Mu bound state at t ¼ 0 Muon g − 2 experiment, currently taking data at is detected as a Mu bound state at a later time Fermilab, is ultimately expected to improve on the ð Þ 2 uncertainty of the muon g − 2 by roughly a factor is proportional to yΦμμyΦee =MΦ. The upper limit quoted by the PSI experiment [43] yields of two [51]. 2 A subset of the bounds estimated here is summarized in ðyΦμμyΦ Þ=MΦ ≲ 0.002G or ee F Fig. 11. Not surprisingly, if all couplings of interest are of order one, constraints from μ → 3e are the strongest and jyΦμμy j Φee ≤ 2 5 10−8 −2 ð Þ 2 . × GeV ; 3:13 translate into MΦ values that exceed hundreds of TeV.CLFV MΦ observables do not constrain, directly, mζ or MΣ, while − − searches for μ → e -conversion are sensitive to both MΦ which implies MΦ ≥ 6.3 TeV for Oð1Þ couplings. and mζ. Since both ζ and Σ are colored, we expect LHC (5) Lepton–lepton scattering: Φ-exchange will also mediate intermediate and high-energy scattering searches for exotic fermions or scalars to constrain, con- þ − þ − servatively, mζ, MΣ ≳ 500 GeV. We return to this issue processes including e μ → e μ , e e → e e , þ − þ − briefly in Sec. IV. Putting it all together, if all new-physics and e e → μ μ .IfMΦ is much larger than the couplings are of order one, searches for CLFV imply upper center-of-mass-energies of interest, the following μ− → þ tree-level effective Lagrangian applies: bounds on the rate for e -conversion that are much stronger than the sensitivity of next-generation experiments. yΦ μy Most of the CLFV bounds can be avoided, along with L ¼ e Φeμ cμc ¯c μ¯c ð Þ eμ 2 e e 3:14 those from 0νββ, if the flavor-structure of the new physics MΦ is not generic. In particular, in the limit where yΦeμ is much larger than all other yΦαβ couplings, most of the constraints Measurements of σðeþe− → μþμ−Þ and þ − þ − above become much weaker. This can be understood by σðe e → e e Þ at LEP [44] can be translated into − þ noting that μ → e -conversion preserves an Lμ − L constraints on the effective scale of the operator e (muon-number minus electron-number) global symmetry above, while the physics processes μ → 3e, μ → eγ, μ → e - conversion, and 0νββ all violate Lμ − Le by two units, yΦeμyΦ μ 4π yΦ y 4π e ≤ and ee Φee ≤ ; ð3:15Þ − − 2 2 Λ2 2 2 2Λ2 while Mu Mu-oscillations violate Lμ Le by four units. MΦ μ MΦ e c c In other words, if only the Φμ e -coupling yΦμe is nonzero, the new-physics portion of the Lagrangian respects an where Λμ ≈ 9.3 TeV and Λe ≈ 8.9 TeV. These Lμ − L global symmetry and all CLFV bounds vanish to a translate into MΦ ≳ 2.5 TeV, given Oð1Þ couplings. e (6) Anomalous magnetic moments: There is a well- very good approximation. The flavor-diagonal constraints − 2 known discrepancy between the experimental value from LEP and the muon g do, however, apply, but are [45] and the SM prediction [46,47] of the anomalous of order 1 TeV for yΦμe of order one, much less severe. This magnetic moment of the muon, 10.1 × 10−10 < is a property of all new-physics scenarios that contain the exp SM −10 Φ-field since, in these scenarios, the only coupling of the aμ − aμ < 42.1 × 10 at the 2σ level. The leptons to the new degrees-of-freedom is the one to Φ. doubly charged Φ-scalar will contribute to the muon − 2 (g ) at the one-loop level. The corresponding 2. Model χΔΣ Feynman diagrams are quite similar to the middle panel of Fig. 6 with the external electron replaced by Here, the SM particle content is augmented by a couple Φ χ ∼ ð6¯ 1Þ χc ∼ ð6 1Þ a muon. In the limit is much heavier than muons of vectorlike fermions ; −1=3 and ; 1=3, and Σ ∼ ð6 1Þ Δ ∼ ð6 1Þ and electrons, the resulting contribution is [48] (see two colored scalars ; 4=3 and ; −2=3. The also [49,50]) most general renormalizable Lagrangian is 2 mμðyΦ μyΦ μ þ yΦμμyΦμμÞ L ¼ L þ L þ Σ c c þ Δ c c þ Δχ¯ χ¯ Δ ¼ − e e ð Þ χΔΣ SM kin yΣu u u yΔd d d yΔχ aμ 2 2 : 3:16 6π MΦ c c ¯ c c c þ yΔχc Δχ χ þ yΣαΣχ lα þ yΔαΔχlα c The negative sign of the contribution indicates that þ mχχχ þ Vð0; Σ; ΔÞþH:c:; ð3:17Þ this type of new physics will not help alleviate L L the discrepancy. We can, nonetheless, derive a limit where SM is the SM Lagrangian, kin contains the kinetic- from the g − 2 measurement by requiring the energy terms for the new particles, and Vð0; Σ; ΔÞ is the absolute value of the contribution to be less than most general scalar potential involving the scalars Δ, Σ,

