ULB-TH/16-19

Interpretation of Flavor Violation

Julian Heeck∗ Service de Physique Th´eorique,Universit´eLibre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium

The observation of a charged-lepton flavor violating process would be a definite sign for beyond the , but would actually only prove that one particular linear combination of lepton numbers is violated. We categorize lepton-flavor-violating processes by their quantum num- bers and show how their discovery can be interpreted model independently, studying in particular which processes are required to establish that the entire flavor group is broken. We also comment on total , seeing as lepton number violation practically implies lepton flavor violation as well.

INTRODUCTION than collider signatures. The next decade will see signif- icant improvement of these limits or even a discovery, The classical Lagrangian of the Standard Model (SM) with MEG-II, , , COMET and DeeMe prob- of physics features the global group ing µ eγ, 3e and µ e conversion; LHCb, BES-III and Belle-II→ probing CLFV→ decays of taus and ; U(1)B U(1)Le U(1)Lµ U(1)Lτ associated with the conservation× of × number× B and the three lepton and the (HL-)LHC probing CLFV decays of the Brout– numbers Le,µ,τ . Defining the total lepton number L Englert–Higgs h among other channels. (Limits ≡ Le + Lµ + Lτ , this global symmetry group can also be on LFV Z decays could be improved by many orders of written in the following basis, magnitude at future e+e− colliders, not listed in our ta- bles.) It is thus timely to study how a possible discovery U(1)B+L U(1)B−L U(1)L −L U(1)L +L −2L . 1 × × µ τ × µ τ e can be interpreted. In the following, we will make an (1) effort to study CLFV model independently by focusing In principle this is useful because the linear combina- on the quantum numbers of the various processes. tion B + L is actually broken at the quantum level by non-perturbative processes by six units, specifically 1 3 ∆B = ∆Le = ∆Lµ = ∆Lτ = 1 [1]. os- LEPTON FLAVOR VIOLATION cillations have proven furthermore that lepton flavor, U(1)L −L U(1)L +L −2L , is not conserved in , µ τ × µ τ e Following standard convention, we define CLFV as leaving at most U(1)B−L as an unbroken symmetry [2]. processes that conserve total lepton number L (and B) Practically speaking, however, the B +L violating pro- but violate U(1)Lµ−Lτ U(1)Lµ+Lτ −2Le and do not in- cesses are much too suppressed (at zero temperature) to volve . The× decays in Tabs. I–II have been ever be observable, and the same goes for charged-lepton sorted into (six) groups according to the lepton num- flavor violation (CLFV) induced by non-zero neutrino bers that are violated; this already enables us to make mj and a non-trivial leptonic mixing matrix U. model-independent qualitative predictions: if one pro- For example, Dirac neutrinos lead to ∆(Lβ Lα) = 2 − cess of a given group is observed, all processes of said CLFV at the one-loop level [3], group unavoidably exist, being at least generated at loop 2 level.2 All processes of one group provide the same 2 Γ(`α `βγ) 3αEM ∆mj1 † quantum-number information. The different branching → Uαj U , (2) Γ(` ` ν ν¯ ) ' 32π M 2 jβ ratios within each group are model dependent, as is the α β α β j=2,3 W → question of whether more than one group is observable. which is smaller than 10−53 for all channels and hence (The chiralities of the involved in CLFV [13] completely unobservable. The Glashow–Iliopoulos– and the impact on lepton-flavor conserving observables Maiani mechanism ensures that all neutrino-induced such as g 2 [14] are also model dependent.) arXiv:1610.07623v2 [hep-ph] 30 Jan 2017 − CLFV processes are suppressed by the sub-eV2 neutrino- It is indeed possible that only one of the groups in 2 -squared differences ∆mij, because degenerate Tabs. I–II is observable, i.e. only one linear combination (Dirac) neutrino masses would render U unphysical and of lepton flavors is violated, while the others are con- thus reinstate lepton flavor symmetry. served (outside of the neutrino sector). A concrete model As a result, the group of Eq. (1) is still an excellent approximate symmetry for charged ; the observa- tion of CLFV would imply physics beyond the SM – and beyond neutrino oscillations – with many models being 1 At present there is one tantalizing ∼ 2.5σ hint for CLFV in the able to saturate current limits (see e.g. Refs. [4, 5] for cur- channel√ h → µτ¯ +¯τµ by CMS [6], yet to be confirmed or excluded by s = 13 TeV data [7] or ATLAS [8, 9]. rent reviews). Processes under experimental investiga- 2 This statement has to be modified if light new-physics modes 0 tion are listed in Tabs. I–II, focusing on rare decays rather such as `α → `β Z [10–12] are included in the list (or observed). 2

