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TeV violate low the to at leptoquarks in interest easrqie h diino tlattwo least at of addition the requires decays sl hoeicePyi ,Uiesta ign Walter-Flex- Universit¨at Siegen, 1, Physik Theoretische + b l → − .INTRODUCTION I. transitions. clν b → rthe or ikn etnnme ilto with violation number lepton Linking clν U 13 and [1–3] 1 b (3 → , 2 sl , + 3 2 B etqak there leptoquark, ) l − nmle h fo- the anomalies b nmle.Pre- anomalies. sa a` n hmsMannel Thomas and Cat`a Oscar → sl + l b − → [4–6] clν lcrwa cl ya by scale electroweak of interplay the requires leptoquarks. it both since nontrivial, agree- mag- quantitative is of This ment order masses. right neutrino the the for with nitude accommodate up to ends one needed anomalies, masses and couplings R ihrpeoeooycnb ece foecon- one if reached violation. be number can phenomenology lepton richer to A process sensitive most ntegnrto fMjrn etiomse ac- masses in anomalies neutrino the for Majorana simultaneously of count generation be the can in masses scalar scale. GUT the the loop, than lighter one much at involves generation place the solution since takes This and, only sector 9]. down-quark the [8, Model by content Standard the version particle of (SM) minimal top on its scalars new two in adding generated radiatively be leptoquarks. on the of depends numbers course quantum of specific operators the these of generation nanme of number a in aietisl nteWibr prtrat At operator Weinberg the expansion. in itself would EFT manifest violation the number lepton be in scale, will electroweak orders the effects different its gener- and at potential, be seen the can in terms violation by number ated lepton then quark, also must effects number. dimension-six lepton conserve that follows ically -etneetv prtr conserve Since operators opera- effective exchanges. 4-lepton leptoquark 4-lepton single anomalies dimension-six from the coming from the- tors to come field effects then effective leading lepto- will of The the language (EFTs). out the ories integrate use and safely quarks can one measured, D ic etioms eeaini elzda the at realized is generation mass neutrino Since nti etrw on u httesaasinvolved scalars the that out point we letter this In can masses neutrino that result well-known a is It oee,i n losfrmr hnoelepto- one than more for allows one if However, ttehdoi cl,weeteaoaisare anomalies the where scale, hadronic the At ( ∗ ) culy yuigtevle fleptoquark of values the using by Actually, . eeaieti su.W n htthere that find We issue. this examine we onic ieo h nmle hc loviolates also which anomalies the of size tae3 -76 ign Germany Siegen, D-57068 3, Straße ieo h otrlvn etnnumber lepton relevant most the of size le,oeatal estergtorder right the gets actually one alies, n cnro ugse,sm happen some suggested, scenarios ent r aitvl eeae.Wt the With generated. radiatively are B adt h xetdsz fthe of size expected the to paid n ffciefil hoisbt tthe at both theories field effective eashv rmtdarenewed a prompted have decays d B ≥ anomalies ffcieoeaos h actual The operators. effective 7 IHP21-3 QFET-2019-03 SI-HEP-2019-03, d prtr ti the is it operator, 5 = B − L R d tgener- it , K and 5 = ( ∗ ) and 2

siders d 7 operators. We briefly comment on ϕ the bounds≥ that the leptoquark scenario would put on lepton number violating processes that could be η χ studied at the LHCb and Belle II. We show that LNV processes are generically well out of reach for c current particle accelerators. νi d d νj ϕ

