Gauged Flavor Group with Left-Right Symmetry
We construct an anomaly-free extension of the left-right symmetric model, where the maximal flavor group is gauged and anomaly cancellation is guaranteed by adding new vectorlike fermion states. We address the question of the lowest allowed flavor symmetry scale consistent with data. Because of the mechanism recently pointed out by Grinstein et al. tree-level flavor changing neutral currents turn out to play a very weak constraining role. The same occurs, in our model, for electroweak precision observables. The main constraint turns out to come from -mediated flavor changing neutral current box diagrams, primarily mixing. In the case where discrete parity symmetry is present at the TeV scale, this constraint implies lower bounds on the mass of vectorlike fermions and flavor bosons of 5 and 10 TeV respectively. However, these limits are weakened under the condition that only is restored at the TeV scale, but not parity. For example, assuming the gauge couplings in the ratio allows the above limits to go down by half for both vectorlike fermions and flavor bosons. Our model provides a framework for accommodating neutrino masses and, in the parity symmetric case, provides a solution to the strong CP problem. The bound on the lepton flavor gauging scale is somewhat stronger, because of Big Bang Nucleosynthesis constraints. We argue, however, that the applicability of these constraints depends on the mechanism at work for the generation of neutrino masses.
One of the long-standing mysteries of physics beyond the Standard Model is the origin of flavor patterns for quarks and leptons. In the Standard Model (SM), they arise from the quark and lepton Yukawa couplings with the SM Higgs boson and are arbitrary, thereby precluding any physical insight as to their origin. Since these flavor patterns may well be the remnants of the breaking of some symmetry, the belief is that pinning down the flavor symmetry at work could provide hints of the underlying dynamics at work. Many possibilities for approaching this important issue from the vantage point of symmetry then present themselves – starting from discrete non-abelian subgroups of these flavor symmetries to continuous global or local ones. The question then arises as to how we determine by low energy observations which particular mechanism is at work and at which scale such a symmetry manifests itself. The hope is that different choices will lead to different characteristic predictions, e.g. a global horizontal symmetry would lead to massless familons at low energies  and discrete symmetries could lead to some relations between observables.
A widely discussed possibility is to study gauged flavor symmetries , which leads to a number of interesting effects such as new gauge bosons and new flavor changing effects mediated by these bosons. The very first test of this possibility is to determine the scale of gauged flavor symmetry. Naïve considerations seem to suggest that this scale is likely to be in the 1000 TeV range; however in specific models this expectation could change drastically. For example, in a recent paper by Grinstein, Redi and Villadoro (GRV) , it has been shown by explicit construction that there are SM extensions with gauged flavor symmetry where this scale could be in the 1 TeV range or even below, compatible with constraints from the hadronic sector. Furthermore this model predicts, by the requirement of anomaly cancellation, new vectorlike quarks, the lightest of which with masses again in the TeV ballpark, hence within the reach of LHC direct searches. The mechanism at work in the GRV model is an inverse, see-saw like, relation between the masses of the quarks and those of the new fermion states  as well as of the flavor gauge bosons, so that the partners of the heaviest quarks are the lightest among the new states. This fact allows to pass all the flavor changing neutral current (FCNC) constraints in a very natural way.
A certain degree of model dependence in the idea of gauging the flavor symmetry is the choice of new fermionic states added in order to cancel the gauge anomalies. To achieve this, the simplest option is to include new, vectorlike quark partners. This choice may however appear to be at odds with that of quark states that need sit in chiral representations. This simple consideration motivates us to pursue here an alternative possibility, where the maximal gauging of flavor symmetry is carried out within a left-right symmetric extension of the SM . This implies that the flavor symmetry group is itself left-right symmetric. The new gauge anomalies resulting from the larger gauge group cancel with the introduction of vectorlike new fermionic partners of the quarks . Besides the known virtues inherent in left-right symmetric extensions of the SM, e.g. the possibility to justify the hypercharge quantum numbers, there appear to be the following advantages in our approach: (i) it provides a natural way to include neutrino masses; (ii) in the quark Lagrangian, it features only three free parameters in the gauge and Yukawa sector, after the rest are fixed by data on quark masses and mixings; (iii) the model provides a simple solution to the strong CP problem without the need for an axion, in a manner similar to that discussed in Refs. [6, 7].
