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

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


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 a e-mail: ymuller@cbpf.br (corresponding author) b e-mail: helayel@cbpf.br flavor Changing Neutral Currents (FCNC) decays and contributions for Lepton flavor 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 analyze the lepton sector through the LFV decays μ → e + γ , τ → μ + γ and τ → e + γ . In the final comments we discuss our results.

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 ξ ν AB and ρ ν AB are LSV parameters, 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 fami- , σ I refers to Pauli's matrices. Our analysis will begin in the quarks sector, where we expect to analyze the relationship between the Lorentz violation and the CP violation that is represented by the CKM Matrix. In the sequence we will analyze the lepton sector, where the absence of Right neutrinos and the universality of weak interactions in this sector demand a thorough analysis.

The Quark sector and FCNC:
Taking into account the flavor structure, the new covariant derivative will act in the quarks sector as follows: where A, B refers to the respective quark family (i.e., flavor).
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: where c W = cos θ W , s W = sin θ W and θ W is the Weinberg angle. Rewriting the Lagrangian (3) in terms of the photon and Z-boson fields, 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]: Here The Interaction Lagrangian can be written explicitly as follows: where and we omit flavor 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. The set of vertex are shown in Table 1. From the Eq. (42), 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. This approximation is based on the fact that non-diagonal elements will allow C decays (c → u + γ /Z for instance), highly suppressed in the SM. From this structure, after the rotation induced by the CKM matrix, we obtain: andξ ji =ξ * i j 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 Interestingly, while c 5 and c 6 remain diagonal in flavor 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 flavor Changing Neutral Current (FCNC) in a treelevel 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 flavor exchange through photon coupling, a phenomenon that does not occur in the SM. Analyzing in detail the interaction that brings us the possibility of FCNC processes given by the Eq. (21), after some algebraic manipulations we can rewrite it as follows: . Now we can calculate the Decay Rate of some possible processes in the non-diagonal sector in the flavor space.
Neutral Kaon Decay: If there is a contribution to the neutral current with FCNC then the neutral K mesons (sd and ds ) could easily decay into a pair of muons, as is shown in Fig. 1.
The standard method to work in meson decays is given by the following parametrization 0|sγ μ γ 5 d|K 0 (q) = i F 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 ) . 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 = 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 B R(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, using M B 0 = 5279, 6 MeV, F B 0 = 186 MeV [18]: and using the latest data from [18] we reach the following bound: Finally, for the B 0 s meson (sb) we find, using M B 0 s = 5366.9 MeV, F B 0 s = 224 MeV [18]: and therefore we have: In summary, the bounds obtained are organized in the Table 3. In the next section we will analyze the lepton sector.

The lepton sector and LFV
Analyzing 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 lepton sector can be written as follows [12,19]: here (L L ) A = ((ν L ) A , ( L ) A ) T and ( R ) B are SU (2) L doublet and singlet respectively, and mass terms from Yukawa interactions are omitted as LSV terms do not influence them. Lagrangian (37) can be split into two components L = L ,SM + L L SV 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: where we define for convenience The Lorentz violation implemented into the Right lepton 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 A , (ν L ) A = ν A , with A and ν A Dirac spinors, we are able to rewrite the Lagrangian as follows: Fig. 2 Hypothetical decay of lepton A into B and a photon due to LSV where 1 4 c W ρ μ and the flavor indexes are omitted for simplicity. The above Lagrangian is a generalization of the model seen in Refs. [12,19], since in this work the flavor structure are take into account.
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 B R(μ → e + γ ) < 4.2 × 10 −13 (90% C.L.) [14]. In the same way as the lepton tau decays we have B R(τ → eγ ) < 3.3 × 10 −8 e B R(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: , m A > m B and are no sum over A, B. Using the identity k μ,k * ν,k = η μν − q μ q ν q 2 and calculating the traces over gamma matrices we get the following expression: where y = m B /m A and we hide the flavor 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 MeV −1 , and for the tau-lepton τ τ = 4.4 × 10 8 MeV −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 B R(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 B R(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 flavor indexes. The bounds are grouped in Table 4 and the region plots are shown in Fig. 5a.
The decays μ → e + γ and τ → μ + γ are drastically suppressed in the SM; this is very important to distinguish between the neutrino flavors. Now, with LSV, these decays , though very tiny, can occur and they signal a very meaningful aspect of LSV in particle physics.

