What if the Masses of the First Two Quark Families are not Generated by the Standard Higgs?

We point out that, in the context of the SM, $|V^2_{13}| + | V^2_{23}|$ is expected to be large, of order one. The fact that $|V^2_{13}| + |V^2_{23}| \approx 1.6 \times 10^{-3}$ motivates the introduction of a symmetry S which leads to $V_{CKM} ={1\>\!\!\!\mathrm{I}} $, with only the third generation of quarks acquiring mass. We consider two scenarios for generating the mass of the first two quark generations and full quark mixing. One consists of the introduction of a second Higgs doublet which is neutral under S. The second scenario consists of assuming New Physics at a high energy scale , contributing to the masses of light quark generations, in an effective field theory approach. This last scenario leads to couplings of the Higgs particle to $s\overline s$ and $c \overline c$ which are significantly enhanced with respect to those of the SM. In both schemes, one has scalar-mediated flavour- changing neutral currents which are naturally suppressed. Flavour violating top decays are predicted in the second scenario at the level $ \mbox{Br} (t \rightarrow h c ) \geq 5\times 10^{-5}$.

The recent discovery of the Higgs particle at LHC, rendered even more urgent to understand the mechanism responsible for the generation of fermion masses and mixing. In the framework of the Standard Model (SM), fermion masses arise exclusively through Yukawa interactions and the Brout-Englert-Higgs mechanism is responsible for both gauge symmetry breaking and the generation of fermion masses. Some of the outstanding questions one may ask, include : i) Two of the salient flavour features in the quark sector are the strong hierarchy of quark masses and the fact that the V CKM matrix is close to the identity. In the framework of the SM, can one conclude that these two features are related in some way? How can one understand small quark mixing in the SM ?
ii) In the SM, all fermion masses are generated through the vacuum expectation value (vev) of the Standard Higgs. Alternatively, one may consider a scenario where the Standard Higgs only gives mass to the third generation, while the masses of the two first generations originate from another source. A crucial question is : how can this alternative scenario be tested at LHC and future accelerators?
In this paper, we address the above two questions. With respect to (i), we show that actually in the SM the "natural" value of (|V 2 13 | + |V 2 23 |) is large, of order one. In order to address this question, we study in detail quark mixing in the extreme chiral (EC) limit, where only the third generation of quarks acquires mass, while m d , m s , m u , m c remain massless. We do the analysis in the context of the Standard Model (SM) and some of its extensions. We will show that in the SM in the EC limit the generic situation is having non-trivial mixing parametrised by an angle with a free value, not fixed in the SM context. Without loss of generality, one can identify this angle with the V 23 entry. Therefore, the fact that experimentally |V 23 | = 4.09×10 −2 , is entirely unnatural within the framework of the SM. In fact, the smallness of |V 23 | may be interpreted as a hint from experiment, indicating that one should find a symmetry or a principle which may account for the smallness of |V 23 |.
In our analysis, we start with the most general rank one matrices M u , M d , taking into account that in the SM the flavour structure of the Yukawa couplings generating the up and down quark mass matrices are entirely independent. The appearance of a non-trivial mixing even in the EC limit case, corresponds to a misalignment of the two mass matrices M u , M d , in flavour space. We define a dimensionless weak basis (WB) invariant denoted A which provides a measure of this misalignment. In the EC limit, this invariant A varies from 0 to 1 , with 0 corresponding to exact alignment and 1 to total misalignment.
With respect to question (ii) we consider the possibility that in leading order the SM Higgs only gives mass to the third generation. This is achieved in a natural way through the introduction of a discrete symmetry S which leads to quark mass matrices of rank one, aligned in flavour space. We then conjecture that the generation of the mass of the first two generations arises from a different source. If this new source is just another Higgs doublet and if one assumes that the new doublet is neutral with respect to the symmetry S, then one is led to a flavour structure analogous to what one encounters in a class of the BGL-type models [1], [2], which have been extensively analysed in the literature [3], [4], [5] [6], [7], [8], [9]. If, on the other hand, the new contribution arises in the framework of an effective field theory where the New Physics (NP) particles have been integrated out, then assuming that this NP contribution is of order m s and m c in the down and up sectors, one can estimate the couplings of the Standard Higgs to tt, bb, cc, ss. It turns out that the couplings to tt, bb do not differ much from those in the SM, but the couplings to cc, ss are significantly enhanced with respect to those in the SM.
