Lorentz-violating nonminimal coupling contributions in mesonic hydrogen atoms and generation of photon higher-order derivative terms

We have studied the contributions of Lorentz-violating CPT-odd and CPT-even nonminimal couplings to the energy spectrum of the mesonic hydrogen and the higher-order radiative corrections to the effective action of the photon sector of a Lorentz-violating version of the scalar electrodynamics. By considering the complex scalar field describes charged mesons (pion or kaon), the non-relativistic limit of the model allows to attain upper-bounds by analyzing its contribution to the mesonic hydrogen energy. By using the experimental data for the $1S$ strong correction shift and the pure QED transitions $4P \rightarrow 3P$, the best upper-bound for the CPT-odd coupling is $<10^{-12}\text{eV}^{-1}$ and for the CPT-even one is $<10^{-16}\text{eV}^{-2}$. Besides, the CPT-odd radiative correction to the photon action is a dimension-5 operator which looks like a higher-order Carroll-Field-Jackiw term. The CPT-even radiative contribution to the photon effective action is a dimension-6 operator which would be a higher-order derivative version of the minimal CPT-even term of the standard model extension.


I. INTRODUCTION
The possibility of Lorentz and CPT violation in quantum electrodynamics (QED) has been studied intensely in the last years. The principal motivation is the possibility of occurring the spontaneous breaking of both symmetries at very high energy, i.e., the Planck scale [1]. The standard model extension (SME) [2] is the main framework proposed to study the possible effects of Lorentz violation into the standard model of the fundamental particles and their interactions. In the minimal SME the Lorentz-violating (LV) coefficients concerning to photonic and fermionic sectors are added in such a way to preserve gauge symmetry and renormalizability and, some of they have strong experimental upper bounds [3]. Detailed studies about the LV coefficients of the fermionic sector can be founded in Refs. [4], for the ones of the CPT-odd photonic sector in Refs. [5][6][7] while for the CPT-even ones in Refs. [8][9][10]. The introduction of Lorentz violation by means of operators with dimensions higher than 4 was first considered in Ref. [11] by studying the effects on particle dispersions relation of a dimension-five operator. Others applications of such a proposal were performed in Refs. [12]. On the other hand, higher order operators are considered in the nonminimal SME, for example, the photon and fermion sectors were analyzed in Refs. [13,14]. Some implications are studied in Refs. [15]. The introduction of Lorentz violation in systems with scalar and/or fermion and gauge fields can be made via nonminimal couplings which could * Electronic address: rodolfo.casana@gmail.com † Electronic address: josberg.silva@ufma.br ‡ Electronic address: frederico.santos@ufma.br introduce higher order operators terms [16][17][18][19][20]. Some consequences or effects of that non-renormalizable coefficients have been investigated in several distinct scenarios [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].
The quantum field theory describing the interaction between spinless charged particles and the electromagnetic field is known as scalar quantum electrodynamics (sQED). The Lorentz-violating extension of sQED has not received same attention like its fermionic version. The reason could be the fact do not exist in standard model a fundamental spinless charged particle or because it is not so rich phenomenologically like its fermionic version. As an example, a CPT-odd power-counting renormalizable term is only possible in photonic and fermionic sectors. Even so, some Lorentz-violating of the scalar electrodynamics has been studied in recent years. For example, aspects like causality, unitarity and spontaneous symmetry breaking of a LV and CPT-odd sQED were analyzed in Ref. [36]. The Higgs mechanism in the context of a Lorentz-violating and CPT-even sQED was studied in Ref. [37]. At classical level many studies about existence of vortices BPS in Lorentz-violating scalar electrodynamics were performed in Refs. [38].
The aim of this manuscript is to study the 1-loop radiative generation of a nonminimal Lorentz-violating and CPT-odd coupling between spinless charged particles and the electromagnetic field in a scalar electrodynamics whose gauge sector includes the Carroll-Field-Jackiw (CFJ) term [39] which breaks both the Lorentz and CPT symmetries. Our study is organized as follows: In Sec. II we present the Lorentz-violating and CPT-odd scalar electrodynamics in which our proposed is based and we establish the respective Feynman rules. In Sec. III, we have computed at 1-loop order the first-order Lorentzviolating corrections to the 3-and 4-point vertex functions composed by scalar and photon fields. The 3-point vertex function allows to impose theoretical upper-bound for the generated coupling constant. In Sec. IV we have established the non-relativistic limit and used the experimental data from the strong shift of the ground state of the pionic (kaonic) hydrogen to impose upper-bounds, too. Finally, we summarize our conclusions and perspectives in Sec. V.

