UWThPh200629
A renormalizable GUT scenario
with spontaneous CP violation
Abstract
We consider fermion masses and mixings in a renormalizable SUSY GUT with Yukawa couplings of scalar fields in the representation . We investigate a scenario defined by the following assumptions: i) A single large scale in the theory, the GUT scale. ii) Small neutrino masses generated by the type I seesaw mechanism with negligible type II contributions. iii) A suitable form of spontaneous CP breaking which induces hermitian mass matrices for all fermion mass terms of the Dirac type. Our assumptions define an 18parameter scenario for the fermion mass matrices for 18 experimentally known observables. Performing a numerical analysis, we find excellent fits to all observables in the case of both the normal and inverted neutrino mass spectrum.
1 Introduction
The group is a favourite candidate for grand unified theories (GUTs) [1] because its 16dimensional irreducible representation (irrep), the spinor representation, contains all chiral fermions included in a Standard Model (SM) family plus an additional neutrino SM gauge singlet. Moreover, such theories allow for type I [2] and type II [3] seesaw mechanisms (see also [4]) for the light neutrino masses. In the construction of theories, there are two options [5], either using lowdimensional scalar irreps but accepting nonrenormalizable terms in the Lagrangian, or one sticks to renormalizable terms, then one has to accept highdimensional scalar irreps according to [6, 7]
(1) 
where the subscripts “S” and “AS” denote, respectively, the symmetric and antisymmetric parts of the tensor product.
In this paper, we deal with the second option. A special renormalizable model is the socalled “minimal SUSY GUT” (MSGUT) [8], which uses, for the Yukawa couplings, one scalar in the and one in the irrep in order to account for all fermion masses and mixings; it contains, in addition, one and one scalar irrep, in order to perform the suitable symmetry breakings. This model has builtin the gaugecoupling unification of the minimal SUSY extension of the Standard Model (MSSM). Detailed studies of this minimal theory have been performed [9, 10], also with small effects of the 120plet [11]. Though the MSGUT works very well in the fermion sector, there is a tension between the scale of the light neutrino masses and the GUT scale. The reason is that the natural order of the neutrino masses in GUTs is eV, where we have used GeV for the electroweak scale and a GUT scale of GeV. This neutrino mass scale is too low because eV where is the atmospheric neutrino masssquared difference—for reviews on the status of neutrino masses and mixing see [13, 14]. Studies of the heavy scalar states [12], together with studies of the fermion mass spectrum, have shown that the MSGUT is too constrained [15], and the tension between the scale of light neutrino masses and the GUT scale cannot be overcome: if one has a good fit of the fermion masses, which requires a seesaw scale below , then the gauge coupling unification of the MSSM [16, 17] is spoiled.
A natural step for supplying additional degrees of freedom to the MSGUT is to add the 120plet of scalars [18] which appears anyway in Eq. (1)—for early works in this direction see [19, 20, 21, 22].^{9}^{9}9We stress that alone does not give a good fit in the charged fermion sector [23]. The disadvantage is that this step adds a considerable number of parameters and reduces the predictability of the theory. Adding the leads to a resurgence of the type I seesaw mechanism [10], as a consequence of the collapse of the seesaw scale with the GUT scale, because type I seesaw allows to enhance the neutrino masses through small Yukawa couplings of the [18, 24, 25]; without the , i.e. in the MSGUT, this process leads to the contradictions mentioned above.
The has electrically neutral components only in its four doublets with respect to the SM gauge group. These contribute to the Higgs doublets , of the MSSM, which are assumed to be the only light scalar degrees of freedom and the only ones which acquire VEVs at the electroweak scale. Thus the MSGUT enlarged by the inherits, from the MSGUT, the scalar fields responsible for spontaneous symmetry breaking above the electroweak scale. In [24] we took this into account by explicitly making the identification
(2) 
where , which defines the seesaw scale, is the vacuum expectation value (VEV) of , with the usual notation for multiplets of the Pati–Salam subgroup [26] of . Furthermore, in that paper we reduced the number of parameters in the fermion mass matrices by assuming a horizontal symmetry and spontaneous CP violation, i.e. real Yukawa couplings, with CP violation stemming from the phases of the VEVs. We showed numerically that this scenario can excellently reproduce the known fermion masses and mixings.
Recently, the role of spontaneous CP violation has been upgraded. A “New MSGUT” (NMSGUT) was proposed [27], defined by extending the MSGUT by the and spontaneous CP violation. It was shown that the requirement of spontaneous CP violation not only has the virtue of reducing the number of parameters of the theory but it has an important impact, via threshold effects, on the unification scale as well; it tends to raise the unification scale and with it the masses of all heavy multiplets, thereby suppressing baryon decay.
