Physics Letters B. Type II seesaw mechanism for Higgs doublets and the scale of new physics

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1 Physics Letters B 674 (009) 7 Contents lists available at ScienceDirect Physics Letters B wwwelseviercom/locate/physletb Type II seesaw mechanism for Higgs doublets and the scale of new physics W Grimus a LLavoura b BRadovčić ac a University of Vienna Faculty of Physics Boltzmanngasse 5 A-090 Vienna Austria b Technical University of Lisbon Centre for Theoretical Particle Physics Lisbon Portugal c University of Zagreb Faculty of Science Department of Physics PO Box 33 HR-000 Zagreb Croatia article info abstract Article history: Received 0 February 009 Received in revised form 0 March 009 Accepted March 009 Available online 4 March 009 Editor: A Ringwald We elaborate on an earlier proposal by Ernest Ma of a type II seesaw mechanism for suppressing the vacuum expectation values of some Higgs doublets We emphasize that by nesting this form of seesaw mechanism into various other seesaw mechanisms one may obtain light neutrino masses in such a way that the new-physics scale present in the seesaw mechanism the masses of scalar gauge-su() triplets scalar SU() doublets or right-handed neutrinos does not need to be higher than a few 0 TeV We also investigate other usages of the type II seesaw mechanism for Higgs doublets For instance the suppression of the vacuum expectation values of Higgs doublets may realize Froggatt Nielsen suppression factors in some entries of the fermion mass matrices 009 Elsevier BV All rights reserved Introduction The type I seesaw mechanism [] is a favourite with highenergy physicists for explaining why the neutrino masses are so tiny Unfortunately in the usual realization of that mechanism the scale m R of the Majorana masses of the right-handed neutrinos ν R should be 0 3 GeV (assuming that the natural scale of the neutrino Dirac mass matrix is the electroweak scale) As a consequence the possibility of direct tests of the seesaw mechanism seems very remote Lowering m R to the TeV scale although desirable from the point of view of experimental tests of the type I seesaw mechanism apparently contradicts the aim with which it was invented since it would require artificially suppressing the Yukawa couplings of the ν R to values of order 0 5 Another mechanism for explaining the smallness of the neutrino masses is the type II seesaw mechanism [] which suppresses the vacuum expectation values (VEVs) of the neutral components of scalar gauge-su() triplets in such a way that the lefthanded neutrinos ν L which acquire Majorana masses from their Yukawa couplings to those neutral components are extremely light Just like the type I seesaw mechanism the type II seesaw mechanism requires a very high mass scale which now occurs in the mass terms of the scalar triplets Those large mass terms make the scalar triplets extremely heavy and therefore the type II seesaw mechanism like the type I seesaw mechanism is very difficult to test experimentally * Corresponding author addresses: waltergrimus@univieacat (W Grimus) balio@cftpistutlpt (L Lavoura) bradov@phyhr (B Radovčić) In the general case for instance in Grand Unified Theories based on the gauge group SO(0) both type I and type II seesaw mechanisms are present [3] Several proposals have been made to bring the high mass scale of the seesaw mechanism(s) down to the TeV range so that they might be experimentally testable for instance at the Large Hadron Collider at CERN The most straightforward possibility is to have cancellations within the type I seesaw mechanism such that m R may be relatively low without the need to excessively suppress the Yukawa couplings [4]; the general conditions for this to happen were given in [5] Cancellations between the type I and type II seesaw contributions to the neutrino masses have also been considered [6] In the inverse seesaw mechanism [7] there is both a high scale in the TeV range and a low scale in the kev range Other proposals include radiative neutrino masses generated by three-loop diagrams [8] or a specific type of mirror fermions [9] In this Letter we develop a proposal originally made in [0] We elaborate on its two separate ideas: i A type II seesaw mechanism suppresses the VEVs of some Higgs doublets ii A nesting of that type II seesaw mechanism inside some other seesaw mechanism (which may be of any type) allows one to lower the high mass scale of that seesaw mechanism The aim of this Letter is to generalize the proposal of [0] in several directions: We describe (in Section ) the general mechanism for suppressing Higgs-doublet VEVs and give several examples thereof /$ see front matter 009 Elsevier BV All rights reserved doi:006/jphysletb