Final comments
In this work, we have focused on the analysis of the proposed new couplings (in flavor space) present in the model [12] in the quark and lepton sectors. In the quark sector, based on the approximation given by Eq. (14), we assume that non-diagonal LSV components will affect the sector of the u-, c-and t-quarks, generating, for instance, C = 0 decays, i.e., D + → π + + μ + + μ − . These events are highly suppressed in the SM, and this supports our approximation (14). This type of decay is a good candidate to confirm if our assumption is solid. It would also allow us to find bounds on the spatial components of non-diagonal LSV parameters and could be a further step towards a more complete analysis of the model. We find that the CKM rotation could generate FCNC processes, even if the Lorentz violation parameters were diagonal in flavor 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 considering the experimental FCNC bounds.
As it can be seen in [19], we need to use the Sun-Centered Frame (SCF) in order to choose a reference system better than the Earth frame. So, in the SCF the modulus of the vector V = (V x , V y , V z ) is found: where χ is the co-latitude of the laboratory and V SC F = (V X , V Y , V Z ). In the case of the LHC Collaborations, we have χ ≈ 44 • .
As we have shown, the FCNC decays impose bounds on the spatial components of the non-diagonal(in flavor space) LSV parameters ρ μ AB . On the other hand, the lepton sector gives us bounds that depend on the time components ρ 0 AB . Analyzing from the SCF perspective, we have ρ 0 AB ≈ ρ T AB , where ρ T AB is the time component of ρ μ AB,SC F . Therefore, we sector, and this coupling can be compared with the coupling used in [20]. In the aforementioned paper, some bounds are fixed through QED processes and the best bounds found are given by c T 1 < 10 −6 MeV −1 ; the allowed region given by this limit is depicted in Fig. 5a. By analyzing it, we can state that our choice of considering the LFV sector gives us stronger bounds, 7 order of magnitude, in the case of B-mesons, up to 10 orders of magnitude in the Kaon decay case (Table 3).
Finally, we can see that, in the electron sector, the interaction Lagrangian given by¯ 1 (c μ 1 γ ν + c μ 2 γ ν γ 5 ) 11 1 ∂ [μ A ν] ⊃ c 0 2 σ · E, generates an Electric Dipole Moment (EDM) to the electron (in fact for all fermions). Since the ACME experiment reveals that the magnitude of the electron's EDM has the upper bound d e < 8.7 × 10 −29 e · cm = 1.3 × 10 −16 MeV −1 [21], we attain the following bound: Besides the strong limit above, no other lepton-EDM can be used to impose limits on LSV parameters, since neither muon nor tau EDM experiments have been carried out so far. In the quark sector, we proceed in a similar way. By taking the best bounds on the up-and down-quarks EDMs, obtained through the proton and neutron EDMs [22], we find the bounds d u < 1.1x10 −27 e.cm = 1.5 × 10 −15 MeV −1 and d d < 6.5 × 10 −28 e.cm = 9.9 × 10 −16 MeV −1 , respectively. Thus, the following bounds can be set: and ρ T 11 + 0.05(ρ T 22 − ρ T 11 ) < 9.9 × 10 −16 MeV −1 As it can be seen, the limit obtained by the u-quark EDM gives us a bound on ρ T 11 a little better than in the case we adopt the electron EDM. By taking the d-quark EDM, we can find a bound on the ρ T 22 LSV parameter. These limits may be seen in Table 4 and in Fig. 5b. Due to experimental limitations in measuring the EDMs of other SM particles, there are no other possible EDM bounds until the moment.
As a step forward, a more complete analysis of the neutrino sector should be studied, particularly the neutrino-photon sector. The new coupling shown in Table 2 could be limited by astrophysical measurements and could bring interesting insights about the relation between LSV and flavor physics in neutrino interactions.