There are two possible realizations of this idea while keeping the gauge group to be at the TeV scale: (a) the discrete parity symmetry is maintained down to the TeV scale or (b) it is broken at some very high scale  so that, at the TeV scale, the two gauge couplings as well as the left and right Yukawa couplings are in general different from each other. We will consider both alternatives below. Within the second alternative, the solution to the strong CP problem mentioned above is not obvious.
Concerning the constraints on the model outlined above, we find that, similarly as in the GRV model, tree-level FCNCs mediated by the flavor gauge bosons are tamed automatically by the hierarchy of their masses. Also, in our case, electroweak (EW) precision tests are automatically fulfilled in the bulk of the parameter space.
The strongest constraint comes from -mediated FCNC box diagrams, primarily mixing. The implied bounds on the scale of the new vectorlike quarks as well as on the flavor gauge bosons, which will be discussed in detail in secs. 3.2, 3.3 and 3.4, depend however on the scale at which parity is broken.
Compatibly with these bounds, the new effects, accessible at the current generation of TeV hadron colliders, include the lightest among the new particles’ masses and various deviations in top-physics observables, like top production and decays. Such deviations are due to the fact that the top (in particular, the right-handed one) mixes non-negligibly onto its new fermionic partner, whereas mixing is tiny to absent for the rest of the quark states. This in turn explains why no deviations are to be expected in the production or decays of any other quark than .
For an overview of the organization of this paper we refer the reader to the table of content on page 1.
2 Flavor Symmetry within Left-Right Models
In the SM, once the Yukawa couplings are set to zero, the maximal flavor symmetry group is . If the weak gauge group is extended to that of the left-right symmetric model, the flavor group becomes which is more economical and, unlike the SM, also simultaneously explains neutrino masses.111For other horizontal symmetry extensions of left-right models, see . Furthermore, gauged flavor symmetries have also been discussed in the context of the Pati-Salam GUT in ref. , see appendix therein.
We will therefore start with the gauge group , where represents the flavor gauge symmetries respectively in the left- and right-handed quark sector, and the corresponding ones for the lepton sector. The particle content and its transformation properties under fundamental representations of the group are reported in table 1.
One can clearly note the one-to-one correspondence between the quark and the lepton multiplets, differing only in the behavior under . It is easy to verify that this field content makes completely anomaly-free, separately in the quark and lepton sectors.
We next discuss the quark Yukawa couplings. We will ignore for the moment the leptonic ones since they do not have any effect on the final results for the quark sector. (The leptonic flavor symmetries are discussed in sec. 4.) In writing the quark Lagrangian at the TeV scale, we will generally assume that the gauge symmetry is restored at that scale. Even under this assumption, one has still to specify where the parity symmetry is broken. As anticipated in the Introduction, one can either suppose that parity is restored at the TeV scale, or else that its restoration takes place at some much higher scale . Let us first focus on the former case, namely of TeV-scale parity. In this case, the Lagrangian for the quark sector reads
where we have written explicitly only the Yukawa interactions. We note at this point that, since under parity and (and similarly for ), parity symmetry requires and the as well as couplings to be real.222In particular, concerning , one can note that there is one single such coupling for either of the up-type or down-type quark interactions with heavy fermions. Hence, one can remove possible phases in by absorbing them in the and fields, respectively. Parity will thus be broken only by the different vevs of (the tilde on these fields in eq. (1) indicates for both L and R).
In the case where parity is broken at a scale much higher than the TeV , the interactions in eq. (1), obtained from each other by the parity operation defined in the previous paragraph, have in principle different couplings. For example
because of different RGE running beneath the scale . Hence, similarly as in eq. (2), in the case of no TeV-scale parity we will distinguish the left and right instances of each gauge, and couplings by an L or R subscript.
Concerning the breaking of the gauge groups, the flavor gauge group is broken spontaneously by the vevs of and while the group by the vevs of the Higgs doublets, , as already mentioned. In particular, we adopt the following vev normalization
while diagonal vevs will be denoted henceforth as .
