Gauge Anomalies and Neutrino Seesaw Models

Lu´ıs M. Cebola Under Supervision of Ricardo GonzálezFelipe CFTP, Departamento de F´ısica, Instituto Superior Técnico (IST), Lisboa (Dated: December 10, 2013) Adding right-handed neutrino singlets and/or fermion triplets to the particle content of the Standard Model allows for the implementation of the seesaw mechanism to give mass to neutrinos and, simultaneously, for the construction of anomaly-free gauge group extensions of the theory. In this project, after briefly reviewing the SM and some theoretical aspects of anomalies, we discuss the anomaly cancellation and electric charge quantization in three popular (type I, II and III) seesaw mechanisms. We then study the seesaw realization of viable two-zero textures for the effective neutrino mass matrix, considering Abelian extensions based on an extra U(1)X gauge symmetry. By requiring gauge anomalies cancellation, we perform a detailed analysis in order to identify the charge assignments under the new gauge symmetry that lead to neutrino phenomenology compatible with current experiments. In particular, we study how the new symmetry can constrain the flavour structure of the Majorana neutrino mass matrix, leading to two-zero textures with a minimal extra fermion and scalar content. The possibility of distinguishing different gauge symmetries and seesaw realizations at colliders is also briefly discussed.

Keywords: Charge Quantization, Gauge Anomalies, Gauge Symmetries, Neutrino Physics, Seesaw Mechanism.

I. STANDARD MODEL one obtain for the covariant derivative 2 † µ v 2 1 1µ 2 2µ The Standard Model (SM) of particle physics (DµH) (D H) = gW Wµ W + Wµ W 8 is a theory concerning three of the fundamental 2 v 3 3µ µ forces/interactions (electromagnetic, weak and strong + gW Wµ gY Bµ gW W gY B + , forces) in Universe, which predicts very accurately 8 − − ··· (6) many of the experimentally verified phenomena. In July/2012, LHC experiments at CERN announced the and for the potential discovery of a Higgs-like particle [1, 2], which is one of the most important discoveries in particle physics 1 3 1 V (H) = + µ2 + λv2 h2 = 2µ2 h2 . and a triumph for the SM predictive power. ··· 2 2 · · · − 2 From a theoretical point of view, SM is a quantum (7) field theory (QFT) invariant under the gauge group Rotating the weak bosons to the mass eigenstates (bosons W , Z and the photon A) we get for the com- SU(3)C SU(2)L U(1)Y , (1) plete mass spectrum ⊗ ⊗ p 2 v which is broken spontaneously by the scalar Higgs field mh = 2µ , mW = gW , − 2 H when it acquires a non-zero vacuum expectation q (8) v 2 2 value (VEV). This Higgs field realizes the electroweak mZ = gW + gY , mA = 0 , spontaneous symmetry breaking (SSB) and generates 2 the mass of weak gauge bosons through the so-called and for the covariant derivative Higgs mechanism [3–5]. The relevant parts of the SM + + − − Dµ = ∂µ igW W T + W T iQeAµ Lagrangian for this mechanism are − µ µ − gW 3 2 (9) † µ i T sW Q Zµ , Higgs = (DµH) (D H) V (H) , (2) − cW − L − i i where the scalar potential V (H) is where T = σ /2 and gY gW 2 s = , c = , 2 † † W p 2 2 W p 2 2 V (H) = µ H H + λ H H . (3) gW + gY gW + gY g g (10) e = Y W ,Q = T 3 + Y. The covariant derivative acting on H is p 2 2 gW + gY i i DµH = ∂µ igW W T igY BµYH H, (4) The operator eQ is the electric charge operator since, − µ − when the electroweak gauge group is spontaneously where gW and gY are, respectively, SU(2)L and U(1)Y broken to the QED one, eQ is coupled to the massless coupling constants and YH is the Higgs field hyper- gauge boson Aµ. charge. In the physical basis, where It is clear that the Higgs mechanism is the connect- ing piece between the electroweak gauge group and 1 0 massive bosons, allowing the unified theory of weak H = , (5) √2 h + v and electromagnetic forces to be consistent. 2

Fields λ λ H qLα `Lα uRα dRα eRα q q γλγ5 γλγ5 U(1)Y 1/2 1/6 −1/2 2/3 −1/3 −1 p q p p q p − − SU(2)L 2 2 2 1 1 1 Tµνλ : γν γµ + γµ γν SU(3)C 1 3 1 3 3 1 p k p k − 1 − 2

TABLE I. Representations under the SM gauge group and k2 k1 k1 k2 hypercharge assignments of the Higgs boson and the SM fermions. q q γ5 γ5 p q p p q p − − In the SM, fermion masses are also generated γν γµ + γµ γν through couplings with the Higgs ﬁelds since mass Tµν : p k1 p k2 − − terms as mf ff = mf fRfL + fLfR are forbid- − − k2 k1 k1 k2 den by the SU(2)L U(1)Y symmetry. Therefore, one can include Yukawa⊗ interactions between fermions and the Higgs ﬁeld and demand that the SSB mechanism FIG. 1. Triangle diagrams with vertices vector-vector- gives the mass terms for fermions. The correct gauge axial and vector-vector-pseudoscalar. invariant Yukawa Lagrangian is