II. THE MINIMAL SCENARIO FIG. 1: Radiative generation of the Weinberg operator. There have been numerous proposals to reproduce the observed discrepancies in RD(∗) and RK(∗) . De- spite efforts to correlate both anomalies to a sin- nations gle new physics particle (see e.g. [10, 11]), the cur- η (3, 2, 1 ); χ (3¯, 1, 1 ) (1) rent consensus is that global data fits favor solutions ∈ 6 ∈ 3 with at least two leptoquarks. In most of these solu- tions, each leptoquark carries the bulk of one of the and anomalies (charged or transitions). η (3, 2, 1 ); χ (3¯, 3, 1 ) (2) Although the space of models is rather constrained, ∈ 6 3 ∈ 3 there are different viable scenarios with leptoquarks Here we will focus on a scenario where the SM is ex- and reducing further the number of possibilities re- tended just with η and χ, which can account for both quires additional information. the b s and b c anomalies. The scenario with As opposed to other new physics scenarios, the → → η and the electroweak triplet χ3 is more restricted presence of leptoquarks immediately opens the way and will be discussed later on. for lepton and violation. Baryon Including the leptoquark fields η and χ, the SM number is constrained by the stringent bounds on interactions (in the weak basis) get enlarged by the , but lepton number violation could be following operators: the origin of neutrino masses, if we assume that they are Majorana particles. = η†( + m2 )η χ∗( + m2 )χ + µχ(ϕ†η) L − η − χ One could therefore ask whether any leptoquark ij ij c ij c λ ℓ¯iηd˜ j λ q¯ ǫℓjχ λˆ u¯ ejχ scenario that accounts for the B anomalies can also − 1 − 2 i − 2 i account for neutrino mass generation in a success- ij c ∗ ˆij c ∗ λ3 q¯i ǫ qj χ λ3 u¯i dj χ (3) ful way. The existence of such a scenario is strongly − − constrained by quantum numbers, and there is no where hermitean conjugate operators are implicitly guarantee a priori that the size of the B anomalies understood and the shorthand notation ψ¯ ǫ ψ′ ¯ ab ′ ij ≡ could also generate neutrino masses at the right or- ψaǫ ψb has been used. The different λn are dimen- der of magnitude. sionless flavor matrices, with i, j generation indices Since the leptoquark masses hover around the TeV and a,b SU(2)L indices. scale, it is clear that quantitative agreement with The previous Lagrangian includes the most gen- neutrino masses can only happen if they are radia- eral renormalizable interactions of χ and η with SM tively generated. The generation of neutrino masses fields. In eq. (3) we have only omitted quartic terms at one loop was first studied systematically in [9]. in the potential, e.g. (η†η)(ϕ†ϕ) or χ∗χ(η†η), which Among the three different mechanisms that were will not be needed for our discussion. identified, there is only one which extends the SM In order not to violate the bounds on proton decay, with two new particles. The topology is depicted in we will assume that baryon number is a conserved fig. 1, where d stands for a generic down-type quark. 1 quantity and choose B[η] = B[χ] = 3 . The di- It is interesting to note that there is no quark interactions of the last− line will therefore be between the up-type and down-type sector for one- absent. loop neutrino mass generation: a one-loop topol- The Lagrangian of eq. (3) can still induce lep- ogy with up-type quarks running into the loop can ton number violating processes whenever the scalar only happen if additional fields are introduced (see operator µχ(ϕ†η) contributes. In other words, the e.g. [12]). The down-type sector is therefore natu- µ-term is a soft breaking of lepton number: when rally selected by the argument of simplicity. µ 0 the leptoquarks decouple from each other The quantum numbers of the new scalar particles and→ lepton number conservation is restored. This can then be fixed by going through the diagram of in particular implies that the one-loop diagram of fig. 1. This selects the two scalar leptoquark combi- fig. 1 will generate a finite Weinberg operator. If 3

the µ-term is the only source of lepton number vi- provided they hover around (10−2) for a low-TeV O olation, as we will assume, then the interactions of leptoquark [16]. Regarding RD(∗) , one needs the potentially heavier masses cannot affect our results. minimal structure Notice that, beyond the neutrino mass diagram, the ue λ = λcτ (7) form of the µ-term implies that LNV processes are 2 { } both dν only induced when leptoquarks are involved. λ = λsντ , λbντ (8) 2 { } Thus, processes affected by only one of the lepto- ue λˆ = λˆcτ (9) quarks, regardless of the value of µ, necessarily con- 2 { } serve lepton number (though not necessarily lepton The