Limits on non-minimal Lorentz violating parameters through FCNC and LFV processes

In this work we analyse a non-minimal Lorentz-violating extension of the electroweak theory in the fermionic sector. Firstly we analyse the relation between the CKM rotation in the quark sector and possible contributions of this new coupling to flavour changing neutral currents (FCNC) processes. In sequel we look for non-diagonal terms through possible leptonic flavour violation (LFV) decays. Strong bounds are presented to the Lorentz violating parameters of both the quark and the leptonic sectors.

In this work we analyse a non-minimal Lorentz-violating extension of the electroweak theory in the fermionic sector. Firstly we analyse the relation between the CKM rotation in the quark sector and possible contributions of this new coupling to flavour changing neutral currents (FCNC) processes. In sequel we look for non-diagonal terms through possible leptonic flavour violation (LFV) decays. Strong bounds are presented to the Lorentz violating parameters of both the quark and the leptonic sectors.

I. INTRODUCTION
Despite its great success, the Standard Model of Particle Physics should not be the final description of nature and it has been shown that in some of its extensions, string theory, for example, is possible that Lorentz symmetry is violated [1,2]. Observation of any, albeit small, sign of Lorentz symmetry violation (LSV) would represent a major paradigm shift and would require re-examination of the very basis of modern physics, i.e., relativity theory and quantum field theory [1][2][3][4].
In the present study we investigate the case proposed in [12] of a non-minimally coupling with a constant 4-vector background with a specific SM sector, the fermion -electroweak Bosons interacting sector. More specifically, we follow two main paths which are; possible contributions for flavour Changing Neutral Currents (FCNC) decays and contributions for Lepton flavour Violation (LFV) decays [13,14]. Studies involving mesons and LSV can be seen in [15,16]. In the first path we use the strong bounds in the FCNC mesons decays given by K 0 → µ + µ − , B 0 → µ + µ − and B 0 s → µ + µ − to find constrains involving the LSV parameters. Going further, we analyse the leptonic sector through the LFV decays µ → e + γ, τ → µ + γ and τ → e + γ. In the final comments we discuss our results.

