Classical Lagrangians for the nonminimal spin-nondegenerate Standard-Model Extension at higher orders in Lorentz violation
aa r X i v : . [ h e p - t h ] F e b Classical Lagrangians for the nonminimal spin-nondegenerateStandard-Model Extension at higher orders in Lorentz violation
Jo˜ao A.A.S Reis a,b , Marco Schreck a, ∗ February 2021 a Departamento de F´ısica, Universidade Federal do Maranh˜aoCampus Universit´ario do Bacanga, S˜ao Lu´ıs – MA, 65085-580, Brazil b Departamento de F´ısica, Centro de Educa¸c˜ao, Ciˆencias Exatas e NaturaisUniversidade Estadual do Maranh˜ao, Cidade Universit´aria Paulo VI, S˜ao Lu´ıs – MA, 65055-310, Brazil
Abstract
We present new results for classical-particle propagation subject to Lorentz violation. Our analysis is dedicated to spin-nondegenerate operators of arbitrary mass dimension provided by the fermion sector of the Standard-Model Extension. In particular,classical Lagrangians are obtained for the operators ˆ b µ and ˆ H µν as perturbative expansions in Lorentz violation. The functional de-pendence of the higher-order contributions in the background fields is found to be quite peculiar, which is probably attributed toparticle spin playing an essential role for these cases. This paper closes one of the last gaps in understanding classical-particlepropagation in the presence of Lorentz violation. Lagrangians of the kind presented will turn out to be valuable for describingparticle propagation in curved backgrounds with di ff eomorphism invariance and / or local Lorentz symmetry explicitly violated. Keywords:
Lorentz violation, Standard-Model Extension, Modified fermions, Classical Lagrangians
1. Introduction
Theories of physics at the Planck scale such as strings [1],loop quantum gravity [2], noncommutative spacetime struc-tures [3], and spacetime foam [4] as well as nontrivial space-time topologies [5] and Hoˇrava-Lifshitz gravity [6] predict vio-lations of Lorentz invariance. Planck-scale physics is presentlynot testable by direct means, but the low-energy fingerprints ofLorentz violation could show up in feasible experiments per-formed at much lower energies. To be able to translate theabsence of signals for Lorentz violation into constraints onmeaningful physical quantities, the Standard-Model Extension(SME) was constructed [7, 8] as an e ff ective field theory frame-work that parameterizes deviations from Lorentz symmetry.The SME can be decomposed into a nongravitational [7, 9–11] and a gravitational part [8, 12] where each of them on itsown consists of a minimal and a nonminimal sector. The mini-mal SME incorporates Lorentz-violating field operators of massdimensions 3 and 4, whereas the nonminimal SME comprisesall higher-dimensional operators. The nonminimal contribu-tions for photons, neutrinos, and Dirac fermions without gravityare neatly classified in [9, 10] where [11] provides a more gen-eral classification that respects the gauge structure of the Stan-dard Model (SM). The most recent work [12] clarifies variousaspects of Lorentz and di ff eomorphism violation in gravity and ∗ Corresponding author
Email addresses: [email protected] (Jo˜ao A.A.SReis), [email protected] (Marco Schreck) introduces a method to construct operators of arbitrary massdimension that are invariant under general coordinate transfor-mations. At large, investigations of Lorentz violation in gravitycan be intricate, which is why linearized gravity is the base ofa series of papers such as [13, 14]. Searches for
CPT -violationare sometimes considered as more important than searches forLorentz violation. Since
CPT violation implies Lorentz viola-tion in e ff ective field theory according to a theorem by Green-berg [15], the SME automatically comprises all CPT -violatingoperators.One of the great successes of the SME over the past twodecades is the large number of tight constraints on both min-imal and nonminimal Lorentz violation in the nongravitationalpart [16]. Lorentz violation in the gravity sector has also beenconstrained, but most of these bounds are significantly weakerthan those of the nongravitational SME. The reason is undoubt-edly that gravitational experiments of high precision are morechallenging to perform in comparison to experiments that areinsensitive to gravity. After all, the experimental value of New-ton’s constant has the largest experimental uncertainty in com-parison to the remaining constants of nature.Therefore, Lorentz violation may well hide in the gravita-tional sector and may have remained unnoticed, so far. Also,there are certain types of Lorentz violation that cannot be ob-served in Minkowski spacetime even if they are enormous, asthey can be removed from the Lagrange density by a redefini-tion of the physical fields. However, such a redefinition losesits validity in the presence of gravity, whereupon the enormousvalue would be suppressed by the weakness of the gravitational
Preprint submitted to Elsevier February 2021 nteraction [17].Earthbound experiments searching for deviations from thelaws of General Relativity include tests of the weak equiva-lence principle performed with test masses of di ff erent mate-rials in drop-tower experiments as well as torsion pendulumand gravimeter experiments. Furthermore, space-born experi-ments such as Gravity Probe B or laser ranging experiments canset bounds on violations of local Lorentz invariance in gravity(see [18] for a compilation of such tests). Lorentz violationin the gravity sector can also be constrained via the absenceof gravitational vacuum Cherenkov radiation that would implyenergy losses of ultra-high-energetic particles by the emissionof gravitons [19]. The announcement of the direct detection ofgravitational waves in 2016 [20] opened another window forsearches for Lorentz violation in gravity, as the latter wouldmodify the emission and propagation properties of gravitationalwaves [14, 21].Earth-based experiments in the gravitational sector involveextended test masses that are beyond the field-theory descrip-tion of the SME. Hence, to test local Lorentz invariance in grav-ity, it is desirable work with an SME-equivalent that parameter-izes Lorentz violation for classical, relativistic, pointlike parti-cles. An algorithm that provides such a description was pro-posed in [22] and since then it has been applied to various setsof Lorentz-violating operators [23–28]. Some of these classi-cal Lagrangians are known to be connected to Finsler struc-tures [29] that give rise to