Anomaly-free constraints in neutrino seesaw models

D. Emmanuel-Costa,1,* Edison T. Franco,2,1,† and R. Gonzalez Felipe3,1,‡

1Departamento de Fısica and Centro de Fısica Teorica de Partıculas (CFTP), Instituto Superior Tecnico,Avenida Rovisco Pais, 1049-001 Lisboa, Portugal

2Instituto de Fısica Teorica, Universidade Estadual Paulista,Rua Doutor Bento Teobaldo Ferraz 271 - Bloco II, 01140-070 - Sao Paulo, SP, Brazil

3Area Cientıfica de Fısica, Instituto Superior de Engenharia de Lisboa,Rua Conselheiro Emıdio Navarro 1, 1959-007 Lisboa, Portugal

(Received 13 February 2009; published 3 June 2009)

The implementation of seesaw mechanisms to give mass to neutrinos in the presence of an anomaly-

free Uð1ÞX gauge symmetry is discussed in the context of minimal extensions of the standard model. It is

shown that type-I and type-III seesaw mechanisms cannot be simultaneously implemented with an

anomaly-free local Uð1ÞX, unless the symmetry is a replica of the well-known hypercharge. For combined

type-I/II or type-III/II seesaw models it is always possible to find nontrivial anomaly-free charge

assignments, which are however tightly constrained, if the new neutral gauge boson is kinematically

accessible at LHC. The discovery of the latter and the measurement of its decays into third-generation

quarks, as well as its mixing with the standard Z boson, would allow one to discriminate among different

seesaw realizations.

DOI: 10.1103/PhysRevD.79.115001 PACS numbers: 12.60.Cn, 11.15.Ex, 14.60.Pq

I. INTRODUCTION

The smallness of neutrino masses is elegantly explainedin the context of seesaw models. Since in the standardmodel (SM) neutrinos are massless, new physics is re-quired beyond the model to account for their tiny masses.The simplest possibility consists of the addition of singletright-handed neutrinos, which seems natural once all theother fermions in the SM have their right-handed counter-parts. In this framework, known as a type-I seesaw mecha-nism [1], neutrinos acquire Majorana masses at tree levelvia a unique dimension-five effective operator.Alternatively, neutrino masses can be generated by atype-II seesaw [2] through the tree-level exchange ofSUð2Þ-triplet scalars, which are color singlets and carryunit hypercharge. A third possibility, often referred as type-III seesaw [3], consists of the introduction of color-singletSUð2Þ-triplet fermions with zero hypercharge.

Adding to the theory new fermions with chiral couplingsto the gauge fields unavoidably raises the question ofanomalies, i.e. the breaking of gauge symmetries of theclassical theory at the quantum level. For consistency,anomalies should be exactly canceled. In the SM thiscancellation occurs for each fermion generation with thehypercharge assignment. On the other hand, in many SMextensions and, particularly, in various grand unified andstring models, extra Uð1Þ factors naturally appear, thusbringing additional anomaly cancellation constraints [4].These extra factors also typically lead to a richer phenome-

nology that might have definite signatures at the forth-coming Large Hadron Collider (LHC) experiments.In this paper we address the above question in the

context of minimal extensions of the SM, based on thegauge structure SUð3ÞC � SUð2ÞL �Uð1ÞY �Uð1ÞX andcoexisting with a seesaw-type mechanism for neutrinomasses. In this framework, the heavy (singlet or triplet)fermions responsible for the seesaw will acquire largeMajorana masses, after the new gauge symmetry is sponta-neously broken by singlet scalar fields with nontrivialUð1ÞX charges. As it turns out, the axial-vector [5] andmixed gravitational-gauge [6] anomaly-free conditions arequite constraining. Furthermore, Yukawa interactions, nec-essary to give masses to SM fermions as well as to imple-ment the seesaw mechanism, restrict the number of Higgsdoublets required by the theory.