From eq. (1) one can read off the up-type fermion mass Lagrangian to be , with , each of the and fields carrying a generation index. The mass matrix reads
For the time being, we assume the parameters and to be much smaller than any of the . (With the subscript in we shall henceforth label the diagonal entries of the flavon vev matrices.) Then, to leading order in an expansion in the parameters , and analogous for the down sector, the above mass matrices assume the following diagonal form
From eq. (5) it is evident that off-diagonalities in the light-quark Yukawa couplings are inherited from off-diagonalities in the flavon vevs . We note first that, even below the scale, it is always possible to have one of the flavon vevs in diagonal form through an appropriate redefinition of the and basis (see eq. (1)). We choose to be that particular flavon multiplet. This amounts to three parameters, fixed by the down-type quark masses. will then be chosen to have a vev pattern of the form
with unitary. Note that, as already mentioned, in eq. (6) follows from the vev pattern being hermitian and hence parity symmetric. This amounts to six real parameters and three phases. Two of these phases can be absorbed as relative phases of two up-type quark fields relative to the third one. This gives the six real parameters and one phase to fit up-type quark masses and the CKM matrix.333We also note incidentally that, from the point of view of our discussion, the couplings can in principle be absorbed into the definition of the vevs respectively. This effectively leaves as free parameters only , besides the scale of the vev and the coupling , making the model very economical.
In the basis of eq. (6), and again in the temporary approximation , the Yukawa couplings of the SM (defined from the interactions and ) read
One can now rotate the fields as
In the hatted basis, is diagonal and is moved to the interaction. Therefore, can be interpreted as the CKM matrix, . As already stated, in our case of left-right symmetry we have strictly at the scale . However, since we are interested only in values not very far from the electroweak symmetry breaking scale , the radiative corrections to the above relation between left and right CKM’s are expected to be small. Therefore, we will henceforth generally identify
with caveats to be commented upon more below in the analysis. Since quark masses are given by , we can draw some conclusions from the approximate relations (7):
For a given value of and of the couplings, eqs. (7) or the corresponding exact expressions in sec. 3.4 allow to univocally fix the entries. Since the vevs set also the mass scale for the flavor gauge bosons (see below in this section for details), the inverted hierarchy mentioned in item (i) implies a similar hierarchy in new flavor changing neutral current effects: the lighter the generations, the more suppressed the effects . This is arguably one of the most attractive features of the model. We will return to this quantitatively in sec. 3.1.
In the exact parity case, the mass matrices , see eq. (4), lead to arg det=0, implying that the strong CP parameter at the tree level vanishes. The one loop calculation for a more general case of this type was carried out in Ref. . Using this result, we conclude that the model solves the strong CP problem without the axion.
Since the condition may in general not hold for all flavors, we need to give for the quark masses a more exact relation than eq. (5). In fact, the fermion mixing matrices (4) can be diagonalized exactly. This will be discussed in sec. 3.4, along with its phenomenological consequences.
Flavor gauge boson masses
The masses of the gauge bosons are obtained from the kinetic terms of and in the Lagrangian, , where the covariant derivatives are
The relevant mass terms read
where is a vector containing the 16 fields in . The are rotated by an orthogonal matrix such that , where are mass eigenstates.
One interesting point to note is that all the flavor gauge boson masses are determined by basically only one vev, namely the largest of the vevs . In fact, for the lightest among the , is multiplied by two powers of the Cabibbo angle (in the limit , one gets at least one massless ) and is larger than the second-largest vev contribution, .
In the subsections of this section we will discuss the various observables that are expected to provide a constraint (or else the possibility of a signal) for the model. Since in some cases – starting from the model spectrum – the model predictions vary in a wide range, we found it useful to explore these predictions with a flat scan of the model parameters. In the case where parity is assumed to be restored at the TeV scale, ranges have been chosen as follows:
and such that TeV (the bound on from applicable constraints is taken into account afterwards).
, , see discussion below eq. (20).
