Revisiting Large Neutrino Magnetic Moments
RRevisiting Large Neutrino Magnetic Moments
Manfred Lindner, ∗ Branimir Radovˇci´c, † and Johannes Welter ‡ Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: May 14, 2018)Current experimental sensitivity on neutrino magnetic moments is many orders of mag-nitude above the Standard Model prediction. A potential measurement of next-generationexperiments would therefore strongly request new physics beyond the Standard Model. How-ever, large neutrino magnetic moments generically tend to induce large corrections to theneutrino masses and lead to fine-tuning. We show that in a model where neutrino masses areproportional to neutrino magnetic moments. We revisit, discuss and propose mechanismsthat still provide theoretical consistent explanations for a potential measurement of largeneutrino magnetic moments. We find only two viable mechanisms to realize large transitionmagnetic moments for Majorana neutrinos only.
1. INTRODUCTION
The neutrino magnetic moment (NMM) in the Standard Model (SM) is of the order 10 − µ B [1–5], where µ B = e m e is the Bohr magneton. At the same time reactor, accelerator and solarneutrino experiments as well as astrophysical observations are lacking many orders of magnitudein sensitivity in order to test the small SM prediction (for a recent review see [6]). The bestcurrent laboratory limit is given by GEMMA, an experiment measuring the electron recoil ofantineutrino-electron scattering near the reactor core. It constrains the effective magnetic momentto be less than 2 . · − µ B [7]. A recent study by Ca˜nas et al. [8] showed that results of thesolar neutrino experiment Borexino give similar limits. They obtain for the individual Majoranatransition moments in the mass basis | Λ | ≤ . · − µ B , | Λ | ≤ . · − µ B , | Λ | ≤ . · − µ B .On the other hand, the smallness of the SM prediction imply that a non-zero measurement ofNMM would be a clear indication for new physics beyond the SM. In view of upcoming experiments,that are able to further increase the sensitivity on the NMM, it is worthy to ask what kind of newphysics could explain large NMM. In other words, we want to address the question of how togenerate large NMM in a theoretically consistent way.The paper is organized as follows. In section 2 we review model independent bounds on theNMM from corrections to the neutrino mass. In section 3 we consider a model with light mil-licharged particles. In section 4 we explicate the generic difficulty to obtain a large NMM withoutfine-tuning neutrino masses in a particularly insightful model. In section 5 we revisit and updateconstraints on existing models that successfully avoid fine-tuning. We discuss and conclude insection 6. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] In the pure SM neutrinos are massless and therefore the NMM is zero. Here we refer to the extensions of the SMallowing for neutrino masses. a r X i v : . [ h e p - ph ] J un
2. NATURALNESS BOUNDS2.1. New physics above the electroweak scale
Since neutrinos are neutral, the leading contribution to the NMM is given by quantum correc-tions. Consider a theory with new physics at the scale Λ and new couplings G that introducesthe NMM at 1-loop. The Feynman diagram generating the NMM µ ν for Majorana neutrinos isdepicted in Fig. 1(a). Removing the photon line will directly result in a radiative neutrino masscorrection δm ν from the diagram in Fig. 1(b). With the new physics above the electroweak scale,the effective NMM operator in the case of Majorana neutrino is of dimension seven and the effectivemass operator is of dimension five. The generic estimate thus gives µ ν ∼ QGv H Λ , δm ν ∼ G v H Λ (1)leading to δm ν . ∼ (cid:15) (cid:18) µ ν − µ B (cid:19) (cid:18) ΛTeV (cid:19) , (2)where v H is the vacuum expectation value of the Higgs and (cid:15) = Q/e is the charge of the particlesrunning inside the loop in units of the electron charge. To avoid fine-tuning, the radiative neutrinomass correction should not be larger than the measured neutrino masses, δm ν (cid:46) m ν . Usingreasonable numbers, m ν ∼ . ∼ TeV and (cid:15) ∼ µ ν (cid:46) − µ B . (3)For Dirac neutrinos the 1-loop effective NMM and neutrino mass operators are of dimension sixand four respectively. With diagrams similar to Fig. 1 this leads to µ ν ∼ QGv H Λ , δm ν ∼ Gv H . (4)By taking the ratio δm ν /µ ν we get the same constraint as in Eqs. (2) and (3).The current best laboratory experimental limit for the NMM is at µ ν ∼ . · − µ B [7], whileneutrino masses above 0 . δm ν low, requires a significant amount of fine-tuning. To reach values µ ν (cid:38) − µ B ,which will be probed in future experiments [10–13], fine-tuning of seven orders of magnitude isrequired. Λ ν νγ HH (a) Λ ν νHH (b) Figure 1: Feynman diagrams generating the NMM and the radiative neutrino mass for Majorana neutrinosinduced by new physics above the electroweak scale.