II. ANOMALY CONSTRAINTS ON SEESAW

We consider a simple extension of the SM with minimalextra matter content, so that neutrinos obtain seesawmasses and all fields are nontrivially charged under anew Uð1ÞX gauge symmetry. We include singlet right-handed neutrinos �R and color-singlet SUð2Þ-triplet fermi-ons T with zero hypercharge to implement type-I and type-III seesaw mechanisms, respectively. To give masses toquarks and leptons, 4 Higgs doublets ðHu;Hd;H�;HeÞ arein general required, while another 2 scalar doublets, H�

and HT , and one SUð2Þ-triplet scalar � are necessary togenerate Yukawa terms for type-II and type-III seesawmechanisms. Finally, a singlet scalar field � is introducedto give Majorana masses to �R and T.To render the theory free of anomalies the following set

of constraints must be satisfied:

*david.costa@ist.utl.pt†edisontf@ift.unesp.br‡gonzalez@cftp.ist.utl.pt

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A1 ¼ Ngð2xq � xu � xdÞ ¼ 0; (1a)

A2 ¼ Ngðx‘=2þ 3xq=2Þ � 2NTxT ¼ 0; (1b)

A3 ¼ Ngðxq=6þ x‘=2� 4xu=3� xd=3� xeÞ ¼ 0; (1c)

A4 ¼ Ngðx2q � x2‘ � 2x2u þ x2d þ x2eÞ ¼ 0; (1d)

A5 ¼ Ngð6x3q þ 2x3‘ � 3x3u � 3x3d � x3eÞ � NRx3� � 3NTx

3T ¼ 0; (1e)

A6 ¼ Ngð6xq þ 2x‘ � 3xu � 3xd � xeÞ � NRx� � 3NTxT ¼ 0; (1f)

where xi denote the fermion charges under Uð1ÞX (thesubscript i refers to the field), Ng is the number of gen-erations, NR is the number of right-handed neutrinos, andNT is the number of fermionic triplets. Equations (1a)–(1e)arise from the requirement of the cancellation of the axial-vector anomaly, while Eq. (1f) results from the cancella-tion of the gravitational-gauge anomaly.

Making use of Eqs. (1a)–(1c) and (1f) we can expressthe charges xq, x‘, xu, and xd in terms of xe, x�, and xT as

xq ¼ �Ngxe þ NRx� � 5NTxT6Ng

;

x‘ ¼Ngxe þ NRx� þ 3NTxT

2Ng

;

xu ¼ � 2Ngxe � NRx� � NTxT3Ng

;

xd ¼ Ngxe � 2NRx� þ 4NTxT3Ng

:

(2)

Substituting these expressions into Eq. (1d) for the A4

anomaly, we derive the condition

A4 ¼ � 4NRNTx�xTNg

¼ 0; (3)

which, clearly, has only the trivial solutions NR ¼ 0, NT ¼0, x� ¼ 0, or xT ¼ 0. Thus, we conclude that it is notpossible to have an anomaly-free local Uð1ÞX and, simul-taneously, type-I (NR � 0) and type-III (NT � 0) seesawmodels, unless Uð1ÞX is proportional to the hyperchargeUð1ÞY , in which case no new gauge symmetry is obtained.Of course, both types of seesaw could coexist if new extramatter content is added to the theory [7]. In such a case, atleast two extra singlets charged under Uð1ÞX are needed tocancel the A4 and A5 anomalies.

Once A4 ¼ 0 is enforced, the anomaly condition A5

given in Eq. (1e) can be written in terms of x� and xT as

A5 ¼ 1

N2g

½NRðN2R � N2

gÞx3� þ 3NTðN2T � N2

gÞx3T�: (4)

A. Type-I seesaw case with NR ¼ Ng

If no fermion triplet is introduced and the type-I seesawmechanism is responsible for neutrino masses, the anomalyconstraint A5 ¼ 0 implies NR ¼ Ng (the solution x� ¼ 0

leads to the hypercharge). Thus, one right-handed neutrinoper generation is required. In this case, all Uð1ÞX chargeassignments can be expressed in terms of the quark doubletcharge xq and the charge ratio � � �xe=xq, as presented

in Table I, leading to an infinite class of anomaly-free localUð1ÞX symmetries. Note that, in this parametrization, � ¼6 corresponds to the hypercharge and � ¼ 3 gives theusual Uð1ÞB�L gauge symmetry. Any other value of �would correspond to a new anomaly-free gauge symmetry.