Finsler geometries. Finsler geometryis less restrictive than Riemannian geometry [30, 31], as it doesnot necessarily rely on quadratic path length functionals [32–34].Further studies in this realm involve the reverse process froma Lagrangian back to the Hamiltonian of a field theory [35],methods of removing singularities of Finsler spaces [36] ac-cording to Hironaka’s theorem [37], applications within classi-cal mechanics and electrostatics [38] as well as investigations ofscalar fields in Finsler spaces [39]. Results on Finsler structuresthat occur for Lorentz-violating photons in the eikonal limit arealso available [40].To complete the picture, a systematic treatment of the spin-nondegenerate fermion operators is still missing. Although re-sults for nonminimal operators at first order in Lorentz violationare already available [26], nothing is known about the structureof higher-order contributions. This gap shall be closed with thecurrent article.Our paper is organized as follows. Section 2 gives a summaryon the most important characteristics of the modified Diracfermion sector of the nonminimal SME that will be of impor-tance in the subsequent analysis. Section 3 summarizes the pro-cedure of how to map a field theory to the description of a clas-sical, relativistic, pointlike particle in terms of a Lagrangian. InSec. 4 some intermediate results are obtained that allow us todetermine the perturbative expansion of the Lagrangians con-sidered. An application of the method in Sec. 5 provides thecovariant Lagrangian for dimension-5 b and H coe ffi cients atsecond order in Lorentz violation. Finally, all findings are con-cluded on in Sec. 6. The appendix presents worthwhile compu-tational details that are of lesser interest to be shown in the main body of the paper. Natural units will be used with ~ = c =
2. Dirac field theory modified by spin-nondegenerate oper-ators
The Dirac fermion sector of the SME describes modifiedspin-1 / L = ψ (i ✁ ∂ − m ψ + ˆ Q ) ψ + H.c. , (1)where ψ is a Dirac spinor field, ψ ≡ ψ † γ the Dirac-conjugatedfield, m ψ the fermion mass, and the identity matrix in spinorspace. All fields are defined in Minkowski spacetime withmetric η µν of signature ( + , − , − , − ). Furthermore, ✁ ∂ ≡ γ µ ∂ µ with the standard Dirac matrices satisfying the Cli ff ord algebra { γ µ , γ ν } = η µν .The operator ˆ Q comprises all contributions that are in accor-dance with the spinor structure of Dirac theory. It consists of aspin-degenerate part that involves the operators ˆ a µ , ˆ c µ , ˆ e , ˆ f , andˆ m . Spin-degenerate Lorentz violation does not lift the twofolddegeneracy of the fermion dispersion relation, i.e., the energy-momentum dependence for spin-up fermions is the same as thatfor spin-down fermions. Apart from these operators, ˆ Q containsa spin-nondegenerate part including the operators ˆ b µ , ˆ d µ , ˆ H µν ,and ˆ g µν . Dirac fermions that interact with background fieldsof these types have propagation properties dependent on theirspin projection. In other words, the dispersion relation of spin-up fermions di ff ers from that for spin-down fermions. Eachof these operators mentioned can be decomposed into a sumof controlling coe ffi cients contracted with a certain number offour-derivatives that successively increases by 2. The mass di-mension d of these controlling coe ffi cients depends on the num-ber of four-derivatives that occur in the operator. Coe ffi cientsof negative mass dimensions are contained in the nonminimalSME and the corresponding operators are power-counting non-renormalizable.In general, the structure of the spin-nondegenerate operatorsis way more involved than that of the spin-degenerate ones,which is a property that will also become evident in the forth-coming sections of the paper. The underlying reason is theirdependence on the spin projection, which is absent for the spin-degenerate operators. For example, the complicated structureof the spin-nondegenerate operators shows up in their disper-sion relations, the spinor solutions of the modified Dirac equa-tion [10, 41, 42] as well as the plethora of di ff erent behaviors inunusual particle physics processes such as vacuum Cherenkovradiation [43].
3. Classical kinematics
To evaluate data from experimental tests of local Lorentzinvariance in gravity, it is desirable to have a framework pa-rameterizing Lorentz violation for classical, pointlike particles.2irst of all, gravity is dominant for macroscopic test bodies,whereupon their behavior is dominated by the laws of classicalphysics. Second, with the law of motion of a pointlike particleat hand, at least the translational behavior of a macroscopic ob-ject consisting of these particles is obtained from integrationsover suitable mass distributions.The SME itself is based on field theory and does not describeLorentz violation for classical particles. Therefore, the modi-fied Dirac fermion sector of Eq. (1) must be mapped suitably tothe Lagrangian L = L ( u ) of a relativistic pointlike particle mov-ing with four-velocity u µ . A reasonable map was constructedaround ten years ago in [22] and it is governed by a set of fiveordinary, nonlinear equations: D ( p ) = , (2a) ∂ p ∂ p i = − u i u , i ∈ { , , } , (2b) L = − p µ u µ . (2c)Equation (2a) is the dispersion equation of the modified Diractheory in Eq. (1). The latter depends on the four-momentum p µ and follows from the requirement that the modified Diracequation have nontrivial spinor solutions. Equations (2b) saythat the centroid of a wave packet constructed from plane-wavesolutions of the modified Dirac equation moves with a groupvelocity equal to the three-velocity u / u of the correspondingclassical particle. The minus sign on the right-hand side takesinto account the di ff erent positions of the spatial index i on bothsides of the equations. Finally, the Euler equation (2c) holdsfor a Lagrangian L that is positively homogeneous of degree 1: L ( λ u ) = λ L ( u ) for λ > p µ = − ∂ L ∂ u µ . (3)The five equations (2) depend on the four-momentum com-ponents, the four-velocity components, and the classical La-grangian to be determined. They should allow us to expressthe four-momentum completely in terms of the four-velocityand to state the Lagrangian as a function of the four-velocity.However, what is expected to work in theory, is challengingin practice, since the equations are both nonlinear and cou-pled. Over the past decade, di ff erent techniques were appliedto successfully obtain