outcome of different papers [10, 14] shows that flavor). for a low-TeV leptoquark the anomalies can be ac- The parameter µ has mass dimensions and is nat- commodated and constraints on other flavor pro- urally expected to be close to the electroweak scale cesses respected if the relevant matrix entries in the v 246 GeV. More precisely, in order not to upset flavor matrices λ are of (10−1) for the left-handed ∼ µ < πm j the Higgs mass at loop order, 4 h. In the couplings and (10−2) forO the right-handed ones. following we will assume that µ ∼v. O In order to avoid conflict with∼ direct detection at the LHC, the leptoquarks have to be at least around III. NEUTRINO MASS GENERATION the TeV scale, i.e. heavy as compared to the elec- troweak scale. This means that they can always be One can now calculate explicitly the diagram of treated as virtual particles and an EFT language, fig. 1. The result is the Weinberg operator [17], where their effects are integrated out, is very conve- nient. (5) ij c ∗ † = C (ℓ¯ ϕ˜ )(ϕ ˜ ℓi) , (10) Integrating out the leptoquarks at tree level gener- Leff 5 j ates effects at the d = 6 level. The resulting effective where C5 depends on which down-type quark runs theory reads: inside the loop. For b quarks one finds ij ∗kn (6) λ1 λ1 µ m2 = d¯ γ d ℓ¯ γ ℓ ij 3 ∗ib bj µλb χ eff 2 n i j µ k C = (λ λ ) log (11) L − 2mη 5 (4π)2 1 2 m2 m2 m2 χ − η η λˆij λˆ∗kn + 2 2 u¯ γµu e¯ γ e where λ is the bottom Yukawa. The neutrino mass m2 n i k µ j b 2 χ matrix then takes the form ij ∗kn λ2 λ2 n µ i k j µv + ǫabǫdf q¯ γ q ℓ¯ γµℓ ij 3 ib ∗bj 2 a d b f m = (λ λ )mb (12) 2mχ ν 2 1 2 2 (4π) √2 mX λij λˆ∗kn 1 2 2 ǫ unqi ekℓj unσ qi ekσµν ℓj U 2 ab ¯ a ¯ b ¯ µν a ¯ b which can be diagonalized with a matrix ν . In or- − 2mχ  − 4  der to simplify our results we have assumed that the (4) leptoquarks have comparable masses, mχ mη = ∼ where Fierz transformations have been performed. mX . The previous operators are written in the weak basis. The contribution of light quarks is qualitatively The rotation to the mass basis gives raise to the different. Because of confinement, it is dominated by following flavor matrices: nonperturbative physics. On dimensional grounds the result takes the form λνd U T λ V λed U T λ V 1 = ν 1 d; 1 = e 1 d; (5) µ ss¯ Cij √ λisλ∗sj ue T dν T ˆue T ˆ 5 2 1 2 h2 i2 (13) λ2 = Uu λ2Ue; λ2 = Ud λ2Uν ; λ2 = Vu λ2Ve ∼− vmηmχ As expected, the presence of leptoquarks selects di- The contribution to the neutrino mass matrix thus rections in flavor space which are not dictated by reads the CKM matrix. The effective operators in eq. (4), is ∗sj when runned down to hadronic scales, provide the ij λ1 λ2 µv δmν 4 ss¯ (14) leading effects to deviations from the SM values for ∼− √2 mX h i RD(∗) [10, 13, 14] and RK(∗) [15]. Specifically, the anomaly in R (∗) can be ac- Given the size of the quark condensate, qq¯ K h i ∼ counted for with the following non-zero matrix en- (250 MeV)3, the ratio − tries δmν 2 qq¯ ed (4π) h2 i (15) λ = λµs, λµb (6) m ∼− m m 1 { } ν b 4 ϕ gives a negligible (10−5) relative correction from light quark exchange.O The bulk of the neutrino masses is thus given by b exchange. q ℓ¯ Using that mν < 0.1 eV, eq. (12) can be written as ∼ χ η ℓc d 2 −3 νb ∗bν 1 TeV < −10 mν 10 λ1 λ2 GeV 10 GeV ∼  mX  ∼ FIG. 2: Generation of d = 7 operators through η and (16) χ exchange. Integration of the leptoquarks leads to the operators of eq. (17). The bound can be saturated with leptoquark masses −2 −3 in the low TeV range and λj (10 10 ). This is precisely the order of magnitude∼ O for− masses leptoquarks, the resulting effective operators read and couplings needed to reproduce the B anomalies. (7) µ ij ∗kn c = λ λ (ℓ¯iϕd˜ j )(¯qk ǫℓ ) Notice however that the entries of the flavor matrices Leff m2m2 1 2 n needed for neutrino masses are not the same as for η χ µ ij ∗kn c R (∗) and R (∗) . Their values will be constrained λ λˆ ℓ¯ ϕd u e , K D + 2 2 1 2 ( i ˜ j )(¯k n) (17) instead by processes like b sνν¯ , which currently mηmχ → have rather loose bounds. which, as expected, violate lepton number by two The successful