Natural Alignment of Quark Flavors and Radiatively Induced Quark Mixings
The standard model does not provide an explanation of the observed alignment of quark flavors i.e. why are the up and down quarks approximately aligned in their weak interactions according to their masses? We suggest a resolution of this puzzle using a combination of left-right and Peccei-Quinn (PQ) symmetry. The quark mixings in this model vanish at the tree level and arise out of one loop radiative corrections which explains their smallness. The lepton mixings on the other hand appear at the tree level and are therefore larger. We show that all fermion masses and mixings can be fitted with a reasonable choice of parameters. The neutrino mass fit using seesaw mechanism requires the right handed mass bigger than 18 TeV. Due to the presence of PQ symmetry, this model clearly provides a solution to the strong CP problem.
Introduction The standard model (SM), in spite of its spectacular successes, leaves many questions unresolved such as the gauge hierarchy problem, the strong CP problem, neutrino masses, and dark matter. One rarely discussed puzzle is: why are the up and down quark flavors approximately aligned in their weak interactions according to their masses, i.e., the top quark is aligned with the bottom quark, both being the heaviest in each sector and similarly the charm and up quarks approximately align with strange and down quarks respectively? Note that the flavor is defined by the interaction prior to symmetry breaking and the masses and mixings of fermions are determined by the Yukawa couplings. It is not apriori guaranteed that the quarks that appear in the gauge interaction will remain dominantly coupled to each other after symmetry breaking. However, we observe that quarks are approximately aligned and we will call this the “flavor alignment” puzzle. Since the Yukawa coupling matrices that determine the alignment are arbitrary with no correlation to masses, flavor alignment is not explained in the SM. This is the situation in most of the extensions of the SM where we simply adjust the Yukawa matrices to fit the observed pattern. It is therefore interesting to search for models where flavor alignment arises naturally for quarks. As far as the lepton sector is concerned, the question of analogous alignment is not clear yet. For example, if the neutrino mass hierarchy is normal, it will be similar to the case of quarks although deviations from alignment are substantial due to larger mixing angles.
There has been several proposals to address this issue: one is within the framework of Randall-Sundrum models where the quarks of different flavor are located anarchically within the bulk kaustubh () and the warping helps to cause alignment; another extends the standard model by an extra local symmetry only for the third generations Alex (), so that and align naturally as a result of the gauge symmetry. In this note, we propose an alternative solution which combines two familiar symmetries studied extensively in the literature namely the symmetry PQ () and the left-right symmetry LR () which resolves this puzzle for all three generations, while at the same time solving the strong CP problem. Flavor alignment in this model emerges for all the generations in a natural manner. The quark mixings remain zero at the tree level and arise at the one loop level bala (); others () providing an explanation for their smallness. Left-right symmetry is essential for our framework since it puts the right handed up and down quarks together and at the same time explains the neutrino masses and mixings. The neutrino mixings arise at the tree level via the seesaw mechanism which explains why their mixings are “large”. One must however, go beyond the minimal LR model since natural alignment emerges only if there is an extra symmetry like symmetry PQ () or super symmetry. We pursue the alternative since it also solves the strong CP problem and provides the axion as a dark matter candidate.
Getting quark mixings out of one loop effects requires additional colored scalars in the model bala () with masses in the 10 TeV range. The colored scalars could either be triplets or sextets. We pursue the triplet alternative here, which in some sense is more minimal, although most of our discussion also applies to the case with sextets The assignment of PQ charges guarantees that the proton decay is forbidden despite the presence of low mass color triplets. Since the color triplet scalar connects the up quark to the down quark, the flavor changing neutral current constraints on their couplings is much weaker. We also give a fit for the lepton sector and find that fitting the neutrino oscillation observations and the charged lepton mass spectrum requires the right handed mass to be in the 20-30 TeV range
The model The left-right symmetric model is based on the gauge group with fermion assignments as doublets of the left and right ’s as
where represents the family index. We choose the following Higgs multiplets: bi-doublet ; and , as in the minimal LR model MS (). In addition, we add the following new multiplets to generate quark mixings at one loop and to implement the PQ symmetry. They are color triplet fields , and gauge neutral fields . The PQ charges and gauge assignment for the all the fields are presented in Table 1.
The most general gauge invariant Yukawa couplings allowed by there PQ charges are given as
Note that the conjugate bi-doublet field has PQ charge and is therefore forbidden from coupling to the quarks instead it (and not ) couples to leptons. As a result, the quark Yukawa matrix can be diagonalized by a change of basis. This feature of our model plays a crucial role in determining the flavor alignment for quarks and is different from the minimal LRSM where both and couple to fermions, thereby spoiling alignment. The also provides the additional advantage of solving the strong CP problem and yielding a dark matter candidate of the universe. Note that the coupling matrices are general anarchic complex matrices which at the one loop level provide quark mixings as well as the CKM CP phase, as we see below. Similarly being anarchic provides the lepton mixings at the tree level via the seesaw mechanism.