Lagrangians for various sets of control-ling coe ffi cients. The first Lagrangians were derived in [22]probably by directly manipulating the system of equations. Bydoing so, the authors obtained Lagrangians for both the spin-degenerate and the spin-nondegenerate fermion sector of theminimal SME. Further results followed in [23].It was soon realized that classical Lagrangians in the con-text of the nonminimal SME were more challenging to derive. Nonminimal operators come with additional powers of four-momentum components increasing the nonlinearity of Eqs. (2).It seemed that Lagrangians exact in Lorentz violation may behighly nontransparent and too unwieldy to be used in applica-tions [24]. Therefore, the focus changed to obtaining such La-grangians at first order in Lorentz violation only. This was doneto simplify computations, but such results were also thoughtto be su ffi cient from a practical viewpoint, as Lorentz viola-tion (at least in Minkowski spacetime) is already tightly con-strained [16].The first Lagrangians of the nonminimal SME at leading or-der in Lorentz violation were derived in [25] by solving Eqs. (2)with the help of Gr¨obner bases. Due to the complexity of spin-nondegenerate operators, the latter analysis only involves clas-sical Lagrangians for spin-degenerate operators. Round abouttwo years later, these findings were complemented in [26] byincluding spin-nondegenerate operators, which provided thefull classical-particle equivalent to the SME at leading orderin Lorentz violation. The experience on the form of such La-grangians gained in the years before greatly contributed to ob-taining these results. Finally, a powerful method to derive per-turbative series of Lagrangians in the Lorentz-violating coe ffi -cients was presented in [27] and applied to a modified scalarfield theory to obtain such series to third order in Lorentzviolation. A subsequent analysis [28] led to equivalent re-sults for the spin-degenerate operators of the nonminimal Diracfermion sector. So far, an analogous investigation for the spin-nondegenerate operators has not been carried out.The intention of the current paper is to establish ties to thespin-nondegenerate operators. We will restrict the analysis tocontrolling coe ffi cients b ( d ) µα ...α ( d − , d ( d ) µα ...α ( d − that are totallysymmetric and H ( d ) µνα ...α ( d − , g ( d ) µνα ...α ( d − that are antisymmet-ric in the first two indices and totally symmetric in the remain-ing ones. Similar assumptions were taken to obtain perturbativeseries of Lagrangians for the spin-degenerate operators in [28].These restrictions are minor in comparison to how they sim-plify the computations. In many cases the coe ffi cients with thedescribed symmetries are dominant, whereas the others are sup-pressed (compare to the leading-order results of [26, 27] thatonly involve the totally symmetric sets of coe ffi cients).The perturbative method first proposed in [27] and appliedto the spin-degenerate fermion operators in [28] shall now beadopted suitably such that it can lead to perturbative series ofLagrangians for the spin-nondegenerate operators ˆ b µ , ˆ d µ , ˆ H µν ,and ˆ g µν . To avoid couplings between di ff erent types of coef-ficients or coe ffi cients of di ff erent mass dimensions, we onlyconsider a particular coe ffi cient type and a fixed mass dimen-sion at a time. The crucial di ff erence to the spin-degenerateoperators is that the dispersion equation (even for the minimalframework) is no longer quadratic in the four-momentum, butquartic, at least. For ˆ b µ and ˆ H µν they can be cast into the form (cid:12)(cid:12)(cid:12) p − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ − m ψ (cid:12)(cid:12)(cid:12) = Υ b , (4a) | p + X − m ψ | = Υ H , (4b)3ith the valuable observer Lorentz scalars Υ b ≡ q ( b ( d ) ⋄ ) − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ p , (4c) Υ H ≡ q X p − H ( d ) ν ⋄ ( H ( d ) ) ⋄ ν − ˆ Y , (4d)ˆ X ≡ H ( d ) µν ⋄ ( H ( d ) ) ⋄ µν , ˆ Y ≡ H ( d ) µν ⋄ ( ˜ H ( d ) ) ⋄ µν , (4e)and the dual of ˆ H µν in momentum space:˜ H ( d ) µν ⋄ ≡ ε µν̺σ ( H ( d ) ) ⋄ ̺σ . (4f)Here, ε µν̺σ is the totally antisymmetric Levi-Civita symbol infour spacetime dimensions with ε =
1. We employ the ⋄ no-tation that was introduced in [28] for convenience. It indicatescoe ffi cients suitably contracted with four-momenta (as opposedto four-velocities).Comparing Eqs. (4a) and (4b) with each other, it is evidentthat they share certain similarities, but there are also crucial dif- ferences. The observer scalar ˆ Y is nonzero only when there isat least one nonzero purely spatial component operator ˆ H i j anda nonzero mixed one ˆ H i . If ˆ Y =
0, we can directly identify( b ( d ) ⋄ ) ↔ − H ( d ) ν ⋄ ( H ( d ) ) ⋄ ν , b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ ↔ − X , (5)which reveals that there is a certain correspondence betweenthese operators at the level of the dispersion equation.
4. Basic results for perturbative expansion
The procedure to obtain a perturbative expansion for a classi-cal Lagrangian starts with a computation of the implicit deriva-tive of the dispersion equation with respect to p µ and to useEq. (2b). A contraction of the result with the spatial momen-tum components p j and a subsequent application of Eq. (2c) aswell as taking into account the general form of ˆ b µ , ˆ d µ , ˆ H µν , andˆ g µν for a fixed mass dimension d implies the following four-velocities as functions of the four-momentum: u µ | b = − L ± Υ b h p µ − ( d − b ( d ) µν ⋄ ( b ( d ) ) ⋄ ν i + b ( d ) ν ⋄ ( b ( d ) ) ⋄ ν p µ − b ( d ) ⋄ b ( d ) µ ⋄ + ( d − h b ( d ) µν ⋄ ( b ( d ) ) ⋄ ν p − b ( d ) ⋄ b ( d ) µ ⋄ i ± Υ b h p − ( d − b ( d ) ρ ⋄ ( b ( d ) ) ⋄ ρ i − ( d − Υ b , (6a) u µ | H = − L ± Υ H [ p µ + d −
3) ˆ X µ ] − Xp µ − H ( d ) µν ⋄ ( H ( d ) ) ⋄ ν + ( d − − X µ p − p ν H ( d ) ν̺µ ⋄ ( H ( d ) ) ⋄ ̺ + Y ˆ Y µ ) ± Υ H [ p + d −
3) ˆ X ] − ( d − Υ H + ( d −
4) ˆ Y , (6b)where, for convenience, we additionally defined the useful ob-server four-vectorsˆ X ̺ ≡ H ( d ) µν̺ ⋄ ( ˆ H ( d ) ) ⋄ µν , ˆ Y ̺ ≡ H ( d ) µν̺ ⋄ ( ˜ H ( d ) ) ⋄ µν . (7)Calculational details on how to obtain these results explicitlyare relegated to Appendix A. Analogous results for ˆ d µ and ˆ g µν follow from Eqs. (6) via the replacements b ( d ) µ ⋄ d ( d + µ ⋄ and H ( d ) µν ⋄ g ( d + µν ⋄ , respectively, and suitable adaptations of themass dimension d . However, in contrast to results at first or-der in Lorentz violation (cf. [26]), it is not possible to performthe replacements ˆ b µ