αβ αβ Y ukawa = Y qLαHue Rβ Y qLαHdRβ II. GAUGE ANOMALIES L − u − d (11) αβ Ye `LαHeRβ + H.c. , − Another possible framework to discuss the hyper- 2 ∗ charge assignments is the consistency of the SM, i.e. where He = iσ H and Yu,d,e are respectively the up quark, down quark and charged-lepton Yukawa being free of gauge anomalies. Anomalies appear couplings matrices. To diagonalize these matrices we when symmetries of the classical Lagrangian are not make the unitary transformations invariant of the functional integral or the path integral formulation of the theory. If this is a gauge or local 0 0 symmetry, we have then a gauge anomaly. Since the fL f = Lf fL , fR f = Rf fR , (12) → L → R SM fermions are chiral, we need to discuss the chiral so that gauge anomaly, and hence, the possibility of violation of the Ward Identities (WI)/non-renormalizability of † the model [7]. Lf mf Rf = df = diag (mf1 , mf2 , mf3 ) . (13) The Abelian case is known as the Adler-Bell-Jackiw Under these transformations the neutral currents re- (ABJ) anomaly or the Abelian chiral anomaly [8, main unchanged, however the charged current (ne- 9]. When we calculate the amplitude ( (k) = µ M glecting the H.c. terms) becomes µ(k) (k), µ(k) is the polarization vector of the gaugeM boson) for some physically possible scattering gW 0 µ 0 0 µ 0 + process of an Abelian theory, one expect that the WI CC = uLαγ dLα + νLαγ eLα Wµ µ L √2 ⇔ (kµ (k) = 0) holds. However, computing the exam- Mλ gW αβ µ µ + ple q Tµνλ presented in Fig. 1 we obtain an anomalous CC = uLα V γ dLβ + νLαγ eLα Wµ , 1 L √2 CKM term. The axial current conservation reads (14) λ λ ∂z Gµνλ = 2miGµν q Tµνλ = 2mTµν , (16) where ⇔   where Vud Vus Vub † VCKM = L Ld = Vcd Vcs Vcb (15) u   G = 0 T j (x)j (y)ψγ γ ψ 0 , V V V µνλ µ ν µ 5 td ts tb (17) Gµν = 0 T jµ(x)jν (y)ψγ5ψ 0 , is the Cabibbo-Kobayashi-Maskawa quark mixing matrix [6]. It follows that, at tree level, ﬂavour can only and Tµνλ, Tµν are their respective Fourier transforma- be changed by charged currents. Therefore, there are tions in momentum space. Using Feynman rules, we no ﬂavour changing neutral currents (FCNC) at tree level in the SM. The particle group representations is another important aspect that allows us to construct gauge invariant terms in the SM Lagrangian. Since the Higgs 1 It is clear that the axial current is only conserved in the boson does not couple to the photon, from Eq. (10), it massless case, nevertheless, as we shall see, the anomalous 1 term appears even if the particle is massless. In the same must have YH = 2 . The hypercharge assignments of way, the quantum analogy of this relation, i.e. the axial WI the SM fermions and their representations are summa- (AWI), holds solely in the massless case. For our purpose, we rized in Table I, which allows the Yukawa Lagrangian shall consider that non-conservation exclusively occurs when to be invariant. an anomaly (anomalous term) is present. 3

Due to the fact that fermions with opposite chiral- qλ qλ c c γλγ5t γλγ5t ity contribute with opposite sign to the anomalous p q p p q p term [11], the anomaly-free condition is abc − − T : a b b a µνλ γνt γµt + γµt γνt a b c p k p k trR t , t t = 0 , (23) − 1 − 2