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αβ written out explicitly in Appendix A. The operator Os is realized at the tree level with topology 2, and the effective scale is given by

gαβ yΣ yΣαyΔβy ≡ u Δd ð Þ 5 2 2 : 3:18 Λ MΣMΔmχ

The μ− →eþ-conversion rates are proportional to j þ j2 0νββ FIG. 8. Feynman diagrams (box-diagrams) contributing to the yΣeyΔμ yΣμyΔe , while those for are proportional CLFV process μ− → e−-conversion, in model χΔΣ. 2 to jyΣeyΔej . Like the previous example, this model also allows for a μ− → − rich set of CLFV processes. The CLFV observables of (3) e -conversion in nuclei: In this model, μ− → − interest are: e -conversion also occurs at the one-loop (1) μþ → eþγ decay: This is generated at the one-loop level, as depicted in Fig. 8. The effective Lagrangian level, as depicted in the left panel of Fig. 7. There is a can be estimated as Δ χc similar diagram with and in the loop. The   y yΔ y yΔ y y yΣ yΣ effective Lagrangian for this process is L ¼ Δμ d Δd e þ Σμ Σu u e μ→e 2 2 2 2 16π ΛΔχ 16π ΛΣχ ð2 Þ yΔμyΔe e yμ ¯ αβ c ðμc cÞð c cÞðÞ Lμþ→ þγ ¼ ðLHÞσ e Fαβ; ð : Þ × e d d 3:21 e 16π2Λ2 3 19 where the subscripts on Λ denotes the dependence where Λ is a function of MΔ and mχ. The bounds on the masses of the new particles. As in the for this model are similar to the ones calculated previous model, this process can also proceed in Eq. (3.8). through the transition-magnetic-moment channel, (2) μ → eee∓ decay: Unlike the previous model, where the photon emits a quark-antiquark pair. here μ → 3e only occurs at the one-loop level. One The bounds arising on the effective scale for Oð1Þ contribution is obtained from the diagram in the left couplings are similar to ones obtained in the pre- panel of Fig. 7, where the photon is off-shell and can ζΦΣ “ ” þ − vious model (model ). There exists a dimension- decay into an e e pair. As far as this contribution c c c c c c μ → 3 ten operator ðd d d d l ½σ · ∂l Þ, which can be is concerned, the rate for e is suppressed μ− → − relative to that for the μ → eγ decay. There are dressed as the process n ne . The relevant also box-diagrams, including the one depicted in the amplitude is right panel of Fig. 7, which could also contribute yΔ y y Δy significantly. Figure 7(right) gives rise to the effec- A ¼ d Δd e μΔ ð c c c c lc½ σlcÞð Þ 4 2 d d d d p· 3:22 tive Lagrangian MΔmχ

yΔμyΔeyΔeyΔe c c c c where p is the typical four-momentum associated Lμ→3 ¼ ðμ e Þðe e Þ; ð3:20Þ e 16π2Λ2 to the process. Note that there is an analogous contribution to pμ− → pe−. This will also mediate − − where Λ is a function of MΔ and mχ. Using μ → e -conversion in nuclei. However, this is an Eq. (3.4), current data constrain Λ ≥ 23 TeV assum- effective operator of very high energy-dimension ing order one couplings. A similar box-diagram and hence suppressed. exists with χc and Σ in the loop; its contribution turn (4) Muonium-antimuonium oscillations and lepton scat- out to be of the same order. tering: Unlike the previous model (model ζΦΣ), this model does not allow for tree-level muonium- antimuonium oscillation, or lepton–lepton scatter- ing. One can, of course, have these processes at the one-loop level through diagrams like the right panel of Fig. 7. The bounds arising from these processes are not expected to be competitive with the other leptonic bounds. (5) Anomalous magnetic moments: there is a new- FIG. 7. Feynman diagrams for the CLFV processes μ → eγ physics contribution to the anomalous magnetic (left) and μ → 3e [box-diagram] (right), in model χΔΣ. moment of the muon and the electron at one-loop

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(e.g., a Δ, χ loop). The situation here is very similar Using Eq. (3.23) [53], to the one discussed in model ζΦΣ. 2 A subset of the bounds estimated here are summarized in yΣ y mΣΔ h¯j j i¼ u Δd h¯jð c c cÞ2j i Fig. 11. As in the previous model, in the absence of flavor- n Heff n 4 2 n u d d n MΔMΣ structure in the new-physics sector, CLFV constraints, 2 yΣ y mΣΔ along with those from 0νββ-searches, overwhelm the ¼ u Δd Λ6 ; ð : Þ − þ 4 2 QCD 3 25 sensitivity of future searches for μ → e -conversion. In MΔMΣ this model, it is also possible to consistently assign Lμ − Le charges to the heavy fields and therefore eliminate the where we estimate the nucleon matrix-element Λ Oð1Þ processes listed above. For example, if we assign charge to be of order QCD. Assuming couplings, Λ ¼ 180 ∼ ∼ ∼ Λ þ1 to χ and charge −1 to χc, only μc couples to χ and only QCD MeV, and mΣΔ MΣ MΔ , this ec couples to χc. This can automatically prevent the above translates into processes from taking place with a sizable rate. Note that this charge assignment will render some of the other new- Λ ≳ 350 TeV: ð3:26Þ physics couplings zero, e.g., yΔχ and yΔχc . Unlike model ζΦΣ, here baryon number is explicitly (2) BNV processes with LNV: The model also allows violated. We note that the Lagrangian Eq. (3.17) has an for BNV processes that violate lepton number accidental Z2 symmetry under which all lepton-fields, related to the effective dimension-twelve operator ¯ 2 2 along with χ and χc, are odd. This implies that nucleon ðdcdcdclcÞ and ðucucdclcÞ , including nn → þ þ − − þ þ decays into leptons are not allowed (e.g., p → π0 þ eþ or π π e e , and pp → e e . These are expected n → π0 þ ν) and, for example, the proton is stable. There to be more suppressed given the high energy- are, nonetheless, a few relevant baryon-number-violating dimension of the effective operator. We qualitatively estimate that existing experimental bounds on pp → (BNV) constraints: þ þ (1) Neutron-antineutron ðn − n¯Þ oscillations: at the tree e e [5] translate into Λ ≳ 1 TeV. – The n − n¯-oscillation bound also outshines the sensitiv- level, the model mediates neutron antineutron os- − þ cillations, which violate baryon number by two ity of future μ → e -conversion experiments and cannot units, as depicted in Fig. 9. The effective Lagrangian be avoided by allowing a non-trivial flavor structure to the for such a process is the dimension-nine operator new-physics since we are especially interested in first- generation quarks. We do note that tree-level BNV proc- 2 esses vanish in the limit mΣΔ → 0 and hence can be yΣuyΔdmΣΔ c c c 2 L −¯ ¼ ðu d d Þ : ð3:23Þ suppressed if mΣΔ is smaller than the other mass-scales n n M4 M2 Δ Σ in the theory. The reason is as follows. If we assign baryon number þ2=3 to Σ and Δ and 1=3 to χ, χc (in units where Here mΣΔ is a parameter in the scalar potential, see the quarks have lepton number 1=3), baryon number is Appendix A. The Institut Laue-Langevin (ILL) violated by the interactions proportional to yΣα, yΔα—by experiment at Grenoble yields the best bounds on one unit—and mΣΔ—by two units. Furthermore, if we free n − n¯ oscillations using neutrons from a reactor assign lepton-number zero to all the new-physics fields, source [52]. The bounds are typically quoted on the lepton number is violated by yΣα, yΔα—by one unit. This transition matrix element of the effective Hamilto- − ¯ δ ¼h¯j j i means that if mΣΔ is zero n n-oscillation requires one to nian, m n Heff n and are rely on the interactions proportional to yΣα, yΔα, which also 1 create or destroy leptons. Since there are no leptons in τ ≡ ≳ 108 ð Þ n − n¯-oscillation, these interactions contribute to it only at n−n¯ jδ j sec : 3:24 m the loop level. In this case, we still expect strong bounds on Λ ≳ 100 TeV, similar to the one-loop contribution discussed in the next model (model ψΔΦ). These can be ameliorated by judiciously assuming a subset of new- physics couplings is small.