II. NON-MINIMAL COUPLING IN THE ELECTROWEAK SECTOR
Based on the non-minimal coupling used in [12] one can extend the idea of a non-minimal coupling in the SU (2) × U (1) sector of the Standard Model. Starting from the implementation of the following covariant derivative proposed we have where Y = 2(Q − T 3 ), g and g are the U (1) Y and SU (2) L coupling constants, respectively, B µ and W I µ are the U (1) Y and SU (2) L Gauge bosons, the indexes A, B refers to Standard Model fermionic families, , * Electronic address: ymuller@cbpf.br † Electronic address: helayel@cbpf.br σ I refers to Pauli's matrices. Our analysis will begin in the quarks sector, where we expect to analyse the relationship between the Lorentz violation and the CP violation that is represented by the CKM Matrix. In the sequence we will analyse the leptonic sector, where the absence of Right neutrinos and the universality of weak interactions in this sector demand a thorough analysis.
A. The Quark sector and FCNC: Taking into account the flavour structure, the new covariant derivative will act in the quarks sector as follows: where A, B refers to the respective quark family (i.e., flavour). Here we use ( The Lagrangian above brings us the following new interaction terms in the Left sector: Since this coupling acts in the interaction sector of the fermionic sector with the electroweak bosonic fields, there will be no changes in the fermion masses neither in the gauge boson mass. As we know, the relationship between the Bosons B, W 3 and A, Z is given from the mass matrix diagonalization after the spontaneous symmetry breaking of the Higgs potential, and from this diagonalization appears the very known relationship: Rewriting the Lagrangian (3) in terms of the photon and Z-boson fields A, Z, we reach: where and Similarly, the couplings in the Right sector gives us the following interaction Lagrangian: The equation above is justified by the fact that that the Right sector is singlet under SU (2) L group transformations. Rewriting the above equation on the same {A, Z} basis we have: Finally, introducing the CKM matrix we have that with the standard parametrization choice, d A = V AB d B where d A represent the physical quarks. The CKM matrix expansion in terms of standard parameters λ, A, ρ e η (λ ≈ |V us | ≈ 0.23, where s 12 = λ, s 23 = Aλ 2 , s 13 e −iδ = Aλ 3 (ρ − iη)), with A ≈ 0.83, ρ ≈ 0.12 and η ≈ 0.35, gives the following CKM matrix approximation [17]: So we can rewrite the new Lagrangian and interaction with the Left and Right sectors. L total LSV = L Lef t LSV + L Right LSV and it is given by: The Interaction Lagrangian can be written explicitly as follows: where and we omit flavour indexes for simplicity. We shall pay attention to c 8 parameter, since it will be the main parameter in which contributes to the process we are interested.
From the equation (40), we can see that the CKM Matrix will not only appear in the charged currents as in the Standard Model, but now we will also get Lorentz violation parameters which can be complex due to the CP violation parametrized by the δ complex phase.
We assume Lorentz violation vectors such that families do not mix, but show dependence on the family. In other words, let's say ξ µ AB is given by: where ξ µ 11 = ξ µ 22 = ξ µ 33 , in principle. We assume that ρ shares the same characteristics. From this structure, after the rotation induced by the CKM matrix we obtain: andξ ji =ξ * ij for i = j. Note that the complex characteristic resides in the non-diagonal components of Lorentz violation parameters, but depends on the difference in magnitudes expressed in our assumption (14). In the low energy limit, that is, in the limit where the massive gauge bosons decay fast enough, the Z decay generates the following new effective interaction between neutral currents in the D-quark sector: Interestingly, while c 5 and c 6 remain diagonal in flavour space, c 7 and c 8 contain contributions from the CKM matrix, so they will not be diagonal, assuming Eq. (14). The c 3 and c 4 parameters also share this property. Importantly, the terms c 7 and c 8 generate the so-called Flavour Changing Neutral Current (FCNC) in a tree-level process. In SM this processes are forbidden in a tree level process, so it is an important result to be discussed. In addition, it is also important to point out that c 3 and c 4 can generate flavour exchange through photon coupling, a phenomenon that does not occur in the SM.
Analysing in detail the interaction that brings us the possibility of FCNC processes given by the equation (21), after some algebraic manipulations we can rewrite it as follows:  The standard method to work in meson decays is given by the following parametrization 0|sγ µ γ 5 d|K 0 (q) = iF K 0 q µ [12]. The vector current will be responsible for the vector mesons (e.g. , K * ), and a similar analysis can be done. Thus, the scattering matrix of the Kaon decay is given by: where (N ) µν 12 = (η µν q.(c 8 ) 12 − q µ (c 8 ) ν 12 ). Therefore, summing over spins of the square of the scattering matrix we reach: where J κ =μ(p)γ κ (v µ − a µ γ 5 )µ(q − p). After simplifications and using the rest frame of the neutral kaon where q = (M K 0 , 0) we reach : where we ignore terms proportional to v µ = 1 − 4s 2 W << 1 and c = c 8,12 = 1 4 c Wρ12 . By the use of the golden rule for two body decay we have: where p and p are the 4-momentum of the final states. Thus, the integrals are given as follows: where P = | p|, κ = (1 − 4y 2 ) and y = mµ M K 0 . Defining c = | c|(sin θ c cos φ c , sin θ c sin φ c , cos θ c ) with φ c , θ c generic angles, the angular integral can be calculated and it are given by: Therefore: So, applying the value of the constants we found the decay rate as follows: where a µ ≈ 0.5, G F = 1.16 × 10 −11 MeV −2 , M K 0 = 497.61 MeV and F K 0 = 164 MeV [18]. According to [18], in the Standard Model neutral Kaons with long half-lives K 0 L have a Decay Rate given by: such way that the Branching Ratio BR(K 0 L → µ + µ − ), experimentally limited to a value less than 6.8 × 10 −9 [18], will be given by : Therefore, the contribution to this decay arise from a possible Lorentz violation must be smaller or in the order of contribution from the Standard Model. From this statement we find the following bound for the spacial components of the background vectors: Similarly we can calculate the Branching Ratio for the B 0 (db) meson: and using the latest data from [18] we reach the following bound: Finally, for the B 0 s meson (sb) we find: and therefore we have: In summary, the bounds obtained are organized in the table II. In the next section we will analyse the leptonic sector.