B. Type-III seesaw case with NT ¼ Ng

In the absence of right-hand neutrinos (NR ¼ 0) andinvoking a type-III seesaw to give masses to neutrinos,the anomaly constraint A5 ¼ 0 yields NT ¼ Ng (the solu-

tion xT ¼ 0 would lead once again to the hypercharge).Therefore, a fermion triplet per family is required to cancelall the anomalies. The corresponding Uð1ÞX charge assign-ments are displayed in Table I as a function of the ratio �.Although in this case the hypercharge solution can berecovered by choosing � ¼ 6, there is no value of � thatleads to the Uð1ÞB�L gauge symmetry. The choice � ¼ 3and xq ¼ 1=3 implies xu ¼ 11=5, xd ¼ x‘ ¼ �1=5, xe ¼�1, and xT ¼ 1=5, which obviously do not correspond tothe correct (B� L) charges. This peculiar type of solutionwas first found in Ref. [8] and, more recently, it has beenthe subject of several studies [9].

C. Type-I seesaw case with NR � Ng

Neutrino oscillation data do not demand the existence ofthree right-handed neutrinos to successfully implement atype-I seesaw mechanism. In fact, the presence of just two�R fields can account for the large leptonic mixing and the

TABLE I. Uð1ÞX fermion charges (normalized with xq) as afunction of the ratio � � �xe=xq in different minimal seesaw

realizations.

Type I Type III Type I

Uð1ÞX charge NR ¼ Ng NT ¼ Ng Ng ¼ 3, NR ¼ 2

xu �� 2 15 ð2þ 3�Þ �� 2

xd 4� � 15 ð8� 3�Þ 4� �

x‘ �3 15 ð9� 4�Þ �3

x� �� 6 4ð�� 6ÞxT

15 ð6� �Þ

xS 5ð6� �Þ

EMMANUEL-COSTA, FRANCO, AND FELIPE PHYSICAL REVIEW D 79, 115001 (2009)

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solar and atmospheric mass-squared differences. This is apredictive scenario since the lightest neutrino is massless,while the other two neutrino masses are fixed by theobserved solar and atmospheric mass-squared differences.It is therefore legitimate to ask ourselves whether such atheory admits an anomaly-free Uð1ÞX gauge symmetry.This would require one to relax the condition NR ¼ Ng

imposed by the A5 anomaly constraint, which is onlypossible if new matter content is added to the theory. Thesimplest possibility is to assume a ‘‘sterile’’ right-handedsinglet �S, which does not participate in the seesaw mecha-nism and has a Uð1ÞX charge xS different from the chargex� of the ‘‘active’’ �R neutrinos. Clearly, an additionalscalar singlet field, �S, would be necessary to generatemass for the sterile right-handed neutrino. In this frame-work, the anomaly conditions given in Eqs. (1e) and (1f)must be modified. Assuming three generations, two activeand one sterile right-handed neutrinos, the anomaly-freesystem of equations implies the charge assignments givenin the last column of Table I. For � ¼ 6, the hypercharge isreproduced, while � ¼ 3 does not lead to B� L sincex‘;e ¼ �1, xu;d;q ¼ 1=3, x� ¼ �4, and xS ¼ 5 [10].

III. QUARK AND LEPTON MASSES

We now consider the constraints coming from theYukawa terms responsible for the quark and lepton masses.The most general Lagrangian terms to generate fermionmasses, and which are compatible with type-I, type-II, andtype-III seesaw models for Majorana neutrinos, are givenby

Yu �qLuRHu þ Yd �qLdRHd þ Ye�‘LeRHe þ Y�

�‘L�RH�

þ YT�‘Li�2THT þ YI�

TRC�R�þ YII‘

TLCi�2‘L�

þ YIIITrðTTCTÞ�þ ��H�H��þ H:c: (5)