2 8 W Grimus et al / Physics Letters B 674 (009) 7 We show (in Section 3) that this type II seesaw mechanism for Higgs doublets may be nested inside various seesaw mechanisms We propose in particular a type II seesaw mechanism for Higgs doublets inside the usual type II seesaw mechanism for scalar triplets We develop (in Section 4) a multiply nested type II seesaw mechanism for many Higgs doublets which may mimic the Froggatt Nielsen mechanism [] This suggests new ways of explaining the relative smallness of some charged-fermion masses without the need for new heavy fermions as in the seesaw mechanism for Dirac fermions [3] In summary the message that we want to convey in this Letter is that by using the nesting of seesaw mechanisms a heavy mass scale m H many orders of magnitude larger than the electroweak scale m ew 00 GeV is not compelling; an m H just two or three orders of magnitude above m ew may suffice Type II seesaw mechanism for Higgs doublets Consider a model with several Higgs doublets φ j = (φ + j φ0 j )T j = n h The VEVs of the Higgs doublets are of the form ( ) 0 φ j 0 = () v j We assume that v m ew Our aim is to produce a seesaw mechanism to suppress v Wewritethescalarpotentialas n h V = μ j φ j φ j + (V l + V l ) + V r () j= We assume that μ < 0 and that μ m ew in order to generate a spontaneous symmetry breaking leading to v m ew Onthe other hand we assume that μ > 0 and that μ = m H m ew In Eq () V l represents some terms linear in φ which we assume to be present in V All the remainder of V ie everything but the mass terms for the Higgs doublets and the terms V l and V l linear in φ and φ respectively is denoted V r; in the simplest cases V r will consist only of quartic terms Inserting the VEVs into the potential one has n h V 0 = μ j v j + Av + A v + V r 0 (3) j= where A has the dimension of the cube of a mass Then despite the positiveness of μ a non-vanishing VEV v is induced approximately given by v A μ (4) The quantity A depends on the specific model It has to contain at least one v j v and this v j will in general be of order m ew If we assume that μ is the only parameter in the scalar potential of order m H then we expect A m3 ew In this case v m 3 ew /m H is suppressed by two powers of m ew over m H where m H is the scale of new physics Two Higgs doublets and a softly broken symmetry: In the original proposal [0] of the type II seesaw mechanism for Higgs doublets there were only two Higgs doublets and no other scalar multiplets A U() symmetry φ e iα φ (5) was softly broken in the scalar potential by V l = μ φ φ (6) Then v μ v μ (7) The VEV v alone must produce the W ± and Z 0 masses therefore v 74 GeV m ew We assume that μ = m H m ew and that μ m H where the symbol means not much larger than We may assume that μ m ew and then v is suppressed by two powers of m ew /m H relative to v 3 General two-higgs-doublet model: Actually one could dispense with any symmetry and consider the general two-higgs-doublet model employing the same assumptions as in the previous paragraph Then in V l not only the term of Eq (6) is present but also (φ φ )(φ φ ) (8) Therefore one has two sources which induce a non-zero v As discussed in [5] one obtains a suppression factor of v of the same order of magnitude as before Two Higgs doublets and a scalar singlet: If we dislike soft symmetry breaking the simplest alternative is to introduce into the theory a complex scalar gauge singlet χ with VEV v χ TheU() symmetry (5) becomes φ e iα φ χ e iα χ (9) Then V l = mφ φ χ (0) v m v χ v μ () There is a large degree of arbitrariness in the orders of magnitude of m and of the VEV of χ but we may conservatively assume them to be of order m ew Then once again v /v (m ew /m H ) Symmetry Z instead of U(): Instead of the U() symmetry (5) originally used in [0] one may employ the weaker symmetry φ φ χ χ () In this case χ may as a matter of fact be a