generation of Majorana neutrino units. If we define ∆Qℓ as the difference of lepton masses with the parameters that reproduce the charge, they induce processes with ∆L =2,∆B =0 anomalies in R (∗) and R (∗) allows one to reverse K D and ∆Qℓ =0, 1, i.e. di dj νν or di uj lν. the argument: if one believes that are Ma- At the hadronic level,→ the operators→ in eq. (17) ap- jorana particles, and that the dynamical origin of its pear as 4-lepton operators, but with a relative sup- mass is much below the GUT scale, then the simplest pression factor of scenario is that containing two scalar leptoquarks. If one further assumes that the flavor couplings have µv 2 (18) only a mild hierarchy, anomalies in both charged and mX neutral currents in B physics should be generated at the level found experimentally. with respect to those coming from eq. (4). On top of this, since the processes they induce do not inter- fere with the Standard Model ones, their effects will generically be very suppressed. IV. OTHER LEPTON NUMBER Consider for instance the ∆L = 2 process b VIOLATING PROCESSES cτ −ν. Compared with b cτ −ν¯, one gets a correc-→ tion → In the previous section we have already discussed 8 b→cτ −ν −3 bν ∗cτ 2 1 TeV that the Weinberg operator is the leading operator B 10 (λ1 λ2 ) (19) − ∼  mX  which violates lepton number. Phenomenologically, Bb→cτ ν¯ however, this operator is rather limited and accounts which optimistically would hover around 10−8. basically for neutrino mass generation. It is there- For the corresponding b s transitions the ratio fore interesting to explore which other processes in- is substantially larger, because→ the process to com- η χ duced by the leptoquarks and would violate lep- pare to is loop-suppressed in the Standard Model. ton number and could in principle be detected at Thus, LHCb and Belle II. 8 Having imposed baryon number conservation, all b→sνν 3 bν ∗sν 2 1 TeV the d = 6 operators that can be generated conserve B 10 (λ1 λ2 ) (20) b→sνν¯ ∼  mX  lepton number as well. At the electroweak scale, B operators that violate lepton number and conserve Since the current experimental bounds are roughly baryon number will appear next at d = 7. For a factor 4 above the Standard Model prediction, bν ∗sν < −2 the model we are considering, these effective opera- the constraint is satisfied for (λ1 λ2 ) 10 with tors are the result of configurations where both lep- mX 1 TeV. ∼ toquarks are exchanged. The relevant topology is Given≃ that neutrinos are not distinguished from shown in fig. 2, which can be constructed by opening antineutrinos at colliders, the previous estimate in- up the loop diagram of fig. 1 and thus adding two dicates that there could be a potentially sizeable lep- extra external fermions. After integrating out the ton number violating contribution to the b s +2ν → 5

d µ− ciated with the running of scales, the leading contri- bution comes as a d = 10 operator, which takes the η W d¯ form η 4GF vµ ij ∗kl c c c µ ϕ u λ λ Vnp ∂µ(¯eiu )(d¯ e ) u¯nγ dp (23) χ m2 m4 1 2 j k l h i χ η   c u µ− To get an estimate of how big ∆Qℓ = 2 effects in B decays can be, consider B− π+µ−µ−. Com- pared to the related lepton-flavour-conserving→ pro- FIG. 3: Generation of the dominant ∆Q = 2 processes cess B− π−µ+µ−, one finds through W exchange. → 12 − − − B →π+µ µ 2 1 TeV −18 B (λ1λ2) 10 (24) − − + − ≃  mX  measurement. However, for the same reason that BB →π µ µ neutrinos and antineutrinos cannot be separated −8 Since experimentally − − + − 10 , one ex- apart, excesses in b sνν¯ would point at new B →π µ µ pects the branching ratioB for the LNV∼ process at physics but could not be→ ascribed unambiguously to 10−26, clearly out of Belle II reach. violations of lepton number. Hadronic τ decays like τ − π+e−e−ν or τ + Instead, processes with ∆L = 2 and ∆Q = 2, i.e. ℓ µ−π+π+ are also generated at→d = 10. For the for-→ with two charged of the same sign, would mer decay mode there are no limits available. The leave very distinct signatures and directly test lep- latter has a current experimental bound at 10−8. In ton number