The following relations then hold for the zi boson chargesunderUð1ÞX: xq ¼ xu þ zu ¼ xd þ zd, x‘ ¼ xe þ ze, x‘ ¼x� þ z�, x‘ ¼ xT þ zT , 2x� þ z� ¼ 0, 2x‘ þ z� ¼ 0,

2xT þ z� ¼ 0, and 2z� þ z� ¼ 0. The solutions of these

equations are presented in Table II, for the three minimalseesaw models considered above. We conclude that, for atype-I seesaw scenario with a right-handed neutrino perfamily, only one Higgs doublet Hu is required to givemasses to all fermions. On the other hand, for a minimaltype-I seesaw solution with Ng ¼ 3 and NR ¼ 2, while the

same Higgs doublet Hu is sufficient to give mass to thequarks and charged leptons, a second one, H�, is needed toimplement the type-I seesaw and give mass to light neu-trinos. If a type-II seesaw is also present, then we mustintroduce an extra Higgs doublet H�. Of course, for thestandard hypercharge solution (� ¼ 6) all three Higgsdoublets coincide, Hu ¼ H� ¼ H�.

For the type-III seesaw case, one can see from theTable II that a single Higgs doublet Hu can simultaneously

give masses to the quarks and to the light neutrinos throughthe seesaw. Yet, another Higgs doubletHe must account forthe charged lepton masses. When the type-II seesaw is alsointroduced, one more Higgs doublet H� is needed. Oncemore, for the standard hypercharge solution all three Higgsdoublets can be identified as a unique doublet.We remark that the analysis presented above should be

taken just as a generic statement on the minimal number ofHiggs doublets required to give mass to fermions, whilekeeping the theory free of anomalies. This, of course, doesnot preclude the presence of different Higgs doubletscharged under the Uð1ÞX symmetry.

IV. PHENOMENOLOGICAL IMPLICATIONS

One of the attractive features of theories with extra Uð1Þgauge symmetries is their richer phenomenology, whencompared with the SM. Indeed, the spontaneous breakingof the symmetry leads to new massive neutral gauge bo-sons X (often called Z0) which, if kinematically accessible,could be detectable at LHC. In the latter case, its decaysinto third-generation quarks, pp ! X ! b �b and pp !X ! t�t could be used to discriminate between differentmodels [11]. In particular, the branching ratios of quark to�þ�� production,

Rb� ¼ �ðpp ! X ! b �bÞ�ðpp ! X ! �þ��Þ ’ 3Kq

x2q þ x2dx2‘ þ x2e

; (6a)

Rt� ¼ �ðpp ! X ! t�tÞ�ðpp ! X ! �þ��Þ ’ 3Kq

x2q þ x2u

x2‘ þ x2e; (6b)

have the advantage of reducing the theoretical uncertain-ties [11]. Here Kq �Oð1Þ is a constant which depends on

QCD and electroweak correction factors. Notice that theseratios depend on the quark and charged lepton Uð1ÞXcharges. Therefore, an analysis of the Rt� � Rb� parame-

ter space would allow one to determine these charges and,in turn, to distinguish the models. Using the charge valuesgiven in Table I we obtain

TABLE II. Uð1ÞX boson charges (normalized with xq) as afunction of the ratio � � �xe=xq in different minimal seesaw

realizations.

Types I/II Types III/II Types I/II

Uð1ÞX charge NR ¼ Ng NT ¼ Ng Ng ¼ 3, NR ¼ 2

zu 3� � 35 ð1� �Þ 3� �

zd �� 3 35 ð�� 1Þ �� 3

ze �� 3 15 ð9þ �Þ �� 3

z� 3� � 21� 4�

zT35 ð1� �Þ

z� 2ð6� �Þ 25 ð�� 6Þ 8ð6� �Þ

z� 6 25 ð4�� 9Þ 6

z� �3 15 ð9� 4�Þ �3

zS 10ð�� 6Þ

ANOMALY-FREE CONSTRAINTS IN NEUTRINO SEESAW . . . PHYSICAL REVIEW D 79, 115001 (2009)

115001-3

Rb� ’ 3ð17� 8�þ �2Þ9þ �2

; Rt� ’ 3ð5� 4�þ �2Þ9þ �2

;

(7)

for the type-I seesaw cases, and

Rb� ’ 3ð89� 48�þ 9�2Þ81� 72�þ 41�2

;

Rt� ’ 3ð29þ 12�þ 9�2Þ81� 72�þ 41�2

;

(8)

for the type-III seesaw case. As can be seen from Figs. 1and 2, the above branching ratios indeed allow one todiscriminate among the models.

