Symmetry constrained two Higgs doublet models
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Abstract
We study twoHiggsdoublet models (2HDM) where Abelian symmetries have been introduced, leading to a drastic reduction in the number of free parameters in the 2HDM. Our analysis is inspired in BGL models, where, as the result of a symmetry of the Lagrangian, there are treelevel scalar mediated FlavourChangingNeutralCurrents, with the flavour structure depending only on the CKM matrix. A systematic analysis is done on the various possible schemes, which are classified in different classes, depending on the way the extra symmetries constrain the matrices of couplings defining the flavour structure of the scalar mediated neutral currents. All the resulting flavour textures of the Yukawa couplings are stable under renormalisation since they result from symmetries imposed at the Lagrangian level. We also present a brief phenomenological analysis of the most salient features of each class of symmetry constrained 2HDM.
1 Introduction
 (i)
it has potentially dangerous scalar mediated Flavour Changing Neutral Currents (FCNC) at tree level,
 (ii)
it leads to a large increase in the number of flavour parameters in the scalar sector, parametrised by two arbitrary \(3\times 3\) complex matrices, which we denote by \(N_d\) and \(N_u\).
In a separate development, which addresses simultaneously the above two problems of 2HDM, it was shown [8] by Branco, Grimus and Lavoura (BGL) that one may have a scenario where there are tree level FCNC, but with \(N_d\) and \(N_u\) fixed entirely by the elements of the Cabibbo–Kobayashi–Maskawa (CKM) matrix. In some BGL models, the suppression of FCNC couplings resulting from the smallness of CKM elements, is such that the new neutral scalars need not be too massive in order to conform with experiment. BGL models have been studied in the literature [9, 10] and their phenomenological consequences have been analysed in the context of the LHC [11, 12, 13]. A generalisation of BGL models has been recently proposed in the framework of 2HDM [14].
Regarding symmetries, Ferreira and Silva [15] classified all possible implementations of Abelian symmetries in 2HDM with fermions which lead to nonvanishing quark masses and a CKM matrix which is not block diagonal (see also [16]).
In this paper we study in a systematic way scenarios arising from different implementations of Abelian symmetries in the context of 2HDM which can lead to a natural reduction in the number of parameters in these models. In the search for these scenarios, we were inspired by BGL and generalised BGL (gBGL) models where the coupling matrices \(N_d\), \(N_u\) (see Eqs. (8)–(9)) can be written in terms of the quark mass matrices and projection operators. Thus we classify the different models according to the structures of \(N_d\), \(N_u\). We identify the symmetry leading to each of the models and the corresponding flavour textures of the Yukawa couplings. These textures are stable under renormalisation, since they result from symmetries of the Lagrangian.
The organisation of the paper is the following. The notation is set up in Sect. 2. We then present our main results in Sects. 3 and 4, obeying what we denote the Left and Right conditions introduced in Eqs. (13) and (16), respectively. We show that, besides BGL and gBGL there is a new type of model obeying Left conditions and that there are six classes of models obeying Right conditions which, as far as we can tell, are presented in full generality here for the first time. For definiteness, we concentrate on the quark sector. Some of the most salient phenomenological implications are presented in Sect. 5, and our conclusions appear in Sect. 6. We defer some technical details to Appendix A. In particular, we present in Appendix A.4 conditions for the identification of the various models which are invariant under basis transformations in the spaces of lefthanded doublets and of uptype and downtype righthanded singlets.
2 Generalities and notation
 (a)The Yukawa coupling matrices are required to obey Left conditionswith$$\begin{aligned} N_d^{0}=\mathrm{L_{d}^0}\,M_d^0,\quad N_u^{0}=\mathrm{L_{u}^0}\,M_u^0, \end{aligned}$$(13)where \(\ell _j^{[q]}\) are, a priori, arbitrary numbers. Here and henceforth we shall often use the index q to refer to matrices in the up (\(q=u\)) or down (\(q=d\)) sectors. We have used the projection operators \(\mathrm{P}_{\!i}\) defined by \([\mathrm{P}_{\!i}]_{jk}=\delta _{ij}\delta _{jk}\) (no sum in j). In matrix form:$$\begin{aligned} \mathrm{L_{q}^0}=\ell _1^{[q]}\mathrm{P}_{\!1}+\ell _2^{[q]}\mathrm{P}_{\!2}+\ell _3^{[q]}\mathrm{P}_{\!3}, \end{aligned}$$(14)These projection operators satisfy \(\mathrm{P}_{\!i}\mathrm{P}_{\!j}=\delta _{ij}\mathrm{P}_{\!i}\) (no sum in i) and \(\sum _i\mathrm{P}_{\!i}=\mathbf {1}\).$$\begin{aligned} P_1 = \left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) , \ \ P_2 = \left( \begin{array}{ccc} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) , \ \ P_3 = \left( \begin{array}{ccc} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1 \end{array} \right) . \end{aligned}$$(15)
 (b)The Yukawa coupling matrices are instead required to obey Right conditionswith$$\begin{aligned} N_d^{0}=M_d^0\,\mathrm{R_{d}^0},\quad N_u^{0}=M_u^0\,\mathrm{R_{u}^0}, \end{aligned}$$(16)where \(r_j^{[q]}\) are, againa priori, arbitrary numbers and, as in Eq. (14), \(\mathrm{P}_{\!i}\) are the projection operators in Eq. (15).$$\begin{aligned} \mathrm{R_{q}^0}=r_1^{[q]}\mathrm{P}_{\!1}+r_2^{[q]}\mathrm{P}_{\!2}+r_3^{[q]}\mathrm{P}_{\!3}, \end{aligned}$$(17)
All the resulting models, that is all 2HDMs obeying Eq. (12) and either Left or Right conditions are analysed in Sects. 3 and 4, respectively.