Mixing in the EC limit: We analyse quark mixing in the EC limit, where the quark mass matrices M d and M u are rank one matrices generated by two independent Yukawa coupling matrices Y d , Y u . Therefore, M d , M u can be written: One does not loose generality by considering the specific ordering of m b , m t in Eq. (1), since a permutation changing these positions can always be included in the unitary matrices U d,u L,R . The quark mixing matrix appearing in the charged weak interactions is given by V 0 = U u L † U d L and it is at this stage an arbitrary mixing matrix. Taking into account that in the EC limit the first two generations are massless, one can make an arbitrary redefinition of the light quark masses through a unitary transformation of the type: where X u,d are 2 × 2 unitary matrices. Under this transformation V 0 transforms as One has the freedom to choose X u,d at will to diagonalize the 2 × 2 upper left sector of V leading to |V 12 | = |V 21 | = 0. Unitarity of V leads then to the constraint V * 13 V 23 = 0. One can then choose, without loss of generality, V 13 = 0 and V CKM becomes then an orthogonal matrix, with mixing only between the second and third generation, characterised by an angle α, with |V 23 | = |V 32 | = | sin α|. The important point that we wish to emphasise is that this mixing in the EC limit of the SM, is arbitrary. The smallness of |V 13 | 2 + |V 23 | 2 in the SM, in general, cannot be related to the smallness of the mass ratios m 2 i /m 2 3 where i = 1, 2. Therefore, in the framework of the SM the observed smallness of |V 23 | ≈ 10 −2 , provides a hint for the presence of a flavour symmetry.
An Invariant Measure of Alignment: Experimentally one encounters in the quark sector V CKM ≈ 1I which corresponds to an alignment of the quark mass matrices in flavour space. It is useful to have an invariant measure of the mixing defined in terms of the mass matrices when written in an arbitrary weak basis. This can be done by defining the following weak basis invariant [10]: where the build blocks are the two matrices: By construction, one has trh d = trh u = 1. Given the two rank one matrices M d,u , described before, corresponding to the EC limit one obtains: The result of Eq. (5) is exact in the EC limit. The invariant A still gives a measure of the size of mixing when the first two generations acquire mass, and in this case we Obtaining Small Mixing Through a Symmetry: As stated before, mixing in the EC limit is parametrised by an arbitrary mixing angle involving two generations. In general, in the SM there is no reason to assume that this mixing angle is either close to zero or maximal, in fact it can take any value. It is possible to introduce a symmetry which leads to the vanishing of this mixing. Without loss of generality, this angle can parametrise mixing between the second and the third generations. Let us consider the following symmetry S, in the context of the particle content of the SM, with only one Higgs doublet.
where Q 0 Lj is a left-handed quark doublet and Φ is the Higgs doublet. All other fermions transform trivially under S. This symmetry leads to the following pattern of texture zeros for the Yukawa couplings: which clearly lead to V CKM equal to the identity. The matrices of Eq. (7) are written in the WB chosen by the symmetry. The matrix Y d can be written in the same form as Y u by means of a rotation of the right-handed down quarks, which simply corresponds to a different choice of WB. Next we present possible ways of extending this scenario in order to generate the masses of the first two generations of quarks, as required experimentally, without generating large mixing and keeping the Higgs mediated flavour changing neutral currents (HFCNC) under control.
Generating the Masses of the First Two Quark Generations: At this stage, one has to address the question of the origin of the masses of the first two generations. The discovery of the Higgs particle at LHC and the study of its production and decay has shown that the vev of the Higgs field gives the dominant contribution to the masses of the fermions of the third generation, namely to the top and bottom quarks, as well as the τ -lepton. It is conceivable that the masses of the quarks of the first two generations arise from a different source, [11], [12], [13], [14], [15], [16], [17], so that the quak mass matrices have the form: where M (0) is generated by the vev of the standard Higgs φ and M (1) may arise from the vev of a second Higgs φ or from other unspecified source. In either case, the fact that there are two different sources giving contributions to the mass of quarks of a given charge, leads to scalar mediated flavour-changing neutral currents (FCNC). These currents are naturally suppressed in both of the scenarios we consider below, once the experimental values of the V CKM entries are taken into account.
