Type II seesaw mechanism for Higgs doublets and the scale of new physics

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(2) triplets, scalar SU(2) doublets, or right-handed neutrinos - does not need to be higher than a few 10 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.


Introduction
The type I seesaw mechanism [1] is a favourite with high-energy 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 10 13 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 10 −5 .
Another mechanism for explaining the smallness of the neutrino masses is the type II seesaw mechanism [2], which suppresses the vacuum expectation values (VEVs) of the neutral components of scalar gauge-SU (2) triplets, in such a way that the left-handed 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.
In the general case, for instance in Grand Unified Theories based on the gauge group SO (10), 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 [10]. 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 [10] in several directions: • We describe (in section 2) the general mechanism for suppressing Higgs-doublet VEVs and give several examples thereof.
• We show (in section 3) that this "type II seesaw mechanism for Higgs doublets" may be nested inside various seesaw mechanisms. 1 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 [12]. 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 [13].
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 ∼ 100 GeV is not compelling; an m H just two or three orders of magnitude above m ew may suffice.
2 Type II seesaw mechanism for Higgs doublets Consider a model with several Higgs doublets φ j = (φ + j , φ 0 j ) T , j = 1, . . . , n h . The VEVs of the Higgs doublets are of the form We assume that |v 1 | ∼ m ew . Our aim is to produce a seesaw mechanism to suppress |v 2 |.
We write the scalar potential as We assume that µ 2 1 < 0 and that |µ 2 1 | ∼ m 2 ew in order to generate a spontaneous symmetry breaking leading to |v 1 | ∼ m ew . On the other hand, we assume that µ 2 2 > 0 and that µ 2 2 = m 2 H ≫ m 2 ew . In equation (2) V l represents some terms linear in φ 2 which we assume to be present in V . All the remainder of V , i.e. everything but the mass terms for the Higgs doublets and the terms V l and V † l linear in φ 2 and φ † 2 , 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 where A has the dimension of the cube of a mass. Then, despite the positiveness of µ 2 2 , a non-vanishing VEV v 2 is induced, approximately given by The quantity A depends on the specific model. It has to contain at least one v j = v 2 and this v j will in general be of order m ew . If we assume that µ 2 2 is the only parameter in the scalar potential of order m 2 H , then we expect |A| ∼ m 3 ew . In this case |v 2 | ∼ m 3 ew /m 2 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 [10] of the type II seesaw mechanism for Higgs doublets there were only two Higgs doublets and no other scalar multiplets. 2 was softly broken in the scalar potential by Then, The VEV v 1 alone must produce the W ± and Z 0 masses, therefore |v 1 | ≈ 174 GeV ∼ m ew . We assume that µ 2 2 = m 2 H ≫ m 2 ew and that |µ 2 | m 2 H , where the symbol means "not much larger than". We may assume that |µ 2 | ∼ m 2 ew and then v 2 is suppressed by two powers of m ew /m H relative to v 1 . 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 equation (6) is present but also Therefore, one has two sources which induce a non-zero v 2 . As discussed in [15], one obtains a suppression factor of v 2 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 . The U(1) symmetry (5) becomes Then, 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 2 A related scenario with the assumption µ 2 1 = 0 was proposed in [14]. 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 2 ≪ m 2 ew , as was done in [10], and then |v 2 | would be even smaller.
Symmetry 2 instead of U (1): Instead of the U(1) symmetry (5) originally used in [10] one may employ the weaker symmetry In this case χ may as a matter of fact be a real field. The symmetry (12) allows for a richer scalar potential, with extra terms (φ † 1 φ 2 ) 2 and χ 4 and their Hermitian conjugates.
Three Higgs doublets: A more complicated model has three Higgs doublets and a symmetry.
Note that we now assume µ 2 j < 0 and µ 2 j ∼ m 2 ew for both j = 1, 3. Then In this case |v 1 | and |v 3 | are necessarily of order m ew (or smaller) and one needs no extra assumption to conclude that |v 2 | ∼ m 3 ew /m 2 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 2 has Yukawa couplings only to the ν R , and φ 1 to all charged fermionic gauge-SU (2) singlets. Then with the small VEV v 2 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 2 | ∼ m 3 ew /m 2 H ∼ 1 eV, where we assume 1 eV to be the scale of the light-neutrino masses, obtaining the estimate m H ∼ √ 10 33 eV = 10 7.5 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 1 to the up-quark singlets and φ 2 to the down-quark singlets in the Yukawa couplings. Assuming we find for the mass of the heavy Higgs doublet m H ∼ 6 m ew .
3 Nesting of seesaw mechanisms

