Structure Of Flavor Changing Goldstone Boson Interactions

General flavor changing interactions for a Goldstone boson (GB) to fermions due to a spontaneous global $U(1)_G$ symmetry breaking are discussed. This GB may be the Axion, solving the strong QCD CP problem, if there is a QCD anomaly for the $U(1)_G$ charge assignments for quarks. Or it may be the Majoron in models from lepton number violation in producing seesaw Majorana neutrino masses if the symmetry breaking scale is much higher than the electroweak scale. It may also, in principle, play the roles of Axion and Majoron simultaneously as far as providing solution for the strong CP problem and generating a small Majorana neutrino masses are concerned. Great attentions have been focused on flavor conserving GB interactions. Recently flavor changing Axion and Majoron models have been studied in the hope to find new physics from rare decays in the intensity frontier. In this work, we will provide a systematic model building aspect study for a GB having flavor changing neutral current (FCNC) interactions in both the quark and lepton sectors, or separately in the quark, charged lepton and neutrino sectors with sources of FCNC interactions identified in detail. We provide a general proof of the equivalence of using physical GB components and GB broken generators for calculating GB couplings to two gluons and two photons, and some issues related to models having spontaneous CP violation are discussed. We will also provide some details for obtaining FCNC GB interactions in several popular models, such as the Type-I, -II, -III seesaw and Left-Right symmetric models, and point out some special features in these models.

Some of the well motivated models having a GB are the Axion and Majoron models. Many of the searches depend on how GB interacts with SM fermions. GB couplings to fermions not only have flavor conserving interactions, but also flavor changing ones. GB interaction with fermions is in derivative form and usually parameterized as the following where f stands for a quark or a charged lepton or a light active neutrino, and i , j are the generation indices. f a is the GB decay constant which sets the scale of U (1) G symmetry breaking. The sizes of c V,A are model dependent.
If neutrinos are Dirac particles, the GB interactions with neutrinos will be the same in form as given above. If neutrinos are Majorana particles, the form may be modified. Also if right-handed neutrinos ν R are introduced to facilitate the seesaw mechanism, ν L and ν c R will have different masses, this also modifies the form of the interaction. The Lagrangian L ν int of GB interaction with neutrinos as appearing in seesaw models will have the following form The flavor changing GB interactions with fermions have a lot of interesting phenomena which can be used to discover a GB. These can be from rare decays of particles containing b, c and s quarks, τ , µ charged lepton decays, neutrino decays, and B-, D-, K-meson and muonium oscillations, and also g-2 of charged leptons [15][16][17][18][19][20][21][22][23][24][25][26]. In this work, we will not repeat to obtain the stringent constraints from various data, but to investigate some interesting features of FCNC GB interactions from a general U (1) G global symmetry break down beyond SM and some related issues. This GB can be an Axion, a Majoron or a mixture of them. In a concrete model, there are usually other Higgs doublets besides the SM one. In general the additional Higgs may also mediate FCNC interactions [27]. These new Higgs bosons all have masses and some of them can be much larger than the electroweak scale. For a massless GB, its FCNC effects will be different. So we will concentrate on FCNC structure of a GB. Note here that FCNC processes can also be generated at loop level where the strength is suppressed. So in this paper we only consider the tree level interactions.
The paper is arranged in the following way. In section II, we provide a systematic model building aspect study for GB interactions in both the quark and lepton sectors with a simple way to identify GB components, and to obtain GB-fermion interactions. For neutrino sector, we take Type-I seesaw as the prototype of model to study. In section III, we discuss under what conditions the general GB can be viewed as the usual Axion or Majoron. We provide a general proof of the equivalence of using physical GB components and GB broken generators for calculating Axion couplings to two gluons and two photons. In section IV, we identify in details the sources for FCNC GB interactions, and discuss how spontaneous CP violation may affect GB-fermion interactions. In section V, we discuss some interesting features of GB interactions with fermions in Type-II, -III seesaw models and Left-Right symmetric models. In section VI, we provide our conclusions.

II. A GENERAL GLOBAL U (1)G MODEL AND ITS GOLDSTONE-FERMION INTERACTIONS
Let us assume that a model has a general U (1) G global symmetry in addition to the SM gauge symmetry. This can be the Peccei-Quinn symmetry for solving the strong CP problem or lepton number (LN) symmetry in connection with Majoron models or some other flavor symmetries depending on how the U (1) G charges are assigned. Fermions in the model transform under U (1) G as f L,R are the fermions in the SM with SU , and for leptons, f i L is L i L : (1, 2, −1/2)(X li L ), f i R can be E i R : (1, 1, −1)(X ei R ). Since X qi L and X li L contain u i L , d i L and ν i L , e i L , we indicate their individual U (1) G charges as X ui L = X di L = X qi L and X νi L = X ei L = X li L for conveniences. If there are right handed neutrinos, f i R is ν i R : (1, 1, 0)(X νi R ). The quantum numbers in the brackets correspond to SU (3) C , SU (2) L , U (1) Y and U (1) G , respectively. The diagonal matrix diag(X f 1 L,R , X f 2 L,R , X f 3 L,R ) in flavor space will be indicated by a diagonal matrix X f L,R . In general there are several Higgs doublets H u,d,e,ν ij transforming as (1, 2, 1/2)(X q,l i L − X u,d,e,ν j R ) which couple to fermions, In component form When the Higgs bosons develop vacuum expectation values (vev), v u,d,e,ν ij , the electroweak symmetry SU (2) L × U (1) Y is broken down to electromagnetic symmetry U (1) em , and at the same time the U (1) G is also broken. Non-zero vevs will give the masses of fermions, and gauge bosons W , Z.
If at the same time the singlets S ij are introduced with U (1) G charge −(X νi R + X νj R ), one can also have the terms Here the superscript c indicates the charge conjugated field.
. If the vevs of v s ij / √ 2 are non-zero and larger than v u,d,e,ν ij , the Type-I seesaw [28][29][30][31][32][33] mechanism will be in effective to provide small Majorana masses for light neutrinos. The singlets can also play the role of making possible dangerous GB interactions invisible as in the DFSZ invisible Axion model [34,35].
As mentioned before, the non-zero vevs of scalars H a ij and S ij not only break the electroweak symmetry to provide the longitudinal components of weak gauge bosons W and Z, but also break the global U (1) G symmetry resulting in a massless GB. The vector z "eaten" by Z boson, in the basis I = (I u ij , I d ij , I e ij , I ν ij , I s ij ), is given by and the U (1) G broken generator vector A is given by The physical GB in this model should be the linear combination a = a · I T , which is orthogonal to z = z · I T . The corresponding vector form is a = α z + A. The requirement that a · z T = 0 dictates α ∼ − A · z T / z · z T . Therefore a is given by [36] where N α is a normalization constant to ensure a · a T = 1, and Expressing the physical GB, a = a · I T , in terms of I a ij , we have In the above, i, j, and k, l are summed over flavor spaces in each sector, and a, b are summed over u, d, e, ν. Here sign(b) takes "−" for b = u, ν and "+" for b = d, e.
The above shows that I b ij and I s ij contain the GB a with amplitude (1/N α )((X ak respectively. The Yukawa couplings of GB a to fermions along with the mass terms are given by where M ij l are mass matrices for up quark M u , down quark M d , charged lepton M e and neutrino M ν . They are given by It is understood that the above mass matrices should be summed over contributions from different pieces of each vev v l ij for each "l" type of fermions. Note that here i and j are not summed. From the above Yukawa couplings, we can identify the fermion current interacting with derivative form of a, L Y → L af = ∂ µ aj µ af , with the help of the equations of motion as We identify 1/f a = v 2 /N α . Note that the following relation holds, where X ν is a 6 × 6 diagonal matrix with non-zero entries to be (