++ Group Process Current Future (Note that L is conserved if we assign L(k ) = 2, so the above Lagrangian by itself does not lead to (Majo-− µ → eγ 4.2 × 10−13 [15] 4 × 10−14[16] rana) neutrino mass.) The two other ∆(Lα +Lβ 2Lγ ) = µ → eee¯ 1.0 × 10−12 [17] 10−16 [18] − ) = 2 6 groups of Tab. II can be generated analogously by im- µ µ → e conv. O(10−12) [19] 10−17 [20, 21] L posing U(1) on Eq. (3). It is not difficult to con- −4 −4 Lα−Lβ

− h → eµ¯ 3.5 × 10 [22] 2 × 10 [23]

e struct other models that generate only one of the groups −7 L Z → eµ¯ 7.5 × 10 [24] – of Tabs. I–II by making use of the symmetries involved, −12 −12 ∆( had→ eµ¯(had) 4.7 × 10 [25] 10 [26] but this should suffice as a proof of principle. −8 −9 τ → eγ 3.3 × 10 [27] 10 [28] The above discussion is meant to illustrate the some- τ → eee¯ 2.7 × 10−8 [29] 10−9 [28] what trivial point that the observation of one CLFV

) = 2 τ → eµµ¯ 2.7 × 10−8 [29] 10−9 [28] τ process only proves that one linear combination of lep- L τ → e had O(10−8) [30] 10−9 [28] ton numbers is broken, while the other(s) might still be − −3 −3 e h → eτ¯ 6.9 × 10 [22] 5 × 10 [23] conserved. Using without loss of generality the basis of L Z → eτ¯ 9.8 × 10−6 [31] – Eq. (1), the observation of one CLFV process means that ∆( −6 0 had→ eτ¯(had) O(10 ) [32, 33] – U(1)Lµ−Lτ U(1)Lµ+Lτ −2Le is broken to a U(1) ZN × × τ → µγ 4.4 × 10−8 [27] 10−9 [28] subgroup. The observation of a second CLFV process 0 τ → µee¯ 1.8 × 10−8 [29] 10−9 [28] (from a different group) implies that U(1) ZN is fur- ther broken down to × ) = 2 −8 −9