k2 k1 k1 k2 when summed over all fermions of the theory. In the SM the hypercharges assignments and parti- FIG. 2. Triangle diagrams with vertices vector-vector- cles representations lead to a consistent theory since axial for non-Abelian gauges. all gauge anomalies vanish. Rather than checking if the SM is anomaly free, given a set of hypercharges, we could impose the can compute Tµνλ as anomaly-free conditions and verify whether or not these constraints lead to the uniqueness of hyper- Z d4p i i charges. If there is a single solution, then electric Tµνλ = i tr γλγ5 − (2π)4 p/ m p/ /q m× charge (hypercharge) is quantized [12, 13]. In this − − − case, the relevant (generalized) anomaly-free condi- ! (18) i k1 k2 tions are γν γµ + ↔ . p/ k/1 m µ ν − − ↔ nG 3 X 3 3 3 3 3 [U(1)Y ] : 6Y + 2Y 3Y 3Y Y = 0 , qi ì − ui − di − ei Thus, the axial WI (AWI) is written as i=1 n λ G q Tµνλ = 2mTµν + µν , (19) 2 X A U(1)Y [SU(2)L] : (3Yqi + Yì ) = 0 , i=1 and conservation occurs only if µν = 0. When cal- nG culating this anomalous term, oneA should use a reg- 2 X U(1)Y [SU(3)C ] : (2Yq Yu Yd ) = 0 , i − i − i ulator that preserves gauge invariance (e.g. Pauli- i=1 Villars regularization [10]) in order to obtain the phys- n XG ical/correct value (i.e. the physical value does not de- U(1)Y : (6Yq + 2Y` 3Yu 3Yd Ye ) = 0 , i i − i − i − i pend on the regularization scheme used and thus, we i=1 should compute it with a gauge invariant regulator). (24) After a laborious algebra manipulation we obtain containing fifteen free parameters. Even with the con- 1 α β straints that arise from the gauge invariant Yukawa µν = εµναβk1 k2 , (20) A 2π2 part of the SM Lagrangian, given in Eq. (11), and hence, AWI does not hold. If we consider gauge Yq YH + Yu = 0 , Yq + YH + Yd = 0 , bosons (massless case) as external legs coupled to the − i − i − i i (25) Y` + YH + Ye = 0 , i = 1, 2, 3 , triangle diagram depicted in Fig. 1, the result would − i i be the same: there is an anomaly/anomalous term. To address the same problem in non-Abelian gauge there are only thirteen equations, which do not yield a theories (e.g. in the SM), we can reproduce this cal- unique solution for the sixteen hypercharges (one also culation including the non-Abelian generators in the needs to include YH ). If we instead consider family vertices (from Feynman rules, the group generator ta universal charges, one obtains a unique solution and modifies the vertex Γµ to Γµta). The respective dia- charge quantization. Thus in the SM, electric charge gram is presented in Fig. 2 and we get is quantized only if family universal charges are assumed. Z 4 abc d p i c i Tµνλ = i tr γλγ5t − (2π)4 p/ m p/ /q m× −  − − III. NEUTRINOS AND SEESAW k1 k2 MECHANISMS a i b ↔ γν t γµt +  µ ν  . p/ k/ m  ↔  − 1 − a b Neutrino oscillation experiments have firmly estab- ↔ (21) lished the existence of neutrino masses and lepton mixing. Since neutrinos are strictly massless in the Since the gamma matrices commute with group gen- SM, this implies that new physics beyond the SM is µ a a erators ([γ , t ] = [γ5, t ] = 0), we can factorize required to account for these observations [14, 15]. c a b trR t t t , which stands for the trace of the group The main concept of these oscillations is very simi- generators in the representation of the fields. This lar to the transitions that change quark flavour. When modification leads to a change inR the anomalous term neutrinos take part in weak interactions they are cre- Aµν given in Eq. (20), ated with a specific lepton flavour (νe, νµ, ντ ), al- though there is a non-zero probability of being in a abc 1 α β a b c different flavour state when they are measured. For = εµναβk k trR t , t t . (22) Aµν 4π2 1 2 three generations of neutrinos, the mismatch between 4 interaction basis and mass basis is given by simple possibilities consist of the addition of singlet right-handed neutrinos (type I seesaw), colour-singlet 3 2 0 X SU(2)L-triplet scalars (type II) or SU(2)L-triplet ν = Uαi νi , (26) | αi | i fermions (type III). i=1 To generate type I seesaw, nR right-handed neu- where U is a 3 3 unitary matrix similar to the quark trino fields νRi with the respective gauge group rep- × resentations and hypercharge assignments (1, 1, 0) mixing matrix [16] and νi is a mass eigenstate. ∼ The fact that neutrino| massesi are tiny constitutes are introduced [18–22]. The respective Lagrangian a puzzling aspect of nowadays particle physics. Since (neglecting the H.c. terms) is neutrinos are massive, the SM should be considered i 1 an effective theory and it is necessary to extend it αi ij c I = SM + νRi∂ν/ Ri Yν `LαHνe Ri MR νRiνRj , with non-renormalizable terms that generate neutrino L L 2 − − 2 (33) masses through new physics. The lowest order non- where Y is a 3 n complex Yukawa coupling matrix renormalizable operator, which generates Majorana ν R and M is a n × n symmetric matrix. Integrating neutrino masses after SSB, is the unique d = 5 Wein- R R R out the heavy states× since they decouple from the low berg operator [17] energy theory, we can obtain αβ T z αβ W einberg = `LαHe C `LβHe +H.c. , (27) z −1 T αβ L − Λ Yν M Y . (34) Λ ' − R ν αβ 2 αβ where mν = v z /Λ is the 3 3 effective neutrino mass matrix, zαβ are complex constants× and Λ is the Directly from Eq. (27), we then get the desired effec- m new high-energy physics cutoff scale, expected to be tive mass matrix of the light neutrinos, with D = Y high enough to generate tiny masses. In this context, v ν , we can write the lepton mass terms in the interaction 2 −1 T −1 T mν = v Yν M Y = mDM m . (35) basis as − R ν − R D

αβ 0 0 1 αβ 0 0c In order to generate the effective neutrino mass matrix lep.mass = me eLαeRβ mν νLα νLβ + H.c. . L − − 2 within the context of type III seesaw, one includes nΣ (28) fermion triplets ΣRi to the SM particle content [23]. In order to diagonalize these matrices we use the uni- The respective gauge group representations and hy- tary transformations given in Eq. (12), percharge assignments are (1, 3, 0). Usually, one ∼ 0 αβ 0 αβ 0 αi defines the field ΣRi as eLα = Le eLβ , eRα = Re eRβ , νLα = Lν νLi , (29) 0 + ! ΣRi √2ΣRi which lead to ΣRi = − 0 , (36) √2ΣRi ΣRi † − LemeRe = de = diag (me, mµ, mτ ) , (30) † ∗ and write the respective type III Lagrangian as Lν mν Lν = dn = diag (m1, m2, m3) ,