III. GOLDSTONE BOSON AS AXION OR MAJORON
As mentioned before, the GB may or may not be a usual Axion or Majoron. Here we make a rough distinction among them depending on their primary role in addressing some physics problems. The massless GB will become massive if the relevant U (1) G charge assignments have SU (3) C anomalies, then this model can be used to solve the strong CP problem. The GB in such models can be viewed as an Axion and the U (1) G can be identified as a variant of the U (1) P Q . The condition is to have This can be understood from a possible GB-gluon coupling aG aµν G a µν by calculating the triangle diagram using the current in Eq. (13). We have [11][12][13] where g 3 is the SU (3) C gauge coupling constant, and T (q) is the generator of SU (3) C for color triplet quarks defined Here the superscripts indicate the contributions from up-and down-type quarks running in the loop of the triangle diagram. They are given by As long as is not zero, there is a color anomaly. This makes the GB to be massive and play the role of the usual Axion.
Here we would like to make a comment on the relation of a couplings to two gluons and two photons. GB coupling to two photons of the type aF µν F µν will be generated by just replacing gluons by photons in the above mentioned triangle diagram. We have The superscripts indicate the contributions from quarks and charged leptons running in the loop. They are given by Here N q c = 3 and N e c = 1 are the effective number of color for quarks and charged leptons, respectively. The above method is referred as calculation using the physical GB.
In the literature for Axion models, the GB-two-photon coupling is usually written as [13] Where g 0 . This method is referred as calculation using the broken generators.
The above gives the same result as Eq. (18) . This condition is actually one of the gauge anomaly free conditions [37,38] Here I i 3 is the value of the third weak isospin component of the "ith" fermion. Therefore this condition is guaranteed for a gauge anomaly free theory to eliminate the term proportional tov 2 related to the component "eaten" by the Z-boson, which results in the some results obtained as using broken PQ generator. The above provides a general proof as discussed in Ref [36]. The results are completely fixed by the U (1) G charges X i L,R and the kind of colored, and charged particles in the model.
Note that if there is no color anomaly for U (1) G , that is N (X) = 0 as in the Majoron models in Ref. [10,26], the situation will be different. In this case to avoid that N (X) appears in the denominator of g 0 aγ = (α em /2πf a )E(X)/N (X), it is better to use g 0 aγ = (α em /4π)E(X) directly. Majoron is also another commonly studied GB which results from spontaneous break down of lepton number, like in the Type-I seesaw model [10]. Therefore there is no color anomaly for GB produced by lepton number breaking.
From our discussion in previous section, the GB can in general have color anomaly and also break lepton number, therefore the GB can be viewed as an Axion and Majoron simultaneously. The GB also exists other names [39], such as Familon [15,16], and Arion [40], which can be considered as special cases discussed here. But whichever name the GB has, it results from a global U (1) symmetry breaking.

IV. FLAVOR CHANGING GOLDSTONE BOSON INTERACTIONS
We now discuss how FCNC GB interactions with fermions emerge. The relevant information is contained in the GB current in Eq. (13). The flavor changing nature of the interaction can be easily seen in the mass eigen-state basis. The mass matrices for fermions can be diagonalized by bi-unitary transformation to the diagonal ones, In the mass-eigen basis, the GB interaction current j µ ac with quarks and charged leptons is given by Here X i L,R are diagonal matrices with the diagonal entries given by (X i1 L,R , X i2 L,R , X i3 L,R ). f m indicates the mass eigenstates. We will drop the superscript "m" to keep notation simple unless stated otherwise. It is clear that when X i L,R are not proportional to unit matrix the GB current is not diagonal in the mass eigen-state basis and therefore flavor changing interaction emerges.
The GB decay constant f a is identified by the relation 1/f a = v 2 /N a . The off-diagonal elements for c V and c A in . For the diagonal elements, ±(v 2 /v 2 ) needs to be added to c ij A entries with "-" for up-quarks, and "+" for down-quarks and charged leptons. If X f L,R entries are order O(1) and have no accidental cancellations, c V,A can be order O(1).
Similarly, GB couplings to neutrinos can be worked with some modifications. We provide some details here. The mass matrix M ν for neutrinos is diagonalized by a 6 × 6 unitary matrixM we have the current j µ aν for neutrinos given by Again f a is identified by the relation 1/f a = v 2 /N a . Compared with Eq. (2), we have From the above, we see that there are more possibilities that FCNC interaction can emerge due to seesaw mass matrix diagonalization. For example, as V ν LL are not unitary in general, FCNC interaction exists in aν L ν L interaction with amplitude proportional to V ν LL V ν † LL . Since V ν LL should be close to the unitary V P MN S matrix, the FCNC interaction is naturally small. The FCNC interaction can also occur, similar to the quarks and charged leptons if X ν L,R are not proportional to unit matrix. Even, X ν L and X ν R are separately proportional to unit matrix, FCNC interactions can still occur if the 6 × 6 diagonal matrix X ν is not proportional to a 6 × 6 unit matrix.
One observes that if X f L,R are set to be unit matrix, there exists only FCNC interaction of a with neutrinos but no interaction with quarks and charged leptons, because V ν LL,RR,LR,RL are separately not unitary. Working in the basis where M e and M R are diagonalized, one can approximate [41,42] Global fit finds that the matrix elements in ǫ are O(10 −3 ) [43]. Therefore, the couplings V ν LL V ν LL † are allowed at the level of 10 −3 . If there are more than one singlet with different lepton numbers and different right-handed neutrinos are assigned with different lepton numbers, one would need to change the Majoron couplings to light neutrinos to V ν LL X ν R V ν LL † with X ν R a diagonal matrix but different diagonal entries. The individual off-diagonal couplings can be much larger than 10 −3 . In general, the off-diagonal entries are arbitrary and should therefore be constrained by data. There are also constraints from mixing between heavy and light neutrinos. However, they can be independent from light neutrino mixings and need to be constrained using data [44].
Before closing this section, we would like to make a comment about theories with spontaneous CP violation and how to identify the GB in the model. Spontaneous CP violation requires more than one Higgs doublet. When a global U (1) G is imposed, there may need more Higgs bosons to construct a model consistent with data [45,46]. In this case it may be more complicated in identifying the physical GB. If spontaneous CP violation exists, the vevs can be complex, that is, v l ij becomes v l ij e iθ l ij . In this case the z and A become in the basis −i(h l ij + iI l ij ), The physical GB field is now This leads to the same j µ af as discussed before. We therefore conclude that no new CP violation phases in GB interactions with fermions arise. Some special cases of this type of models have been discussed in Ref. [47][48][49].