violation. In the model we are consid- both cases the effects predicted by the leptoquark ering, these processes are generated through the di- scenario considered in this paper are unobservable. agram of fig. 3. Once the leptoquarks are integrated The previous estimates for ∆Q = 2 processes are out, one ends up with the following d = 9 effective ℓ based on a specific leptoquark construction. How- operators (up to hermitean conjugation) at the elec- ever, one can show that the statements about their troweak scale: undetectability are rather generic by using an EFT (9) µ ij ∗kl c † µ c c a argument. At the electroweak scale, ∆Ql = 2 op- = λ λ Dµ (ℓ¯i ǫq )ϕ D d¯ (ǫℓ ) Leff m2 m4 1 2 j a k l erators first appear at d = 7, and are restricted± χ η     µ to [18] + λij λˆ∗klD (¯e uc)ϕ† Dµ d¯c (ǫℓc)a m2 m4 1 2 µ i j a k l χ η     (dγ¯ µu)ℓ¯c ǫ (D ℓ) and (dγ¯ µu)ℓ¯c ǫ σ (Dν ℓ) (25) (21) µ µν Based on dimensional grounds, these operators are In order to induce LNV processes with violation of weighted by a coefficient that scales like Λ−3, where lepton charge by two units such as the ones in fig. 3 Λ is the new-physics scale. However, it is easy to one needs to pull a W boson out of the covariant realize that such operators cannot be generated from derivatives above. the tree level exchange of a heavy particle. Given At the electroweak scale, a representative pro- a UV model, these operators will, at the most, be cess described by the operators of eq. (21) is t generated at the one-loop level. Therefore, at the bW −µ+µ+. In order to get an approximate order→ of hadronic scale, the ∆Q = 2 part of, e.g., the first magnitude, the decay rate can be estimated by fac- ℓ operator in eq. (25) will be of the form torizing the 4-body phase space and assuming that the final-state particles are massless. One then finds C G ijklpr F (d¯ γ u )(d¯ γµu )(¯ec e ) (26) that (4π)2 Λ3 i µ j k l p r 12 − + + −14 2 1 TeV where Λ should be understood as the geometric Γ[t bW µ µ ] 10 (λ1λ2) GeV → ∼  mX  mean of the new-physics masses and Cijklpr is (22) a flavor matrix. Assuming that Λ (few TeV), one could, e.g., enhance the decay∼ rate O for Since this process is background-free, it would re- B− π+µ−µ− to quire at least 1016 top decays for detection, a number → which is out of the capabilities of current detectors. − + − − BB →π µ µ < 10−12 (27) Other representative processes concern B decays. − − − B →π µ+µ ∼ In this case, it is convenient to work with operators B defined at the hadronic scale, obtained by integrat- This is an optimistic upper bound. A model with all ing the W and Higgs fields. Neglecting effects asso- new particles at the low TeV scale is hardly realistic. 6

In practice, compliance with flavor constraints will moment [23] compatible with the discrepancy ob- push some of the masses up and thus lower this ratio. served experimentally [24]. 10−12 is therefore to be understood as a generous Regarding lepton number violating processes, upper bound, which is clearly too suppressed to be there will be qualitatively no changes if χ is re- detected at B factories. placed by χ3. Neutrino masses will be generated by the t3 = 0 component of the triplet and the for- mulas given above will get modified trivially with V. SCENARIOS WITH A LEPTOQUARK the appropriate replacements of the λ matrices. The TRIPLET lepton number violating processes depicted in fig. 2 will be generated with two of the components of χ3. The criteria we used to build our model was its abil- This will add an extra diagram, but will generate ity to generate neutrino masses. We have thus fo- the same effective operators at the hadronic scale. 1 cussed our attention on the leptoquark pair η(3, 2, 6 ) Similarly, for ∆Qℓ = 2 processes, there will also be ¯ 1 an extra topology to the one of fig. 3, with the W and χ(3, 1, 3 ). However, we already pointed out that, regarding neutrino mass generation, an alter- exchanged between the different components of χ3. 1 The bounds that we found would only get affected native scenario would be to consider η(3, 2, 6 ) and χ ¯, , 1 by (1) effects. 