II. THE THEORETICAL FRAMEWORK
The basic framework of our investigation is a scalar electrodynamics with a massive photon whose kinetic term possess the CPT-odd and Lorentz-violating (LV) Carroll-Field-Jackiw (CFJ) term. The LV lagrangian density representing this model is given by where L φ is the charged scalar field part, L A the gauge part, and L I the interaction part. Explicitly: The covariant canonical quantization of the CPT-odd and Lorentz violating massive electrodynamics given in Eq. (3) was performed in [40,41]. The small photon mass term is included through the Stueckelberg mechanism and it is required for a consistent quantization. It must be observed that we have not included the CPT-odd term because it can be eliminate by means of the following field redefinition φ → e −iκ µ xµ φ which transforms (5) in It implies that LV and CPT-odd vector background κ µ can be absorbed by means of a mass redefinition therefore the interaction (5) is not a true LV term.
In the remain of the manuscript we will consider only the 1-loop corrections to the vertex interactions given by the Eq. (4) produced at first-order by the CFJ vector background.
The Feynman rules, in the Feynman gauge, we will use to attain our goal, are: • Photon propagator where S µν is the first order LV modification to photon propagator given by • Tree level scalar-photon 3-vertex it is read as • Tree level scalar-photon 4-vertex it is read as

III. 1-LOOP CORRECTIONS TO THE SCALAR-PHOTON VERTEX INTERACTIONS
The 1-loop radiative corrections we are interested are those containing only contributions at first-order in the Lorentz-violating background (K AF ) µ . The antisymmetric character of the LV tensor S µν precludes the firstorder LV corrections at 1-loop order for vacuum polarization tensor and pure scalar Green functions. However, the 3-and 4-vertex functions which mix the gauge and scalar fields have it. By using the Feynman rules enumerated before, we mount the associated amplitude to be In the following, we extract the first-order LV contributions which have no ultraviolet divergences so a regularization process is no necessary. Thus, the relevant Lorentz-violating integrals are We now introduce the momentum variables then by regarding the forward scattering process we will consider s 2 ≈ m 2 and only the first-order contributions in t, such that the Eq. (13) becomes After apply a set of known techniques to solve Feyn-man integrals, the Eq. (15) is reduced to be with where due to m γ m, we have considered only the more relevant contribution. By using it we rewrite the Eq. (16) in a simple form: where we have introduced the coupling constant gw ν defined by It is clear that in the limit where the photon has null mass appears a infrared divergence such as it was observed in a CPT-odd quantum electrodynamics [42]. Recently, in Refs. [40,41], the consistent analysis of the Cherenkov radiation including the CFJ term necessarily requires the use of a massive photon whose mass satisfies the condition m 2 γ > K 2 AF , i.e., the existence of CPT violation precludes the photon mass be null. Such a condition is in agreement with the current bounds for photon mass and CPT violation: By considering the mass of the complex scalar field m > m γ and K AF a space-like vector, we can show So, for K AF along the z-axis, we can rewrite the Eq. (19) as The maximum value of the right-hand side allows to impose a theoretical upper-bound for the constant gw ν , where e represents the magnitude of electron charge. Now we consider the boson particle to be the pion, the theoretical upper-bound becomes where m π = 139.57018(35)10 6 eV [43]. By considering the kaon, we attain a similar upperbound, where m K = 493.677(16)10 6 eV [43].
The Lorentz-violating contribution of the 3-point vertex function provides the following dimension-five operator to the effective action the CPT-odd coupling gw µ plays a role analog to an anomalous magnetic moment. At first sight this operator is no gauge invariant due to the absence of the covariant derivative, however the terms to turn it gauge invariant arise from the LV contributions to the 4-point vertex function.

B. The 4-point vertex function
The LV contribution to the 4-point vertex function is obtained from the following Feynman graph, It provides the amplitude Similarly to the 3-point case, we take into account only the first-order LV contributions, so we have In order to extract the relevant contributions for our purpose we again use the variables s and t defined previously in Eq. (14) and by considering the forward scattering limit (we set s 2 ≈ m 2 and consider only the smaller contributions of t and q), we obtain After some algebraic manipulations to solve the Feynman integrals, the LV contribution to the 4-point vertex function is By considering only the relevant contribution (17) of I (m γ /m) we can rewrite the expression (30) where we have using the coupling constant (19). It provides the following contribution to the effective action: Here it is necessary to make an observation about the absence of terms proportional to t in Eq. (30), the major reason is that in despite of they be non null their contributions to the effective action vanish identically. The gauge invariant contribution to the effective action is obtained by summing the terms (26) and (32), i.e., L ef f = L (3) + L (4) , This interaction could be useful in the study of Lorentz and CPT violation in scalar charged boson systems despite of it be a nonrenormalizable one.