in the models of Sect. 3, obtained by imposing the Left conditions in Eq. (13), each lefthanded doublet \(Q_{Li}^0\) couples exclusively, i.e. to one and only one, of the scalar doublets \(\varPhi _{k}\),

in the models of Sect. 4, obtained by imposing the Right conditions in Eq. (16), each righthanded singlet \(d_{Ri}^0\), \(u_{Rj}^0\), couples exclusively to one scalar doublet \(\varPhi _{k}\).
3 Symmetry controlled models with “Left” Conditions
We present in this section the different models arising from an Abelian symmetry and for which there are matrices \(\mathrm{L_{d}^0}\) and \(\mathrm{L_{u}^0}\) such that Eq. (13) is verified. To this end, we have constructed a program which produces all models satisfying the Abelian symmetries in Eq. (12), and which lead to nonvanishing quark masses and a CKM matrix which is not block diagonal, thus verifying the results in ref. [15].^{3} For each Abelian model, the program then checks if it satisfies in addition Eq. (13). Thus, our final list will be complete. Before addressing the models themselves, it is convenient to make some observations on the effect of rotating into mass bases of the up and down quarks.
3.1 Conditions in the mass basis
3.2 How to determine \(\ell _i\)
Here we show how one determines the coefficients \(\ell _i\) (\(i=1,2,3\)) just by examining the form of the Yukawa matrices \(\varGamma _{1}\) and \(\varGamma _{2}\). For definiteness, we concentrate on the down sector. The reasoning for the up sector follows similar lines and yields the same conclusions.
3.3 Left models
Omitting the trivial cases of type I or type II 2HDMs, for which the transformation properties in Eq. (12) have no flavour dependence (both \(\mathrm{L_{d}}\) and \(\mathrm{L_{u}}\) are in that case proportional to the identity matrix \(\mathbf {1}\)), we now address the different possible models which obey Left conditions.
3.3.1 BGL models
Parametrisation
3.3.2 Generalised BGL: gBGL
3.3.3 jBGL
Properties  