If the contribution to neutrino masses from the diagram in Fig. 1(b) is suppressed for somereason, there are still contributions from higher-loop diagrams induced by the NMM operator likethe one in Fig. 2. In order to derive constraints on the NMM, Bell et al. [14, 15] and Davidsonet al. [16] performed effective operator analyses for Dirac and Majorana neutrinos. Requiring thenaturalness condition δm ν (cid:46) m ν to avoid the fine-tuning they found the model independent boundfor Dirac neutrinos of the order µ ν (cid:46) − µ B , when taking the new physics scale Λ = 1 TeV and δm ν (cid:46) . m ν (cid:46) . µ ν τ ν µ , µ ν τ ν e (cid:46) − µ B , µ ν µ ν e (cid:46) · − µ B [15], which are already worse than current experimentalconstraints. Λ ν ν Figure 2: Higher-loop neutrino mass contribution induced by the presence of the NMM operator.
Now let us assume that the new physics is generated below the electroweak scale. For exampleone could think of a hidden sector, containing light particles. In this case, the effective NMM andneutrino mass operators generated by the Feynman diagrams in Fig. 3 are of dimension five andthree respectively. The naive estimate µ ν ∼ QG Λ , δm ν ∼ G Λ (5)leads to δm ν . ∼ (cid:15) (cid:18) µ ν − µ B (cid:19) (cid:18) ΛGeV (cid:19) . (6) Λ ν νγ (a) Λ ν ν (b) Figure 3: The Feynman diagrams for the NMM and the radiative neutrino mass induced by new physicsbelow the electroweak scale.
Given the estimates of Eqs. (2) and (6) it seems that there are two possibilities for generatinglarge NMM. Either the masses of the new particles are high and one has to find a mechanism thatavoids the naturalness bound or the new particles are light with fractional charge (cid:15) <
1. In thenext section we want to address the latter case, while for the rest of the paper we will assume thatnew physics is above the electroweak scale.
3. NATURAL LARGE NMM VIA MILLICHARGED PARTICLES
Motivated by the estimate of Eq. (6) we are interested in particles with low mass, Λ < (cid:15) ∼ . ∼ . µ ν ∼ − µ B in a technically natural way.In order to investigate this on a more quantitative level, we assume a millicharged scalar s anda Dirac fermion ψ coupling to light Majorana neutrinos in the form L = f i ψ R ν Li s + f (cid:48) j ν Lj ψ L s † + h.c. (7)Such couplings generate both, corrections to the neutrino masses as well as NMMs. In this workwe compute the loop diagrams with the help of package X [17]. For the neutrino mass correctionwe obtain in the limit M ≡ m s = m ψ δm ν i ν j = f i f (cid:48) j + f j f (cid:48) i π M log M µ . (8)The magnetic and electric dipole moments can be extracted from the corresponding form factorsof the effective neutrino-photon interaction Lagrangian L effint = − F jiµ ( q ) ν j iσ µν q ν m ν j + m ν i ν i − i F ji(cid:15) ( q ) ν j iσ µν q ν m ν j + m ν i γ ν i (9)by taking the limit q → µ ν j ν i = F jiµ (0), (10) (cid:15) ν j ν i = F ji(cid:15) (0). (11)Projecting out the corresponding form factors, we get in the limit M ≡ m s = m ψ µ ν j ν i = i(cid:15)e π M Im[ f i f (cid:48) j − f j f (cid:48) i ], (12) (cid:15) ν j ν i = i(cid:15)e π M Re[ f i f (cid:48) j − f j f (cid:48) i ], (13)where (cid:15) is the fractional charge of s and ψ . Assuming no cancellation in the couplings among theflavours one arrives at the relation between (cid:15) and Mµ ν δm ν = (cid:15)e M . (14)Now one can ask the question, which values for mass and millicharge of the new particles arenecessary so that observable NMMs can be generated without fine-tuning. Taking δm ν ∼ . µ ν close to the current experimental sensitivity, we obtain the requiredratio (cid:15)/M . The result is shown in Fig. 4, where we overlay the curves of constant NMM overexcluded regions [18, 19] in the plane of fractional charge and mass of the new particle.There seems to be no room for large NMMs generated by light millicharged particles. RedGiantsCollidersSLACSN1987A BBN N eff CMB N eff - - - - -
20 Log ( M / eV ) L o g ( ϵ ) μ ν = - μ B μ ν = - μ B μ ν = - μ B Figure 4: Lines of constant µ ν for δm ν = 0 . M and fractional charge (cid:15) of themillicharged particle. The constraints are coming from several observables and are taken from Ref. [18]. Seealso the working group report and references therein [19].