real field The symmetry () allows for a richer scalar potential with extra terms (φ φ ) and χ 4 and their Hermitian conjugates Three Higgs doublets: A more complicated model has three Higgs doublets and a symmetry Z 4 : φ φ φ 3 iφ 3 (3) Note that we now assume μ j < 0 and μ j m ew for both j = 3 Then V l = λ(φ 3 φ )(φ 3 φ ) (4) V r = λ (φ φ ) + (5) v λ v v 3 μ (6) The original proposal [0] was a type II seesaw mechanism for Higgs doublets within a type I seesaw mechanism A later suggestion [] was a type II seesaw mechanism for Higgs doublets within a type III seesaw mechanism A related scenario with the assumption μ = 0 was proposed in [4] 3 As a matter of fact since V l in this case breaks softly the symmetry (5) it would be technically natural to assume μ m ew aswasdonein[0] andthen v would be even smaller

3 W Grimus et al / Physics Letters B 674 (009) 7 9 In this case v and v 3 are necessarily of order m ew (or smaller) and one needs no extra assumption to conclude that v m 3 ew /m H Before we proceed to investigate the nesting of seesaw mechanisms we want to mention some simple applications of a seesaw mechanism for Higgs doublets Suppose that φ has Yukawa couplings only to the ν R and φ to all charged fermionic gauge-su() singlets Then with the small VEV v we have the option of a seesaw mechanism for Dirac neutrinos if we dispense with a ν R Majorana mass term We would then use v m 3 ew /m H ev where we assume ev to be the scale of the light-neutrino masses obtaining the estimate m H 0 33 ev = 0 75 GeV We could also try to explain the smallness of the down-type-quark masses as compared to the up-type-quark masses by enforcing the coupling of φ to the up-quark singlets and φ to the down-quark singlets in the Yukawa couplings Assuming v v m ( ) b mew (7) m t m H we find for the mass of the heavy Higgs doublet m H 6m ew 3 Nesting of seesaw mechanisms 3 Type I seesaw mechanism ThetypeIseesawformulais[] M ν = M T D M R M D (8) where M ν is the effective ν L Majorana mass matrix M R is the Majorana mass matrix of the ν R and M D is the Dirac mass matrix connecting the ν R to the ν L This Dirac mass matrix is generated by Yukawa couplings L Yukawa = ν R φ YD L + Hc (9) where Y is a matrix (in flavour space) of Yukawa coupling constants D L = (ν L l L ) T are SU() doublets of left-handed leptons φ is the Higgs doublet whose VEV is suppressed by a type II seesaw mechanism and φ iτ φ In order for the Yukawa couplings of the ν R in Eq (9) to involve only the Higgs doublet φ one needs to suitably extend the symmetries U() Z or Z 4 of the previous section In the case of the U() symmetry one must add D L e iα D L and l R e iα l R to the assignment (9) (the l R are the right-handed charged-lepton singlets) [0] In the case of the Z or Z 4 symmetries one must add ν R ν R to the assignments () and (3) respectively It follows from Eq (9) that M D = v Y hence M ν = v Y T M R Y As before we assume the matrix elements of M ν to be of order ev If we allowed the VEV v to be of order the electroweak scale m ew 00 GeV and assuming the Yukawa coupling constants to be of order unity we would find the scale m R of M R to be of order 0 3 GeV as advertised in the introduction Lowering m R to the TeV scale while keeping v m ew requires (assuming no cancellation mechanism) the Yukawa couplings to be of order 0 5 as also advertised in the introduction But if v m 3 ew /m H is suppressed by a type II seesaw mechanism for Higgs doublets as first proposed in [0] thenev m 6 ew /(m Rm 4 H ) even with Yukawa coupling constants of order unity This represents a fivefold suppression of the neutrino masses Assuming for simplicity m R = m H oneobtains m H ev 6 TeV (0) 3 Type II seesaw mechanism In the type II seesaw mechanism [] a scalar gauge-su() triplet ( δ Δ = + / δ ++ ) δ 0 δ + / () is introduced such that the ν L acquire Majorana masses through the VEV of the neutral component of the scalar triplet: ( ) 0 0 Δ 0 = () v Δ 0 This VEV is induced by the term linear in Δ in the scalar potential and is suppressed by the high mass of the scalar triplet In order for the terms linear in Δ to involve only the Higgs doublet φ whose VEV is suppressed by the type II seesaw mechanism discussed in Section weintroduceaz 4 symmetry 4 : φ iφ Δ Δ (3) We write the scalar potential as V = μ j φ j φ j + μ Δ Tr(Δ Δ) + ( μ φ φ + ) Hc j= + (μ φ Δ φ + Hc) + V q (4) where V q consists only of quartic terms The Z 4 symmetry is softly broken by operators of dimension two Instead of a softly broken symmetry for the type II seesaw mechanism for the VEV of φ one could employ one of the alternatives given in Section Inorderto have a Dirac mass term for charged leptons and a Majorana mass term for ν L generated by VEVs of φ and Δ respectively one needs to extend the Z 4 symmetry in Eq (3) to D L id L and l R il R Now we proceed according to Section On the one hand we assume that μ < 0 and that μ m ew in order to generate a spontaneous symmetry breaking with v m ew On the other hand we require μ > 0 μ Δ > 0 and μ μ Δ m H m ew (5) The terms linear in φ and Δ in Eq (4) generate non-vanishing VEVs v and v Δ respectively Using the result for v of Eq (7) the VEV of Δ is given by v Δ μ v μ Δ μ (μ ) v μ 4 (6) μ Δ As before there is a degree of arbitrariness in the orders of magnitude of μ and μ but we may assume them to be of order m ew Then v Δ is suppressed by six powers of m ew /m H relative to v Keeping the Yukawa coupling constants of order unity this represents a sixfold suppression of the neutrino masses Assuming again the matrix elements of M ν to be of order ev which amounts to v Δ evwitheq(5) we estimate m H ev 7TeV (7) By raising the mass of the φ to 0 TeV one shifts μ Δ below TeV and the δ ++ whose mass is just μ Δ could possibly be within reach of the LHC see for instance [7] and the references therein 4 In [6] a softly broken U() symmetry has been used instead together with assumptions on the soft-breaking parameters in the scalar potential

4 0 W Grimus et al / Physics Letters B 674 (009) 7 4 Multiple nesting of type II seesaw mechanisms for Higgs doublets In this section we show that a multiple nesting of successive type II seesaw mechanisms for several Higgs doublets is able to realize Froggatt Nielsen [] suppression factors by using only Higgs doublets and renormalizable interactions As an example we consider the hierarchy of charged-fermion masses: m t m ew m b m c m τ GeV m s m μ 0 GeV m u m d 0005 GeV m e GeV (8) This hierarchy suggests that the charged-fermion mass matrices may involve a suppression factor ɛ /0 according to the pattern ( ɛ ɛ 3 ) ( ɛ ɛ ɛ 3 ) M u ɛ ɛ 3 M d ɛ ɛ ɛ 3 ɛ ɛ 3 ɛ ɛ ɛ 3 ( ɛ ɛ ɛ 4 ) M l ɛ ɛ ɛ 4 (9) ɛ ɛ ɛ 4 The suppression factors in the various elements of these mass matrices may be explained àla Froggatt Nielsen [] as the result of a spontaneously broken horizontal symmetry We suggest to view them instead as the product of a nested type II seesaw mechanism for Higgs doublets 5 We postulate the existence of six Higgs doublets φ 6 where φ and φ have VEVs of order m ew and Yukawa couplings which generate the first column of M u φ 3 has VEV of order ɛm ew and generates the second column of M u and the first columns of M d and M l φ 4 has VEV of order ɛ m ew and its Yukawa couplings yield the second columns of M d and M l and so on We implement the hierarchy of VEVs in the following way The scalar potential is of the form 6 ( V = j= μ j + λ j φ j φ j ) φ j φ j + j<k(λ jk φ j φ jφ k φ k + λ jk φ j φ kφ k φ j) + V t + V t (30) We assume that μ and μ are both negative and of order m ew while μ 36 are positive and of order m H with(m ew/m H ) ɛ 6 The VEV of φ 3 is induced out of the VEVs of φ and φ via a term κ φ φ φ φ 3 (3) in V t Thisleadstov 3 κ v v /μ 3 Since the coupling constant κ v 3 is of order m 3 ew /m H Afterwards the VEV ofφ 4 is induced by a further term in V t κ φ φ 3φ φ 4 (3) This leads to v 4 κ v v 3 /μ 4 m5 ew /m4 H The VEVs v 5 and v 6 are successively induced by terms 5 A similar idea was already put forward in [8] and subsequently combined with the leptonic model of [0] in a supersymmetric way [9] 6 With ɛ /0 this produces only a slight difference between m ew and m H This certainly constitutes a drawback of the present model κ 3 φ φ 4φ φ 5 (33) κ 4 φ φ 5φ φ 6 (34) respectively in V t 7 InordertomakesurethatthereareinV t no other terms which might induce larger (unsuppressed) VEVs we must impose a symmetry S on the theory For simplicity we assume that symmetry to be Abelian: S : φ j σ j φ j (35) with σ j =for j = 6 We assume of course the six factors σ 6 to be all different In order for the four terms (3) (34) to be allowed we must