Model  Sym.  Tree FCNC  Parameters  Transformation 
G–W  \(\mathbb {Z}_{2}\)  \(\left( N_u\right) _{ij}\propto m_{u_i} \delta _{ij}\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto \varPhi _{2}\) 
\(\left( N_d\right) _{ij}\propto m_{d_i} \delta _{ij}\)  
Type A  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i} (t_\beta \delta _{ij} (t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[u]}i}^{*}{\hat{r}}_{\mathrm{[u]}j}^{\phantom {*}})\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}= m_{d_i} t_\beta \delta _{ij}\)  \({\hat{r}}_{\mathrm{[u]}}^{\phantom {*}}\)(+4)  \(u_{R3}^0\mapsto e^{i\theta }u_{R3}^0\)  
Type B  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i} (t_\beta ^{1}\delta _{ij}+(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[u]}i}^{*}{\hat{r}}_{\mathrm{[u]}j}^{\phantom {*}})\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}= m_{d_i} t_\beta \delta _{ij}\)  \({\hat{r}}_{\mathrm{[u]}}^{\phantom {*}}\)(+4)  \(u_{R1}^0\mapsto e^{i\theta }u_{R1}^0,u_{R2}^0\mapsto e^{i\theta }u_{R2}^0\)  
Type C  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i} t_\beta \delta _{ij}\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}= m_{d_i} (t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[d]}i}^{*}{\hat{r}}_{\mathrm{[d]}j}^{\phantom {*}})\)  \({\hat{r}}_{\mathrm{[d]}}^{\phantom {*}}\)(+4)  \(d_{R3}^0\mapsto e^{i\theta }d_{R3}^0\)  
Type D  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i} t_\beta \delta _{ij}\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}= m_{d_i} (t_\beta ^{1}\delta _{ij}+(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[d]}i}^{*}{\hat{r}}_{\mathrm{[d]}j}^{\phantom {*}})\)  \({\hat{r}}_{\mathrm{[d]}}^{\phantom {*}}\)(+4)  \(d_{R1}^0\mapsto e^{i\theta }d_{R1}^0, d_{R2}^0\mapsto e^{i\theta }d_{R2}^0\)  
Type E  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i} (t_\beta \delta _{ij} (t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[u]}i}^{*}{\hat{r}}_{\mathrm{[u]}j}^{\phantom {*}})\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}= m_{d_i} (t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[d]}i}^{*}{\hat{r}}_{\mathrm{[d]}j}^{\phantom {*}})\)  \({\hat{r}}_{\mathrm{[u]}}^{\phantom {*}}\),\({\hat{r}}_{\mathrm{[d]}}^{\phantom {*}}\)(+8)  \(d_{R3}^0\mapsto e^{i\theta }d_{R3}^0, u_{R3}^0\mapsto e^{i\theta }u_{R3}^0\)  
Type F  \(\mathbb {Z}_{n\ge 2}\)  \(\left( N_u\right) _{ij}= m_{u_i}(t_\beta ^{1}\delta _{ij}+(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[u]}i}^{*}{\hat{r}}_{\mathrm{[u]}j}^{\phantom {*}})\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}, d_{R3}^0\mapsto e^{i\theta }d_{R3}^0\) 
\(\left( N_d\right) _{ij}= m_{d_i} (t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){\hat{r}}_{\mathrm{[d]}i}^{*}{\hat{r}}_{\mathrm{[d]}j}^{\phantom {*}})\)  \({\hat{r}}_{\mathrm{[u]}}^{\phantom {*}}\),\({\hat{r}}_{\mathrm{[d]}}^{\phantom {*}}\)(+8)  \(u_{R1}^0\mapsto e^{i\theta }u_{R1}^0, u_{R2}^0\mapsto e^{i\theta }u_{R2}^0\) 
Notice how, with respect to Eq. (38), the structures of the down Yukawa matrices \(\varGamma _{1}\) and \(\varGamma _{2}\) are interchanged (while the \(\varDelta \) matrices remain the same).
Parametrisation
One can see that BGL is not a particular case of jBGL. Also, BGL is a particular limit of gBGL, and jBGL is a sort of “Flipped” gBGL. One might wonder whether there is some sort of “Flipped” BGL, obtainable from an Abelian symmetry, which arises as a suitable limit of jBGL. It is possible to see by inspection of the symmetry transformations in Eq. (12) that such a case is not allowed.
3.4 Summary of models with Left conditions
Models obeying the Left conditions of Eq. (18). For the uBGL and dBGL models we only show the FCNC corresponding to one case, the top and bottom models, respectively. The first row shows Glashow–Weinberg models without tree level FCNC for comparison. In the “Parameters” column, (+4) is the number of new real parameters corresponding to \({\hat{n}}_{\mathrm{[q]}}\)
Properties  