Adding a Second Higgs Doublet: The simplest possibility to generate masses for the first two generations, is through the addition of a second doublet φ which is neutral under S. In this case, the contribution of φ to the quark mass matrix is of the form: This structure coincides with what one encounters in a class of BGL models [1]. It has been shown that in this model, the full flavour structure only depends on V CKM and thus the model obeys the Minimal Flavour Violation [18], [19], [2] principle. In this model there are FCNC but they are naturally suppressed by small V CKM elements. In this context, there are two types of BGL models: (1) top models described by Eqs. (6), (7) and (9) with FCNC only in the down sector and (2) bottom models with the rôle of up and down quarks interchanged. This second class of models give rise to FCNC's only in the up sector. From low energy flavour data the scale of new physics in top models can be quite light at a few hundred GeV [6]. Bottom like models introduce new scales close to the TeV region [6]. Flavour conserving or flavour blind Higgs observables can be accommodated in both categories because the new couplings, compared to the SM couplings, i.e., the coupling modifiers κ Z , κ W , κ t , κ τ , κ b , κ g and κ γ may deviate from 1 at the percent level. These models have been extensively studied in the literature [5] [6], [7], [8], [9]. In the Higgs sector the most relevant prediction specific to top models is the decay h → bs + sb with branching ratios at most between 10 −3 and 10 −2 [8] The bottom models predict the rare top decay t → hc with a branching ratio of at most 10 −3 [8] In both classes of models these predictions can be correlated with h → µτ + τμ occuring at a branching ratio which can reach at most 10 −2 Generating light quark masses from New Physics at a high energy scale: Here, we consider that only one Higgs doublet is introduced in the framework of the SM and introduce the symmetry S of Eq. (6) which implies that only the third generation of quarks acquire mass, with V CKM = 1I. We shall consider that the quark masses of the first two generations arise from New Physics contributing to the Yukawa couplings in leading order through an effective six-order operator of the form: The Yukawa and quark mass matrices have then the form [13]: The fact that Y d,u are not proportional to M d,u leads to Higgs mediated Flavour-Changing-Neutral -Currents (FCNC). At this stage, it is useful to estimate the size of the new mass scale Λ. From Eq. (11) and taking into account that v = 174 GeV, Note that in the present framework one obtains |V 23 | ≈ O(m s /m b ) but one does not provide an explanation for the smallness of m b /m t . We will show that the potentially dangerous FCNC are naturally suppressed in the present framework. The down quark mass matrix is diagonalised by : where D d ≡ diag(m d , m s , m b ), with an analogous expression for the up sector. In the quark mass eigenstate basis, the Yukawa coupling matrix becomes: At this stage, it is useful to with an analogous expression for the up sector. The strength of the Higgs couplings Y m d is controlled by Eqs. (13), (14) and one has to take into account the very strict bounds on flavour violating scalar couplings, which can be derived from K 0 − K 0 , B d −B d , B s −B s , D 0 −D 0 mixings. These bounds have been recently analysed in [20]. From B s − B s mixing, one derives bounds on |(U * dL ) 32 (U dR ) 33 | and |(U * dL ) 33 (U dR ) 32 | which taking into account that |(U dL ) 33 | ≈ 1 and |(U dR ) 33 | ≈ 1, lead to |(U dL ) 32 | ≤ 1.4×10 −2 . Similarly, one derives from B d −B d mixing the bound |(U dL ) 31 | ≤ 3×10 −3 . It is remarkable that these bounds lead to |(U * dL ) 31 (U dR ) 32 | |(U * dL ) 32 (U dR ) 31 | ≤ 4.4× 10 −4 which guarantees that the Higgs contribution to K 0 − K 0 mixing is sufficiently suppressed, to conform to the strict experimental bound. Flavour-changing scalar couplings in the up-sector are controlled by |(U * uL ) 3i (U uR ) 3j |. On the other hand, U uL is constrained to be in a region which can generate the observed V CKM = (U † uL U dL ). Once these constraints are taken into account, one predicts, in the present framework, the strength of flavour-changing decays of the top quark, namely It is interesting to notice that in this framework the previously analysed new flavour changing Higgs contributions all arise from the third column of the matrices U dL , U dR , U uL and U uR . So far we have only discussed the