Setting the couplings , one effectively absorbs them into the definition of the vevs, respectively (the relevant combination entering fermion mixing is , see eqs. (4)). This assumption is however restrictive for the mass spectrum of the flavor gauge bosons, that depends on , but not on , see eq. (11). Therefore, we have also scanned .
Finally, we have taken for the couplings.
In the other scenario where parity is not a good symmetry at the TeV scale, all the left vs. right couplings can be chosen as different from each other. Concerning the couplings, in  examples have been given of scenarios where for a UV complete theory which conserves parity. Here we therefore limit ourselves to the reference choice , Concerning the other parameters:
With regards to item 2, we have scanned and the rest of the parameters in .
We have further enforced that the left vs. right instances of each coupling do not differ from each other by more than a factor of 5.
Two concluding comments concern the reality of the couplings and the hermiticity of the Yukawa vevs, implicitly assumed in the above items. From the discussion below eq. (1), one can argue that, in the case where parity is not a good TeV-scale symmetry, non-negligible complex phases may be present in (some of) the couplings. This may in turn have an impact on CP violating observables, which are, however, not the main concern in this paper, for the reasons mentioned in sec. 3.2. Finally, departures from hermiticity in the Yukawa vevs correspond (see discussion beneath eq. (6)) to assuming sensible departures from eq. (9). Throughout this paper we neglect such effects. Again, we will comment on this assumption in sec. 3.2.
3.1 Tree-level FCNC effects
The flavor gauge bosons couple to the currents . Similarly as in Ref. , these interactions give rise to new, tree-level, contributions to the 4-fermion operators
with Latin and Greek indices on the quark fields denoting flavor and respectively color, and where . In the quark mass eigenstates basis, the Wilson coefficients of the above operators read
where can be or , and a sum over and in the range is understood. The matrices rotating the fields from the flavor to the mass eigenbasis should be chosen as
compatibly with eq. (8) and in the approximation of neglecting the mixing between quarks and heavy fermion states.
Updated bounds on the Wilson coefficients in eq. (3.1) have been reported by the UTfit collaboration  and usefully tabulated in their table 4 for the different meson-antimeson mixing processes. The contributions, predicted in our model, to the above coefficients have been explored by the random scan mentioned at the beginning of sec. 3. As previously anticipated, these contributions are well within the existing bounds in the bulk of the explored parameter space. As an illustration, we report in Fig. 1 the largest in magnitude among these contributions, that to the mixing coefficient , in the case of TeV-scale parity. The mass is therefore mostly bounded from box diagrams with exchange, as discussed in sec. 3.2. The resulting bound, TeV , is shown in Fig. 1 as a vertical shaded area extending leftwards. On the masses, on the other hand, we will return in sec. 3.3.
3.2 Loop FCNC effects and lower bound on the mass
In ref. , it was pointed out that box diagrams with mixed -quark and -quark exchange provide a severe lower bound on the -symmetry breaking scale in the case of the minimal LR symmetric model with Higgs bidoublets. This happens on account of the constraint from the mass difference, , and the bound reads TeV.444Even more stringent bounds would come from CP violating observables again in the sector, but these bounds are much more model dependent.
In our model, beside the known quarks, also their heavy fermionic partners propagate in the box diagrams, because of fermion mixing. The full amplitude, for a given quark-heavy fermion doublet (e.g. and ), is depicted in Fig. 2. The sum of the contributions turns out to display a GIM-like mechanism of cancellation, as we will shortly explain.
First note that the upper fermion line in any of the box diagrams of Fig. 2 will be proportional to a factor of or depending on whether the fermion is a quark or its heavy partner (in the lower fermion lines one will have the hermitian conjugate of the same expressions). Assuming propagation of a single quark species, it is therefore easy to see that is such that
where we have abbreviated with and dropped the superscript on the masses. One should note at this point that
the mass eigenvalues for quarks and heavy fermions come with opposite signs (cf. eq. (4)),
after factoring out the common mass term mentioned in item b, the diagram is proportional to , as in the GIM mechanism. In our model the mass difference between a quark and its fermionic partner is smallest in the top sector. Interestingly, in most of the meson mixings’ phenomenology of the down-sector, including the CP violating observable , the top contribution is the most important one.