Model  Sym.  Tree FCNC  Parameters  Transformation 
G–W  \(\mathbb {Z}_{2}\)  \(\left( N_u\right) _{ij}\propto \delta _{ij}m_{u_j}\)  \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto \varPhi _{2}\) 
\(\left( N_d\right) _{ij}\propto \delta _{ij}m_{d_j}\)  
uBGL (t)  \(\mathbb {Z}_{n\ge 4}\)  \(\left( N_u\right) _{ij}=\delta _{ij}(t_\beta (t_\beta +t_\beta ^{1})\delta _{j3})m_{u_j}\)  \(V\), \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}=(t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){V_{ti}^*}{V_{tj}^{\phantom {*}}})m_{d_j}\)  \(Q^0_{L3}\mapsto e^{i\theta }Q^0_{L3},u_{R3}^0\mapsto e^{i2\theta }u_{R3}^0\)  
dBGL (b)  \(\mathbb {Z}_{n \ge 4}\)  \(\left( N_u\right) _{ij}=(t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){V_{ib}^{\phantom {*}}}{V_{jb}^*})m_{u_j}\)  \(V\), \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}=\delta _{ij}(t_\beta (t_\beta +t_\beta ^{1})\delta _{j3})m_{d_j}\)  \(Q^0_{L3}\mapsto e^{i\theta }Q^0_{L3}, d_{R3}^0\mapsto e^{i2\theta }d_{R3}^0\)  
gBGL  \(\mathbb {Z}_{2}\)  \(\left( N_u\right) _{ij}=(t_\beta \delta _{ij} (t_\beta +t_\beta ^{1}){\hat{n}}_{\mathrm{[u]}i}^{*}{\hat{n}}_{\mathrm{[u]}j}^{\phantom {*}})m_{u_j}\)  \(V\), \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto \varPhi _{2}\) 
\(\left( N_d\right) _{ij}=(t_\beta \delta _{ij}(t_\beta +t_\beta ^{1}){\hat{n}}_{\mathrm{[d]}i}^{*}{\hat{n}}_{\mathrm{[d]}j}^{\phantom {*}})m_{d_j}\)  \({\hat{n}}_{\mathrm{[q]}}^{\phantom {*}}\)(+4)  \(Q^0_{L3}\mapsto Q^0_{L3}\)  
jBGL  \(\mathbb {Z}_{n \ge 2}\)  \(\left( N_u\right) _{ij}=(t_\beta \delta _{ij} (t_\beta +t_\beta ^{1}){\hat{n}}_{\mathrm{[u]}i}^{*}{\hat{n}}_{\mathrm{[u]}j}^{\phantom {*}})m_{u_j}\)  \(V\), \(t_\beta \), \(m_{q_k}\)  \(\varPhi _{2}\mapsto e^{i\theta }\varPhi _{2}\) 
\(\left( N_d\right) _{ij}=(t_\beta ^{1}\delta _{ij}+(t_\beta +t_\beta ^{1}){\hat{n}}_{\mathrm{[d]}i}^{*}{\hat{n}}_{\mathrm{[d]}j}^{\phantom {*}})m_{d_j}\)  \({\hat{n}}_{\mathrm{[q]}}^{\phantom {*}}\)(+4)]  \(Q^0_{L3}\mapsto e^{i\theta }Q^0_{L3},\quad d^0_{Rj}\mapsto e^{i\theta }d^0_{Rj}\) 
4 Symmetry controlled models with right conditions
In the previous section we have explored 2HDM whose symmetry under the Abelian transformations in Eq. (12) is supplemented by the requirement that the \(M_{q}^0\) and \(N_{q}^0\) obey the relations in Eq. (13), where \(\mathrm{L_{q}^0}\) in Eq. (14) acts on the left. In this section we analyse symmetry based models where we impose the conditions of Eq. (16), \(N_{q}^0=M_{q}^0\,\mathrm{R_{q}^0}\), where \(\mathrm{R_{q}^0}\) in Eq. (17) acts on the right, that is, models which obey Right conditions.
4.1 Conditions in the mass basis
4.2 How to determine \(r_i\)
4.3 Right models
It is obvious that cases in which both \(\mathrm{R_{d}}\) and \(\mathrm{R_{u}}\) are proportional to the identity matrix have been discarded automatically by the discussion of models with Left conditions. But, for Right conditions it is still possible to have either \(\mathrm{R_{d}}\propto \mathbf {1}\) or \(\mathrm{R_{u}}\propto \mathbf {1}\) (but not both). Among the six different types of models which obey Right conditions, the first four have that property.
4.3.1 Type A
4.3.2 Type B
4.3.3 Type C
4.3.4 Type D
4.3.5 Type E
4.3.6 Type F
4.4 Summary of models with Right conditions
We summarize in Table 1 the main properties of the different models discussed in the previous sections, which obey Right conditions.
5 Phenomenology
In the previous sections we have presented different classes of models which include controlled treelevel FCNC; different cases within the same class share the same number of parameters, and this number varies among different classes. This section is devoted to a discussion of aspects related to the phenomenology of the different models.

deviations from SM expectations in the flavour conserving processes involving the 125 GeV Higgslike scalar,

possible sizable FCNC processes involving the 125 GeV Higgslike scalar,

proposed searches for new fundamental scalars.
 1.
how can one fix or extract parameters of a given model?
 2.
how can one tell apart different models?

uBGL, dBGL and types A,B,C and D only have tree level FCNC in one quark sector, up or down, not both. Furthermore, uBGL and dBGL are fixed in terms of the CKM matrix, while the couplings \({\hat{r}}_{\mathrm{[q]}j}^{\phantom {*}}\) in A, B, C and D are free parameters.