In the present paper we retain Eq. (2) as a reference point. We do not employ any horizontal symmetry but we again motivate real Yukawa coupling matrices by spontaneous CP violation. However, we assume that it is of a very specific kind: CP is solely violated by imaginary VEVs of the ; the VEVs of the and are assumed to be real. In this way, the mass matrices of the downquarks, upquarks, charged leptons and the neutrino Diracmass matrix are hermitian. This scenario was originally proposed in [21], its compatibility with sufficiently slow proton decay shown in [22]. However, in [21, 22] it was assumed that the type II seesaw mechanism is dominating. Since this is incompatible with having only one large scale, we have in the present paper type I dominance and neglect possible small contributions of type II, suppressed by . Our scenario gives an excellent fit to all known fermion masses, mixings and the CKM phase , as good as the one in [24], though it is of a rather different type. This shows that the fermion data do not fix the enlarged MSGUT in a unique way and there is considerable freedom in reducing the number of parameters in this theory.
The paper is organized as follows. In Section 2 we discuss CPinvariant Yukawa couplings and lay out our scenario. The method and results of our numerical analysis are discussed in Section 3. In Section 4 we present the conclusions. Appendix A contains a small collection of formulas for the spinor representation, which is helpful for Section 2.
2 An scenario motivated by spontaneous CP violation
Let us a define a transformation
(3) 
where is a fermionic 16plet, is the chargeconjugation matrix, the 45 gauge fields are denoted by (), for and for , , and the are signs. No summation is implied in Eq. (3). The 45 (hermitian) generators of the gauge group in the fermionic are given by
(4) 
For a representation of the operators ( of the Clifford algebra and useful formulas concerning the spinor irrep see Appendix A. One can easily check that the gauge interaction of the fermionic 16plet is invariant under the transformation (3) if [7, 28, 29]
(5) 
Since
(6) 
one finds
(7) 
Denoting the generators of the Lie algebra by , we mention that
(8) 
is the automorphism associated with root reflection, which is the canonical automorphism associated with CP. Such an automorphism exists for all compact Lie groups and is the reason why any gauge Lagrangian, whether for fermions or scalars, is CPinvariant [7, 28, 29].
Now we transfer the CP transformation to the Yukawa couplings given by the Lagrangian
(9)  
The indices , denote the family indices, indices, and is the chargeconjugation matrix. Summation over family and indices is implied in Eq. (9). The matrix , which ensures invariance, is defined in Eq. (A6). The Yukawa coupling matrices have the properties are
(10) 
We define CP transformations (no summations implied)
(11)  
(12)  
(13) 
for the scalar fields of the irreps , and , respectively; the latter two are totally antisymmetric tensor fields, is selfdual in addition.
Now we require invariance of the Lagrangian (9) under the CP transformation given by Eqs. (3), (11), (12) and (13). As an example we take the and obtain
(14) 
The equality sign on the righthand side of the arrow defines the condition of CP invariance: the CPtransformed Yukawa Lagrangian must be identical with its hermitian conjugate. Evaluating Eq. (14) with the help of Eq. (A6), we find a hermitian Yukawa coupling matrix. Performing an analogous computation for the and we arrive at the conclusion that the CP transformation requires
(15) 
Together with Eq. (10) this means that all Yukawa coupling matrices are real.
In order to obtain a nontrivial CKM phase it is necessary to break CP invariance. The scenario we envisage was originally proposed in [21]. In the context discussed here, we assume that

the VEVs of the and are real,

CP is spontaneously broken by the VEVs of the ,

this breaking is maximal, i.e., the VEVs of the are imaginary.
Thus, the mass matrices of the charged fermions and the neutrino Diracmass matrix are given, respectively, by
(16)  
(17)  
(18)  
(19) 
with
(20) 
The mass matrices (16)–(19) are hermitian. The light neutrino mass matrix is given by
(21) 
with scalar triplet VEVs and . The mass Lagrangian of the “light” fermions reads
(22) 
We finish this section with some remarks. models have included the socalled Dparity [30], which is a specific involutory transformation which uses the branching rule
(23) 
under the Pati–Salam group [26] and exchanges the two Pati–Salam irreps. One can combine CP with Dparity and interpret such a transformation as parity in the usual sense [29]. Requiring invariance of the theory under this parity, gives the same restrictions on the Yukawa coupling matrices as CP alone, since the theory is invariant under Dparity anyway.