3. Model ψΔΦ Here, the SM particle content is augmented by a ψ ∼ ð3¯ 1Þ ψ c∼ couple of colored vectorlike quarks ; 4=3 and ð3 1Þ Δ ∼ ð6 1Þ ; −4=3, a colored exotic scalar ; −2=3, and a FIG. 9. Tree-level Feynman diagram that mediates n − n¯ doubly-charged scalar Φ ∼ ð1; 1Þ−2. The Lagrangian is oscillations in model χΔΣ. given by

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L ¼ L þ L þ Φlclc þ Δ c c ψΔΦ SM kin yΦαβ α β yΔd d d ¯ c c c c þ yΦψ Φψ u þ yΔψ Δψu þ mψ ψψ þ VðΦ; 0; ΔÞþH:c: ð3:27Þ

L L where SM is the SM Lagrangian, kin contains the kinetic- energy terms for the new particles, and VðΦ; 0; ΔÞ is the most general scalar potential involving the scalars Δ, Φ, written out explicitly in Appendix A. αβ It is easy to check that this model realizes Os via topology 2 [Fig. 5(right)] and

gαβ yΦαβyΦψ yΔψ yΔ ≡ d : ð3:28Þ − ¯ Λ5 2 2 FIG. 10. One-loop Feynman diagram that mediates n n MΦMΔmψ oscillations in model ψΔΦ.

Here, like in model ζΦΣ, yΦαβ controls the lepton-flavor Λ ≳ 127 ð Þ structure of the model. μ− → eþ-conversion rates are TeV: 3:30 2 proportional to jyΦeμj , while those for 0νββ are propor- 2 As advertised, however, baryon-number violation is pro- tional to jyΦ j . ee portional to λΔΦ¯ and can be suppressed—or eliminated As far as CLFV is concerned, this model is very similar — λ → 0 ζΦΣ completely in the limit ΔΦ¯ , when baryon number is to model since here and there the presence of the a good symmetry of the Lagrangian. Φ doubly-charged scalar determines most of the lepton- As in model χΔΣ, here one can also construct the number conserving phenomenology. Similar to model 2 dimensional-twelve operator ðdcdcdclcÞ which gives rise ζΦΣ, the CLFV bounds can be avoided by assuming the þ þ − − to phenomena like nn → π π e e . Such processes are new-physics couplings are not generic. If the new-physics higher dimensional, and hence expected to be more portion of the Lagrangian respects an Lμ − L global e strongly suppressed. A subset of the bounds, estimated symmetry, all CLFV bounds vanish to a very good here and in the previous subsubsections, are summarized approximation. in Fig. 11. Like model χΔΣ, here baryon number is violated but, χΔΣ “ ”— also like model , there is a Z2 lepton-parity all 4. Model ΦΣΔ lepton-fields are odd and all other fields are even—which implies baryon decays into leptons are not allowed. If we Here, the SM particle content is augmented by assign lepton-number þ2 to Φ, baryon-number þ2=3 to Δ, only scalar fields: a color-singlet doubly-charged scalar c Φ ∼ ð1 1Þ Σ ∼ ð6 1Þ and baryon-number ∓ 1=3 to ψ, ψ , baryon-number- ; −2, and two colored scalars, ; 4=3 λ Δ ∼ ð6 1Þ violating phenomena are proportional to the ΔΦ¯ coupling and ; −2=3. The most general renormalizable in the scalar potential. The same coupling also violates Lagrangian is lepton-number by two units.5 This implies that n − n¯- L ¼ L þ L þ Φlclc þ Δ c c oscillations do not occur at the tree level since BNV ΦΣΔ SM kin yΦαβ α β yΔd d d is always accompanied by LNV. However, at one- þ Σ c c þ ðΦ Σ ΔÞþ ð Þ loop, n − n¯-oscillations can take place, as depicted in the yΣu u u V ; ; H:c: 3:31 Feynman diagram in Fig. 10. It translates into the effective L L Lagrangian where SM is the SM Lagrangian, kin contains the kinetic- energy terms for the new particles, and VðΦ; Σ; ΔÞ is the Φ Δ Σ 2 most general scalar potential involving the scalars , , , y yΦψ yΔψ λ ¯ mΨ L ¼ Δd ΔΦ ð c c cÞ2 ð Þ written out explicitly in Appendix A. n−n¯ 2 4 2 u d d ; 3:29 16π MΔΛ This is the only no-vectors model where the effective αβ operator Os is realized at the tree level through topology 1, where Λ is an effective scalar arising out of the masses of Δ, and the effective scale is given by Φ and ψ. Assuming all couplings are Oð1Þ and all mass gαβ yΦαβyΣ y m scales are of the same order, current experimental bounds ≡ u Δd ΔΣΦ ð Þ 5 2 2 2 : 3:32 translate into Λ MΦMΔMΣ