7→ − ˆ A µ and ˆ H µν
7→ − ˆ T µν with the pseu-doscalar and two-tensor operators ˆ A µ and ˆ T µν , respectively, ofEq. (7) in [10]. Such replacements would induce couplings be-tween di ff erent types of coe ffi cients and Eqs. (6) are not validfor such scenarios.Contracting a four-velocity of Eqs. (6) with u µ and usingEq. (2c) again provides quadratic equations in the correspond-ing Lagrangians, such as for the case of spin-degenerate oper-ators [27, 28]. This finding is interesting, since the dispersionequations for the spin-nondegenerate operators are quartic atleast (as mentioned before). For example, for the case of ˆ b µ weobtain 0 = ζ ± b L b + ψ ± b L b − u , (8a) with ζ ± b = ± Υ b + b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ ± Υ b h m ψ − ( d − b ( d ) ν ⋄ ( b ( d ) ) ⋄ ν i − ( d − Υ b , (8b)and ψ ± b = n ± Υ b h m ψ − ( d − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ i − ( d − Υ b o − × n ( d − ∓ Υ b − b ( d ) ν ⋄ ( b ( d ) ) ⋄ ν − m ψ ) b ( d ) ̺σ ⋄ u ̺ ( b ( d ) ) ⋄ σ + b ( d ) ⋄ h b ( d ) ν ⋄ u ν + ( d − b ( d ) κλ ⋄ u κ p λ io . (8c)Note that we reformulated the denominator in the latter formu-las via Eq. (4a) to eliminate four-momentum components. Thisprocedure has turned out to reduce computation time. Equa-tion (8a) can be solved for the Lagrangian at the cost that itstill partially depends on the four-momentum. In total, thereare four solutions: two for particles and two for antiparticles.These solutions are very powerful, since they permit us to com-pute Lagrangians as perturbative series in the controlling coef-ficients.The procedure is iterative and for particles it starts with thestandard result L ( d )0 = − m ψ u with u ≡ √ u . The latter is ob-tained from Eq. (8a) in the limit of vanishing Lorentz viola-tion. It is linked to a zeroth-order canonical momentum via4 p ) µ ≡ − ∂ L ( d )0 /∂ u µ . Inserting the latter into an appropriate par-ticle solution and keeping all contributions linear in Lorentz vi-olation implies a first-order Lagrangian L ( d )1 and another canon-ical momentum ( p ) µ valid at first order in Lorentz violation.This iteration can be continued successively to arrive at L ( d ) q + valid at ( q + L ( d ) q and ( p q ) µ ≡ − ∂ L ( d ) q /∂ u µ . The corresponding Lagrangiansfor antiparticles follow from the results for particles via the sub-stitution m ψ
7→ − m ψ . In this case, the iterative procedure startsby inserting L ( d )0 = m ψ u into an antiparticle solution of Eq. (8a).
5. Second-order classical Lagrangians
Experience showed that such perturbative computations tothird order in Lorentz violation are feasible for spin-degenerateoperators — independently of their mass dimensions [28]. Thesituation was quickly revealed to be very di ff erent for the spin-nondegenerate cases. First of all, applying the perturbative al-gorithm described before may require significantly more com-putation time for these operators. Secondly, while covariantexpressions were obtained by a straightforward generalizationof particular cases of spin-degenerate coe ffi cients, it turned outto be exceedingly more challenging to accomplish the same forspin-nondegenerate operators. Considering the controlling coe ffi cients ( K ( d ) ) α α ... of aLorentz-violating operator of mass dimension d , we introducethe following dimensionless tensors of rank l via contractionsof the latter with suitable numbers of four-velocities:( ˜ K ( d ) ) α ...α l ≡ m d − ψ ( K ( d ) ) α ...α l α l + α l + ... ˆ u α l + ˆ u α l + . . . , (9)where we employ ˆ u α ≡ u α / u . These tensors play an essentialrole in the construction of covariant forms of Lagrangians. Acrucial observation made both in case of the scalar field the-ory in [27] and for the spin-degenerate fermion operators of theSME [28] is that the tensors that may occur in the Lagrangianshave a maximum rank of 2 — independently of the mass di-mension of the operator. The same seems to hold true for thedimension-5 b coe ffi cients. The situation is slightly di ff erent forthe dimension-5 H coe ffi cients, as the latter are antisymmetricin the first two indices, whereupon a H (5) µν̺σ fully contracted withfour-velocities is identical to zero: ˜ H (5) =
0. Therefore, tensorsof rank 3 at the maximum will be necessary to construct covari-ant Lagrangians for ˆ H µν .In what follows, we will present covariant classicalLagrangians for the dimension-5 b and H coe ffi cients atsecond order in Lorentz violation. Results will only begiven for particles. Appendix B provides details on thecomputational procedure. For the b coe ffi cients we obtained L (5) ± , b = L ± q (˜ b (5) ) − (˜ b (5) ) α (˜ b (5) ) α + b (5) ) − b (5) ) (˜ b (5) ) α (˜ b (5) ) α + b (5) )(˜ b (5) ) α (˜ b (5) ) αβ (˜ b (5) ) β − b (5) ) α (˜ b (5) ) αβ (˜ b (5) ) γβ (˜ b (5) ) γ (˜ b (5) ) − (˜ b (5) ) α (˜ b (5) ) α ! . (10)The latter Lagrangian holds for a totally symmetric choice of b (5) µν̺ , which corresponds to 20 independent coe ffi cients. The sit-uation is more complicated for the H coe ffi cients. First of all,a slight problem may arise with the notation originally chosenin [27] that we wanted to take over for consistency. Some re-sults may be expressed more conveniently in terms of the dualof H ( d ) µνα ...α ( d − , which we will denote by a tilde as usual:( ˜ H ( d ) ) µνα ...α ( d − ≡ ε µν̺σ ( H ( d ) ) ̺σα ...α ( d − . (11)Note that suitable dimensionless contractions of the dual withˆ u µ are now denoted by a double tilde:( ˜˜ H ( d ) ) α ...α l ≡ m d − ψ ( ˜ H ( d ) ) α ...α l α l + α l + ... ˆ u α l + ˆ u α l + . . . . (12)Thus, the latter does not correspond to the dual of the dual inthis paper. As a first step, we focus on the minimal dimension-3 coe ffi cients H (3) µν . The Lagrangian for the full set of six in-dependent coe ffi cients has been unknown, so far. Exact re-sults were obtained for the sectors of mixed coe ffi cients H (3)0 i and the purely spacelike ones H (3) i j separately. As long as these two sectors do not couple to each other, it holds that˜ Y (3) ≡ ( H (3) ) µν ( ˜ H (3) ) µν =
0. The Lagrangian for these casescan be found, e.g., in Eq. (15) of [22].Now, for Y (3) ,
0, the algorithm above is employed to obtaina perturbative form of the Lagrangian whose covariantization ismuch less involved than for the dimension-5 b coe ffi cients. Theoutcome at third order in Lorentz violation reads L (3) ± , H = L ∓ q − ( ˜˜ H (3) ) α ( ˜˜ H (3) ) α − ( ˜ Y (3) )