4. RADIATIVE NEUTRINO MASS MODEL
Let us now explicate the generic difficulty to obtain large NMMs without fine-tuning neutrinomasses in models with new physics above the electroweak scale. We start by adding two scalar SU (2) L doublets η , φ as well as a new charged Dirac fermion Σ = Σ L + Σ R with the quantumnumbers η = (cid:18) η η − (cid:19) ∼ (2 , − / L Li = (cid:18) ν Li l Li (cid:19) ∼ (2 , − / φ = (cid:18) φ − φ −− (cid:19) ∼ (2 , − / − L/R ∼ (1 , − L Li is the SM lepton doublet. Neutrinos are massless at the tree-level and neutrino massesare generated at loop-level via the Yukawa interactions L Y = Y i Σ R ˜ η † L Li + Y (cid:48) j Σ cL φ † L Lj + h.c. (17)From the scalar potential interactions the electroweak symmetry breaking generates the mixingbetween η − and φ − (cid:18) η η (cid:19) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) ηφ (cid:19) , (18)which leads to L Y = Y i Σ R (cos θη − − sin θη − ) ν Li + Y (cid:48) j ν CLj (sin θη +1 + cos θη − )Σ L + h.c. (19)The neutrino mass matrix results from the loop diagram depicted in Fig. 5(a). Note that thecontributions from η and η differ by a relative minus sign, so that the divergencies cancel eachother. We obtain M ν i ν j = Y i Y (cid:48) j + Y j Y (cid:48) i π m Σ sin θ cos θ (cid:34) m η m η − m log (cid:32) m η m (cid:33) − m η m η − m log (cid:32) m η m (cid:33)(cid:35) . (20)We added only one charged Dirac fermion Σ, implying that only two of the eigenvalues of M arenon-zero. Hence the lightest neutrino is massless. ν i ν j η , η Σ (a) ν i ν j η , η Σ γ (b) ν i ν j η , η Σ γ (c) Figure 5: Diagrams for neutrino mass and magnetic moment in the radiative neutrino mass model.
The electric and magnetic dipole moments result from the diagrams depicted in Fig. 5(b), (c)and are computed as in the previous section. The result is µ ν j ν i = − ie sin θ cos θ π m Σ Im (cid:2) Y i Y (cid:48) j − Y j Y (cid:48) i (cid:3) f ( m m , m m ), (21) (cid:15) ν j ν i = − ie sin θ cos θ π m Σ Re (cid:2) Y i Y (cid:48) j − Y j Y (cid:48) i (cid:3) f ( m m , m m ), (22)with the loop function f ( a , a ) = a ( a − log( a ) − ( a − (cid:0) − ( a + 1) a + ( a − a log( a ) + a + a (cid:1) ( a − ( a − . (23)Note that for Majorana neutrinos, we expect µ ν j ν i and (cid:15) ν j ν i to be hermitian and antisymmetric, i.e.to be purely imaginary. In addition, if CP is conserved, either the magnetic or the electric momentis zero. See for example Ref. [6] for more details. Now, what can we learn from this exercise?To answer this question, let us first recognize that in this model the origin of the NMM is thesame as the neutrino mass. There are no other sources of neutrino masses so that fine-tuning is notpossible. Due to this connection it is possible to predict the NMM matrix by using experimentalvalues of the leptonic mixing matrix and the neutrino masses.As an example, we assume all CP-phases of the PMNS-matrix U to be zero. Since in our modelthe lightest neutrino is massless, the masses of the other two are given by the measured mass squaredifferences. We use the results of the global fit from Ref. [20] and obtain the mass matrix from therelation M ν j ν i = U diag(0 , m ν , m ν ) U † . (24)Using Eq. (20) with reasonable numbers for the scalar and fermion masses m = 1 . m = 0 . m Σ = 1 TeV one can solve Eq. (24) for the Yukawa couplings Y Y Y = . ∓ . i . ∓ . i · x · − , Y (cid:48) Y (cid:48) Y (cid:48) = . . ± . i . ± . i · x · − , x ∈ C . (25)In this way we obtain for the Majorana neutrino electric and dipole moment matrices µ ν j ν i = ± i − − .