5 ζΦΣ ψΔΦ From this perspective, the yΦψ -coupling violates lepton Here, like in model and model , yΦαβ controls the number by two units. lepton-flavor structure of the model. μ− → eþ-conversion

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FIG. 11. Summary of the most stringent CLFV, BNV, and lepton-scattering bounds on the effective scale of the all-singlets operator for the different models discussed in the text. These bounds assume that all new physics couplings are of order one and all new physics masses are approximately the same. Other bounds are discussed in the text. Bounds from the observables in the grey area can be softened or eliminated if the new-physics couplings have a very non-generic lepton-flavor structure (e.g., if the new-physics model obeys, at least approximately, an Lμ − Le symmetry, as discussed in the text). Bounds from n − n¯-oscillations, in the blue area, can be softened or eliminated if the new-physics couplings are chosen in a way that baryon number is at least approximately conserved. Note that model ζΦΣ conserves baryon number and hence does not contribute to n − n¯ oscillations. In the limit where the masses of the new particles are heavy, there are independent hadron collider bounds similar to those from LEP. The expected sensitivity of the Mu2e experiment is indicated by the solid line.

2 νc rates are proportional to jyΦeμj , while those for 0νββ are antineutrino fields exist, the following coupling is also 2 μ lcσ¯ μνc proportional to jyΦeej . The CLFV phenomenology here is allowed, for the color-singlet C : Cμ . We return to very similar to the one in model ζΦΣ and model ψΔΦ. the issue of generating neutrino masses in Sec. VI. If we choose to assign lepton-number þ2 to Φ and Quantum field theories with massive vector bosons, baryon-number þ2=3 to both Σ and Δ, all LNV and BNV in general, have severe problems in the ultraviolet. The couplings are in the scalar potential. Some couplings models presented here are no exception. The most general violate only baryon number (e.g., mΣΔ), some violate only “UV-complete” Lagrangians we will be considering are, in 6 lepton number (e.g., mΔΣΦ), while others violate both fact, not really UV-complete as, for example, we expect the (e.g., λΔΦ¯ ). This means that BNV phenomena can occur at scattering of longitudinal vector bosons to violate partial- the tree level, like in model χΔΣ. Indeed, n − n¯-oscillations wave unitarity in the ultraviolet, indicating that a proper occur at the tree level via the Feynman diagram in Fig. 9. UV-completion of the theory is required. As is well known, A subset of the bounds, estimated in the previous sub- there are a few possible ways to UV-complete theories subsections, are summarized in Fig. 11. Here too BNV with massive vector bosons. They could, for example, be phenomena are controlled by a different set of couplings composite objects of some confining gauge theory. In the as LNV ones, and can be “turned off” by imposing baryon scenarios discussed here, since the vector-boson Cμ carries number as a conserved, or approximately conserved, electric-charge (and hypercharge) and, in some cases, color, symmetry. some of the fundamental fields of the UV theory must transform nontrivially under the SM gauge symmetry. μ B. Models with a new vector boson Another possibility is that C is a gauge boson associated to some broken gauge symmetry. The fact that Cμ is As discussed earlier and summarized in Table IV, there charged and potentially colored makes the construction are two different vector bosons capable of realizing the of UV-complete models nontrivial. Below—in model all-singlets operator at the tree level in a way that other NC—we explore in a little more detail the possibility that LNV operators are also avoided. These are a color-singlet Cμ W ð1 1Þ the color-singlet may be the R-boson in left-right with hyper-charge one [ ; 1] or a color-octet with symmetric extensions of the SM. hypercharge one [ð8; 1Þ1]. We will refer to both of them μ All models are listed below. It turns out that, unlike the as Cμ. The only allowed couplings of C to SM fermions no-vectors models, all of them conserve baryon number. c μ c is Cμd σ¯ u (see Table IV). If, however, left-handed Phenomenologically, most of the models give rise to the CLFV processes already discussed before and the bounds 6 αβ Note that the effective coupling of Os , Eq. (3.32),is and challenges one needs to address are very similar to proportional to mΔΣΦ. those of no-vectors models. For this reason, we do not

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L ¼ L þ L þ Φlclc þ cσ¯ μ c elaborate on experimental bounds but, for the most part, ζΦC SM kin yΦαβ α β gCduCμd u concentrate on whatever unique features the different þ Φζc c þ cσ¯ μζ þ ζζc models possess. yΦζc d gCuζCμu mζ þ VðΦ;CÞþH:c:; ð3:35Þ 1. Models ΦC L L where SM is the SM Lagrangian, kin contains the kinetic- Here, the SM particle content is augmented by a charged- energy terms for the new particles, and VðΦ;CÞ is the Φ ∼ ð1 1Þ μ ∼ ð8 1Þ scalar ; −2, and a vector C ; 1. The most vector-scalar potential listed in Eq. (A2) in Appendix A. general renormalizable Lagrangian is The coefficient of the all-singlets operator is