Electroweak precision data also severely constrain anymixing with the ordinary Z boson to the subpercent level[4]. In the models under consideration, the Z� X mixingappears due to the presence of Higgs bosons which trans-forms under the SM gauge group and the new Uð1ÞXAbelian gauge symmetry. Generically, such mixing is pro-portional to the combination gZgX

Piyixiv

2i , where gi are

the gauge couplings, and vi � hHii are the vacuum expec-tation values of the Higgs doublets with

Piv

2i ¼ v2.

Hence, the requirement of vanishing Z� X mixing im-poses an additional constraint on the Uð1ÞX charges. Inparticular, for minimal setups with only two Higgs dou-blets, it is possible to determine � as a function of a singleratio, e.g. ru ¼ vu=v. For type-I seesaw with NR ¼ Ng

together with type-II seesaw, we find

� ¼ 6� 3

r2u; (9)

while a type-I seesaw with only 2 right-handed neutrinosyields

� ¼ 21� 18r2u4� 3r2u

: (10)

Finally, a type-III seesaw realization implies

� ¼ 12r2u � 9

1þ 2r2u: (11)

Once again, the different models are clearly distinguish-able, as can be seen from Fig. 3.

V. CONCLUSION

In summary, we have implemented minimal seesawrealizations in extensions of the SM with a local Uð1ÞXsymmetry. We have shown that type-I and type-III seesawmechanisms cannot be simultaneously realized, unless the

−10 −8 −6 −4 −2 0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

α

Rtµ

, b µ

Rt µ Type I

Rb µ Type I

Rt µ Type III

Rb µ Type III

FIG. 1. Branching ratios of the X decays into quarks andmuons as a function of the charge ratio �.

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

Rt µ

Rb

µ

Type I

Type III

FIG. 2. Rt� � Rb� plane for type-I and -III seesaw realiza-tions.

0 0.2 0.4 0.6 0.8 1−10

−8

−6

−4

−2

0

2

4

6

8

10

vu / v

α

Type I + Type II

Type I with 2 νR

Type III

FIG. 3. The Uð1ÞX charge ratio � as a function of the ratioru ¼ vu=v for vanishing Z� X mixing.

EMMANUEL-COSTA, FRANCO, AND FELIPE PHYSICAL REVIEW D 79, 115001 (2009)

115001-4

Uð1ÞX symmetry is a replica of the standard hypercharge ornew fermionic fields are added to the theory. When com-bined type-I/II or type-III/II seesaw models are considered,we have seen that it is always possible to assign nontrivialanomaly-free charges to the fields.

The LHC era is an exceptional opportunity to exploreand discover physics beyond the SM. The minimal seesawmodels studied here are well-motivated (nonsupersymmet-ric) extensions of the SM. The discovery of a new neutralgauge boson with a mass up to a few TeV and typicalelectroweak scale couplings would be one of the firstevidences of a new TeV scale sector. We have demon-strated how the gauge boson signature at LHC, like itsdecays into third-generation quarks and mixing with theordinary Z boson, could allow one to constrain these see-saw models and discriminate among them. Furthermore,these indirect seesaw signatures have the advantage ofbeing independent of the Yukawa interactions of the heavyneutrino fields with the SM charged leptons. The latterinteractions are expected to be very suppressed and leadto unobservable production rates of the heavy fields, if the

extra Uð1Þ gauge symmetry is broken at TeV energies. Inthis scenario, the seesaw mechanism could still be naturalif some flavor symmetries that correctly reproduce the lightneutrino masses, while keeping sizable Yukawa couplings,are invoked [12]. Clearly, there is much more phenome-nology with the seesaw than the one considered here. Inparticular, when the decays of X to heavy Majorana neu-trinos are kinematically allowed, like-sign dilepton signalsoffer another interesting possibility for studying the phys-ics of new gauge bosons at LHC [13,14].