i αi where de, dn are the diagonal mass matrices and mi III = SM + tr ΣRiD/ΣRi Y `LαΣRiHe L L 2 − T (i = 1, 2, 3) is the mass of the light neutrino νi. With (37) these transformations the charged current (neglecting 1 ij c MΣ tr ΣRiΣRj + H.c. , the H.c. terms) becomes − 2

gW αβ µ †iα µ + where YT is a 3 nΣ complex Yukawa coupling uLα V γ dLβ + νLi U γ eLα W , × CKM PMNS µ matrix and MΣ is a nΣ nΣ symmetric matrix. √2 × (31) This mass term includes Majorana masses for the 0 c 0 ﬁelds ΣRi + ΣRi and Dirac masses for the ﬁelds + c − where ΣRi +ΣRi. Looking closely to the type I Lagrangian   given in Eq. (33), we identify similar terms by simply Ue1 Ue2 Ue3 replacing νRi, Yν , MR by ΣRi, YT , MΣ . There- †   { } { } UPMNS = LeLν = Uµ1 Uµ2 Uµ3 . (32) fore, it is straightforward to conclude that Uτ1 Uτ2 Uτ3 αβ z −1 T αβ YT MΣ YT , The unitary matrix UPMNS is known as the Λ ' − (38) Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic 2 −1 T −1 T mν = v YT MΣ YT = mT MΣ mT , mixing matrix. − −

One of the most appealing theoretical frameworks with mT = vYT . to understand the smallness of neutrino masses is the so-called seesaw mechanism. In this context, the tree- To address the charge quantization problem in the level exchanges of new heavy states generate the ef- context of seesaw models one needs to modify the fective neutrino mass matrix at low energies. Three anomaly-free equations to3 5

nG nΣ nR 3 X 3 3 3 3 3 X 3 X 3 [U(1)Y ] : 6Y + 2Y 3Y 3Y Y 3Y Y = 0 , qi `i − ui − di − ei − σi − νi i=1 i=1 i=1 nG nΣ 2 X X U(1)Y [SU(2)L] : (3Yq + Y` ) 4Yσ = 0 , i i − i i=1 i=1 (39) nG 2 X U(1)Y [SU(3)C ] : (2Yq Yu Yd ) = 0 , i − i − i i=1 n n n XG XΣ XR U(1)Y : (6Yq + 2Y` 3Yu 3Yd Ye ) 3Yσ Yν = 0 , i i − i − i − i − i − i i=1 i=1 i=1