off-diagonal Higgs couplings. In the diagonal couplings, one has to distinguish between the couplings of the third generation (i.e. tth and bbh) and those of the two light generations. Taking into account that |(U * dL ) 33 (U dR ) 33 | ≈ 1 and also |(U * uL ) 33 (U uR ) 33 | ≈ 1 it is clear that the couplings of the third quark generation coincide with those in the SM. On the contrary, for the first two generations, one has a significant enhancement by a factor of 3, leading, for example to: At this stage, the following comment is in order. For the down sector, the experimental constraints from meson mixing are very strict and the prediction of Eq. (16) for q = d, s, is solid. For cc although the enhancement of Eq. (16) holds for most of the allowed parameter space, there are regions of allowed parameter space, where the enhancement is not as strong. So far, we have only discussed the quark sector. Note that the observed lepton flavour mixing is large and therefore there is no motivation to opt for the symmetry S to act in the lepton sector in a way analogous to the quark sector, since this would lead to V P M N S = 1I in leading order, in contrast to experiment. We shall assume that leptons are neutral with respect to S, which leads to couplings of the Higgs particle to leptons which coincide with those of the SM. Taking into account that Γ SM (h →cc)/Γ SM (h → all) ∼ 3%, assuming Eq. (16) and that the other relevant decay channels do not change, we get Γ(h → all) ≈ 1.23Γ SM (h → all). This result gives rise to a definitive prediction for the signal strength parameters µ f [21] in the decay channels f = γγ, ZZ, W W, ττ , bb: compatible with the combined ATLAS CMS analysis [21]. Looking at the coupling modifiers analysis κ f we have as in the SM no modification of the couplings to the relevant channels κ Z = κ W = κ t = κ τ = κ b = κ g = κ γ = 1 (18) but the large enhancement in the undetected cc channel contributes to the so-called beyond the SM branching ratio BR BSM ∼ 18.8% in perfect agreement with the 34% joint upper bound from ATLAS and CMS at 95% C.L. [21].
TeV completion: A possible TeV completion of the present model, can be implemented in the framework of an extension of the SM where one adds three Q = −1/3 vector-like quarks, D α , and three Q = 2/3 vector-like quarks, U β , isosinglets of SU (2), to the spectrum of the SM.
We introduce the symmetry S considered in Eq. (6), with the standard like quarks transforming as before. The symmetry S is spontaneously broken by the vev of the Higgs field and we also allow for a soft-breaking term D Lβ d R3 with a similar softbreaking term for the up sector. The leading higher order operators are the ones in Eq. (10), which, in the down sector, are generated through the diagram of Fig. 1. It can be shown that a realistic quark mass spectrum and a correct pattern of quark mixing can be generated, although a full description goes beyond the scope of this paper [22].
In summary, the crucial points of our paper are: -Contrary to what may be a common belief, in the SM, the natural value of |V 23 | 2 + |V 13 | 2 is of order one. In the SM, without an additional symmetry, the smallness of V CKM mixings cannot be derived from the observed strong hierarchy of quark masses.
-We point out that the fact that |V 23 | 2 +|V 13 | 2 is small may be considered as a hint of Nature suggesting the introduction of a symmetry S. We have given an example of such a symmetry, which leads to V CKM = 1I with only the third quark generation acquiring mass.
-We have suggested two different scenarios to generate the masses of the two lighter quark generations. One of them, consists of the introduction of a second Higgs doublet, which is neutral under S. This framework leads to a BGL type-model [1], [2] which have been analysed in the literature. Another scenario consists of assuming that New Physics at a high energy scale, contributes to the light quark masses in an effective field theory approach. This scenario leads to the following striking predictions which can be tested at LHC-run 2, as well as in other future accelerators: The diagonal Higgs quark couplings of the 3rd generation, i.e. tth and bbh, essentially coincide with those of the SM. The diagonal Higgs couplings of the lighter quarks are enhanced with respect to those of the SM, by about a factor of three, with the most significant effect of this enhancement given by Eq. (16). In this framework one predicts Higgs mediated flavour violating top decays, as indicated in Eq. (15)