B. The Leptonic sector and LFV :
Analysing now the lepton sector we shall recall that there is no CKM mechanism in this sector, so we shall be presenting an analysis based on an assumption of non-diagonal LSV parameters. As we will see, this non-diagonal terms gives us stronger bounds than the diagonal ones showed in the quark sector. Shallowly speaking, we have that after modification of the covariant derivative the Lagrangian corresponding to the leptonic sector can be written as follows [12,19]: are SU (2) L doublet and singlet respectively, and mass terms from Yukawa interactions are omitted as LSV terms do not influence them. Lagrangian (36) can be split into two components L = L ,SM + L LSV and the last component can be written as follows Writing the B µ and W 3 µ fields at the base of the Z and the photon fields the above equation will bring us the following Lagrangian interaction for the Left sector: The Lorentz violation implemented into the Right leptonic sector will be given by Rewriting the above equation on the {A µ , Z µ } basis, we reach: Let's now return to the full LSV Lagrangian. Using the definitions ( L ) A = P L A = 1−γ5 2 A and ( R ) A = P R A = 1+γ5 2 A , (ν L ) A = ν A , with A and ν A Dirac spinors, we are able to rewrite the Lagrangian as follows: 1 4 c W ρ µ and the flavour indexes are omitted for simplicity. The above Lagrangian is a generalization of the model seen in Ref. [12,19], since in this work the flavour structure are take into account.

Interaction
Vertex Table I: Vertex factors obtained from Eq.(40). Here q µ represents the A, W or Z 4-momentum.
As we can see a novel coupling between the photon and the neutral current is generated. Going further, we can also see that a coupling between the electromagnetic current and the Z 0 boson also arises. A possible decay from Lorentz violation will be the so-called neutrino-free muon decay, µ → e + γ. This process is prohibited in the Standard Model, so that experimentally there are strong limits to this decay, with Branching Ratio BR(µ → e + γ) < 4.2 × 10 −13 (90% C.L.) [14]. In the same way as the lepton tau decays we have BR(τ → eγ) < 3.3 × 10 −8 e BR(τ → µγ) < 4.4 × 10 −8 , both with 90% of confidence level [14].
From a momentum conservation perspective, the A → B + γ decay can occur as long as the mass of the A lepton is greater than the B mass. However in the Standard Model this decay is prohibited, so we can use this decay to find bounds for the Lorentz violation parameters.
Directly, the scattering matrix which describes the decay of the Feynman diagram shown in Fig. (2) is given by: where Γ µ AB = q ν (c  where y = m B /m A and we hide the flavour indexes c µ 1,AB and c µ 2,AB for simplicity. With that, we can calculate the decay rate of this process and, using the rest frame of the lepton A , is given by: where So: Using the most recent measurements, we have that the mean life of the muon is given by τ µ = 3.3 × 10 15 M eV −1 , and for the tau-lepton τ τ = 4.4 × 10 8 M eV −1 [18]. Thus we have the Branching ratio for the muon decay is given by [18]: Then we reach the following bound: where ∆ 2 12 = (c 0 1,12 ) 2 + (c 0 2,12 ) 2 . Similarly from the Branching Ratio BR(τ → µ + γ) < 4.4 × 10 −8 [18] we get the bound as follows: where ∆ 2 23 = (c 0 1,23 ) 2 + (c 0 2,23 ) 2 . Finally, in the process τ → e + γ, we have the Branching ratio upper bound given by BR(τ → e + γ) < 3.3 × 10 −8 [18]. Thus we get the third limit which will be given by: where ∆ 2 13 = (c 0 1,13 ) 2 + (c 0 2,13 ) 2 . Rewriting the limits obtained in terms of the initial 4-vectors we have with no sum over flavour indexes. The bounds are grouped in table III and the region plots are shown in Fig. 3.
The decays µ → e + γ and τ → µ + γ are drastically suppressed in the SM; this is very important to distinguish between the neutrino flavours. Now, with LSV, these decays , though very tiny, can occur and they signal a very meaningful aspect of LSV in particle physics.