In this section, we will discuss the symmetry breaking pattern in our model. To do this, we will begin with the general gauge and L-R symmetric, renormalizable Higgs potential which is given as
where terms denote terms of type where is the relevant scalar field and terms denote mixed couplings between different scalar fields of the form . We also define L-R symmetry for fields as while using the usual definitions of parity transformation for other fields. This will force all the couplings in the Higgs potential to be real.
The symmetry breaking in our model happens when the Higgs fields take the following vacuum expectation values (VEV),
The symmetry breaking in this model goes in several steps with the PQ scale , , being the highest at around GeV. The scale is chosen to make the axion invisible and make it compatible with all known astrophysical and cosmological constraints jekim (). The next scale in the model is the right-handed scale , , which is chosen to be of the order of few TeVs to be compatible with rare process constraints. The SM electroweak symmetry breaking takes place when the fields acquires a VEV such that . Getting the electroweak VEV to be of the order of a 100 GeV requires a fine tuning between the and which is the usual fine tuning of the the invisible axion models jekim ().
The value of coupling is taken to be to protect neutrino masses from receiving large type-II contributions. A property of our model is that in the absence of the coupling, the VEV can be set to zero at the minimum of the potential. It only becomes non-zero once . We can look for a solution to the VEVs with the hierarchy by minimizing the potential in eq.(Natural Alignment of Quark Flavors and Radiatively Induced Quark Mixings) . By minimizing the potential with respect to and , we get the conditions
respectively. From eq.(5), we can see that for an , we can get the induced by the term by taking which in turn gives . The question then arises is whether this is an additional fine tuning beyond the usual fine tuning of invisible axion models needed to separate the weak scale from the PQ scale. We find that the choice of , is radiatively stable in our model because in the absence of the term, the model has an accidental symmetry under which and respectively with the rest of the fields unchanged. Since term breaks this discrete symmetry softly down to , whatever value we choose for it at the tree level remains stable under radiative corrections. This is because is multiplicatively renormalized. Thus our choice of parameters is field theoretically natural and we need only two fine tunings; one to get the SM scale and the other to get the LR scale. We will determine below the LR scale needed in this model.
From this discussion above, we find that the axion field has components in all the Higgs fields except the and if . For leading order in , we get
where is used to denote the imaginary part of the corresponding complex field and vevs as given earlier in the text and is the normalization factor for the axion field. This field is orthogonal to all the physical fields in the theory including the longitudinal mode of boson. Due to large hierarchy between and other scales, we find that .
Flavor alignment We are now ready to show that this model naturally leads to the alignment of quark flavors as claimed. Note that without loss of generality, we can choose the basis for the quark fields before symmetry breaking such that the quark Yukawa couplings is diagonal. This implies that the up quarks of different generations are aligned with the down quarks of the corresponding generation. The mass of the top and bottom quarks are given at the tree level by . The second tree level relation is not very well satisfied and one has to invoke the one loop correction. Since the one loop corrections are expected to be small, the flavor alignment remains.
The situation in the lepton sector is slightly different. It follows from the Yukawa couplings that Yukawa matrix can also be diagonalized by the choice of an appropriate basis so that the charged lepton as well as the Dirac mass matrix for neutrinos is now diagonal. However, the neutrino masses arise out of seesaw mechanism which involves the right handed neutrino mass matrix, , which is a general complex symmetric matrix unconstrained by the PQ symmetry. By adjusting the elements of , we can get any flavor of neutrino to go with any flavor of charged lepton without any prior alignment. In other words, this can allow for large departures from alignment which is the case for leptons. In fact, this can also allow for the extreme case of mis-alignment which occurs when there is inverted hierarchy of neutrino masses.
One loop corrections and fits to quark masses and mixings From the Yukawa couplings in eq.(Natural Alignment of Quark Flavors and Radiatively Induced Quark Mixings), it can be seen that the up and down quark mass matrices are proportional to each other at the tree level. This implies that there is no mixing among the quark flavors, i.e., is an identity matrix. After the symmetry breaking, there is a new tree level contribution to the quark mass matrices coming from the term generating an effective coupling of the form but the generation structure of the coupling is same as the tree level coupling and can therefore be absorbed to a redefinition of the tree level term. Thus this also does not lead to any flavor mixing. However, at the one loop level the color triplets, , generate the necessary quark mixing through radiative corrections as shown in fig. 1.