τ τ → µµµ¯ 2.1 × 10 [29] 10 [28] L τ → µ had O(10−8) [30] 10−9 [28] − −2 −3 Z∆(Lµ−Lτ ) Z∆(Lµ+Lτ −2Le) , (5) µ h → µτ¯ 1.2 × 10 [7] 5 × 10 [23] × L Z → µτ¯ 1.2 × 10−5 [34] – ∆( which might contain some redundancies as we will see had→ µτ¯(had) O(10−6) [32, 33] – later (see also Ref. [37] on this topic). All CLFV pro- cesses can be conveniently drawn on this lattice, shown in TABLE I: CLFV with conserved L and B, omitting CP con- Fig. 1, labeled by one representative process (e.g. µ eγ jugate processes. Current limits on the branching ratios are → stands for all ∆(Le Lµ) = 2 processes from Tab. I). Here at 90% C.L. (h/Z decays at 95% C.L.). A full list of CLFV − involving hadrons (had) can be found in the PDG [30]. we included far more processes than listed in Tabs. I–II for the sake of illustration, even though many of them are not testable. In fact, the only realistically testable Group Process Current Future processes not already listed areµe ¯ µe¯ (violating −8 −9 → ∆(Lµ + Lτ − 2Le) = 6 τ → eeµ¯ 1.5 × 10 [29] 10 [28] ∆(Le Lµ) = 4), for which experimental limits from −8 −9 − ∆(Lτ + Le − 2Lµ) = 6 τ → µµe¯ 1.7 × 10 [29] 10 [28] exist [38], and τ µµµe¯e¯. → ∆(Le + Lµ − 2Lτ ) = 6 µe → ττ –– Simple vector addition now allows us to make model- independent interpretations of CLFV from Fig. 1. For TABLE II: CLFV with conserved L and B, omitting CP con- example, the observation of µ eγ = ( 3, 1) and jugate processes. Current limits at 90% C.L. τ µγ = (0, 2) implies that τ→ eγ = (− 3,−1) must exist,→ as well as in fact all other CLFV→ processes,− because each point on the lattice can be written as n( 3, 1) + − − has been put forward in Ref. [35] that only generates the m(0, 2) with n, m Z. Observing two different groups ∆(Lµ Lτ ) = 2 group of Tab. I by extending a two-Higgs- ∈ − of Tab. I is thus sufficient to prove that all CLFV exist, doublet model with a spontaneously broken U(1)Lµ−Lτ i.e. that the flavor group U(1)Lµ−Lτ U(1)Lµ+Lτ −2Le is symmetry. Models for ∆(Le Lµ,τ ) = 2 can be built 3 × − completely broken. It is of course impossible to make in complete analogy. The groups of Tab. II, which con- model-independent statements about the size and ratios sist of at most one observable process, can for example of the different CLFV channels. be obtained by extending the SM by an SU(2)L singlet Two (orthogonal) CLFV processes are however not ++ scalar k with 2, which has the Yukawa always enough to establish the full breakdown of the couplings flavor group. For example, the observation of τ → c ++ µγ = (0, 2) and τ eeµ¯ = ( 6, 0) only implies that a gαβ` PR`β k + h.c., (3) → − L ⊃ α coarser sublattice is generated, which in particular does not include µ eγ or τ eγ (Fig. 1). Formally, with symmetric coupling matrix gαβ and chiral projec- → → U(1)Lµ−Lτ U(1)Lµ+Lτ −2Le is broken to a Z2 subgroup tion operator PR [36]. Imposing a (local or global) ×

U(1)Lµ−Lτ symmetry severely restricts g,

c c c ++ (gµτ µRτR + gτµτ RµR + geeeReR) k + h.c., (4) 3 L ⊃ From Fig. 1 it might seem like U(1)Lµ+Lτ −2Le always has an unbroken Z3 subgroup under which (µ, τ, e) ∼ (1, 1, −2), but thus allowing only for the ∆(Lµ + Lτ 2Le) = 6 decay − this is actually a U(1)L subgroup because (1, 1, −2) = (1, 1, 1) τ eeµ¯ of Tab. II but none of the other CLFV groups. mod 3, which can hence be ignored. → 3

∆(L L ) µ − τ e¯e¯ τ¯τ¯τµ¯ µ¯µ¯ τ¯τ¯ τ µµµe¯e¯ → → → µ¯e¯ τ¯τ¯ τ µµe¯ → → e¯e¯ τ¯τ¯ τ µγ µe¯ µe¯ → → → τ eγ µ¯ eγ¯ → → τ eeµ¯ τ¯ e¯eµ¯ → →

µ eγ τ¯ eγ¯ ∆(Lµ + Lτ 2Le) → → − µe¯ µe¯ τ¯ µγ¯ ee ττ → → → τ¯ µ¯µe¯ µe ττ → → τ¯ µ¯µ¯µee¯ µµ ττ ee τττµ¯ → → →

FIG. 1: CLFV processes (only one representative shown per group) organized by their U(1)Lµ−Lτ × U(1)Lµ+Lτ −2Le breaking.