52 0 − . . . · − µ B , (cid:15) ν j ν i = 0, (26)with values many orders of magnitude below current experimental sensitivity. Since it does not al-low for fine-tuning, this model illustrates the generic problem in generating large NMMs. Therefore,consistent models predicting large NMMs have to include a mechanism that avoids this connectionof neutrino mass and NMM. That is why in well-studied models without such a mechanism, likethe left-right symmetric model [21] and the supersymmetric model [22], the NMM predictions arefar from being detected in next-generation experiments. On the other hand, a recent parameterstudy in the framework of the minimal supersymmetric model found room for large NMM [23], butdoes not solve the fine-tuning problem.
5. NATURALLY LARGE NMM VIA SYMMETRIES
To generate a sizable NMM and to avoid fine-tuning by suppressing neutrino mass loop contri-butions one should rely on some sort of a symmetry. There are two classes of symmetries. First onecould try to build a suppression mechanism using one of the quantum numbers of the photon. Thiswas proposed by Barr, Freire and Zee (BFZ) in Ref. [24–26] using the spin. For the other quantumnumbers, like the parity or charge conjugation we checked all one loop subdiagram possibilitiesand found no such suppression mechanism. Second, there are models exploiting the symmetryproperties of the effective NMM and mass operators. The following were already proposed in theliterature, namely: Voloshin-type symmetry [27, 28] (e.g. SU(2) with ν ↔ ν C ), SU(2) horizontalsymmetry [29, 30] and discrete symmetries [31–35]. In Ref. [24] BFZ proposed the spin-suppression mechanism. The idea is that the loop diagramgenerating the NMM has a sub-diagram involving the scalar h + and the vector W . The neutrinomass contribution diagram has the same sub-diagram with the photon line removed, see Fig. 6.In this case, because of the spin conservation, only the longitudinal degrees of freedom of the W contribute. When the sub-diagram is embeded in the full diagram in Fig. 7 (a) it will beproportional to the Yukawa coupling and the neutrino mass contribution is thus suppressed bypowers of the lepton mass. Note that this mechanism still holds for higher order contributions, i.e.also diagrams of the form of Fig. 2 are suppressed. In this way the naturalness bounds summarizedin the previous section can be avoided. h + W + γ (a) h + W + (b) Figure 6: The sub-diagrams of the BFZ spin suppression mechanism. When removing the photon line,only the longitudinal components of the W will contribute, because of the spin conservation. An essential ingredient for this mechanism is the charged scalar singlet h + with the coupling tothe SM lepton doublet in the form L = f ji h + L cLj iτ L Li . (27) ν i ν j l − W + h + φ b φ + a λ ab (a) ν Ri ν Lj l − W + φ +2 φ +2 φ h φ i (b) Figure 7: (a) Two-loop neutrino mass contribution in the BFZ model. The NMM can be computed byattaching the photon line to any of the charged particles inside the loop. (b) A similar diagram for themodel with Dirac neutrinos.