gBGL, jBGL and types E and F have tree level FCNC in both sectors. However, in gBGL and jBGL, the parameters controlling them, \({\hat{n}}_{\mathrm{[u]}j}^{\phantom {*}}\) and \({\hat{n}}_{\mathrm{[d]}j}^{\phantom {*}}\), are not independent, they are related through CKM, Eq. (44), while that is not the case in models E and F where \({\hat{r}}_{\mathrm{[u]}j}^{\phantom {*}}\) and \({\hat{r}}_{\mathrm{[d]}j}^{\phantom {*}}\) are independent.
6 Conclusions
The recent discovery of a scalar particle prompted the search for more scalars and respurring the study of models with two Higgs doublets. A general two Higgs doublet model has double the number of Yukawa couplings already present in the SM. It would seem that this would lead us even farther away from an understanding of the flavour sector. Moreover, 2HDM typically lead to FCNC, which are tightly constrained by experiment. In this article we entertain the possibility that these two issues are solved in a natural way by the presence of Abelian symmetries. We are inspired by the BGL [8] models, where FCNC are entirely determined by the CKM matrix elements, and by gBGL [14] models, which have a larger parametric freedom.
We show that such models can be obtained by enhancing an Abelian symmetry with the Left condition in Eq. (13). Since Ferreira and Silva [15] had listed all 2HDM models constrained by an Abelian symmetry and consistent with nonzero quark masses and a nondiagonal CKM, we could perform an exhaustive search for all such models, and show there is one, and only one, further class of models obeying the Left condition, which we dubbed jBGL.
We have developed a similar Right condition (16) and again performed an exhaustive search over the set of models with an Abelian symmetry. We identified six new classes of models, named Types A through F. For all cases, the FCNC matrices \(N_d\) and \(N_u\) have been written in terms of masses, \(\tan {\beta }\), CKM entries, and vectors containing all the remaining parametric freedom. All FCNC couplings have the generic form in Eq. (90). Finally, we discussed how one could in principle tell these models apart, by concentrating on the use of charged Higgs decays to disentangle gBGL from jBGL models
Footnotes
 1.
Notice however that the bidiagonalisation of the mass matrices still leaves the freedom to rephase individual quark fields. Together with the CKM matrix, the \(N_d\), \(N_u\) matrices should enter physical observables in rephasing invariant combinations [19].
 2.
The global continuous symmetry transformation in Eq. (12) would give massless NambuGoldstone bosons upon electroweak symmetry breaking: they can be avoided by adding soft breaking terms in the scalar potential.
 3.
In fact, there is a misprint in Eq. (89) of the published version of [15], which however is correct in the arxiv version.
 4.
This is consistent with the fact that BGL can be recovered as a particular limit of generalised BGL models.
 5.
Interpreting the situation the other way around, Eq. (83) would provide a window of sensitivity to the righthanded analog of CKM (for example, in extensions to models with a gauged \(SU(2)_L \otimes SU(2)_R\) symmetry).
 6.
It goes without saying that the more experimental signals are available, the better the identification can be established. On that respect, available information on Higgs production \(\times \) decay signal strenghts can provide some sensitivity to the diagonal couplings in Eq. (89). However, that avenue is not as distinctive as tree level FCNC.
 7.
The sign is irrelevant when considering exclusively off diagonal couplings with neutral scalars.
 8.
Naturally, a tree level calculation would yield \(\varGamma (H^+ \rightarrow \bar{u}_\alpha d_k) = \varGamma (H^ \rightarrow u_\alpha \bar{d}_k)\) even for complex parameters, consistent with the absense of direct CP violation in the presence of a single diagram. Including some loop diagram(s), with different strong and weak phases, would yield a CP violating asymmetry.
 9.
Due to Eq. (103), the change \(1 \leftrightarrow 2\) implies a change \(t_\beta \leftrightarrow  t_\beta ^{1}\) in the parametrization of the \(N_q\) matrices. Thus, models which could seem to differ by such a change, do in fact correspond to the same model.
Notes
Acknowledgements
This work is partially supported by Spanish MINECO under Grant FPA201568318R, FPA201785140C33P and by the Severo Ochoa Excellence Center Project SEV20140398, by Generalitat Valenciana under Grant GVPROMETEOII 2014049 and by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CERN/FISNUC/0010/2015 and CFTPFCT Unit 777 (UID/FIS/00777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. GCB was supported in part by the National Science Centre, Poland, the HARMONIA project under contract UMO2015/18/M/ST00518 (2016–2019). MN acknowledges support from FCT through postdoctoral Grant SFRH/BPD/112999/2015.
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