V. SPECIAL FEATURES FOR SEESAW AND LEFT-RIGHT SYMMETRIC MODELS
We now discuss some interesting features of flavor changing GB interactions with fermions in some of the popular models, the Type-II, -III seesaw, and Left-Right symmetric models.

A. Type-II seesaw model
The simplest realization of Type-II seesaw [33,[50][51][52][53] is by introducing a triplet Higgs field χ : (1, 3, 1)(−2) that couples to the neutrinos to give neutrino mass when χ develops a vev v χ / √ 2 via the termL c L χL L . There is no need of introducing right-handed neutrino ν R as in Type-I seesaw model. To have a GB, the Majoron in this case, one can impose the global lepton number conservation in the potential [54,55]. Since the χ field has a non-zero lepton number, its vev breaks both electroweak symmetry and global lepton number. The Goldstone boson "eaten" by Z boson is given by z = (vI + 2v χ I χ )/ v 2 + 4v 2 χ . The Majoron is the another orthogonal component (2vI − v χ I χ )/ v 2 + 4v 2 χ whose coupling to neutrinos is proportional to the neutrino mass matrix. The mixing will induce Majoron to couple to charged leptons and quarks. Since the vev of χ is constrained to be less than a few GeV from the precise measurement of ρ parameter [14], therefore the couplings of GB to charged leptons and quarks are small, and the couplings to neutrinos are proportional to neutrino masses. There are no FCNC GB interactions. To remedy the problems related to light degrees of freedom in the model, one can introduce a singlet S of the type discussed in Type-I seesaw model which couples to χ and H through the term HχHS. But this still will not induce FCNC GB interactions. If L i have different U (1) G as discussed in the general GB model in section II, there is the need to introduce several χ fields with the U (1) G charges X ij χ = −(X νi L + X νj L ) and also to extend S to S ij . The term HχHS is changed to H i χ kl H j S mn with the indices contracted in all possible ways for SM gauge group and also U (1) G singlets. In this case, following procedures in section II, we obtain the GB-neutrino current Here L is not proportional to unit matrix, FCNC interactions will emerge. In the neutrino mass eigen-state basis, we have where V P MN S is the lepton mixing matrix.
At least two triplet fields χ with different U (1) G charges need to be introduced to have FCNC interaction. If the quark and charged lepton U (1) G charges are also similarly the general model discussed, their corresponding couplings to the GB will be given by Eq. (22) which lead to FCNC GB interaction with fermions in general.