Type I seesaw mechanism
The type I seesaw formula is [1] 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 where Y is a matrix (in flavour space) of Yukawa coupling constants, D L = (ν L , ℓ L ) T are SU (2) doublets of left-handed leptons, φ 2 is the Higgs doublet whose VEV is suppressed by a type II seesaw mechanism andφ 2 ≡ iτ 2 φ * 2 . In order for the Yukawa couplings of the ν R in equation (19) to involve only the Higgs doublet φ 2 , one needs to suitably extend the symmetries U(1), 2 or 4 of the previous section. In the case of the U(1) 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) [10]. In the case of the 2 or 4 symmetries, one must add ν R → −ν R to the assignments (12) and (13), respectively.
It follows from equation (19) As before, we assume the matrix elements of M ν to be of order eV. If we allowed the VEV v 2 to be of order the electroweak scale m ew ∼ 100 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 10 13 GeV, as advertised in the introduction. Lowering m R to the TeV scale while keeping v 2 ∼ m ew requires (assuming no cancellation mechanism) the Yukawa couplings to be of order 10 −5 , as also advertised in the introduction. But if v 2 ∼ m 3 ew /m 2 H is suppressed by a type II seesaw mechanism for Higgs doublets, as first proposed in [10], then 1 eV ∼ m 6 ew / (m R m 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 , one obtains

Type II seesaw mechanism
In the type II seesaw mechanism [2], a scalar gauge-SU (2) triplet is introduced such that the ν L acquire Majorana masses through the VEV of the neutral component of the scalar triplet: This VEV is induced by the term linear in ∆ in the scalar potential and is suppressed by the high mass of the scalar triplet.
In order for the terms linear in ∆ to involve only the Higgs doublet φ 2 whose VEV is suppressed by the type II seesaw mechanism discussed in section 2, we introduce a 4 symmetry 4 : We write the scalar potential as where V q consists only of quartic terms. The 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 2 , one could employ one of the alternatives given in section 2. In order to have a Dirac mass term for charged leptons and a Majorana mass term for ν L generated by VEVs of φ 1 and ∆, respectively, one needs to extend the 4 symmetry in equation (23) to D L → iD L and ℓ R → iℓ R . Now we proceed according to section 2. On the one hand, we assume that µ 2 1 < 0 and that |µ 2 1 | ∼ m 2 ew in order to generate a spontaneous symmetry breaking with |v 1 | ∼ m ew . On the other hand, we require The terms linear in φ 2 and ∆ in equation (24) generate non-vanishing VEVs v 2 and v ∆ , respectively. Using the result for v 2 of equation (7), the VEV of ∆ is given by 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 ∆ is suppressed by six powers of m ew /m H relative to v 1 . 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 ∆ ∼ 1 eV, with equation (25) we estimate By raising the mass of the φ 2 to 20 TeV, one shifts µ ∆ below 1 TeV, and the δ ++ , whose mass is just µ ∆ , could possibly be within reach of the LHC-see for instance [17] and the references therein.

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 [12] suppression factors by using only Higgs doublets and renormalizable interactions.
As an example, we consider the hierarchy of charged-fermion masses: This hierarchy suggests that the charged-fermion mass matrices may involve a suppression factor ǫ ∼ 1/20 according to the pattern The suppression factors in the various elements of these mass matrices may be explained a la Froggatt-Nielsen [12] 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 1,...,6 , where φ 1 and φ 2 have VEVs of order m ew and Yukawa couplings which generate the first column of M u , φ 3 has VEV of order ǫm 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 ǫ 2 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 We assume that µ 2 1 and µ 2 2 are both negative and of order m 2 ew , while µ 2 3,...,6 are positive and of order m 2 H , with (m ew /m H ) 2 ∼ ǫ. 6 The VEV of φ 3 is induced out of the VEVs of φ 1 and φ 2 via a term VEVs v 5 and v 6 are successively induced by terms respectively, in V t . 7 In order to make sure that there are in V 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: with |σ j | = 1 for j = 1, . . . , 6. We assume, of course, the six factors σ 1,...,6 to be all different. In order for the four terms (31)-(34) to be allowed, we must assume Therefore, It follows that the bilinears φ † j φ k (j < k) transform as 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 . This requires σ p 1 = σ p 2 for p = 1, 2, . . . , 14.
Therefore we must choose for S a group n with n > 14. It is enough to choose S = 15 with where ω ≡ exp (2iπ/15). From the list (38) we learn that the full V t is It is easy to check that with this V t VEVs with the right powers of the suppression factor ǫ ∼ (m ew /m H ) 2 are generated. One obtains The other terms in V t generate subdominant (in terms of ǫ) contributions to the VEVs.

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 1 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 [10] 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 . U777-Plurianual. The work of B.R. is supported by the Croatian Ministry of Science, Education and Sport under the contract No. 119-0982930-1016. B.R. 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.