The observation in item c has the potential of substantially weakening the severe bounds on the parity breaking scale coming from , and we reserve to come back to this issue in a separate study. As stated elsewhere, the predictions for CP violating observables are however quite model-dependent, and here we confine our discussion to CP conserving ones, in particular . In this case, the dominant loop contribution comes from the charm sector, hence the mechanism described above is much less effective, since the quark - heavy fermion splitting is very large. On account of this constraint, we find that the lower bound on the mass from mass difference coincides with that in ref. , namely we get TeV. This bound holds in the exact parity case (), that we have been assuming in this discussion.
On the other hand, if parity is broken at a scale much higher than the TeV scale, one expects a splitting in the TeV-scale values of and in eq. (15), and values of provide a further suppression of eq. (15) by the same factor. For example, assuming , this bound scales down to TeV. One can see this by simply noting that, as far as and the gauge couplings are concerned, scales as , and that the calculation in the SM is dominated by loops mediated by the charm quark, whose vectorlike partner is, to first approximation, decoupled. As discussed at the beginning of sec. 3 the choice  will be our reference one for the scenario of no TeV-scale parity.
Two further comments are in order here. First, we note that a choice such as will also affect the collider bound since the production rate will go down by the factor as well. Second, high-scale parity breaking will in general also cause some misalignment of the left and right CKM matrices. In particular, large off-diagonal entries in the right CKM matrix have the potential of correspondingly increasing the contributions to flavor observables, a simple example being, again, that of meson anti-meson mixings. The interest of this example is in the fact that the potential phenomenology of these contributions encompasses not only flavor violation, but also mixing-induced CP violation, namely observables like , and .555On the other hand, we do not expect large effects to these observables to come from the other potential sources of flavor mixing discussed in this paper, namely tree-level FCNCs mediated by flavor gauge bosons (see sec. 3.1) or quark – heavy-fermion mixing (see sec. 3.4.2).
For the aims of the present discussion, we will assume that CKM entries undergo corrections due to RGE that don’t modify their hierarchical structure, hence that the induced misalignment between and is small enough not to grossly alter the main argument of this section. A more detailed answer can be given, we feel, only in the context of specific models.
3.3 Flavor gauge boson mass scale
As mentioned above, from the point of view of flavor violating effects mediated by exchange, the model is compatible with as small as O(TeV), and this represents a potentially interesting new signal. Of course the question arises here, whether there are other model constraints placing a more stringent lower bound on this mass.
From eq. (11), one expects the spectrum to be correlated with the Yukawa vevs, which are fixed, in the combinations and , by the requirement that, after fermion mixing, quark masses have the observed values. Therefore, with fixed ’s, one may expect lower masses to occur for larger couplings. This vs. correlation is, however, completely smeared out by the freedom in the choice of the coupling .
A correlation (albeit again not very sharp) is instead observed between the lowest allowed mass and the -breaking scale , which, in turn, is related to the bound discussed in sec. 3.2. This correlation, and the applicable bound, is shown in Fig. 3 in the two cases of exact TeV-scale parity (meaning ) and of no TeV-scale parity (where we assume, as mentioned ). From the lowermost points, one can see that the minimum mass tends to grow with growing . In particular, the lowermost red (blue) points on the right of the red (blue) vertical line imply allowed values going down to about 3 TeV.
3.4 Fermion mixing and its consequences
The fermion mixing matrix in eq. (4) can be diagonalized via orthogonal transformations acting separately on each generation and each chirality, namely:
where the hats denote mass eigenstates. This eingenvalue problem admits the following analytic solution (we recall again that the diagonal entries of the flavon vev matrices are denoted with )
with rotation angles given by
Provided the couplings are of the same order, we will generally have . Accordingly expanding the above formulae one obtains the following approximate, but accurate solution ()
Here and are to be identified respectively with the quark and heavy partners masses. A completely analogous solution exists in the down sector and is obtained from the above formulae with just the substitution . Note that the quark masses implied by the approximate in eq. (7) can be obtained from the in eq. (3.4) in the limit .