IV. THE MESONIC HYDROGEN ATOMS
The pionic hydrogen is a system where the electron is replaced by a pion. While the standard hydrogen atom can be described by Dirac's equation, the pionic hydrogen can be in principle treated in the context of scalar electrodynamics because the spinless character of the pion. The crucial difference is the fact of the pion is an unstable composite particle constituted by a quark-antiquark pair with mean lifetime around 3.95 × 10 7 eV −1 . Many properties of the pionic hydrogen were studied in Ref. [44] by considering only QED effects. However, there are quantum chromodynamics (QCD) contributions to the binding energies and level widths of the atomic levels whose most notorious effect is presented by the 1S state. It is measured by means of the X-ray transitions by comparing with the pure electromagnetically bound state.
In the context of Lorentz and CPT violation some aspects of low-energy QCD are analyzed in Ref. [45] and for pions and nucleons in Ref. [46]. Both references perform their analysis within the formalism of chiral perturbation theory [47], an effective quantum field theory used to describe some low-energy aspects of QCD. Nevertheless, because of its large mass, the pionic hydrogen can be considered as a nonrelativistic system such that we use the transition 2P→1S to impose an upper-bound for the coupling constant gw µ of the CPT-odd and Lorentzviolating interaction obtained in Eq. (33).
The relevant contribution at nonrelativistic limit of the term (33) is which contributes to the Hamiltonian with the following perturbation, such an interaction is not predicted by pure quantum electrodynamics so it can be used to impose an upperbound for the coupling g π w. The LV coupling g π w is playing the role of a magnetic moment for the pion allowing to study it as a background-orbit interaction. Then, the correspondent correction to pionic hydrogen Hamiltonian is given by where µ is the reduced mass of the proton-pion system and the magnetic field in Eq. (35) was set to be in a similar way occurring in the spin-orbit case.
By considering, the background w along the z-axis, the correspondent shift to the pionic hydrogen energy is Only states with nonnull angular momentum projection L z will receive energy corrections so the energy of the 1S state remains unaltered. All other states gain the following energy correction: We observe the states with lower values of n and receive the most significant Lorentz violating corrections. Consequently, the 2P state is the more affected while 1S no receive LV correction.
In the pionic hydrogen the study of the transitions from excited levels 4P , 3P and 2P to the fundamental state 1S are used to measure the effects of the strong interactions [48]. The difference between standard QED predictions and experimental data are usually considered to compute the shift produced in the 1S state by QCD effects (see Fig. (3)). The experimental value of the shift is ε 1S ≈ (7.086 ± 0.007(stat) ± 0.006(sys))eV (40) to low. Our purpose is to consider such a energy shift by supposing the LV corrections are smaller than strong corrections error. Thus, we attain the following upper-bound, where we use the proton mass to be 938.272081(6)10 6 eV [43]. Such a bound is in according with the theoretical one obtained in Eq. (23). Similar procedure can be used with the kaonic hydrogen whose measured strong shift [49] is, ε 1S ≈ −283 ± 36(stat) ± 6(sys)eV, (42) such that it provides the upper-bound which is not better than the one provided by the pionic hydrogen.

V. REMARKS AND CONCLUSIONS
We have studied the radiative generation of a dimension-5 operator, (i/2)gw µ µναβ F αβ φ † ↔ D ν φ, in a scalar electrodynamics endowed with the Carroll-Field-Jackiw term breaking both the Lorentz and CPT symmetries. Such a term coupling the charged scalar and electromagnetic fields is ultraviolet finite but it could generate infrared divergences for massless photons. The presence of the CFJ background precludes the photon becomes massless because of the constraint m 2 γ > K 2 AF compatible with current bounds for photon mass and CPT violation [40,41]. Such bounds allow to impose theoretical upper-bounds for the coupling constant gw µ by considering the charged scalar particle to be a meson (pion or Kaon), the best one is obtained when consider the Kaon (see Table I).
By regarding the non relativistic contribution of (33) for the energy of a hadronic atom (pionic or kaonic hydrogen) it is observed that the background vector g w interacts with the magnetic field (see Eq. (35)) such that it looks like playing the role of a magnetic moment for scalar particle. By considering it like a spin-orbit interaction, the resulting Hamiltonian (see Eq. (36)) provides the energy correction given in Eq. (38). Such energy correction is compared with the experimental error of the measured 1S strong-shift of the hadronic atom (pionic or kaonic hydrogen) to obtain a upper-bound for the coupling g w. This time, the strong-shift of the pionic hydrogen gives a better upper-bound (see Table I).