B. Type-III seesaw model
In Type-III seesaw model [56], one replaces the right handed neutrinos ν R by the SU (2) L triplet Σ c L = Σ R , the charge conjugation of Σ L , transforming as a (1, 3, 0) under the SM gauge group. It carries a U (1) G charge X ν R as in the Type-I seesaw model. The component fields are as the following We will rename them with ν R = Σ 0 c L , ψ L = Σ − L and ψ R = Σ + c L . The Yukawa interaction terms are given by The GB field is in general given by Eq. (10). The GB couplings to up-and down-type quarks and also to neutrinos are the same as those given in Type-I seesaw model. But the couplings to charged leptons will be modified because of the existence of ψ L,R . We have the mass and GB interaction terms where Using the equations of motion, the GB current j µ e in the interaction ∂ µ a j µ e , can be written as One can easily see that GB will have FCNC interactions with charged leptons too.
We would like to mention a special feature noticed recently in Ref. [26] which can be achieved by just introducing one S to the usual Type-III seesaw model, and normalizing f a to be equal to v s as that in Ref. [26] by choosing X e L,R = X ν R = 1/2. In this casev 2 = 0. Using vector current conservation ∂ µ (Ēγ µ E +ψγ µ ψ) = 0, we have The mass matrix M c can be diagonalized in the form Here V e L(R) are 6 × 6 unitary matrices. Writing V e into blocks of 3 × 3 matrices, we have We then obtain Majoron J interactions with neutrinos and charged leptons in the mass basis as The size of off-diagonal entries is as large as the level of 10 −3 /f J , similar to that in Type-I seesaw model. If there are more than one singlet with different lepton numbers and different right-handed neutrinos are assigned with different lepton numbers, one would need to change the Majoron couplings to light neutrinos to V ν LL X ν R V ν LL † with X ν R a diagonal matrix but different diagonal entries. The individual off-diagonal couplings can be much larger than 10 −3 /f J . In this model, the GB is a typical Majoron whose FCNC interactions with fermions can lead to interesting consequences as shown in Ref. [26].
We note in passing that because of the appearance of new particle ψ in the theory, the GB-two-photon coupling in Type-III seesaw model will be modified compared with that in Type-I seesaw model. One needs to add a new term To have a GB symmetry in the Left-Right symmetric model, at least two bi-doublets φ 1,2 transforming as (1, 2, 2, 0) with different U (1) G charges need to be introduced in order to have phenomenologically acceptable quark mass matrices and mixing. This also implies different generations of quarks and also leptons, some of them, should have different U (1) G charges. We will construct a minimal model which also has triplets ∆ L : (1, 3, 1, 1) and ∆ R : (1,1,3,1) to make effective the seesaw mechanism. It turns out at least two different sets of triplets are needed to make the resulting U (1) G invisible in the sense of DFSZ type [59].
We write the bi-doublets as: φ 1,2 = φ 1,2 ,φ 1,2 , whereφ i = iσ 2 φ * i . Both φ i andφ i are doublets of SU (2) L . Writing in this way enables us to use directly the results obtained before for finding GB field since they both transform the same under SU (2) L . The components of these fields are The Yukawa interactions are given by If there is just one bi-doublet, only one of the κ terms is allowed for the quark and lepton sectors because of the non-zero U (1) G charges. This leads to the up and down sector of quark mass matrices to be proportional each other, which results in unrealistic mass relations without mixing. This is the reason that one needs to have more than one bi-doublet. Because of the U (1) G charges assigned, the κ and Y have the following forms We will assume v Li = 0, the quark mass matrices M u,d and the lepton mass matrices M e and M ν are given by We now work out the GB fields following the method previously used. The vevs of ∆ Ri break SU (2) R and also U (1) B−L , and the vevs of φ 1,2 break both the SU (2) R and SU (2) L , and all of them also break U (1) G . For working out the physical GB, we choose three broken generators I L 3 , B − L and A of I L 3 , B − L and U (1) G symmetries as The physical GB will be the linear combination with its orthogonal to I L 3 and B − L. We have where Expressing a in terms of I i field of the various scalars, we have Note that if there is only one ∆ Ri or both of ∆ Ri have the same U (1) G charge, there is no I Ri in a, then the axion decay constant is order v which is a visible axion type. We obtain the GB currents for charged fermions and neutrinos in the form given in Eqs. (22) and (24) with and for u, d and e replacev 2 /N α and v 2 /N α by −v 2 Rv 2 /N α and−v 2 R v 2 /N α . Also for right handed neutrinos, replace

VI. DISCUSSIONS AND CONCLUSIONS
We have carried out a systematic model building study for FCNC GB interactions in both the quark and lepton sectors, or separately in the quark, charged lepton and neutrino sectors. Several popular models have been discussed.
There are two types of sources for FCNC GB with fermions. One of them is that different generations of fermions have different U (1) G charges, and another due to mass splits of left-and right-handed particles, like neutrino masses in Type-I and Type-III seesaw models. Even all generations have the same U (1) G charges, there are in general FCNC interactions of GB with neutrinos. For Type-III seesaw model, there are also FCNC GB interactions with charged leptons. For Type-II seesaw model, at least two triplets are needed to have FCNC GB interactions with fermions. For Left-Right model, to have a GB with invisible type of FCNC interaction with fermions, at least two bi-doublets plus more than one triplets scalars need to be introduced in the usual model to make GB interactions with fermions to be invisible similar as that in DFSZ model.
We have provided a general proof of the equivalence of using physical GB components and GB broken generators for calculating GB couplings to two gluons and two photons, although they have the different form. The final results only depend on the U (1) G charges X i L,R and the kind of colored, and charged particles in the model. Parameters in the FCNC interactions do not affect GB interactions with two gluons and two photons. We have shown that in models having spontaneous CP violation, there is no new CP violating phase for GB interactions with fermions.
Whether or not fundamental GB exist is of course an experimental issue. Several high luminosity facilities in running, such as the BESIII, LHCb, BELLE-II, will provide us with more information. We eagerly wait for more data to come to test models having FCNC GB interactions with fermions.