ACKNOWLEDGMENTS

This work was partially supported by Fundacao para aCiencia e a Tecnologia (FCT, Portugal) through theProjects No. POCTI/FNU/44409/2002, No. PDCT/FP/63914/2005, No. PDCT/FP/63912/2005, and No. CFTP-FCT UNIT 777, and also partially funded by POCTI(FEDER). The work of E. T. F. was supported byFundacao de Amparo a Pesquisa do Estado de Sao Paulo(FAPESP) under the Project No. 03/13869-3.

[1] P. Minkowski, Phys. Lett. 67B, 421 (1977); T. Yanagida,in Proceedings of the Workshop on Unified Theory andBaryon Number in the Universe, edited by O. Sawada andA. Sugamoto (KEK, Tsukuba, Japan, 1979); S. L.Glashow, in Quarks and Leptons, Cargese 1979, editedby M. Levy et al. (Plenum, New York, 1980), p. 707; M.Gell-Mann, P. Ramond, and R. Slansky, in Supergravity,edited by D. Freedman and P. van Nieuwenhuizen (North-Holland, Amsterdam, 1979), p. 315; R. N. Mohapatra andG. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).

[2] M. Magg and C. Wetterich, Phys. Lett. 94B, 61 (1980); J.Schechter and J.W. F. Valle, Phys. Rev. D 22, 2227 (1980);R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165(1981).

[3] R. Foot, H. Lew, X.G. He, and G. C. Joshi, Z. Phys. C 44,441 (1989).

[4] For a recent review, see P. Langacker, arXiv:0801.1345and references therein [Rev. Mod. Phys. (to be pub-lished)].

[5] S. L. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell and R.Jackiw, Nuovo Cimento A 60, 47 (1969); S. L. Adler andW.A. Bardeen, Phys. Rev. 182, 1517 (1969).

[6] R. Delbourgo and A. Salam, Phys. Lett. 40B, 381 (1972);T. Eguchi and P. G.O. Freund, Phys. Rev. Lett. 37, 1251(1976); L. Alvarez-Gaume and E. Witten, Nucl. Phys.

B234, 269 (1984).[7] For the implementation of type-I and type-III seesaw

mechanisms in the context of grand unified theories, seee.g. E. Ma, Phys. Rev. Lett. 81, 1171 (1998); B. Bajc andG. Senjanovic, J. High Energy Phys. 08 (2007) 014; P.Fileviez Perez, Phys. Lett. B 654, 189 (2007); B. Bajc, M.Nemevsek, and G. Senjanovic, Phys. Rev. D 76, 055011(2007); P. Fileviez Perez, Phys. Rev. D 76, 071701 (2007).

[8] S.M. Barr, B. Bednarz, and C. Benesh, Phys. Rev. D 34,235 (1986); E. Ma, Mod. Phys. Lett. A 17, 535 (2002).

[9] E. Ma and D. P. Roy, Nucl. Phys. B644, 290 (2002); S.M.Barr and I. Dorsner, Phys. Rev. D 72, 015011 (2005); R.Adhikari, J. Erler, and E. Ma, Phys. Lett. B 672, 136(2009).

[10] J. C. Montero and V. Pleitez, Phys. Lett. B 675, 64 (2009).[11] S. Godfrey and T.A.W. Martin, Phys. Rev. Lett. 101,

151803 (2008).[12] J. Kersten and A.Y. Smirnov, Phys. Rev. D 76, 073005

(2007); A. Abada et al., J. High Energy Phys. 12 (2007)061; F. del Aguila and J. A. Aguilar-Saavedra, Phys. Lett.B 672, 158 (2009).

[13] W.Y. Keung and G. Senjanovic, Phys. Rev. Lett. 50, 1427(1983).

[14] F. del Aguila and J. A. Aguilar-Saavedra, J. High EnergyPhys. 11 (2007) 072, and references therein.

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