3 The numerical analysis


As argued in the introduction, with only one large scale, the GUT scale, in the theory, we can neglect the type II seesaw contribution in Eq. (21).^{11}^{11}11A quantitative justification will be given in this section. Then a possible phase of is irrelevant. With
(24) 
we rewrite the mass matrices as
(25)  
(26)  
(27)  
(28)  
(29) 
Without loss of generality we assume to be diagonal. Then all redundant parameters are removed and we arrive at 12 real parameters in , , and six real ratios of VEVs. Thus our scenario has 18 independent parameters for 18 observables: nine chargedfermion masses, three mixing angles and the CP phase in the CKM matrix, the atmospheric and solar neutrino masssquared differences and , and three lepton mixing angles.
Eqs. (25)–(29) are amenable to a numerical analysis, which will, in particular, yield values for and . If we fix the triplet VEV , e.g. by identifying it with the GUT scale—see Eq. (2), this analysis will also yield definite values for and because
(30) 
A reasonable condition on these VEVs is given by [24]
(31) 
This inequality certainly holds at the electroweak scale. Assuming that it holds approximately at the GUT scale as well, we will subject our fit results to this consistency check.
To find a numerical solution for the parameters in Eqs. (25)–(29), we build as usual [17, 23, 24] a function for the 18 observables,
(32) 
whose input values are given in Table 1; these values refer to an MSSM parameter . The letter symbolizes the set of 18 parameters, i.e. the Yukawa couplings and ratios of VEVs. The functions express our theoretical predictions, as functions of the parameter set , for the observables, obtained from Eqs. (25)–(29). As convention for the quark and lepton mixing matrix we use that of the Review of Particle Properties [32]. The function is minimized analytically with respect to . In this way we obtain a function of the remaining 17 parameters, which is minimized numerically by employing the downhill simplex method [33].
In the following we will also consider functions where a specific quantity is pinned down to a given value—for previous uses of such a method see e.g. [17, 24]. If we want to pin down a quantity , which is independent of the 18 observables, to a value , we add in Eq. (32) and minimize the thus obtained . If coincides with one of the observables , the term above is added but at the same time the term has to be removed from the of Eq. (32). In that way, we can study the sensitivity of our scenario to a variation of a quantity .
Normal:  

Inverted: 
A fit in the case of normal neutrino mass ordering:
We search for a solution in the case of the normal ordering of the neutrino masses (, ). In that case we find an excellent fit with the following properties:
(33) 
In the second relation in this equation we have used —see Eq. (2). The corresponding values of the matrix elements of , , are given by
(34)  
where all numerical values are in units of MeV. The fit values of the VEV ratios are listed in Table 2. The of the fit practically comes only from two observables: the pull of (leptonic mixing angle) is 0.45 and the pull of is 0.31.
We can ask the question if our scenario makes some predictions. For the best fit we find . However, this small value misleading, because pinning in shows that in the large range the fit is still very good, with . The worst is about 15 and occurs around , where for instance is not well reproduced and the leptonic becomes too large. As for , Fig. 1 shows that the preferred value is about 0.01. However, we cannot consider this as a prediction since in a wide range around this value the is still acceptable. Only at very small the fit becomes bad, mainly because of , and the atmospheric mixing angle . The quantity measures how hierarchical a normal neutrino mass spectrum is. The as a function of is depicted in Fig. 2. We read off that is preferred and the quickly becomes bad for larger , mainly owing to , and the leptonic . Also for very small the fit worsens, for similar reasons as for large , however, not in a dramatic way. On the other hand, in the MSGUT there is a preferred range and there is a genuine lower bound on as well [17].
With Eq. (30) and the upper bound on in Eq. (33) we see that we are allowed to raise to , without violating the inequality (31).
It has to be checked that our numerical solution given by Eq. (34) and Table 2 respects the perturbative regime in the Yukawa sector. Since the procedure has been explained in detail in [24], we confine ourselves to the essentials. The two Higgs doublets of the MSSM, and , have hypercharges and and VEVs and , respectively. The corresponding Yukawa coupling matrices are given by , etc. It turns out that the largest Yukawa couplings are , where the largest contribution comes from . Using , it is given by . This confirms that the Yukawa couplings are safely in the perturbative regime.
Finally, we want to estimate the size of type II seesaw contributions to . The corresponding mass matrix is given by —see Eqs. (2), (21) and (30). The largest element in is the 33element. With the value of this element from Eq. (34) and from Table 2, the type II mass matrix contributes at most . Since we expect eV, we find the announced suppression with respect to type I seesaw contributions.
A fit for the inverted neutrino mass spectrum:
Searching for a fit by imposing the inverted ordering of the neutrino masses ( as for the normal spectrum, but ), we find a solution which is even better as in the case of normal ordering. It has the following properties:
(35) 
with the matrices
(36)  
where all numerical values are in units of MeV, and the VEV ratios are displayed in Table 2. For all practical purposes the fit is perfect and there is no need to give any pull values.