The realization of spin suppression mechanism in [24] uses three scalar doublets φ a , with theneutral component of one of them, say φ , obtaining a non-zero vacuum expectation value. Fromthe antisymmetric interaction L = ˜ M ab h + ( φ − a φ b − φ − b φ a ) (28)and the quartic term of the scalar potential L = λ ab (cid:104) φ † (cid:105) φ a (cid:104) φ † (cid:105) φ b (29)one obtains the diagram for the NMM, see Fig. 7(a).In order to estimate if the model is still viable, one can derive the following relation betweenthe radiative neutrino mass δm ν i ν j and the NMM µ ν i ν j [24]: δm ν i ν j = (cid:32) m j − m i M W (cid:33) · (cid:18) δM + δM M (cid:19) · (cid:18) M TeV (cid:19) · (cid:18) µ ν i ν j − µ B (cid:19) · . · eV, (30)where m i are the charged lepton masses, M W the W boson mass, M is the scalar mass, assuming M ≡ M ∼ M and δM , δM being the mass differences of the charged and neutral componentsof φ and φ . New charged scalar particles like h + and φ +2 , would have been seen by the LHC ifconsiderably lighter than 1 TeV. See for example SUSY searches for slepton decays [36, 37]. Inthe limit of massless neutralinos the bounds are of the same order of magnitude as for h + due tosimilar decay channels. Let us therefore assume the new particle masses at TeV scale, M ∼ µ ν i ν j ∼ − µ B this yields δm ν e ν µ = (cid:18) δM + δM M (cid:19) eV, (31) δm ν µ ν τ , δm ν τ ν e = (cid:18) δM + δM M (cid:19) · . · eV. (32)In order to satisfy the limit on the upper bound of neutrino masses from various cosmologicalobservations [9], one needs δm ν i ν j (cid:46) m ν i ν j < . δM + δM M < . · − with noneed for fine-tuning. This shows that even though this is a two-loop diagram, the mechanism stillgives sizable NMMs and is in agreement with current experimental bounds.It is interesting to think about a modified version of this model in order to apply the idea toDirac neutrinos. We hence need a scalar connecting the right-handed neutrinos and the left-handedcharged leptons. Beside the Higgs doublet φ , one could introduce an additional scalar doublet φ = ( φ , φ − ) T with the interaction Y L L φ ν R . Then with the term from the scalar potential λφ † φ φ † φ one would obtain the Feynman diagram depicted in Fig. 7(b) leading to a large NMM.However, the potential also contains the coupling λ (cid:48) φ † φ φ † φ which after electroweak symmetrybreaking generates a term linear in φ , i.e. inducing (cid:104) φ (cid:105) (cid:54) = 0. This leads to an additional tree-levelsource of neutrino mass and thus fine-tuning can not be avoided. Therefore, there is no simpleimplementation of the BFZ spin suppression mechanism for Dirac neutrinos. Another suppression mechanism is to impose SU (2) ν symmetry with (( ν R ) C , ν L ) T transformingas a doublet. It contains the transformation ν L → ( ν R ) C , ν R → − ( ν L ) C , so that the mass and theNMM operators transform as [27] ν L ν R → − ν L ν R , (33) ν L σ µν ν R F µν → + ν L σ µν ν R F µν , (34)i.e. the NMM term is invariant under this symmetry, while the mass term is not. Note that forincorporating this idea one needs Dirac neutrinos. In an UV-complete theory ( ν R ) C then needs tobe in the same multiplet with ν L , which is already a part of the SU (2) L doublet. The simplestpossible implementation is to enlarge the electroweak gauge symmetry to SU (3) L × U (1) X fromRef. [28]. The SU (2) ν symmetry can not be exact and the neutrino mass is therefore proportionalto the breaking scale of the new symmetry.The NMM and neutrino mass are generated by diagrams with two charged components η and η from the scalar SU (3) L triplet. They are related by [28] µ ν = δm ν e ∆ m η log m η m τ . (35)We have to take into account the naturalness condition on the squared mass difference ∆ m η = m η − m η , emerging from radiative corrections after symmetry breaking [28]:∆ m η (cid:38) α W π M V , (36)where M V is the mass of the vector boson associated with the SU (2) ν symmetry breaking and α W is the electroweak fine-structure constant.Taking the experimental limits on the SU (3) L gauge boson masses [38] into consideration weset M V ∼ m η ∼ m η (cid:38) · GeV . By setting δm ν (cid:46) . µ ν (cid:46) − µ B . This still implies fine-tuning of four orders of magnitude to reach anobservable NMM of µ ν ∼ − µ B . We thus conclude that within this framework it is not possibleto generate observable NMM in theoretically consistent way. The idea from the Voloshin symmetry can also be applied to Majorana neutrinos, which havezero diagonal NMM. Babu and Mohapatra [29] proposed that a large transition NMM can beachieved while suppressing neutrino mass