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The Lorentz and CPT violating effects on the Z\to l^+ l^- decay

We study the Lorentz and CPT violating effects on the branching ratio BR, the CPT violating asymmetry A_{CPT} and the ratio of the decay width, including only the Lorentz violating effects, to the one obtained in the standard model, for the flavor dependent part of the lepton flavor conserving Z\to l^+ l^- (l=e,\mu,\tau) decay. The inclusion of the Lorentz and CPT violating effects to the standard model contribution is too small to be detected, since the corresponding coefficients are highly suppressed at the low energy scale.

A considerable theoretical work has been done to construct a fundamental theory at higher scales, like the Planck scale, which the standard model (SM) of particle physics is its low energy limit. In such scales, there are hints that the Lorentz and CPT symmetries are broken [1], in contrast to their conservation in the SM. The string theories [2] and the non-commutative theories [3] are the examples of high energy extensions of the SM. Even if the Lorentz and CPT symmetry violations exist in the extended theories given above, the small violations of these symmetries can appear at the low energy level.
The general Lorentz and CPT violating extension of the SM is obtained in [4,5]. In the extension of the SM the Lorentz and CPT violating effects are carried by the coefficients coming from an underlying theory at the Planck scale. These coefficients can arise from the expectation values in the string theories or some coefficients in the non-commutative field theories [3]. The loop quantum gravity [6], the space time foam [7] and cosmological scalar fields [8] are the possible sources of the Lorentz violating coefficients . Furthermore, the space-time varying couplings are also associated with the Lorentz violation, and they affect the photon dynamics [9] .
In the literature, there are various studies in which some of the coefficients are probed, by using the experiments [10,11]. The general Lorentz and CPT violating Quantum Electro Dynamics (QED) extension has been studied in [12,13] and in [13] the one loop renormalizability of this extension has been shown. In [14] the Lorentz and CPT violating effects on the branching ratio (BR) and the CP violating asymmetry A CP for the lepton flavor violating (LFV) interactions µ → eγ and τ → µγ, has been analyzed in the model III version of the two Higgs doublet model (2HDM) and the relative effects of new coefficients on these physical parameters have been studied. The Lorentz and CPT violating effects in the Maxwell-Chern-Simons model have been examined in [15,16] and these effects in the non-commutative space time have been analyzed in [17]. In [18], the theoretical overview of Lorentz and CPT violation has been done; in [19], the possible signals of Lorentz violation in sensitive clock-based experiments has been investigated and in [20], the superfield realizations of Lorentz-violating extensions of the Wess-Zumino model were presented. The threshold analysis of ultra-high-energy cosmic rays can also be used for Lorentz and CPT violation searches. The basis for such threshold has been investigated in [21].
In the present work, we study the Lorentz and CPT violating effects on the BR, the CPT violating asymmetry (A CP T ) and the ratio R of the decay width Γ, including only the Lorentz violating effects, to the one obtained in the SM, for the flavor dependent part of the lepton flavor conserving Z → l + l − (l = e, µ, τ ) decay. The additional contribution, coming from the Lorentz and CPT violating effects, on the physical parameters we study is too small to be detected, since the corresponding coefficients are highly suppressed at the low energy scale.
Our aim is to investigate the relative importance of the coefficients which are responsible for the Lorentz and CPT violating effects on the BR of the decays under consideration. Furthermore, we predict the possible CPT violating asymmetry A CP T which is carried by the limited number of coefficients, e µ and g µνα in the present process. The A CP T is sensitive to the flavor structure of the process, however, it is considerably small, as expected. Finally, we study the ratio R to understand contribution of the Lorentz and CPT violating effects on the flavor structure of the decay and we observe that these effects are too weak to be detected in the present experiments.
The paper is organized as follows: In Section 2, we present the theoretical expression for the decay width Γ, the A CP T and the ratio R, for the lepton flavor conserving Z → l + l − (l = e, µ, τ ) decay, in the case that the Lorentz and CPT violating effects are switched on. Section 3 is devoted to discussion and our conclusions.