To study the effect of this, we do an orthogonal rotation of by an angle to go the mass basis with masses respectively. The mass matrices for up and down quark masses including the one loop contribution (from figure 1) can be written as (in the limit )
where the first terms in both these expressions denote the tree level contribution and the second, the one loop one. It is clear that the up and down quark mass matrices are not proportional to each other any more and generate nonzero CKM angles. Without any loss of generality, one can rotate the hermitian Yukawa matrix into a real diagonal matrix. Similarly, the phases appearing in the diagonal elements of the complex-symmetric Yukawa matrix can be absorbed through a redefinition of quark fields. In the case of color triplets, the one-loop corrections to up-type quarks is proportional to the tree-level masses of down-type quarks. This feature can naturally explain the inverted hierarchy of masses for first generation quarks since the tree level masses of the first generation quarks are very small i.e. MeV and MeV and the dominant contribution to their masses comes from the one loop effect (which has the property of inverting them as observed). For the charm and strange quarks on the other hand, no such inversion takes place since the tree level contribution to charm mass is already close to observed value.
Lepton sector In the lepton sector, the hermitian Yukawa coupling can be rotated into a real diagonal matrix, like in the quark sector. The couplings is now completely determined by the charged lepton masses such that . The neutrino masses are generated through a combination of type-I and type-II seesaw mechanisms and the neutrino mixings arise from the right handed neutrino mass matrix which is completely anarchic. The light neutrino mass matrix can be approximated as
The diagonal phases of the complex-symmetric Yukawa coupling can be absorbed through a re-definition of neutrino fields.
Since the color triplet masses are free parameters, we take TeV and a mixing angle for fitting the fermion masses. This choice is however not essential to get a mass fit. Since the VEVs and are related through the electroweak Higgs VEV, the lepton sector and quark sector is tightly connected through the ratio A point in the parameter space which fits the fermion masses and mixing parameters are given in the table 2.
We find that the neutrino mass fit requires the right-handed scale TeV since we have not found any fits for lower values of . Recalling that and the lowest value of allowed in the LR model is DMZ (), we find that for our model to work, we must have an TeV.
Discussions and conclusion Several points worth noting about the model:
Flavor changing neutral current constraints There are two sources of quark flavor violation in the model:
(i) As in the minimal LR model, there are second neutral Higgs () mediated flavor changing effects, once the one loop effects are included to generate quark mixings. This puts s lower limit on the mass in the range of 10 TeV or higher. This is lower than the scale in our model and does not require beyond LR physics to explain this as would be the case in case for TeV.
(ii) A second source of flavor violation in the model comes from the quark couplings to the color triplet fields . The corresponding case for color sextet fields were analyzed in Ref. fortes (). The difference in our case is that color triplets always connect up quarks to down quarks. This is identical to the constraints from the up-down connecting sextet case analyzed in fortes (). There are two kinds of processes that violate flavor with triplets: at the tree level they lead to flavor violating decays such as , etc. From Ref. fortes (), we find that in the case products such as etc are bounded and for TeV, these bounds are of order one. Since in our case we take TeV, these bounds are weaker and are consistent with our choice of . The second type of color triplet induced FCNC comes from box graphs which lead to processes such as , mixing. Again, the most stringent constraint of this type are: from mixing and from mixing and from mixing. Clearly for our choice of TeV, our choice of parameters are quite consistent with these bounds.
(iii) As far as the leptonic flavor change is concerned, the dominant contributions come from exchange at the one loop level for etc. and at the tree level for and decays. For masses near 50 TeV, our parameter choice is consistent with current bounds from these as well as other processes.
The choice of our PQ charges for the lepton sector is dictated by the requirement that proton decay be forbidden in the model since all scales except the PQ scale are in the multi-TeV range.
In the neutrino sector, the Dirac CP phase is found to be in the 4th quadrant and is (), which is within 3 range of the current NoVA and T2K results Nova ().
The lightest right handed neutrino has a mass around 100 GeV and is coupled to all three charged leptons. However its production rate is suppressed due to heavy right handed mass as well as due to a small heavy-light neutrino mixings, with the largest mixing being the . It is therefore not observable at the LHC.
We have not explored the question of leptogenesis in the model; however we note that there are two right handed neutrinos which are quasi-degenerate in the model, which is a pre-requisite for leptogenesis and also that TeV, which guarantees that all washout effects are small hambye ().
We have given a fit for normal neutrino hierarchy for which case we get eV. This is below the current bound from Planck and other experiments review (). However, this can be tested in forthcoming experiments such as LSST survey and EUCLID mission etc. which are expected to bring the current limits on the neutrino mass sum to the level of 0.02 eV.
In conclusion, we have presented a simple resolution of the “flavor alignment puzzle” of the standard model using a combination of left-right with Peccei-Quinn symmetry. The model also solves the strong CP problem as well as the problem of neutrino masses and mixings. We provide a fermion fit in the model for both the quark and lepton sector.
Acknowledgements.– We thank K. Agashe for discussions and comments on the manuscript. This work is supported by the US National Science Foundation under Grant No. PHY1620074.
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