Observation of charged lepton flavor violation ⇒ Remaining symmetry

∆(Lα − Lβ ) = 2 U(1)Lα+Lβ −2Lγ

∆(Lα + Lβ − 2Lγ ) = 6 U(1)Lα−Lβ

∆(Lα + Lβ − 2Lγ ) = 6 and ∆(Lα − Lβ ) = 2 Z2: `γ → −`γ ∆(Lα + Lβ − 2Lγ ) = 6 and ∆(Lα + Lγ − 2Lβ ) = 6 Z3:(`α, `β , `γ ) ∼ (0, 1, 2)

∆(Lα − Lβ ) = 2 and ∆(Lα − Lγ ) = 2 –

∆(Lα − Lβ ) = 2 and ∆(Lα + Lγ − 2Lβ ) = 6 –

TABLE III: Observation of CLFV (U(1)Lµ−Lτ ×U(1)Lµ+Lτ −2Le breaking) via processes of Tabs. I–II and remaining subgroups after up to two CLFV discoveries. Three observations – at least one from each table – imply full flavor breakdown. under which flip sign. A third CLFV observa- violate flavor. Similarly, the 12 processes ` `(``¯), tion, with an odd number of electrons, e.g. µ eγ or e.g. τ µµe¯ and ee ττ, can be seen as part→ of the τ µµe¯, is thus necessary to establish full flavor→ break- 27 3→ 3¯ 3 3¯, which→ again includes many singlets ing.→ For the convenience of the reader, we provide a sim- not⊂ of interest⊗ ⊗ for⊗ LFV. Baryon multiplets do not have an ple flowchart of possible CLFV in Tab. III, assuming only analogue in LFV because angular and the as- the processes of Tabs. I–II are observable. Since we are sumed conservation forbid processes such only able to experimentally test flavor breaking by very as ` `¯`¯ that would correspond to 3 3 3. We stress few units, the remaining discrete subgroups, i.e. Eq. (5), that→ the similarity of Fig. 1 with the⊗ ⊗ multiplets are surprisingly simple (only Z2 or Z3). is purely formal; SU(3)` is broken badly by the different lepton masses, leaving only the abelian Cartan subgroup Let us make one final remark about the similarity of U(1)Lµ−Lτ U(1)Lµ+Lτ −2Le as a possible symmetry of Fig. 1 with the well-known multiplets of Gell- nature, which× is thus a better starting point for LFV. Mann’s [39]. The latter is a way to orga- nize qq¯ (and qqq) states according to their transforma- tion properties under the (approximate) flavor symme- LEPTON NUMBER VIOLATION try SU(3)f with q = (u, d, s) 3, leading in particular to a meson octet as a result∼ of 8 3 3¯. In Fig. 1 For the study of CLFV we assumed L and B to be con- we are effectively organizing processes⊂ such⊗ as ` `(``¯ ) served, which is a convenient simplification – allowing us according to their transformation properties under→ the to represent CLFV in a two-dimensional plane (Fig. 1) – flavor symmetry SU(3)` with ` = (e, µ, τ) 3. The and could well be true if neutrinos are Dirac and ∼ `α `βγ processes then form a LFV “octet” similar B L is unbroken [2]. Let us loosen this assumption and to the→ meson octet, where we of course neglect the two allow− for lepton number violation (LNV), still keeping B neutral states (corresponding to π0 and η) that do not conserved for simplicity. Even though our decomposition 4