Let us focus on the general solution in eqs. (3.4) and (3.4). For fixed and , one can determine the combination by inverting the equations , with the up-type quark mass values on the r.h.s. These equations admit a real solution in only for sufficiently high . Note in fact that, with fixed values for the other parameters, the maximum of occurs for (see also the first of eqs. (3.4)). Necessary condition for a real solution to exist is therefore
where we have assumed . Since GeV – very close to – the above inequality implies . (From the analogous inequalities in the down-quark sector, one also derives .) Note as well that, for the boundary value , the solution of eq. (20) for is . This explains the lower bound chosen for in our scans, see beginning of sec. 3.
In the case where parity is not a good symmetry at the TeV scale, we would expect the light quark Yukawa couplings to be left-right asymmetric as already noted. In this case, the formulae for quark as well as heavy-fermion masses and mixings change accordingly, i.e. eqs. (3.4) are replaced by
This case becomes very similar to the GRV examples  and as we see from the figures below allows vectorlike quark masses of about 2 TeV, making them, in principle, accessible at the LHC. If , then the vectorlike quark masses could be even lighter, as is clear from eq. (3.4).
The one difference from the TeV-scale parity case is that there is no reason for to be real and therefore the model does not solve the strong CP problem. However, if there is parity restoration at some high scale, at that scale one does have a solution to the strong CP problem and an extrapolation is necessary to estimate how large a is induced at low energy. This kind of analysis is beyond the scope of this paper and we hope to take it up separately.
Bounds on heavy-fermion masses and mixings
The interesting phenomenological question is that of the magnitude of the mixing angles and of the lowest allowed masses and for the heavy up-type and down-type fermion partners. Note that these masses are given by the and solutions in eq. (3.4). According to these equations, the heavy fermion masses are correlated with both the -breaking scale and with the scales of gauge flavor symmetry breaking. In practice the latter correlation is blurred by the dependence on the unknown and parameters. On the other hand, the correlation with still allows to infer the lowest allowed values for and , taking into account the bound discussed in sec. 3.2. The situation is illustrated in Fig. 4, that displays the lightest up-type fermion partner mass vs. . Blue and respectively orange dots refer to the parity vs. no-parity scans, see beginning of sec. 3 for details. The vertical lines represent the corresponding bounds. One can see that, while the exact parity case seems to exclude TeV, in the no parity case masses going down to 2 TeV or even lower are possible. We mention that we found similar values to be possible also for the lightest down-type heavy fermion masses. In fact, as evident already from the first of eqs. (3.4), a large ratio does not necessarily imply a corresponding hierarchy in because can be substantially larger than 1.
Further qualitative information can be obtained from eqs. (3.4) for the mixing angles. For , given the very large values of the , one can expect vanishingly small left- and right-sector mixing angles. For the top case, , the left-sector mixing angle is still generally small. In fact, note that the quantities and are very similar in size, and appear with opposite signs in the numerator of . On the other hand, in the right-sector case, the term appears with reversed sign, and this, depending on the choice of parameters, may result in a non-negligible .
Fig. 5 illustrates the above considerations more quantitatively. The upper panels show the mixing angles in the left-handed top sector against the mass, in the parity (left panel) and in the no-parity case (right panel). One can see that mixing in the left-handed sector is always fairly small – taking into account the bound discussed in sec. 3.2, mixing is such that . The corresponding angle in the right-handed sector is displayed in the lower panels for the case of TeV-scale parity (the case of no parity gives similar results). In this case, the amount of mixing is not affected at all by the mass bound, and we typically find or larger, as also displayed in the histogram. This implies , and may lead to potential effects in observables like FCNC top decays such as and observables sensitive to operators with 4 powers of the top field.
In short, mixing angles anywhere else than in the case are vanishingly small. E.g., for , we find . This bound is relevant to effects in observables like and , on which we will be more quantitative in sec. 3.4.1.
3.4.1 Electroweak precision tests
The decay is an example of the prototype process , with any of the massive vectors, any of the fermions in the model, and . This kind of processes allows to estimate the magnitude of tree-level non-oblique corrections that the model introduces.
The interactions relevant to