2 L ¼ L þ L þ Φlclc þ cσ¯ μ c gαβ yΦαβg m Φ yΦαβgCdugCuζyΦζc ΦC SM kin yΦαβ α β gCduCμd u ≡ Cdu C þ ð Þ 5 2 4 2 2 : 3:36 Λ M M M M mζ þ VðΦ;CÞþH:c:; ð3:33Þ Φ C Φ C Instead, we could add to the particle content of L L Φ ψ ∼ ð3¯ 1Þ where SM is the SM Lagrangian, kin contains the kinetic- model C a pair of vectorlike quarks ; 4=3 and ðΦ Þ ψ c ∼ ð3 1Þ energy terms for the new particles, and V ;C is the ; −4=3. The most general Lagrangian in this case vector-scalar potential listed in Eq. (A2) in Appendix A. is, assuming Cμ is a color-octet vector-boson, This is the simplest model as far as its particle content is c c ¯c μ c concerned. Lepton number can be assigned to the various LψΦ ¼ L þ L þ yΦαβΦlαlβ þ g Cμd σ¯ u μ C SM kin Cdu fields in a way that the term CμC Φ in the vector-scalar þ Φ¯ ψ c c þ ψ¯ σ¯ μ c þ ψψc potential violates it by two units (Φ lepton-number 2, Cμ yΦψ u gCψdCμ d mψ lepton-number zero). þ VðΦ;CÞþH:c:; ð3:37Þ The all-singlets operator is realized at the tree level via L L topology 1. The effective couplings and scale are where SM is the SM Lagrangian, kin contains the kinetic- energy terms for the new particles, and VðΦ;CÞ is the 2 vector-scalar potential listed in Eq. (A2) in Appendix A. gαβ yΦαβg m Φ ≡ Cdu C : ð3:34Þ Clearly, this is very similar to the model in Eq. (3.35), with Λ5 2 4 MΦMC just the charges for the vectorlike quarks different. Here, the coefficient of the all-singlets operator is Here, like all models that include the Φ-field, yΦαβ 2 μ− → þ gαβ yΦαβg m Φ yΦαβgCdug ψ yΦψ controls the lepton-flavor structure of the model. e - ≡ Cdu C þ C d : ð3:38Þ 2 Λ5 2 4 2 2 conversion rates are proportional to jyΦeμj , while those for MΦMC MΦMCmψ 2 0νββ are proportional to jyΦeej . A very similar Lagrangian describes the model where the In both scenarios one can assign lepton number to the gauge boson is a color-singlet, Cμ ∼ ð1; 1Þ . The only new-physics fields such that both mCΦ and the coupling of 1 Φ — — difference is the presence of an extra interaction between the field to the new fermion and a quark yΦζc or yΦψ μ ¯ μ C and the Higgs doublet, proportional to CμHD H. This violate lepton number by two units. In this way, one can interaction is inconsequential for LNV. control which topology contributes most to the all-singlets Λ5 There are strong constraints on the production of charged operator. Note, however, that both contributions to gαβ= ð 2 2 Þ vector bosons that couple to quarks, which will be are proportional to yΦαβ= MΦMC . discussed later, while, as already mentioned, the color- singlet vector also allows couplings to left-handed anti- 3. Model NC ¯c μ c neutrinos ∝ Cμl σ¯ ν . The new vector-boson Cμ ∼ ð8; 1Þ1 can also be used to generate the all-singlets operator at the tree level if there 2. Model ζΦC and ψΦC are color-octet fermions N ∼ ð8; 1Þ0. In this case, the most We can add a new vectorlike fermion to model ΦC in general renormalizable Lagrangian is such a way that more LNV interactions are allowed and one L ¼ L þ L þ lcσ¯ μ þ cσ¯ μ c generates, at the tree level, the all-singlets operator via both NC SM kin gCNαCμ α N gCduCμd u topologies in Fig. 5. This can be done in two different ways. þ mNNN þ Vð0;CÞþH:c; ð3:39Þ We can add to the particle content of model ΦC a pair ζ ∼ ð3¯ 1Þ ζc ∼ ð3 1Þ L L of vectorlike quarks ; −5=3 and ; 5=3. The where SM is the SM Lagrangian, kin contains the kinetic- most general Lagrangian is, assuming Cμ is a color-octet energy terms for the new particles, and Vð0;CÞ is the vector-boson, potential for the vector field listed in Eq. (A2) in