III. FINAL COMMENTS
In this work we analyse the non-diagonal sectors (in the flavour space) of the proposed model [12] in the quark and leptonic sectors. In the quark sector we find that the CKM rotation could generate FCNC processes even if the Lorentz violation parameters were diagonal in flavour space. We realize that, through the FCNC processes, only the spatial components or the 4-vector parameters ξ and ρ contribute to the FCNC decays and we found limits between |10 −8 − 10 −10 | MeV −1 by use of the experimental FCNC bounds.
As can be seen in [19], we need to use the Sun Centered Frame (SCF) in order to took a better inertial frame than the earth frame. So, in the SCF the module of any vector V = (V x , V y , V z ) in the SCF are given by: where χ is the colatitude of the laboratory and V SCF = (V X , V Y , V Z ). In the case of the LHC collaboration we have χ ≈ 44 o .  where V AB = ( ρ AA − ρ BB ) SCF , for A, B = 1, 2, 3, |Ξ B 0 SCF | 2 = 0.98 (W 2 X − 0.12 V 2 13,X ) + 1.02 (W 2 Y − 0.12 V 2 13,Y ) + (W 2 Z − 0.12 V 2 13,Z ) + −(W X W Y − 0.12 V 13,X V 13,Y ) , with W = V 12 − ρ V 13 = ρ 11 − ρ 22 − ρ( ρ 11 − ρ 33 ), ρ ≈ 0.12 and η ≈ 0.35 [17]. As can be seen, the limits are not simple, specially the limit from the B 0 decay. Besides the complexity of the bound obtained from B 0 decay, the bound reached from the K 0 and the B 0 s decays can be visualized as a hyperboloid with width approximately below 10 −10 M eV −1 and 10 −8 M eV −1 , respectively.
In the leptonic sector the bound are functions of the temporal components ρ 0 AB . Where analysed in the SCF perspective we have ρ 0 AB ≈ ρ T AB , where ρ T AB is the temporal component of ρ µ AB,SCF . Therefore, we can affirm that we stronger bounds (|10 −12 −10 −16 | MeV −1 ) are found for the non-diagonal components of the LSV parameters by use of the lepton flavour violation branching ratios. Is important to highlight that our bounds in the leptonic sector are, comparing with our weakest bound, five times times more accurate than the bound reach in Ref. [19] (where ρ T SCF < 8 × 10 −7 M eV −1 ). τ → e + γ |∆13| = 0.19 (ξ T 13 ) 2 + 0.10 ξ T 13 ρ T 13 + 0.03 (ρ T 13 ) 2 < 2.4 × 10 −13 Table III: Limits for Lorentz violation parameters from the experimental limits of the leptonic LFV sector. Figure 3: Plot of the allowed regions in ξ T AB × ρ T AB parameter space. Here A, B = 1, 2 refers to µ → e + γ decay, A, B = 2, 3 refers to τ → µ + γ decay and A, B = 1, 3 refers to τ → e + γ decay. We use θ W = arccos(80/91).