assume σ = σ σ 3 σ = σ 3σ 4 σ = σ 4σ 5 σ = σ 5σ 6 (36) Therefore σ 3 = σ σ 4 = σ 3 σ σ σ 5 = σ 4 σ σ 3 6 = σ 5 σ 4 (37) It follows that the bilinears φ j φ k ( j < k) transform as φ φ : φ φ 3: φ φ 4: φ φ 5: φ φ 6: φ 5 φ 6: σ σ σ φ σ φ 3: σ σ σ 3 φ σ 3 φ σ 4: φ σ 3 φ σ 4 4: σ 4 σ 3 φ σ 3 φ σ 4 5: φ σ 4 3 φ σ 5: φ σ 4 φ σ 6 5: σ 6 σ 5 φ σ 5 φ σ 4 6: φ σ 4 3 φ σ 6 6: φ σ 6 4 φ σ 6: σ σ 8 σ 8 (38) We assume that all the factors in this list are different from unity else there would be (at least) two Higgs doublets transforming identically under S and also different from each other so that there are as few terms as possible in V t Thisrequires σ p σ p for p = 4 (39) Therefore we must choose for S a groupz n with n > 4 It is enough to choose S = Z 5 with φ ωφ φ ω φ φ 3 φ 3 φ 4 ω 4 φ 4 φ 5 ω 3 φ 5 φ 6 ω 6 φ 6 (40) where ω exp(iπ/5) From the list(38) we learn that the full V t is V t = κ φ φ φ φ 3 + κ φ φ 3φ φ 4 + κ 3 φ φ 4φ φ 5 + κ 4 φ φ 5φ φ 6 + κ 5 φ 3 φ φ 3 φ 5 + κ 6 φ 4 φ φ 4 φ 6 + κ 7 φ 3 φ 6φ 4 φ 5 + κ 7 φ 3 φ 5φ 4 φ 6 + κ 8 φ φ 6φ 4 φ 3 + κ 8 φ φ 3φ 4 φ 6 + κ 9 φ φ 5φ 3 φ 4 + κ 9 φ φ 4φ 3 φ 5 (4) It is easy to check that with this V t VEVs with the right powers of the suppression factor ɛ (m ew /m H ) are generated One obtains 7 Instead of the terms (3) (34) we might imagine other possibilities The present text thus constitutes only a proof of the viability of the mechanism

5 W Grimus et al / Physics Letters B 674 (009) 7 v 3 κ v 4 κ v 5 κ 3 v 6 κ 4 v v μ (4) 3 v v 3 μ (43) 4 v v 4 μ 5 v v 5 μ 6 κ 5 v 3 v μ (44) 5 (κ 8 + κ v v 4 v 8 ) 3 (45) μ 6 The other terms in V t generate subdominant (in terms of ɛ) contributions to the VEVs 5 Conclusions The main point in this Letter is the observation that the VEVs of some Higgs doublets may be suppressed by a type II seesaw mechanism in the same way as the VEVs of scalar gauge triplets We have furthermore emphasized that this Higgs-doublet type II seesaw mechanism may be combined with other seesaw mechanisms of any type I II III or even with itself in a multiply nested way If there are only two mass scales at our disposal the electroweak scale m ew and a heavy scale m H m ew one may through this procedure suppress some mass terms by a factor (m ew /m H ) p where the power p can be considerably larger than as in the standard type I seesaw case While the standard seesaw mechanisms are applied to Majorana neutrinos the type II seesaw mechanism for Higgs doublets whether in its simple or in its multiply nested form is able to suppress any Dirac-fermion masses without one having to introduce any new fermionic degrees of freedom in the theory Our aim was not to promote a specific type of seesaw mechanism rather to point out the wealth of possible scenarios It is also beyond the scope of this Letter to check the compatibility of each particular scenario with the experimental data for instance with electroweak precision tests Thus in individual cases the parameter space may have to be restricted or the scenario modified A seesaw mechanism always involves the ad hoc introduction of a heavy scale m H The usual belief is that either the new physics at m H is not directly accessible by experiment because that scale is too high or contrived cancellation mechanisms are needed to lower m H The main message of this Letter is that neither of the two conclusions is compelling As originally demonstrated in a specific case [0] and generalized in this Letter the nesting of the type II seesaw mechanism for Higgs doublets with other seesaw mechanisms or with itself provides a very simple method to lower m H This method requires an extension of the scalar sector and therefore leads to new physics at the scale m H Acknowledgements WG and LL acknowledge support from the European Union through the network programme MRTN-CT The work of LL was supported by the Portuguese Fundação para a CiênciaeaTecnologia through the project U777-Plurianual The work of BR is supported by the Croatian Ministry of Science Education and Sport under the contract No BR gratefully acknowledges the support of the University of Vienna within the Human Resources Development Programme for the selected SEE Universities and the hospitality offered at the Faculty of Physics References [] P Minkowski Phys Lett B 67 (977) 4; 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