Group Process Current Future For example, the discovery of ∆(Lα + Lβ) = 2 would imply that U(1) U(1) could still be a good ∆L = 2 0νββ O(1025 yr) [44] 1026 yr [44] Lα−Lβ Lγ e symmetry for charged leptons,× making necessary further had→ ee had 6.4 × 10−10 [45] 10−12 [26] −11 −12 CLFV or LNV observations to establish full flavor break- ∆Lµ = 2 had→ µµ had 8.6 × 10 [46] 10 [26] ing (a grid similar to Fig. 1 but with axes Lγ and Lα Lβ ∆Lτ = 2 had→ ττ had – – can be drawn to make model-independent studies).− It is −11 −11 ∆(Le + Lµ) = 2 µ → e¯ conv. 3.6 × 10 [47]  10 [48] straightforward but tedious and not particularly illumi- had→ µe had 5.0 × 10−10 [45] 10−12 [26] nating to extend the flowchart of Tab. III to LNV. (The −8 −9 ∆(Le + Lτ ) = 2 τ → e¯had 2.0 × 10 [49] 10 [28] processes in Tab. IV were also studied in connection to had→ τe had – – the Majorana neutrino mass matrix in Ref. [53].) Baryon −8 −9 ∆(Lµ + Lτ ) = 2 τ → µ¯ had 3.9 × 10 [49] 10 [28] number violation of course adds yet another dimension had→ τµ had – – to the parameter space and can be explored completely analogously. TABLE IV: Processes violating total lepton number L by two units (90% C.L. limits), assuming conserved baryon number. CONCLUSION

In summary, the observation of CLFV implies a break- of Eq. (1) suggests LFV (as defined above) and LNV to down of the approximate global flavor symmetry group be unrelated issues, all currently probed LNV processes U(1) U(1) in the charged-lepton sec- (Tab. IV) in fact violate lepton flavor. True LNV, con- Lµ−Lτ Lµ+Lτ −2Le tor. If one process× is observed, one linear combination of serving flavor and baryon number, is a lot harder to come L L and L + L 2L is broken, while the orthogo- by and involves at least six leptons – e.g.e ¯e¯ µµττ plus µ τ µ τ e nal− one might still be− conserved (up to tiny neutrino- charged – because all known fermions→ carry fla- mediated contributions). Depending on the process, vor or baryon number.4 As such, any observation of LNV this could imply the existence of other testable CLFV is practically also an observation of LFV. We list possi- channels (Tab. I), but could also be an isolated process ble LNV processes in Tab. IV, focusing again on decays (Tab. II). If two (orthogonal) CLFV processes are ob- rather than collider signatures. We urge experimental- served, Fig. 1 can be used to predict additional processes, ists to complete Tab. IV by looking for meson decays which might not necessarily be all possible ones due to into τ +`+, e.g. B+ π−τ +`+, with ` e, µ, τ [40]. a possible remaining discrete subgroup. CLFV is hence Similar to CLFV we→ only list LNV by a few∈ units{ because} far more than a yes–no question, with up to three qual- experimental signatures of ∆L > 2 are much more chal- itatively different discoveries required to establish that lenging [41, 42]. (A first preliminary limit on ∆L = 4 e no flavor symmetry exists in the charged-lepton sector. was presented recently by NEMO-3 in the form of a lower The discovery of lepton number violation would practi- limit of 2.6 1021 yr on the neutrinoless quadruple beta cally imply lepton flavor violation as well and thus has decay [41] 150×Nd 150Gd + 4e− [43].) 60 64 to be taken into account when interpreting data. With a Of the processes→ in Tab. IV, neutrinoless double beta large number of experiments exploring untested parame- decay (0νββ) [44, 50] is the only one that is sensitive ter space, it is entirely possible that we see one or more to neutrino-induced LNV/LFV. Since we already know discoveries within the next decade. from neutrino oscillations that lepton flavor is broken, the observation of 0νββ would prove that all three lepton numbers Le,µ,τ are broken in the neutrino sector. This implies in particular that neutrinos are Majorana parti- Acknowledgements cles [51] – and the existence of additional particles such as scalars or heavier Majorana partners as a UV comple- I thank the organizers and participants of the XIIth tion of the Weinberg operator [52] – but is not sufficient Rencontres du Vietnam: Precise theory for precise ex- to tell us anything about charged lepton flavor. periments for an inspiring conference that led to this An observation of a process from Tab. IV other than work. I am especially grateful to Sacha Davidson for 0νββ would on the other hand imply that lepton flavor comments on the manuscript and acknowledge support is definitely broken in the charged-lepton sector, by the F.R.S.-FNRS as a postdoctoral researcher.

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