2( ˜˜ H (3) ) α ( ˜˜ H (3) ) α − ( ˜ Y (3) ) ( ˜˜ H (3) ) α ( ˜˜ H (3) ) αβ ( ˜˜ H (3) ) βγ ( ˜˜ H (3) ) γ h − ( ˜˜ H (3) ) α ( ˜˜ H (3) ) α i / . (13)It is clear that the terms of second and third order must be di-rectly proportional to the quantity ˜ Y (3) such that for ˜ Y (3) = H coe ffi cients at second order in Lorentz vio-5ation. One possibility of writing it up is as follows: L (5) ± , H = L ∓ q − ( ˜˜ H (5) ) α ( ˜˜ H (5) ) α + δ L (2) − δ L (2) c ( ˜˜ H (5) ) α ( ˜˜ H (5) ) α ! , (14a) δ L (2) = S (5)1 − S (5)5 − S (5)6 + S (5)7 + S (5)8 ) − S (5)2 + S (5)3 − S (5)4 , (14b) δ L (2) c =
12 ( ˜ Y (5) ) + S (5)7 + S (5)8 + S (5)9 ) , (14c)with the observer Lorentz scalars expressed in terms of tensorsformed from the coe ffi cients H (5) µν̺σ (instead of those of the dualoperator): S (5)1 = [( ˜ H (5) ) α ( ˜ H (5) ) α ] , (15a) S (5)2 = ( ˜ H (5) ) α ( ˜ H (5) ) α ˜ X (5) , (15b) S (5)3 = ( ˜ H (5) ) α ( ˜ H (5) ) αβ ( ˜ H (5) ) γδβ ( ˜ H (5) ) γδ , (15c) S (5)4 = ( ˜ H (5) ) αβγ ( ˜ H (5) ) αβ ( ˜ H (5) ) δǫγ ( ˜ H (5) ) δǫ , (15d) S (5)5 = ˆ u α ( ˜ H (5) ) β ( ˜ H (5) ) αβγ ( ˜ H (5) ) δǫ ( ˜ H (5) ) δǫγ , (15e) S (5)6 = ˆ u α ( ˜ H (5) ) β ( ˜ H (5) ) αβγ ˆ u δ ( ˜ H (5) ) ǫ ( ˜ H (5) ) ǫγδ , (15f) S (5)7 = ( ˜ H (5) ) α ( ˜ H (5) ) α ˆ u β ( ˜ H (5) ) βγδ ( ˜ H (5) ) γδ , (15g) S (5)8 = ( ˜ H (5) ) α ( ˜ H (5) ) β ˆ u γ ( ˜ H (5) ) γδα ( ˜ H (5) ) βδ , (15h) S (5)9 = ( ˜ H (5) ) α ˆ u β ( ˜ H (5) ) βαγ ( ˜ H (5) ) δ ( ˜ H (5) ) γδ , (15i)˜ X (5) =
14 ( ˜ H (5) ) αβ ( ˜ H (5) ) αβ , (15j)˜ Y (5) =
14 ( ˜ H (5) ) µν ( ˜˜ H (5) ) µν . (15k)The latter Lagrangian is valid for a H (5) µν̺σ symmetric in its lasttwo indices. The quantities ˜ X (5) , ˜ Y (5) correspond to ˆ X , ˆ Y ofEq. (4e). The form of L (5) ± , H is obviously much more involvedthan that of L (5) ± , b .Several remarks are in order with respect to Eqs. (10), (14).First, for b (5) µν̺ = H (5) µν̺σ = L , as expected. Second, the term at first order in Lorentzviolation for ˆ b µ corresponds to that obtained earlier in [26]when d = H µν this is clear, as explainedin Appendix C. Third, the first-order contributions come withdistinct signs, which indicates the spin-nondegenerate nature ofˆ b µ , ˆ H µν . In contrast, the second-order terms come with a singlesign only. In general, modified dispersion relations for spin-nondegenerate operators exhibit an analogous behavior. Thedegeneracy of the fermion energy with respect to the spin pro-jection is lifted by such operators, which means that, e.g., ˆ b µ couples di ff erently to fermions of spin-up in comparison to fermions of spin-down. This di ff erent coupling manifests it-self via distinct signs in front of contributions that contain oddpowers of the Lorentz-violating background. The same is thecase when an expression formed of even powers of controllingcoe ffi cients occurs inside a square root. The distinct couplingdoes not play a role, though, for all contributions involving evenpowers of controlling coe ffi cients.Fourth, the first-order terms in Lorentz violation are smooth,even when the argument inside the square root vanishes. How-ever, the same does not hold for the second-order term. Thedenominator can vanish for certain configurations of the con-trolling coe ffi cients and the four-velocity components, whichmeans that these contributions become singular in these cases(as long as the numerators do not vanish, as well). Hence, forthe perturbative expansion to make sense, su ffi ciently large re-gions around such singularities in parameter space must be dis-regarded. Fifth, as the first-order terms contain square roots,their first partial derivatives with respect to the controlling co-e ffi cients are not smooth when the expression under the squareroot vanishes. In particular, the latter holds for b (5) µν̺ = H (5) µν̺σ =
0, respectively. For the second-order contribution itis the second partial derivatives with respect to the controllingcoe ffi cients that are not smooth for certain configurations suchas for vanishing controlling coe ffi cients. In general, singulari-ties of classical Lagrangians for spin-nondegenerate operatorsare attributed to the fact that particle spin, which plays a crucialrole for these operators, is a quantum property that cannot bedescribed consistently in the setting of classical physics.Singularities are known to occur also in the case of the mini-mal b coe ffi cients. Considering the corresponding Wick-rotatedLagrangian as an algebraic variety, it was explicitly demon-strated that this variety can be desingularized [36] in accordancewith Hironaka’s theorem [37]. Finding a suitable desingulariza-tion procedure for the cases studied here may be interesting, butis beyond the scope of the paper.The second-order term for ˆ H µν can be decomposed into twocontributions. It holds that δ L (2) c = H i or the purely spacelike ones ˆ H i j onlyare considered. When both types of operators are nonzero, δ L (2) c ,
0, in general, and must be taken into account. Thus, δ L (2) c describes the coupling between the mixed sector and thepurely spacelike sector of the dimension-5 H coe ffi cients. Incontrast to ˜ Y (3) for d =
3, ˜ Y (5) is not su ffi cient to include all con-trolling coe ffi cients coupling the two sectors with each other,but additional Lorentz scalars S (5)7 . . . S (5)9 are indispensable.We also see that 2 is the maximum rank of observer tensorsthat occur in L (5) ± , H such as for the dimension-5 b coe ffi cientsand the spin-degenerate operators. Third-rank tensors ( ˜ H (5) ) αβγ play a role, indeed, but their first or second index is alwayscontracted with another ˆ u µ . Note the antisymmetry in the firsttwo indices, which prohibits us to express such contractions interms of ( ˜ H (5) ) αβ .Finally, to demonstrate that Eqs. (6a), (6b) can be adoptedto the d and g coe ffi cients, we obtained the result for the clas-sical Lagrangian of the minimal, symmetric d coe ffi cients inEq. (D.1) of Appendix D. There are some similarities of the6atter Lagrangian with Eq. (10), but additional structures occur.Thus, the Lagrangian for the dimension-4 d coe ffi cients at thislevel is already more involved than that of the dimension-5 b co-e ffi cients. This finding clarifies why it has been that challengingto find classical Lagrangians for the minimal d coe ffi cients asopposed to the minimal b coe ffi cients whose exact result wasalready determined in the very first paper [22] on classical par-ticle propagation in the SME.