which contain 15 + nR + nΣ free parameters. Even textures. Their phenomenological implications have with the gauge invariance constraints that arise from been studied in Refs. [24–28] and we shall not discuss the type I Lagrangian (or type III Lagrangian or even them any further here. in a mixed type I/type III scenario), charge is not To conclude this section we present the possi- quantized. However, if family universal charges are ble two-zero textures of mν . There are fifteen assumed, the electric charge quantization appears in textures which can be classified into six categories the framework of type I, II or III seesaw models. (A, B, C, D, E, F): Despite the simplicity of the seesaw mechanism in explaining the smallness of the neutrino masses,     the corresponding high-energy theory usually contains 0 0 0 0 ∗ ∗ many more free parameters than those required at low A1 : 0  , A2 :   ;  ∗ ∗ ∗ ∗ ∗ energies. We recall that the effective neutrino mass 0 matrix mν can be written in terms of only nine phys- ∗ ∗ ∗ ∗ ∗ ical parameters: 3 light neutrino masses and 3 mix-     ing angles + 3 phases, that parametrize the PMNS 0 0 mixing matrix. For instance, the type I seesaw La- ∗ ∗ ∗ ∗ B1 :  0  , B2 : 0  , grangian given in Eq. (33) with n right-handed neu- ∗ ∗  ∗ ∗ R 0 0 trino fields, contains altogether 7nR 3 free parame- ∗ ∗ ∗ ∗ ters. The same parameter counting holds− for the type III seesaw with the replacements nR nΣ, Yν YT     → → 0 0 and MR MΣ. ∗ ∗ ∗ ∗ → B3 : 0 0  , B4 :   ; It then becomes clear that for a high energy seesaw  ∗ ∗ ∗ ∗ theory to be predictive the number of free parameters 0 0 ∗ ∗ ∗ ∗ should be somehow reduced. A well-motivated framework is provided by the so-called zero textures of the       Yukawa coupling matrices. In some cases, such zeros ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ also propagate to mν , implying relations among the C :  0  ; D1 :  0 0 , D2 :  0 ; ∗ ∗ ∗  ∗ ∗  neutrino observables. These textures can be obtained, 0 0 0 0 for instance, in the presence of flavour symmetries or ∗ ∗ ∗ ∗ ∗ additional local gauge symmetries.       The neutrino mass matrix mν is a symmetric ma- 0 0 0 trix with six independent entries. There are 6!/[n!(6 ∗ ∗ ∗ ∗ ∗ ∗ − E1 :  0  , E2 :   , E3 :  0 ; n)!] different textures, each containing n independent ∗ ∗ ∗ ∗ ∗ ∗ ∗  0 0 texture zeros. Since each matrix entry is a complex ∗ ∗ ∗ ∗ ∗ ∗ ∗ number, there are 2n constraints. It can be shown that any pattern of mν with more than two indepen-       0 0 0 0 dent zeros (n > 2) is not compatible with current ∗ ∗ ∗ ∗ ∗ neutrino oscillation data. Clearly, one-zero textures F1 : 0  , F2 : 0 0 , F3 :  0 ;  ∗ ∗  ∗  ∗ ∗  in mν have much less predictability than the two-zero 0 0 0 0 ∗ ∗ ∗ ∗ ∗ the symbol “ ” denotes a nonzero matrix element. In the flavour basis,∗ where the charged-lepton mass ma- 2 Since for our purposes, namely the study of anomaly-free trix me is diagonal, i.e. me = de, only seven pat- gauge extensions of the SM and their connection with the terns, to wit A1,2, B1,2,3,4 and C [29], are compatible flavour structure of mν , the type II seesaw mechanism does not lead to relevant constraints (due to its bosonic character), with the present neutrino oscillation data [30]. Since we shall not consider it in this work. any ordering of the charged leptons in the flavour ba- 3 The complete discussion of quantized charges in type I, II and sis is allowed, any permutation transformation acting III seesaw extensions to the SM is presented on the thesis. on the above patterns is permitted, provided that it 6 leaves me diagonal. In particular, the following per- By requiring cancellation of gauge anomalies, we mutation sets can be constructed: study the allowed charge assignments under the new gauge symmetry, when two or three right-handed neu- 1 (A1, A2, B3, B4, D1, D2), P ≡ trino singlets or fermion triplets are added to the SM 2 (B1, B2, E3), particle content. We then discuss the phenomeno- P ≡ (40) logical constraints on these theories, requiring consis- 3 (C, E1, E2), P ≡ tency with current neutrino oscillation data. In par- 4 (F1, F2, F3). P ≡ ticular, by extending the SM with a minimal extra fermion and scalar content, we study how the new Starting from any pattern belonging to a particular gauge symmetry can lead to predictive two zero tex- set, one can obtain any other pattern in the same set tures in the effective neutrino mass matrix. We also by permutations. briefly address the possibility of distinguishing different charge assignments (gauge symmetries) and neutrino textures at collider experiments. IV. ANOMALY-FREE GAUGE SYMMETRIES IN NEUTRINO SEESAW FLAVOUR MODELS A. Anomaly Constraints on the Extended Gauge Group Abelian symmetries naturally arise in a wide va- riety of grand unified and string theories. One of We consider a renormalizable theory containing the the interesting features of such theories is their richer SM particles plus a minimal extra fermionic and scalar phenomenology, when compared with the SM. In par- content, so that light neutrinos acquire seesaw masses. ticular, the spontaneous breaking of additional gauge We include nR singlet right-handed neutrinos νR and symmetries leads to new massive neutral gauge bosons nΣ color-singlet SU(2)-triplet fermions Σ to imple- which, if kinematically accessible, could be detectable ment type-I and type-III seesaw mechanisms, respec- at the Large Hadron Collider (LHC). Clearly, the ex- tively. Besides the SM Higgs doublet H that gives perimental signatures of these theories crucially de- masses to quarks and leptons, a complex scalar sin- pend on whether or not the SM particles have non- glet field S is introduced in order to give Majorana trivial charges under the new gauge symmetry. As- masses to νR and Σ. suming that the SM fermions are charged under the We assume that each fermion field f have a charge new gauge group, and that the new gauge boson has a xf under the new U(1)X gauge symmetry. For quarks, mass around the TeV scale, one expects some effects a family universal charge assignment is assumed, while on the LHC phenomenology. leptons are allowed to have non-universal X charges. In this section (discussed in detail in Ref. [31]), we As we have seen in previous sections, in the presence consider Abelian extensions of the SM based on an ex- of extra fermion degrees of freedom, the anomaly con- P tra U(1)X gauge symmetry, with X a B bαLα ditions may change. Furthermore, when we extend ≡ − α being an arbitrary linear combination of the baryon the gauge group, for instance by including a U(1)X number B and the individual lepton numbers Lα. Abelian symmetry, extra conditions should be satis- Then, we consider all possible gauge symmetries to fied to render the theory free of the U(1)X anomalies. perform a systematic study, thus complementing pre- Following the same line of reasoning of Section II, we vious works on several aspects. obtain the system of constraints

2 U(1)X [SU(3)C ] : nG (2 xq xu xd) = 0 , − − nG nΣ 2 3nG 1 X X U(1)X [SU(2)L] : xq + xì 2 xσi = 0 , 2 2 − i=1 i=1 nG 2 xq 4xu xd X xì U(1)X [U(1)Y ] : nG + xei = 0 , 6 − 3 − 3 2 − i=1 nG 2 2 2 2 X 2 2 (41) [U(1)X ] U(1)Y : nG x 2x + x + x + x = 0 , q − u d − ì ei i=1 nG nR nΣ 3 3 3 3 X 3 3 X 3 X 3 [U(1)X ] : nG 6 x 3 x 3 x + 2 x x x 3 x = 0 , q − u − d ì − ei − νi − σi i=1 i=1 i=1 n n n XG XR XΣ U(1)X : nG (6xq 3xu 3xd) + (2 xì xei) xνi 3 xσi = 0 . − − − − − i=1 i=1 i=1 7

nR nΣ Anomaly constraints Symmetry generator X 0 2 0 bi + bj = 3a, bk = 0 B − 3Lj − bi(Li − Lj )

bi + bj = 0, bk = 0 Li − Lj

0 2 bi + bj = 0, bk = 0 Li − Lj 0 2 1 bi + bj = 3a, bk = 0 B − 3Lj − bi(Li − Lj )

bi + bj = 0, bk = 0 Li − Lj 0 1 2 bi + bj = 0, bk = 3a B − 3Lk − bi(Li − Lj )

bi + bj = 0, bk = 0 Li − Lj 0 0 3 0 bi + bj + bk = 3a (B − L) + (1 − bi)(Li − Lj ) + (1 − bk)(Lk − Lj ) 0 0 bi + bj + bk = 0 −bi(Li − Lk) − bj (Lj − Lk)