contribution by using horizontal flavour SU (2) H symme-try. In their model the electron and muon SU (2) L doublets together form the SU (2) H doublet Ψ L ,while the tau doublet is a SU (2) H singlet Ψ L . Also the right-handed electron and muon togetherform a SU (2) H doublet Ψ R .0 For this mechanism to work, Babu and Mohapatra introduce in addition to the Higgs doublet φ s the following new scalars: one bidoublet φ (i.e. doublet under SU (2) H as well as under SU (2) L ),one SU (2) H doublet η = (cid:0) η +1 η +2 (cid:1) and two SU (2) H triplets σ , . The latter are responsiblefor breaking the horizontal symmetry in such a way that there is no tree-level mixing betweengeneration-changing horizontal gauge bosons and the generation-diagonal ones, for more details werefer to Ref. [29].Introducing this set of particles lead among others to the Yukawa couplings f ηiτ Ψ cL iτ Ψ L and f (cid:48) tr(Ψ L φ ) τ R . Together with the interaction µ κ s ( η +1 φ +1 + η +2 φ +2 ) coming from the cubic term fromthe scalar potential, where κ s is the vacuum expectation value of the SM Higgs, one arrives at the ν e − ν µ transition NMM µ ν e ν µ = 2 e f f (cid:48) π m τ µ κ s m η − m φ (cid:32) m η − m φ (cid:33) , (37)with m η = m η = m η and m φ = m φ = m φ . The horizontal symmetry is spontaneously brokenby the vacuum expectation values of the scalar triplets. The breaking induces a mass splittingbetween the charged components of φ and η and thus leads to non-zero neutrino mass δm ν e ν µ = f f (cid:48) π m τ µ κ s (cid:32) m φ − m η log m φ m η − m φ − m η log m φ m η (cid:33) . (38)Assuming ∆ m η = m η − m η (cid:28) m η and ∆ m φ = m φ − m φ (cid:28) m φ as well as ∆ m η /m η = ∆ m φ /m φ one obtains (cid:18) µ ν e ν µ − µ B (cid:19) = 2 (cid:18) δm ν e ν µ eV (cid:19) (cid:18) GeV ∆ m η (cid:19) (cid:32) m η m φ − (cid:33) log m η m φ . (39)This shows that one can obtain NMM of the order 10 − µ B without fine-tuning, if the mass splitting∆ m η is at GeV scale. The ∆ m η can be small and technically natural because it emerges from asoft cubic interaction with the triplet σ that breaks the SU (2) H .The model can accommodate SU (2) H breaking in the charged lepton m e and m µ masses. Italso predicts additional neutrino mass contributions m ν e ν τ and m ν µ ν τ . Demanding that their valuesare less than 0 . µ ν e ν µ ∼ − µ B .One could think of including the τ flavour instead of e or µ flavour in SU (2) H , or extendingthe horizontal symmetry to all three generations, e.g. using SU (3) H . Both of which would notallow for an extra source of the horizontal symmetry breaking in the coupling of the Higgs bosonto charged leptons, since h → τ τ decays have been observed by the LHC [39, 40]. This mechanismtherefore can only give a large ν e - ν µ transition moment.
6. DISCUSSION AND CONCLUSION
SM predictions for the NMM are many orders of magnitude lower than current experimentalsensitivity. With large NMMs generated by millicharged particles below the electroweak scale onecan in principle avoid fine-tuning of the neutrino masses, but it would be in strong tension withcosmological observations. As we have showed in a very insightful model, theories with new physicsabove the electroweak scale predicting observable NMMs generically lead to large neutrino masscorrections, thus requiring fine-tuning of several orders of magnitude. We reviewed models proposed1in literature that avoid the resulting naturalness bounds and suppress the neutrino mass correctionby a symmetry. It turned out that building a model with large Dirac NMM in a technically naturalway does not seem to be possible anymore. On the other hand, for Majorana neutrinos, using a SU (2) H horizontal symmetry one can only realize a large ν e - ν µ transition moment. In the BFZmodel, which relies on the spin-suppression mechanism, it is also possible to generate sizable ν e - ν µ as well as ν e - ν τ and ν µ - ν τ transition moments.In Ref. [41] Fr`ere, Heeck and Mollet derive inequalities between the transition moments forMajorana neutrinos. They argue that a possible measurement of µ ν τ at SHiP [42] would hint tothe Dirac nature of the neutrino. However, in this work we have shown that NMMs of observablesize can not be generated by models with Dirac neutrinos in a theoretically consistent way. Acknowledgments
We are thankful for very helpful discussions with Evgeny Akhmedov, Hiren Patel and StefanVogl. BR acknowledges the support by the Alexander von Humboldt Foundation. [1] K. Fujikawa and R. Shrock, Phys. Rev. Lett. , 963 (1980).[2] P. B. Pal and L. Wolfenstein, Phys. Rev. D25 , 766 (1982).[3] R. E. Shrock, Nucl. Phys.