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Appendix A. One can assign lepton number to the new TABLE V. Fields in model NC assuming Cμ is the right-handed ð2Þ ð2Þ ð1Þ fields, −1 for N, zero for Cμ, such that the Majorana masses WR-boson of an SU L × SU R × U B−L gauge theory. of the color-octet fermions control LNV. The operator ð ð2Þ ð2Þ ð1Þ Þ αβ Particles SU L; SU R; U B−L O is generated at the tree level—topology 2—and its s ≡ ð Þð2 1 1 6Þ coefficient is QL uL;dL ; ; = QR ≡ ðuR;dRÞð1; 2; 1=6Þ 2 ψ ≡ ðν ;e Þð2; 1; −1=2Þ gαβ g g αg β L L L ≡ Cdu CN CN : ð3:40Þ ψ ≡ ðν ;e Þð1; 2; −1=2Þ Λ5 M4 m R R R C N ΔL ≡ scalar (3,1,1) ΔR ≡ scalar (1,3,1) Here, the lepton-flavor structure of the all-singlets Φ ≡ ð2 2 0Þ − þ LR scalar ; ; operator is governed by the couplings gCNα. The μ → e - 2 conversion rates are proportional to jgCNegCNμj , while those for 0νββ are proportional to jg2 j2. ð2Þ ð2Þ ð1Þ CNe SU L × SU R × U B−L. This model requires an ð2Þ ð1Þ → Similar to many of the previous models, CLFV process extended Higgs sector to break the SU R × U B−L are ubiquitous here. However, since μc and ec couple to the ð1Þ U Y. To avoid bounds from collider experiments, this c μ same fields through the operators Cμl σ¯ N, and since the breaking needs to happen at a higher scale so that the new − þ rate for μ → e -conversion requires both gCNe, gCNμ to be gauge bosons WR, ZRs have large-enough mass. Candidate ð2Þ relevant, it is not possible to choose new physics couplings charge-assignments of the particles under SU L × ð2Þ ð1Þ such that most CLFVobservables are relatively suppressed. SU R ×U B−L are listed in Table V, where we associate In this scenario, given several existing experimental con- N to the conjugate of the right-handed neutrino νR. − þ Δ ð2Þ straints, the rates for μ → e -conversion are outside the The vev of R, the SU R scalar triplet, gives Majorana reach of the next-generation experiments. However, it is masses to the right-handed neutrinos, while that of the ð2Þ Δ possible to slightly modify the model to suppress CLFV. of the SU L scalar triplet L contributes to the Majorana Instead of introducing one field N, one can introduce the masses of the left-handed neutrinos. One can construct c Φ pair N and N with the Lμ − Le charges þ1 and −1 Yukawa interactions involving the Higgs bidoublet LR, respectively; in other words, the Lagrangian will include which leads to the LHN Yukawa interaction. In this ana- μþ c μ c μ c μ ≡ l¯cσ¯ μ ¯cσ¯ μ c terms Cμe σ¯ N and Cμμ σ¯ N . The Majorana mass term logy, C WR , and the interactions Cμ N;Cμd u would be forbidden by the global symmetry and replaced are gauge interactions. with the Dirac mass term proportional to NNc. ∼ ð1 1Þ A similar scenario arises with Cμ ; 1 and a gauge- IV. COLLIDER BOUNDS singlet fermion N ∼ ð1; 1Þ0. In this case, a neutrino Yukawa interaction LHN is also allowed and the model is nothing Here we briefly discuss interesting signatures and con- more than the type-I seesaw model [54–59] plus a charge- straints we expect from collider experiments; a detailed one vector boson. This scenario violates the requirements collider study of all models listed in the previous section is we introduced earlier: here, the Weinberg operator ðLHÞ2 is beyond the scope of this paper. All new physics particles generated at the tree level, as in the type-I seesaw model. introduced in the different models are listed in Table IV. It should be pointed out that it is possible to suppress the They include colored vectorlike fermions, charged and tree-level contribution to the Weinberg operator by choos- colored scalars, and charged and colored vector-bosons. Φ þ − → ing very small neutrino Yukawa couplings. In this case, As mentioned earlier, the -scalar will mediate e e þ − þ − → μþμ− the phenomenology is similar to the one discussed in e e or e e in the t-channel. In the limit where Φ the previous models.7 the mass is larger than the center-of-mass energy of the As discussed before, models with a heavy vector-boson collider, these interactions are already constrained by require extra care in order to be rendered consistent in the measurements at LEP [44]. For lighter masses, different, ∼ ð1 1Þ stringent constraints on the new-physics couplings are ultraviolet. In the case of Cμ ; 1, this can be achieved by þ − appreciating that it acts like the right-handed W-boson expected. Future e e colliders under consideration, like W in left-right symmetric models [58,60–63].Infact, the ILC [64], FCC-ee [65], and CEPC [66], would be R sensitive to much higher effective mass-scales. The ILC, for the Lagrangian for Cμ ∼ ð1; 1Þ and the gauge-singlet fer- 1 1000 −1 N ∼ ð1; 1Þ example, with an integrated luminosity of fb ,is mion 0 is a subset of the left-right symmetric Λ Lagrangian, where the SM gauge group is extended to capable of probing new physics scales that are roughly below 75 TeV [64,67] (or MΦ ≲ 20 TeV for order one couplings) through the process eþe− → μþμ−. The exact 7 c c c Here, dimension-seven operators like LμHe u d are also sensitivity would depend on the polarization of the electron generated. These yield large contributions to neutrino masses if the Yukawa couplings are not small. In the case of the color-octet and positron beams as well as systematic uncertainties − − Φ Majorana fermion N ∼ ð8; 1Þ0, Yukawa couplings to the charged at the ILC. An e e collider would be sensitive to leptons do not exist and therefore this is not an issue. s-channel exchange and the properties of Φ could be