0 3 bi + bj = 0, bk = 0 Li − Lj 0 3 1 bi + bj = 3a, bk = 0 B − 3Lj − bi(Li − Lj )

bi + bj = 0, bk = 0 Li − Lj

1 3 bi + bj = 0, bk = 0 Li − Lj

2 2 bi + bj = 0, bk = 0 Li − Lj

TABLE II. Anomaly-free solutions for minimal type I and/or type III seesaw realizations and their symmetry generators. 0 In all cases, i 6= j 6= k and bi ≡ bi/a. Cases with a = 0 correspond to a purely leptonic symmetry.

Since the anomaly equations are nonlinear and con- For instance, when nR = 3 and nΣ = 0, the max- tain many free parameters, some assumptions are usu- imal anomaly-free Abelian gauge group extension is ally made to obtain simple analytic solutions. We U(1)B−L U(1)Le−Lµ U(1)Lµ−Lτ [32]. PnG × × shall consider models where X a B i=1 biLi is an arbitrary linear combination of≡ the baryon− number B and individual lepton numbers Li, simultaneously al- B. Phenomenological Constraints lowing for the existence of right-handed neutrinos and fermion triplets that participate in the seesaw mech- For our study, besides the usual SM Yukawa inter- anism to generate Majorana neutrino masses. Un- actions, the relevant Lagrangian terms in the context der the gauge group U(1)X , the charge for the quarks of (minimal) type I and type III seesaw models are qL, uR, dR, is universal, xq = xu = xd = a/3 while the charged leptons `Li, eRi have the family non-universal Yu qLuRHe + Yd qLdRH + Ye `LeRH + Yν `LνRHe charge assignment xì = xei = bi, with all bi differ- 1 T T T ∗ − + mRν CνR + Y1 ν CνRS + Y2 ν CνRS ent. The latter condition guarantees that the charged 2 R R R lepton mass matrix is always diagonal (i.e. it is de- 1 T fined in the charged lepton flavour basis), assuming + mΣTr Σ C Σ + YT `Liτ2Σ H 2 that the SM Higgs is neutral under the new gauge T T ∗ + Y3 Tr Σ C Σ S + Y4 Tr Σ C Σ S + H.c. , symmetry. The right-handed neutrinos νR and/or the triplets Σ are allowed to have any charge assignment (43) bk, where k = 1 . . . nG. − We assume that the SM Higgs doublet is neutral un- Substituting the U(1)X charge values into the der the U(1)X gauge symmetry, and that the complex anomaly equations (41), we obtain the constraints singlet scalar field S has a U(1)X charge equal to xs. Here Y1,2 are nR nR symmetric matrices, while Y3,4 X × bk = 0, are nΣ nΣ symmetric matrices. × k≤nΣ Notice that, in general, the U(1)X symmetry does n XG X not forbid bare Majorana mass terms for the right- bi = bj = nG a, (42) handed neutrinos and fermion triplets. For matrix i=1 j≤nR entries with X = 0, such terms are allowed. In turn, nG entries with X = 0 are permitted in the presence of X X X b3 b3 3 b3 = 0 . the singlet scalar6 S, charged under U(1) . The latter i − j − k X i=1 j≤nR k≤nΣ gives an additional contribution to the Majorana mass terms once S acquires a VEV. The solutions of this system of equations and the Since a universal U(1)X charge is assigned to corresponding symmetry generators X are presented quarks, the new gauge symmetry does not impose in Table II, for minimal type I and type III seesaw real- any constraint on the quark mass matrices. How- izations with nR+nΣ 4. We note that in the absence ever, our choice of a non-universal charge assignment ≤ of right-handed neutrinos only purely leptonic (a = 0) for charged leptons, with all bi different, forces the gauge symmetry extensions are allowed. This is a di- charged lepton mass matrix to be diagonal. Thus, rect consequence of the second constraint in Eq. (42). leptonic mixing depends exclusively on the way that Given the charge assignments, one can identify the neutrinos mix. As discussed in Section III, the effec- maximal gauge group corresponding to each solution. tive neutrino mass matrix mν is obtained after the 8

Symmetry generator X |xs| MR mν

B + Le − Lµ − 3Lτ 2 D2 B + 3L − L − 5L 2 e µ τ A1 B + 3Le − 6Lτ 3 (MR)11 = (MR)23 = (MR)33 = 0

B + 9Le − 3Lµ − 9Lτ 6

B + Le − 3Lµ − Lτ 2 D1 B + 3L − 5L − L 2 e µ τ A2 B + 3Le − 6Lµ 3 (MR)11 = (MR)22 = (MR)23 = 0