Now we come to the predictions of our scenario in the case of the inverted neutrino mass spectrum. Concerning CP violation in neutrino oscillations, our best fit gives . However, this value has no meaning because as a function of is flat for all practical purposes. The same is true for the leptonic quantities and in the physically relevant ranges. However, there is a definite prediction for the neutrino mass spectrum: hierarchy is strongly preferred—see Fig. 2. When the quantity becomes large the fit turns bad; however, there is no clearcut reason, it is mostly the downquark masses and the topquark mass which are not well reproduced and the fit value of the leptonic is around its experimental upper bound.
In the second relation of Eq. (35) we have again used our reference value (2). Now inequality (31) is respected for , i.e. there is more freedom for than in the normal case.
As before, large Yukawa couplings in and are induced by . But now, because is so big, a slightly larger coupling is , still in the perturbative regime. The discussion of the smallness of type II seesaw contributions to proceeds as for the normal spectrum.
In Eq. (36) the elements , and are rather small. This might suggest to set them zero, which is achieved by the horizontal symmetry , . However, this is untenable because it would lead to vanishing . The reason is that this horizontal can be combined with the CP transformation of Section 2 to a new symmetry CP, under which the vacuum state of our scenario is invariant.^{12}^{12}12Under CP, the VEVs do not change sign! Consequently, in that case there is no CP violation [34, 35] and one can show—as it must be in such a case—that [35].
4 Conclusions
In this paper we have investigated fermion masses and mixings in an SUSY scenario,^{13}^{13}13In our analysis SUSY enters only via the input parameters whose values we need at the GUT scale. In the evolution of these parameters from the electroweak to the GUT scale we assume the renormalization group equations of the MSSM. originally proposed in [21], where the Yukawa coupling matrices of the scalars in the irreps , and are real and CP violation is induced only by imaginary VEVs of the . This gives a scenario with 18 real parameters in the fermion mass sector. Recent results from the MSGUT require a single heavy breaking scale, which is then cogent for the MSGUT extended by the as well.
There are the following differences between Ref. [21] and the present paper: Firstly, our is induced by the type I seesaw mechanism and type II is negligible, whereas in [21] it was assumed that type II dominates. Secondly, we use a purely numerical method, employing the minimization of a function, whereas Ref. [21] uses an approximate semianalytical method.
We have found excellent fits to fermion masses and mixings for both types of neutrino mass spectra. We want to emphasize this in particular for the inverted mass spectrum, for which the system of fermion mass matrices in the MSGUT—which has no —does not allow an acceptable fit [17], though complex Yukawa couplings and VEVs and contributions to from both seesaw types are admitted.^{14}^{14}14The MSGUT system has 13 absolute values and 8 phases. The fits presented in this paper have the following features. The diagonal Yukawa coupling matrix of the is strongly hierarchical and is responsible, in the chargedfermion mass spectra, for the hierarchy between 2nd and 3rd families. The correct size of the neutrino masses is reproduced by a cooperation of two effects: rather large contributions to the neutrino Diracmass matrix from the couplings of the and , where is the Yukawa coupling matrix of the , and a moderately small coupling matrix of the , which enters with its inverse in the type I seesaw formula. The contribution of the to the chargedfermion masses and to the CKM matrix is rather small, whereas in introduces large leptonic mixing angles. Similar features were found in the previous sample fit of [24], though the assumptions concerning the fermion mass matrices in that paper are quite different from those in the present paper, apart from the use of spontaneous CP violation in both scenarios.
Unfortunately, our scenario is not very predictive. However, it does have one clearcut prediction, namely hierarchy for both the normal and inverted neutrino mass spectrum. This is quantified by the observable in Fig. 2 from where we read off .
Apparently, extending the MSGUT by the leads to an ambiguous situation concerning fermion mass matrices: quite different assumptions can result in excellent fits. Whether these fits are compatible with the NMSGUT [27], where the VEVs are subject to certain relations, remains to be checked. One aspect seems to emerge: spontaneous CP violation plays in important role in both the fermion mass matrices [24] and the spontaneous breaking [27] of .
Acknowledgment: W.G. is grateful to L. Lavoura for useful discussions.
Appendix A The spinor representation of
A possible representation—on the space of the fivefold tensor product of —for the Clifford algebra associated with the Lie algebra is given by [36]
(A1) 
where a superscript denotes the fold tensor product. The matrices () are the Pauli matrices and denotes the unit matrix. It is easy to check that the () fulfill
(A2) 
It is well known that the matrices
(A3) 
have precisely the same commutation relations as the basis elements
(A4) 
of . The generate the spinor irrep of on the 16dimensional space
(A5) 
Note that anticommutes with all ().
For the Yukawa couplings one needs the matrix and its following properties:
(A6) 
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