6. Conclusions and outlook
In this article we have studied the propagation of a classi-cal, relativistic, pointlike particles in the presence of Lorentzviolating-operators of the spin-nondegenerate, nonminimalSME fermion sector. In particular, we applied the algorithmintroduced in [27] to the operators ˆ b µ and ˆ H µν to obtain aquadratic equation in the Lagrangian that can be solved per-turbatively in Lorentz violation. By doing so, we were able toderive covariant Lagrangians at second order in Lorentz viola-tion for the totally symmetric dimension-5 b coe ffi cients andthe dimension-5 H coe ffi cients symmetric in the last two in-dices. The computations were involved, but the results obtaineddemonstrate their feasibility.It was surprising to find the following generic behaviorfor ˆ b µ : L (5) ± , b = L ± q − gram(˜ b , ˆ u ) − f (5) b (˜ b , ˜ b (˜ b · ˜ b ) , ˜ b (˜ b · ˜ b · ˜ b ) , ˜ b · ˜ b · ˜ b · ˜ b )gram(˜ b , ˆ u ) , (16)where we dropped the mass dimension from ˜ b for convenienceand introduced an intuitive short-hand notation for scalar andmatrix products, e.g. ˜ b · ˜ b · ˜ b ≡ ˜ b α ˜ b αβ ˜ b β , etc. Furthermore, weemployed the Gram determinant gram(˜ b , ˆ u ) ≡ ˜ b · ˜ b − ˜ b (see thefirst paper of Ref. [29]) and f (5) b is a function characteristic forthe dimension-5 b coe ffi cients. It is expected that the behaviorfor a general mass dimension d is analogous with f (5) b replacedby f ( d ) b . The parameters of the linear combination of Lorentzscalars contained in f ( d ) supposedly depend on the mass dimen-sion d , but the overall structure of the function does probablynot change. Note that f (3) b = d =
3, can be reproduced.Now, the generic form of the Lagrangian for ˆ H µν reads L (5) ± , H = L ∓ q − gram( ˜˜ H , ˆ u ) − f (5) H (( ˜ H · ˜ H ) , ( ˜ H · ˜ H ) ˜ X , . . . )gram( ˜˜ H , ˆ u ) , (17)where we again dropped the mass dimension from ˜ H , ˜˜ H forbrevity. Such as before, an intuitive short-hand notation forscalar and matrix products is used as well as the Gram determi-nant gram( ˜˜ H , ˆ u ) ≡ ˜˜ H · ˜˜ H . Furthermore, f (5) H is another function valid for the dimension-5 H coe ffi cients that involves variousLorentz scalars formed from ˜ H and ˆ u . It is again reasonable toassume that the generic form of the function f ( d ) H is analogousto that of f (5) H , but that the parameters of the linear combinationsof Lorentz scalars depend on the mass dimension.The results of this paper show that perturbative computationsof classical Lagrangians for spin-nondegenerate operators are,in principle, feasible despite of them being challenging. Subsetsof coe ffi cients that imply simple special cases from Eqs. (10),(14) do not seem to exist. Computations for the d and g co-e ffi cients were not performed explicitly (except for the case ofthe minimal d coe ffi cients considered in Appendix D). How-ever, the basic results stated in Eqs. (6a), (6b) are expected tobe taken over conveniently to the case of ˆ d µ and ˆ g µν , respec-tively, simply by adapting the mass dimension and the numberof indices (as described in the paragraph below Eq. (7)). Theresults presented here may be of interest for mathematiciansalike, as they pose the base for constructing Finsler structuresbeyond those investigated in, e.g., [27, 29]. It could also beworthwhile to study cases of Lorentz-invariant operators of theSM e ff ective field theory (see, e.g., Tab. V in [11] and Tab. XXin [12] in Minkowski spacetime).Further questions that can still be tackled in the context ofclassical Lagrangians are as follows: i) What are the results forcontrolling coe ffi cients that are not totally symmetric? ii) Howdo the parameters of f ( d ) b , H depend on the mass dimension d ? iii)What is the form of higher-order contributions in Lorentz viola-tion for spin-nondegenerate Lorentz violation? iv) How to treatcases with di ff erent types of coe ffi cients coupled to each other?However, these problems are rather specific and are probably oflesser interest to the scientific community working on Lorentzviolation. Therefore, our conclusion is that with the results ofthe current paper the problem of classical Lagrangians describ-ing pointlike particles subject to Lorentz violation parameter-ized by the SME can be considered as solved around 10 yearsafter it was proposed originally in [22]. A compilation of theLagrangians obtained in [22, 23, 26–28] forms a framework toparameterize Lorentz violation for classical pointlike particles.The latter could be coined the “point-particle Standard-ModelExtension.” This framework may play a valuable role for theo-ries that lie beyond the hexagon presented in Fig. 2 of the recentpaper [12]. Acknowledgments
The authors thank V.A. Kosteleck´y and B. Edwards forvaluable discussions. M.S. greatly acknowledges support viathe grants FAPEMA Universal 01149 /
17, FAPEMA Universal00830 /
19, CNPq Universal 421566 / / / Finance Code 001.