B + 9Le − 9Lµ − 3Lτ 6

B − Le + Lµ − 3Lτ 2 B4 B − L + 3L − 5L 2 e µ τ B3 B + 3Lµ − 6Lτ 3 (MR)13 = (MR)22 = (MR)33 = 0

B − 3Le + 9Lµ − 9Lτ 6

B − Le − 3Lµ + Lτ 2 B3 B − L − 5L + 3L 2 e µ τ B4 B − 6Lµ + 3Lτ 3 (MR)12 = (MR)22 = (MR)33 = 0

B − 3Le − 9Lµ + 9Lτ 6

TABLE III. Anomaly-free U(1) gauge symmetries that lead to phenomenologically viable two-zero textures of the neutrino mass matrix mν in a type I seesaw framework with 3 right-handed neutrinos. In all cases, the Dirac-neutrino mass matrix mD is diagonal and the charge assignment xνi = xì = xei = −bi is verified. The solutions belong to the permutation set P1. For a mixed type I/III seesaw scenario with nR = 3 and nΣ = 1 only the solutions with |xs| = 3 remain viable. decoupling of the heavy right-handed neutrinos and with 6 zeros (i.e. permutations of the diagonal ma- fermion triplets. In the presence of both (type I and trix) and their respective charge assignments. Thus, type III) seesaw mechanisms it reads as all together there exist 96 viable solutions. No other anomaly-free solutions are obtained in our minimal −1 T −1 T mν mD M m mT M m , (44) setup. Solutions leading to M = D , D , B , B ' − R D − Σ T R 1 2 3 4 have been recently considered in Ref. [32]. The re- where, according to Eq. (43), maining solutions, to our knowledge, are new in this context. For a mixed type I/III seesaw with nR = 3 ∗ mD = Yν H , MR = mR + 2Y1 S + 2Y2 S , and nΣ = 1, only the set of solutions with xs = 3 h i h i h ∗i | | mT = YT H , MΣ = mΣ + 2Y3 S + 2Y4 S . in Table III are allowed, since the anomaly equations h i h i h i (45) imply that the bk coefficient associated to the fermion triplet charge is always zero. In what follows we restrict our analysis to minimal Notice also that, starting from any pattern given in seesaw scenarios with nR + nΣ 4. The requirement Table III, other patterns in the table can be obtained ≤ that charged leptons are diagonal (b1 = b2 = b3) im- by permutations of the charged leptons. For instance, 6 6 poses strong constraints on the matrix textures of mD starting from the symmetry generators that lead to and mT . Indeed, considering either a type I or a type the A1 pattern, those corresponding to A2 and B3 are III seesaw framework, only those matrices with a sin- obtained by µ τ and e µ exchange, respectively. ↔ ↔ gle nonzero element per column are allowed. Further- Similarly, the B4 texture can be obtained from A2 more, matrices with a null row or column are excluded through the e τ exchange. since they lead to a neutrino mass matrix with deter- ↔ minant equal to zero, not belonging to any pattern of those given in Section III, however mixed type I/III seesaw mechanisms can relax this constraint. C. Gauge Sector and Flavour Model Discrimination We look for anomaly-free U(1)X gauge symmetries that lead to phenomenologically viable two-zero textures of the neutrino mass matrix mν , namely to pat- For the effects due to the new gauge symmetry to terns A1,2, B1,2,3,4 and C given in Section III. Solu- be observable, the seesaw scale should be low enough. tions were found only within a type I seesaw frame- One expects a phenomenology similar to the case with work with three right-handed neutrinos, or in a mixed a minimal B L scalar sector [33]. Nevertheless, by −0 type I/III seesaw scenario with three right-handed studying the Z resonance and its decay products, one neutrinos and one fermion triplet. In Table III we could in principle distinguish the generalized U(1)X show the allowed solutions, for the cases when the models from the minimal B L model. − Dirac-neutrino mass matrix mD is diagonal, which Due to their low background and neat identifica- implies the charge assignment xνi = bi. All the solution, leptonic final states give the cleanest channels − tions belong to the permutation set 1 [see Eq. (40)]. for the discovery of a new neutral gauge boson. In P We remark that, for each pattern of mν , there are the limit that the fermion masses are small compared 0 0 another 20 solutions corresponding to matrices mD with the Z mass, the Z decay width into fermions is 9

0 2 10 10

A1, B3 101 10−1 µ µ A B 0 1,2 , 3,4 / b/

τ 10 R R

A2, B4 10−2 A2, B4 A −1 1 10 A A2, B3,4 2 B 3 B 4 −3 −2 10 10 10−3 10−2 10−1 100 10−3 10−2 10−1 100 Rt/µ Rt/µ

FIG. 3. Rt/µ − Rb/µ branching ratio plane for the FIG. 4. Rt/µ − Rτ/µ branching ratio plane for the anomaly-free solutions of Table III, leading to neutrino anomaly-free solutions of Table III, leading to neutrino mass matrix patterns of type A1,2 and B3,4. mass matrix patterns of type A1,2 and B3,4. approximately given by approximately given in our case by