075033-16 ACCESSIBLE LEPTON-NUMBER-VIOLATING MODELS AND … PHYS. REV. D 100, 075033 (2019) studied—or constrained—on-resonance if the collider LNV phenomena can also be probed at colliders. The all- energy were high enough. singlets operator will mediate uc uc → ececdc dc scattering, The colored scalars Σ and Δ,andCμ (both the color-singlet as discussed briefly in [24]. Up to color factors and and the color-octet) can be produced at hadron colliders like symmetry factors, the cross section for this process is the LHC through the quark or the gluon channels. For σ ∝ g2s4=Λ10. This can lead to interesting signatures at the example, the dijet channel qq → ΣðΔÞ → qq can be used large hadron collider (LHC) or the international linear 2 2 c c c c to probe the contact interaction ðy =MΣðΔÞÞq q q q ,which collider (ILC) (exchanging the role of the charged-leptons are a valid description of colored-scalar exchange in the limit and the up-quarks). The latter is similar to searches for the − − − − where the scalar masses are beyond the reach of the collider. LNV process e e → W W at lepton colliders, except for Recent dijet studies at ATLAS and CMS [5,68,69] translate the fact that the final-state dijet invariant masses are not into a lower bound on the mass of scalar diquarks [70] and, in related to the W-boson mass [76]. Similar studies could our case, imply masses for Σ, Δ,andCμ that exceed around also be pursued with a muon collider [77]. 5 TeV, for order one couplings. Σ and Δ will also mediate, at the tree level, processes like gg → ΣΣðΔΔÞ → 4q.The V. SUMMARY: VIABILITY OF LARGE μ − → e + corresponding signature is a pair of dijet resonances and can CONVERSION RATES be used to constrain the properties of the colored scalars. The Concerning the viability of different models to mediate corresponding bounds, however, are expected to be weaker − þ observable μ → e -conversion in nuclei at next-generation than those of dijet searches as long as the couplings between experiments, our main results, illustrated in Fig. 11 with the the new bosons and the quarks are order one. Note that the assumptions of a universal mass scale Λ for new particles few TeV upper bound does not trivially apply for smaller and Oð1Þ couplings, can be summarized as follows: couplings and lower masses for Σ and Δ and Cμ, see, for (1) For all the models considered, CLFV and 0νββ example, [68,69]. Relatively light bosons that couple to provide the most stringent bounds on the effective quarks relatively strongly are know to survive collider scale of the all-singlet operator. These bounds, constraints, see e.g., [71]. A detailed analysis of this very Oð10 − 100Þ TeV, are much stronger than the sen- rich topic, as mentioned above, is beyond the scope of sitivity of next-generation μ− → eþ-conversion ex- this paper. periments, Oð10Þ GeV. However, depending on The literature on searches for vectorlike exotic quarks— “ ”— the lepton-flavor structure of the models considered, including octet neutrinos is also large and diverse. it is possible to avoid most of these constaints. One Bounds, many of which are listed and briefly discussed possibility discussed here is that if the new-physics in the particle data book [5], hover around 500 GeV. Lagrangian respects an Lμ − L (muon-number A more detailed discussion of exotic quark searches in the e minus electron-number) global symmetry, then all LHC can be found, for example, in [72]. Existing bounds the CLFV and 0νββ are significantly weakened. depend rather strongly on the decay properties of the exotic (2) For models which explicitly violate baryon number, colored fermions. Model-independent bounds are much the n − n¯-oscillation bound—Oð100Þ TeV—also weaker, as summarized, for example, in [5]. − þ outshines the sensitivity of future μ → e - New colored (and/or charged) particles that couple to the conversion experiments and cannot be avoided by SM Higgs boson will modify the Higgs production rate via allowing a nontrivial flavor structure to the new gluon fusion and the decay rate into two photons, i.e., physics. Even in these cases, one can get remove gg → H → γγ. The doubly-charged scalar Φ also contrib- these bounds by postulating that baryon number is a utes to the decay process H → 4l at tree level. Precision global symmetry of the Lagrangian. measurements of Higgs production and decay will translate (3) The new interactions predicted by the models are into bounds on the properties of Φ, Δ, Σ, and Cμ.In also tightly constrained by LEP and other collider addition, one should also worry about electroweak preci- experiments, which also probe scales (Oð1Þ TeV) sion tests of the SM, although corresponding constraints beyond the sensitivity of future μ− → eþ-conversion might be weaker than direct searches at the LHC. The scalar Φ experiments. These bounds cannot be alleviated by and the vector Cμ, for example, will modify (via triangle taking advantage of symmetry arguments. They can, loop diagrams) the partial decay widths of the Z-boson into however, be weakened by judiciously choosing Φ quarks and leptons. Moreover, if only couples to the pair different couplings (mass-scales) to be relatively μ → μμ ττ e , the universality of Z ee, and will be violated. small (large), as we discuss in the some concrete Finally, as all new charged particles listed in Table IV are scenarios. ð2Þ singlets under SU L, there are no contributions to the oblique parameters (S, T, U) [73,74], as demonstrated in, VI. DISCUSSIONS AND CONCLUDING REMARKS e.g., [75], where contributions from vectorlike down-type quarks to the oblique parameters were shown to vanish if Lepton number and baryon number are accidental ð2Þ these do not mix with the SM SU L quark doublets. global symmetries of the classical SM Lagrangian (and

075033-17 DE GOUVÊA, HUANG, KÖNIG, and SEN PHYS. REV. D 100, 075033 (2019) baryon-number–minus-lepton-number is an accidental scale of the all-singlets operator is lower than the strongest global symmetry of the quantum SM Lagrangian). LNV lepton-number conserving bounds, depicted in Fig. 11. can be probed in a variety of ways, ranging from rare Another important point is that, for example, the LEP nuclear processes to collider experiments. So far, there is no bounds apply to y2=M2 in the limit where M is outside direct evidence for LNV. Nonzero neutrino masses are the direct reach of LEP. The coefficient of Os,however,is often interpreted as evidence for LNV. In most scenarios proportional to y=M2 [see, for example, Eq. (3.2), propor- 2 where this is the case, because neutrino masses are tiny, tional to yΦμe=MΦ, versus Eq. (3.15)], proportional to the rates for LNV processes are way out of the reach of 2 2 yΦμ =MΦ). For smaller coupling and mass and fixed experimental probes of LNV, except for searches for 0νββ. e y2=M2, y=M2 is relatively larger. Finally, strictly speaking, Here, we concentrated on identifying and discussing all estimates here rely on effective theories. For light-enough models where this is not the case and asked whether there new particles and smaller couplings, constraints are, in some are UV-complete models where the rate for μ− → eþ- cases, significantly weaker once translated into the effective conversion in nuclei is close to the sensitivity of next- scale of the all-singlets operator O . generation experiments. All models identified here violate s All of the scenarios discussed here fail, by design, to lepton number at energies scales around one TeV (or lower) explain the observed active neutrino masses. CLFV con- and are best constrained by searches for CLFV, BNV, and straints alone imply that the contribution of these new- 0νββ. BNV bounds are sometimes strongly correlated, physics models to Majorana active neutrino masses are tiny, sometimes not, to the LNV physics. LNV scales that are smaller than what is required by observations by at least low enough so one approaches the sensitivity of future two or three orders of magnitude. In order to accommodate searches for μ− → eþ-conversion in nuclei—along with large active neutrino masses, more degrees-of-freedom, other LNV process we did not discuss, like rare meson different from the ones discussed here, need to be added to decays (e.g., D− → Kþμ−μ−)—require a nongeneric, but the SM particle content. One possibility is to postulate that, often easy to impose, lepton-flavor structure for the new other than the new-physics that leads to the all-singlets physics. In these cases, high-energy hadron and lepton operator at the tree level, there are other sources of LNV, colliders also offer interesting constraints and opportunities perhaps at a much larger energy scale. The high-scale type- for future discovery. I seesaw, with gauge-singlet fermions νc with Majorana In more detail, we identified all UV-complete models masses much larger than the weak scale would do the trick, that realize, at low-energies, the all-singlets dimension-nine for example. Most other models constructed to “explain” O ¼ cμc c c c c operator s e u u d d , identified in [19], and do not small active neutrino Majorana masses should also work realize any other LNV effective operator with similar out fine. In some cases, the two sources of LNV may strength. All new particles—scalars, fermions, and vector “ ” — interfere, as would be the case of the type-I seesaw with bosons are listed in Table IV. Different models consist of any of the models that contain the color-singlet vector the most general renormalizable Lagrangian of the SM plus μ boson C ∼ ð1; 1Þ1. different combinations of two or three of these particles. Another possibility is to postulate that the physics Given a concrete Lagrangian, we estimate the rates for and responsible for the all-singlets operator is the only source existing constraints from many low-energy observables. of LNV. In this case, small neutrino masses can be The bounds presented here are rough estimates. For the accommodated by adding gauge-singlet fermions νc without most part, we assume new-physics couplings to be order a Majorana mass and tiny Yukawa couplings to L and H. one, and assume all new mass scales are of the same order. The absence of the Majorana masses for the left-handed Given the various bounds estimated here, it is fair to ask antineutrinos is natural in the t’Hooft sense: if the LNV whether, for any of the models identified, it is reasonable to μ− → þ parameters in the models discussed here vanish, lepton assume that the rate for e -conversion is within reach number is a good symmetry of the Lagrangian. In this case, of next-generation experiments. The answer, we believe, is neutrinos are pseudo-Dirac fermions since the left-handed affirmative as long as the lepton-flavor structure of the neutrinos and the left-handed antineutrinos both acquire model is not generic and, in some cases, if BNV phenom- small Majorana masses8 on top of the dominant Dirac ena are more suppressed than naively anticipated, i.e., BNV masses. These scenarios are constrained, quite severely, couplings are relatively small. At face-value, flavor- — — — — by solar neutrino experiments see [78,79] since they independent bounds see, e.g., Fig. 11 appear to be mediate neutrino-oscillation processes with long oscillation strong enough to render μ− → eþ-conversion out of exper- imental reach for the foreseeable future. This need not be the case, for a few reasons. One is that the different bounds 8 usually apply only to the masses of a subset of the new- In the case of the right-handed neutrinos, their loop-induced Majorana masses are proportional to the neutrino Yukawa physics particles, while the coefficient of the all-singlets couplings. Since the Yukawa couplings are very small, so are operator depends on the mass of all new degrees-of- the Majorana masses, in spite of the fact that they are associated freedom. If one saturates all existing bounds carefully, the to a relevant operator.