3(3 3 3 ). It is instructive to highlight the differ- O ences between both scenarios for the phenomenolog- ical applications discussed in this paper. Imposing baryon number conservation, there is VI. CONCLUSIONS only one operator coupling χ3 to fermions, and the Lagrangian reads The tensions observed in semileptonic B decays, if confirmed, would be a clear signal of lepton univer- = η†( + m2)η χ†( + m2 )χ + µ(ϕ†χ η) sality violation. However, the fact that the most L − η − 3 χ 3 3 ij ij c natural scenarios to accommodate the discrepancies λ ℓ¯iηd˜ j λ q¯ iτ χ ℓj (28) − 1 − 2 i 2 3 involve leptoquarks also suggests that lepton num- ber could be violated. Below the TeV scale the leptoquarks can be inte- In this paper we have entertained this idea and grated out. The leading lepton number conserving explored its consequences. With only one lepto- processes appear at the d = 6 level. Focusing on χ 3 quark, lepton number violation would imply baryon exchange, one obtains the following effective opera- number violation. With two leptoquarks, which tors: is what data currently seems to favor (among the ij 2 ∗kn one-leptoquark solutions, only the U1(3, 2, ) model (6) λ2 λ2 c c µ 3 = (¯q ǫℓj)(ℓ¯kǫq ) (¯qnγµqi)(ℓ¯kγ ℓj) Leff m2 i n − is not excluded), one can accommodate neutrino χ   masses while having a stable proton. We have (29) only considered minimal extensions of the Standard Model that can account for the anomalies in both From the previous operators it is clear that χ3 con- b s and b c transitions, namely scenarios with tributes to b s transitions [19], and not signifi- → → cantly to b →c. The phenomenology of this opera- two scalar leptoquarks. Interestingly, there is a sin- → (∗) (∗) tor at the hadronic scale has actually been studied gle scenario that can explain RK and RD and also in [20] and shown to be able to accommodate the violate lepton number, with the following set of lep- RK(∗) anomaly. toquarks: This means that a scenario with η(3, 2, 1 ) and 6 η(3, 2, 1 ); χ(3¯, 1, 1 ) (30) ¯ 1 6 3 χ3(3, 3, 3 ) will generate neutrino masses but can only account for discrepancies in the neutral b s The dominant lepton number violating effect is the → transitions. This scenario has already been inves- generation of Majorana neutrino masses at the one- tigated in [21, 22], where the motivation for having loop level. It is remarkable that the present size of these two leptoquarks was to accommodate neutrino the deviation in RK(∗) and RD(∗) give the right order 1 masses and RK simultaneously. of magnitude for neutrino masses. Since χ(3¯, 1, ) ¯ 1 3 The advantage of a scenario with χ(3, 1, 3 ) and also bears a contribution to the anomalous 1 η(3, 2, 6 ) is that it is far more encompassing: one magnetic moment able to explain the current dis- can describe simultaneously the anomalies in b c crepancy with the Standard Model prediction, this and b s transitions and at the same time have→ scenario is rather attractive. a mechanism→ for neutrino mass generation. Addi- Beyond neutrino masses, the effects of the lep- ¯ 1 tionally, it has been shown that χ(3, 1, 3 ) can also toquarks on lepton number violation processes are generate an effect on the muon anomalous magnetic extremely suppressed, currently at an undetectable 7 level. One could have (1) deviations in b sνν the low-TeV scale. transitions, but since neutrinosO and antineutrinos→ cannot be distinguished at the detector level, this potential deviation of Standard Model physics would not be a conclusive signal of lepton number violation. Acknowledgements If the deviations in the b s and b c tran- sitions persist and this model→ is taken seriously,→ it would then indicate that with the B anomalies we We thank Thorsten Feldmann for reading the are actually probing the scale of lepton number vio- manuscript and for very stimulating discussions. lation, which would also be the scale of flavor lepton This work is supported in part by the Deutsche universality breaking and would turn out to be at Forschungsgemeinschaft (DFG FOR 1873).

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