Appendix A. Computation of four-velocities
Here we demonstrate explicitly how to derive the four-velocities given in Eq. (6). The derivations make extensive use7f Eqs. (2) as well as of the property that objects like b ( d ) ⋄ , H ( d ) ⋄ ,etc. are positively homogeneous of a certain degree in the four-momentum. Appendix A.1. Operator ˆ b µ The dispersion equation for the operator ˆ b µ reads0 = h p − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ − m ψ i + b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ p − b ( d ) ⋄ ) . (A.1)Its implicit derivative is given by0 = h p − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ − m ψ i p ∂ p ∂ p j + p j − b ( d ) µ ⋄ ∂ ( b ( d ) ) ⋄ µ ∂ p j + b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ p ∂ p ∂ p j + p j ! + b ( d ) µ ⋄ ∂ ( b ( d ) ) ⋄ µ ∂ p j p − b ( d ) ⋄ ∂ b ( d ) ⋄ ∂ p j . (A.2)Using Eq. (2b) results in0 = h p − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ − m ψ i p u j − u p j + u b ( d ) µ ⋄ ∂ ( b ( d ) ) ⋄ µ ∂ p j + b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ h p u j − u p j i − u b ( d ) µ ⋄ ∂ ( b ( d ) ) ⋄ µ ∂ p j p + u b ( d ) ⋄ ∂ b ( d ) ⋄ ∂ p j . (A.3)A contraction of the latter derivative with p j in conjunction with p j ∂ ( b ( d ) ) ⋄ µ ∂ p j = ( d − (cid:20) ( b ( d ) ) ⋄ µ − p · uu ( b ( d ) ) ⋄ µ (cid:21) , (A.4a) p j ∂ b ( d ) ⋄ ∂ p j = ( d − (cid:20) b ( d ) ⋄ − p · uu ( b ( d ) ) ⋄ (cid:21) , (A.4b)as well as Eq. (2c) implies:0 = h p − b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ − m ψ i h p ( − p u − L ) − u p j p j + ( d − b ( d ) µ ⋄ (cid:16) u ( b ( d ) ) ⋄ µ + L ( b ( d ) ) ⋄ µ (cid:17)i + b ( d ) µ ⋄ ( b ( d ) ) ⋄ µ h p ( − p u − L ) − u p j p j i − d − b ( d ) µ ⋄ (cid:16) u ( b ( d ) ) ⋄ µ + L ( b ( d ) ) ⋄ µ (cid:17) p + d − h ( b ( d ) ⋄ ) u + Lb ( d ) ⋄ ( b ( d ) ) ⋄ i , (A.5)which can be solved for u . A covariantization providesEq. (6a). Appendix A.2. Operator ˆ H µν The dispersion equation has the form0 = ( p − m ψ + X ) − X p + H ( d ) ν ⋄ ( H ( d ) ) ⋄ ν + Y , (A.6) with ˆ X and ˆ Y defined in Eq. (4e). Its derivative with respect to p j is given by0 = ( p − m ψ + X ) p ∂ p ∂ p j + p j + ∂ ˆ X ∂ p j ! − ∂ ˆ X ∂ p j p − X p ∂ p ∂ p j + p j ! + H ( d ) ν ⋄ ∂ ( H ( d ) ) ⋄ ν ∂ p j + Y ∂ ˆ Y ∂ p j . (A.7a)with ∂ ˆ X ∂ p j = ∂ ( H ( d ) ) ⋄ µν ∂ p j H ( d ) µν ⋄ , (A.7b) ∂ ˆ Y ∂ p j = ∂ ( H ( d ) ) ⋄ µν ∂ p j ˜ H ( d ) µν ⋄ + ( ˆ H ( d ) ) ⋄ µν ∂ ( ˜ H ( d ) ) µν ⋄ ∂ p j . (A.7c)Inserting Eq. (2b) leads to0 = ( p − m ψ + X ) p u j − u p j − u ∂ ˆ X ∂ p j ! + u ∂ ˆ X ∂ p j p − X (cid:16) p u j − u p j (cid:17) − u H ( d ) ν ⋄ ∂ ( H ( d ) ) ⋄ ν ∂ p j − u ˆ Y ∂ ˆ Y ∂ p j . (A.8)Multiplication of the latter with p j gives0 = ( p − m ψ + X ) h p ( − p u − L ) − u p j p j − d − u ˆ X + L ˆ X ) i + d − u ˆ X + L ˆ X ) p − X h p ( − p u − L ) − u p j p j i − d − u H ( d ) ν ⋄ ( H ( d ) ) ⋄ ν + L h d − p ν H ( d ) ν̺ ⋄ ( H ( d ) ) ⋄ ̺ + H ( d )0 ν ⋄ ( H ( d ) ) ⋄ ν i − d −
3) ˆ Y ( u ˆ Y + L ˆ Y ) , (A.9)where we employed p j ∂ H ( d ) µ ⋄ ∂ p j = − p · uu h ( d − H ( d ) µν ⋄ p ν + H ( d ) µ ⋄ i + ( d − H ( d ) µ ⋄ , (A.10a) p j ∂ H ( d ) µν ⋄ ∂ p j = ( d − (cid:18) H ( d ) µν ⋄ − p · uu H ( d ) µν ⋄ (cid:19) , (A.10b) p j ∂ ˆ X ∂ p j = p j ∂ ( H ( d ) ) ⋄ µν ∂ p j H ( d ) µν ⋄ = d − (cid:18) ( H ( d ) ) ⋄ µν − p · uu ( H ( d ) ) ⋄ µν (cid:19) H ( d ) µν ⋄ = d − (cid:18) ˆ X − p · uu ˆ X (cid:19) , (A.10c)8 j ∂ ˆ Y ∂ p j = p j ( H ( d ) ) ⋄ µν ∂ p j ˜ H µν ⋄ + ( H ( d ) ) ⋄ µν p j ∂ ˜ H µν ⋄ ∂ p j = d − (cid:20)(cid:18) ( H ( d ) ) ⋄ µν − p · uu ( H ( d ) ) ⋄ µν (cid:19) ˜ H ( d ) µν ⋄ + ( H ( d ) ) ⋄ µν (cid:18) H ( d ) µν ⋄ − p · uu ˜ H ( d ) µν ⋄ (cid:19)(cid:21) = d − (cid:18) ( H ( d ) ) ⋄ µν ˜ H ( d ) µν ⋄ − p · uu ( H ( d ) ) ⋄ µν ˜ H ( d ) µν ⋄ (cid:19) = d − (cid:18) ˆ Y − p · uu ˆ Y (cid:19) , (A.10d)as well as Eq. (2c) and the dispersion equation (4b). Here, ˆ X and ˆ Y are the zeroth-order components of the vector-valuedquantities defined in Eq. (7). Note that Eq. (A.10a) has aslightly di ff erent form compared to the other relations that fol-low from the homogeneity of the expressions considered. Thereason is that H ( d ) µν ⋄ is antisymmetric in the first two indicesand completely symmetric in the remaining indices only. Fi-nally, Eq. (A.9) can be solved for u . Covariantization resultsin the four-velocity of Eq. (6b). Appendix B. Covariantization of specific Lagrangians