0 + − 02 σ(pp Z τ τ ) 0 g 2 2 Rτ/µ = → 0 → + − Γ(Z ff) mZ0 xfL + xfR , (46) σ(pp Z µ µ ) → ' 24π 2→ 2→ 2 (49) x`3 + xe3 b3 Kτ 2 2 Kτ 2 . where xfL and xfR are the U(1)X charges for the ' x`2 + xe2 ' b2 left and right chiral fermions, respectively. More- over, the decays of Z0 into third-generation quarks, Clearly, this ratio can be used to distinguish models 0 0 with generation universality (R 1) from models pp Z b b and pp Z t t can be used to dis- τ/µ ' criminate→ → between different→ models,→ having the advan- with non-universal couplings, as those given in Ta- tage of reducing the theoretical uncertainties [34, 35]. ble III. The Rt/µ Rτ/µ branching ratio plane is depicted in Fig. 4. In− this case, the neutrino mass matrix In particular, the branching ratios Rb/µ and Rt/µ of quarks to µ+µ− production, patterns exhibit a clear discrimination in the plane, having overlap of two solutions in just three points. 0 0 2 2 In conclusion, by studying the decays of the Z bo- σ(pp Z b b) xq + xd R = → → 3Kb , son into leptons and third-generation quarks at col- b/µ σ(pp Z0 µ+µ−) ' x2 + x2 → → `2 e2 (47) lider experiments, it is possible to discriminate differ- 0 2 2 σ(pp Z t t) xq + xu ent gauge symmetries and the corresponding flavour R = → → 3Kt , t/µ σ(pp Z0 µ+µ−) ' x2 + x2 structure of the neutrino mass matrix. → → `2 e2 could serve as discriminators. The Kb,t (1) factors incorporate the QCD and QED next-to-leading-∼ O V. CONCLUSIONS order correction factors. Substituting the quark and charged-lepton U(1)X charges, we obtain The recent discovery of a Higgs-like particle at the LHC reinforces the great success of the SM as the 2 2 effective low energy theory for the electroweak inter- Kb a Kt a R ,R , (48) actions. In spite of this, there remain a few aspects b/µ 3 b2 t/µ 3 b2 ' 2 ' 2 that cannot be explained within the SM. In particular, neutrino oscillation experiments have confirmed yielding R R . Fig. 3 shows the R R b/µ ' t/µ t/µ − b/µ that neutrinos have non-vanishing masses and mix. branching ratio plane for the anomaly-free solutions The well-known seesaw mechanism is an appealing given in Table III, which lead to the viable neutrino and economical theoretical framework to explain the mass matrix patterns A1,2 and B3,4, with two inde- tiny neutrino masses. In this context, the addition of pendent zeros. As can be seen from the figure, the new heavy particles (fermions or bosons) to the the- solutions split into five different points in the plane, ory allows for the generation of an effective neutrino which correspond to the allowed values of the b2 coef- mass matrix at low energies. As is well known, theo- ficient, b2 = 1, 3, 5, 6, 9, assuming a = 1. The allowed | | ries that contain fermions with chiral couplings to the mν patterns are shown at each point. gauge fields suffer from anomalies and, to make them + − The ratio Rτ/µ of the branching fraction of τ τ to consistent, the chiral sector of the new theory should µ+µ− has also proven to be useful for understanding be arranged so that the gauge anomalies cancel. One models with preferential couplings to Z0 [35]. It is attractive possibility is to realize the anomaly cancel- 10 lation through the modification of the gauge symme- ity of our approach would be lost, since rotating the try. charged leptons to the diagonal basis would destroy, in most cases, the zero textures in the neutrino mass We have considered extensions of the SM based on matrix. Abelian gauge symmetries that are linear combina- Working in the charged lepton flavour basis, we tions of the baryon number B and the individual lep- have found that only a limited set of solutions are vi- ton numbers Le,µ,τ . In the presence of a type I and/or able, namely those presented in Table III, leading to type III seesaw mechanisms for neutrino masses, we two-zero textures of the neutrino mass matrix with a have then looked for all viable charge assignments minimal extra fermion and scalar content. All allowed and gauge symmetries that lead to cancellation of patterns were obtained in the framework of the type gauge anomalies and, simultaneously, to a predictive I seesaw mechanism with three right-handed neutri- flavour structure of the effective Majorana neutrino nos (or in a mixed type I/III seesaw framework with mass matrix, consistent with present neutrino oscilla- three right-handed neutrinos and one fermion triplet), tion data. Our analysis was performed in the physical extending the SM scalar sector with a complex scalar basis where the charged leptons are diagonal. This singlet field. implies that the neutrino mass matrix patterns with Finally, we briefly addressed the possibility of dis- two independent zeros, obtained via the seesaw mech- criminating the different charge assignments (gauge anism, are directly linked to low-energy parameters. symmetries) and seesaw realizations at the LHC. We We recall that, besides three charged lepton masses, have shown that the measurements of the ratios of there are nine low-energy leptonic parameters (three third generation final states (τ, b, t) to µ decays of the neutrino masses, three mixing angles, and three CP new gauge boson Z0 could be useful in distinguishing violating phases). Two-zero patterns in the neutrino between different gauge symmetry realizations, as can mass matrix imply four constraints on these parame- be seen from Figs. 3 and 4. This analysis provides a ters. Would we consider charge assignments that lead complementary way of testing flavour symmetries and to nondiagonal charged leptons, then the predictabil- their implications for low-energy neutrino physics.

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