B206 , 359 (1982).[4] M. Dvornikov and A. Studenikin, Phys. Rev.
D69 , 073001 (2004), hep-ph/0305206.[5] M. S. Dvornikov and A. I. Studenikin, J. Exp. Theor. Phys. , 254 (2004), hep-ph/0411085.[6] C. Giunti and A. Studenikin, Rev. Mod. Phys. , 531 (2015), 1403.6344.[7] A. G. Beda, V. B. Brudanin, V. G. Egorov, D. V. Medvedev, V. S. Pogosov, M. V. Shirchenko, andA. S. Starostin, Adv. High Energy Phys. , 350150 (2012).[8] B. C. Canas, O. G. Miranda, A. Parada, M. Tortola, and J. W. F. Valle, Phys. Lett. B753 , 191 (2016),[Addendum: Phys. Lett.B757,568(2016)], 1510.01684.[9] C. Patrignani et al. (Particle Data Group), Chin. Phys.
C40 , 100001 (2016).[10] C. Giunti, K. A. Kouzakov, Y.-F. Li, A. V. Lokhov, A. I. Studenikin, and S. Zhou, Annalen Phys. ,198 (2016), 1506.05387.[11] T. S. Kosmas, O. G. Miranda, D. K. Papoulias, M. Tortola, and J. W. F. Valle, Phys. Rev.
D92 ,013011 (2015), 1505.03202.[12] T. S. Kosmas, O. G. Miranda, D. K. Papoulias, M. Tortola, and J. W. F. Valle, Phys. Lett.
B750 , 459(2015), 1506.08377.[13]
CONUS: The COhernt NeUtrino Scattering experiment , in preparation.[14] N. F. Bell, V. Cirigliano, M. J. Ramsey-Musolf, P. Vogel, and M. B. Wise, Phys. Rev. Lett. , 151802(2005), hep-ph/0504134.[15] N. F. Bell, M. Gorchtein, M. J. Ramsey-Musolf, P. Vogel, and P. Wang, Phys. Lett. B642 , 377 (2006),hep-ph/0606248.[16] S. Davidson, M. Gorbahn, and A. Santamaria, Phys. Lett.
B626 , 151 (2005), hep-ph/0506085.[17] H. H. Patel, Comput. Phys. Commun. , 276 (2015), 1503.01469.[18] H. Vogel and J. Redondo, JCAP , 029 (2014), 1311.2600.[19] R. Essig et al., in
Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics:Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013 (2013),1311.0029, URL http://inspirehep.net/record/1263039/files/arXiv:1311.0029.pdf .[20] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, and T. Schwetz, JHEP , 087(2017), 1611.01514.[21] M. Nemevsek, G. Senjanovic, and V. Tello, Phys. Rev. Lett. , 151802 (2013), 1211.2837.[22] M. Gozdz, W. A. Kaminski, F. Simkovic, and A. Faessler, Phys. Rev. D74 , 055007 (2006), hep-ph/0606077.[23] A. Aboubrahim, T. Ibrahim, A. Itani, and P. Nath, Phys. Rev.
D89 , 055009 (2014), 1312.2505. [24] S. M. Barr, E. M. Freire, and A. Zee, Phys. Rev. Lett. , 2626 (1990).[25] S. M. Barr and E. M. Freire, Phys. Rev. D43 , 2989 (1991).[26] K. S. Babu, D. Chang, W.-Y. Keung, and I. Phillips, Phys. Rev.
D46 , 2268 (1992).[27] M. B. Voloshin, Sov. J. Nucl. Phys. , 512 (1988), [Yad. Fiz.48,804(1988)].[28] R. Barbieri and R. N. Mohapatra, Phys. Lett. B218 , 225 (1989).[29] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. , 228 (1989).[30] M. Leurer and N. Marcus, Phys. Lett. B237 , 81 (1990).[31] D. Chang, W.-Y. Keung, S. Lipovaca, and G. Senjanovic, Phys. Rev. Lett. , 953 (1991).[32] G. Ecker, W. Grimus, and H. Neufeld, Phys. Lett. B232 , 217 (1989).[33] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. , 1705 (1990).[34] D. Chang, W.-Y. Keung, and G. Senjanovic, Phys. Rev. D42 , 1599 (1990).[35] H. Georgi and L. Randall, Phys. Lett.
B244 , 196 (1990).[36] V. Khachatryan et al. (CMS), Eur. Phys. J.