The algorithm described in Sec. 4 is usually applied for acertain set of controlling coe ffi cients that are chosen to be thesame. For example, considering the isotropic part of the opera-tor ˆ b (5) µ that is characterized by the single dimensionless coe ffi -cient x ≡ m ψ b (5)000 we obtain the following Lagrangian at thirdorder in Lorentz violation: L ± b , = L (cid:16) ± ξ (1) b x + ξ (2) b x ± ξ (3) b x + . . . (cid:17) , (B.1a) ξ (1) b = u | u | ( u ) / , (B.1b) ξ (2) b = − u u ( u ) , (B.1c) ξ (3) b = u | u | ( u + u )( u ) / , (B.1d)where u is the spatial part of u µ . Note that this result only holdsin a single observer frame where all controlling coe ffi cients b (5) µν̺ vanish except of b (5)000 . Analogous results follow for other setsof coe ffi cients. However, it is highly desirable to join all theseresults to obtain a Lagrangian in covariant form.It has turned out a formidable task to covariantize the findingsthat are valid in particular observer frames only. The reason forthis is connected to a hitherto unexpected covariant form of thespin-nondegenerate Lagrangians at second order in Lorentz vi-olation. In fact, the covariant second-order contribution wasfound to be the ratio of an expression at fourth order and an ex-pression at second order in Lorentz violation. Thus, already atsecond order, suitable observer scalars at fourth order must betaken into account to construct a covariant expression, whichrenders computations much more challenging than for the spin-degenerate cases. By obtaining a certain number of additional Lagrangians in particular observer frames, we observed that thesecond-order terms in the denominator correspond to the radi-cand of the square root that occurs in the contributions at firstorder in Lorentz violation. Therefore, these terms are of thegeneric form ξ (2) b = f (5) b ((˜ b (5) ) , . . . )(˜ b (5) ) − (˜ b (5) ) α (˜ b (5) ) α , (B.2a)for the b coe ffi cients and ξ (2) H = f (5) H (( ˜ H (5) ) α ( ˜ H (5) ) α , . . . )( ˜˜ H (5) ) α ( ˜˜ H (5) ) α , (B.2b)for the H coe ffi cients. The functions f (5) b and f (5) H correspondto linear combinations of observer scalars formed from fourcopies of the controlling coe ffi cients suitably contracted withfour-velocities.Fortunately, there are certain tools available that turned outto be of great use for constructing observer scalars from a givenset of tensors. One of these is the Mathematica package xTras that is an extension of the package xTensor [44]. The latter al-lows us to define tensors on a manifold endowed with a certainmetric that can be interpreted as the Minkowski metric in thiscase. For the case of the operator ˆ b (5) µ we define the second-ranktensor ˜ b (5) µν ≡ b (5) µν̺ u ̺ that is assumed to be symmetric in its in-dices. For ˆ H (5) µν we define the third-rank tensor ˜ H (5) µν̺ ≡ H (5) µν̺σ u σ that is taken as antisymmetric in the first two indices. xTras provides the command AllContractions that is applied on adirect product of four objects ˜ b (5) µν (and ˜ H (5) µν̺ ) and a suitablenumber of four-velocities to form all possible observer scalars.There are 20 possibilities for the case of the dimension-5 b co-e ffi cients and 280 for the H coe ffi cients, respectively.For each specific observer frame these Lorentz scalars arecomputed and a generic linear combination is formed thatis inserted into Eqs. (B.2) where these are then mapped tothe second-order terms obtained via the perturbative algorithm(such as Eq. (B.1c)). Repeating this procedure for a su ffi cientnumber of reference frames, leads to a linear system of equa-tions in the parameters of the generic linear combination. Itturned out that the majority of parameters remains free andthese are set equal to zero, whereas a very restricted set ofparameters has nonzero values. The nonzero parameters forˆ b (5) µ are { , − , , − } and they are multiplied with the Lorentzscalars found in the second-order term of Eq. (10). The numberof nonzero parameters is higher for ˆ H (5) µν , which is not a surprise,as the overall number of coe ffi cients is much higher. They canbe read o ff the second-order term in Eq. (14) where they arelinked to the Lorentz scalars of Eq. (15). Appendix C. Peculiarity for ˆ H µν in initial algorithmic step A peculiarity arises in the first step of the perturbative methodapplied to ˆ H µν . Inserting ( p ) µ = m ψ ˆ u µ into the quadratic equa-tion for L , the latter has the following form at first order in the9ontrolling coe ffi cients:0 = (1 + υ ± ) Lm ψ ! + υ ± √ u Lm ψ − u , (C.1a) υ ± = X ( d ) + ( d − H ( d ) ) α ( ˜˜ H ( d ) ) α ∓ q − ( ˜˜ H ( d ) ) α ( ˜˜ H ( d ) ) α . (C.1b)By solving this quadratic equation, one obtains the standard re-sult L = − m ψ u for particles instead of a first-order Lagrangiansuch as for ˆ b µ . Thus, without already knowing the first-orderLagrangian, the perturbative algorithm does not seem to work.Fortunately, first-order results for ˆ H µν are already availablefrom the Ansatz -based method of Ref. [26]. So we had to em-ploy the findings from the latter paper to obtain the second-order Lagrangian of Eq. (14). Whether or not a physical reasonis attributed to this peculiar cancelation at first order in Lorentzviolation remains unknown at this moment.
Appendix D. Minimal d coe ffi cients For demonstration purposes, we employed the adoptedEq. (6a) to obtain a Lagrangian for the minimal d coe ffi cientsat second order in Lorentz violation: L ( ± )2 , d = L ± q ( ˜ d (4) ) − ( ˜ d (4) ) α ( ˜ d (4) ) α − f (4) d ( ˜ d (4) ) − ( ˜ d (4) ) α ( ˜ d (4) ) α , (D.1a) f (4) d =
12 [( ˜ d (4) ) −
3( ˜ d (4) ) ( ˜ d (4) ) α ( ˜ d (4) ) α +
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