τ ) decay with the addition of the Lorentz and CPT violating effects
In this section, we study the Lorentz and CPT violating effects on the BR, the CPT asymmetry and the ratio R for the leptonic Z decay. In the SM, this process is allowed at tree level and the where Here the coefficients c αµ , d αµ , e µ , f µ and g λνµ are responsible for the Lorentz violation. Even if the U(1) charge symmetry and renormalizability does not exclude the part of lagrangian including the coefficients e µ , f µ and g λνµ , they are not compatible with the electroweak structure of the SM extension. However, the possible nonrenormalizable higher dimensional operators respecting the electroweak symmetry and including Higgs field with vacuum expectation value can create these highly suppressed terms (see [4] for details). In our analysis we also take these terms into consideration since they are sources of the CPT violation ( [13]). Now, we would like to present the additional vertex due to the Lorentz and CPT violating effects for the Z → l + l − decay: where Our aim is to calculate the decay width of the Z → l + l − process including the Lorentz violating effects. It is known that the invariant phase-space elements in the presence of Lorentz violation are modified [16]. In the conventional case where there are no Lorentz violating effects, the well known expression for decay width in the Z boson rest frame reads dΓ = (2π) 4 1 In the case of the existence of the new Lorentz violating effects lying in the part −ψM ψ where M = m+M 1 , M 1 = a µ γ µ + b µ γ 5 γ µ + 1 2 H µν σ µν (see [13] for details), the modified Dirac equation becomes (γ µ q µ − m − M 1 + Γ µ 1 q µ ) ψ = 0 not take into account these tiny additional effects 2 and use the conventional expression for the decay width in the Z boson rest frame (see eq. (4)). With the inclusion of the Lorentz violating effects in the matrix element, the Lorentz violating part of the decay width Γ(Z → l + l − ) is obtained as Here s l = m 2 Z 4 − m 2 l , the parameters c 00 and d 00 are the zeroth components of the coefficients c αβ and d αβ and the last term g is g = ǫ ijk g ikj , where i, j, k = 1, 2, 3. Notice that we take only the additional part of the decay width which is linear in the Lorentz violating coefficients.
The eq. (6) shows that Γ LorV io depends on the CPT even (c 00 and d 00 ) and the CPT odd g coefficients. Using Γ LorV io it is easy to calculate the Lorentz violating part of the BR (BR LorV io ) as: where the Γ Z is the total decay width of the Z boson and its numerical value is Γ Z = 2.490 (GeV ).

This equation shows that the A CP T depends on the flavor part of the decay under consideration
and becomes larger for the heavier lepton pair decay.
Finally, we study the ratio R = where Γ f lavor LorV io (Γ f lavor SM ) is the flavor dependent part of the decay width including only the Lorentz violating (the SM without Lorentz violating) effects. This ratio is sensitive to the lepton mass and it is dominant for the light lepton pair decay.

Discussion
The SM is invariant under the Lorentz and CPT transformations, however, the small violations of Lorentz and CPT symmetry, possibly coming from an underlying theory at the Planck scale, can arise in the extensions of the SM. In this section, we analyze the Lorentz and CPT violating effects on the BR and the A CP T for the Z → l + l − (l = e, µ, τ ) decays, in the SM extension.
Furthermore, we study the ratio R = The natural suppression scale for these coefficients can be taken as the ratio of the light one m l to the one of the order of the Planck mass. Therefore, the coefficients which carry the Lorentz and CPT violating effects are roughly in the range of 10 −23 − 10 −17 [11]. Here the first (second) number represent the electron mass m e (m EW ∼ 250 GeV ) scale. We take the numerical values of the coefficients |d|, |c|, |e|, |g| at the order of the magnitude of 10 −23 − 10 −17 .
In Fig. 1 (2), we present the magnitude of the coefficient dependence of the Lorentz violating part of the BR (BR LorV io ) for the decay Z → τ + τ − (µ + µ − ). Here solid (dashed, small dashed) line represents the dependence to the coefficient c 00 (d 00 , g), in the case that the other coefficients have the same numerical value 10 −20 . Notice that, in the figures, the parameter χ denotes to the size of c 00 , d 00 and g for different lines. It is observed that the BR is more sensitive to the coefficient c 00 compared to the others and the contribution of the new effects to the BR is at the order of the magnitude of 10 −19 for the large values of the coefficient c 00 ∼ 10 −17 . With the increasing values of the coefficient g the BR decreases for Z → τ + τ − decay and this effect is weak for the light lepton pair case, namely Z → µ + µ − decay. Notice that the Lorentz violating coefficient dependence of the BR LorV io for the decay Z → e + e − is almost the same as the one for the decay Z → µ + µ − . Now, we analyze the CPT violating asymmetry A CP T for the decays under consideration.
The coefficient g and the lepton flavor are responsible for this violation as seen in the eq. (8).
The A CP T enhances with the increasing lepton mass.
Finally, we study the ratio R = At this stage we would like to summarize our results: We analyze the Lorentz and CPT violating effects on the BR, A CP T and the ratio R and we study the relative behaviors of different coefficients by taking their numerical values at the order of the magnitude of 10 −20 − 10 −17 : • The contribution of the Lorentz and CPT violating part to the BR of the decays Z → l + l − (l = e, µ, τ ) is at most at the order of the magnitude of 10 −19 for the large values of the coefficients c 00 , d 00 and g and these numbers are too small to be detected.
• We predict the numerical value of A CP T at the order of 10 −20 (10 −21 , 10 −23 ) for the large values of the coefficient g ∼ 10 −17 for the decay Z → τ + τ − (µ + µ − , e + e − ). This physical parameter is driven by the coefficient g and the lepton flavor. It enhances with the increasing values of lepton mass.
• We study the ratio R =