075033-18 ACCESSIBLE LEPTON-NUMBER-VIOLATING MODELS AND … PHYS. REV. D 100, 075033 (2019) lengths. A more detailed analysis is beyond the scope of APPENDIX: SCALAR AND VECTOR-SCALAR this paper. POTENTIALS In summary, UV models which induce Os at the tree- μ− → þ The most general potential involving all the Higgs level can yield a e -conversion rate that is accessible Φ ∼ ð1 1Þ Δ ∼ ð6 1Þ and the new scalars ; −2, ; −2=3 and to future experiments if (i) the UV physics respects, at least Σ ∼ ð6 1Þ ; 4=3, in the no-vectors models, is given by approximately, a lepton-flavor symmetry, such as Lμ − Le, in order to avoid LFV constraints, (ii) the UV physics ðΦ Σ ΔÞ¼μ2 jΦj2 þ μ2 jΣj2 þ μ2 jΔj2 þ λ jΦj4 respects, at least approximately, baryon-number conserva- V ; ; Φ Σ Δ Φ 4 4 2 2 tion, in order to evade BNV bounds, and (iii) the UV model þ λΣjΣj þ λΔjΔj þ λHΦjHj jΦj contains relatively small couplings, especially those that 2 2 2 2 2 2 þ λ ΣjHj jΣj þ λ ΔjHj jΔj þ λΦΣjΦj jΣj govern lepton-flavor-conserving observables, in order to H H 2 2 2 2 2 avoid constraints like those from LEP. þ λΦΔjΦj jΔj þ λΣΔjΣj jΔj þ mΣΔΣΔ ¯ 3 ¯ 2 þ λΔΦ¯ Δ Φ þ mΔΣΦΔΣΦ þ λΔΣΦΔΣ Φ: ACKNOWLEDGMENTS ðA1Þ We would like to thank Jeff Berryman and Kevin Kelly for useful discussions. The work of A.dG. was supported in In the text, we also refer to VðΦ; Σ; 0Þ, Vð0; Σ; ΔÞ, and part by DOE Grant No. de-sc0010143. W. C. H. acknowl- VðΦ; 0; ΔÞ. These are given by Eq. (A1) where the field edges funding from the Independent Research Fund labeled 0 is set to zero. Denmark, Grant No. DFF 6108-00623. J. K. would like Similarly, the most general potential in the vector models to thank Claudia Hagedorn and the Department of Physics, involving the Higgs, the scalar Φ and the vector Cμ is Chemistry and Pharmacy, as well as the Study Travel Fund ðΦ μÞ¼μ2 jΦj2 þ μ2 j j2 þ λ jΦj4 þ λ j j4 of the Faculty of Science at the University of Southern V ;C Φ C Cμ Φ C Cμ Denmark, for their support for his visit at North- þ λ j j2jΦj2 þ λ j j2j j2 western University. He would also like to thank HΦ H HC H Cμ 2 2 μ Northwestern University and the particle theory group þ λΦCjΦj jCμj þ mCΦCμC Φ: ðA2Þ for their hospitality during his stay. M. S. acknowledges support from the National Science Foundation, Grant In the text, we also refer to Vð0;CμÞ. This is given by No. PHY-1630782, and to the Heising-Simons Eq. (A2) where the Φ field is set to zero. Foundation, Grant No. 2017-228. The CP3-Origins centre Throughout, we assume the parameters of the various is partially funded by the Danish National Research scalar and scalar-vector potentials are such that none of the Foundation, Grant No. DNRF90. new-physics scalar fields acquire vacuum expectation values.

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