Tests of Lorentz and CPT symmetry with hadrons and nuclei
We explore the breaking of Lorentz and CPT invariance in strong interactions at low energy in the framework of chiral perturbation theory. Starting from the set of Lorentz-violating operators of mass-dimension five with quark and gluon fields, we construct the effective chiral Lagrangian with hadronic and electromagnetic interactions induced by these operators. We develop the power-counting scheme and discuss loop diagrams and the one-pion-exchange nucleon-nucleon potential. The effective chiral Lagrangian is the basis for calculations of low-energy observables with hadronic degrees of freedom. As examples, we consider clock-comparison experiments with nuclei and spin-precession experiments with nucleons in storage rings. We derive strict limits on the dimension-five tensors that quantify Lorentz and CPT violation.
pacs:11.30.Cp, 11.30.Er, 12.39.Fe, 14.20.Dh
Lorentz symmetry Ein05 ; Wig39 ; Wei89 , the covariance of the laws of physics under rotations and boosts in four-dimensional spacetime, plays a central role in physics and is at the basis of the standard model (SM) of particle physics and general relativity. In particle physics, it is closely related to the invariance under the combined transformations of charge conjugation, parity, and time reversal (CPT). In quantum field theories, with mild assumptions, Lorentz symmetry implies CPT invariance, while CPT violation implies Lorentz violation (LV) Wes89 ; Gre02 . Nowadays, research into the breaking of Lorentz symmetry is strongly motivated by theories that attempt to unify quantum mechanics and general relativity Lib13 ; Tas14 . Some of these theories contain mechanisms that naturally lead to Lorentz violation qgmodels . The intriguing possibility exists that remnants of LV at high energy are detectable at energies that are in reach of present-day experiments. The detection of the corresponding signals would be a revolutionary discovery and could point us to the correct theory of quantum gravity. LV, in fact, is one of the few possibilities to get an experimental handle on quantum gravity.
In particle physics, the consequences of LV at low energy are conveniently studied within an effective field theory (EFT), which allows for a systematic and model-independent framework. The pertinent operators are built from SM fields coupled to fixed-valued Lorentz tensors (sometimes called “background fields”), while keeping many desirable SM features, such as gauge invariance and the SM gauge-group structure, energy and momentum conservation, micro-causality, and observer Lorentz covariance sme . The tensors parametrize LV, which presumably originates from more fundamental Lorentz-tensor fields that obtained a vacuum expectation value through spontaneous symmetry breaking at high energy. This approach has led to the standard-model extension (SME) sme , which is the most general and widely-used framework for theoretical and experimental considerations of Lorentz and CPT violation in particle physics.
At low energy, LV results in unique experimental signals that are in principle easily distinguished from Lorentz-invariant physics beyond the SM, in particular frame dependence of observables and a dependence on sidereal time. Experimental constraints can be characterized and classified in terms of bounds on the components of the LV tensors in the SME. An overview of the existing experimental bounds can be found in Ref. datatables . Most experimental bounds on LV have been obtained in the area of quantum electrodynamics, while recently progress also has been made in the weak sector Noo13 ; Alt13 ; Dia13 ; Vos15 . However, most precision tests of Lorentz and CPT symmetry take place at low energies where quantum chromodynamics (QCD) is nonperturbative. This complicates the study of LV operators that contain quark or gluon fields, to the extent that only a relatively small number of direct bounds exists for the strong sector datatables .
In this paper, therefore, we explore the use of chiral perturbation theory (PT), the low-energy EFT of QCD Wei79 ; Gas84 (for reviews, see e.g. Refs. Wei96 ; Ber06 ; Sch12 ), to investigate the consequences of several higher-dimensional LV operators with quark and gluon fields. We construct, in Section II, chiral Lagrangians that describe LV interactions between pions, nucleons, and photons. The large nucleon mass is treated in the heavy-baryon approach Jen91 . Our approach is similar in spirit to previous studies of the breaking of parity Kap93 and time reversal Mer10 from dimension-six operators, as applied, for example, to P-odd Mae00 and P- and T-odd Hoc05 electromagnetic form factors of the nucleon. Within this framework, it becomes possible to study various LV observables for hadronic and nuclear degrees of freedom. In Section III we first construct the LV Hamiltonian, and next we identify in Section IV observables for clock-comparison experiments with nuclei in atoms and ions and storage-ring experiments with nucleons. We obtain bounds on our LV tensors from existing experiments and identify opportunities to further constrain the parameter space. We end with a summary and outlook in Section V. In Appendix A we briefly review the construction of the chiral Lagrangian and the use of naive dimensional analysis. Appendix B is devoted to the use of field redefinitions to reduce the number of effective operators.
Ii The Lorentz-violating chiral Lagrangian
ii.1 Operators with quarks and gluons
We start with a set of operators relevant below the electroweak scale GeV, but above the scale of chiral-symmetry breaking 1 GeV in QCD, where MeV is the pion decay constant. LV is associated with a high-energy scale beyond , presumably to be identified with the Planck scale. Many LV operators have been discussed elsewhere in the literature. In Ref. sme all possible LV operators compatible with the SM gauge structure and of mass-dimension 3 and 4 are given. This restriction to power-counting renormalizable operators is sometimes called the minimal Standard-Model Extension (mSME). A characterization of nonminimal, higher- (5-, 6-,) dimensional operators exists for electrodynamics, neutrinos, and free fermions Kos09 .
In an EFT framework, higher-dimensional operators are expected to be suppressed by powers of some high-energy scale, . In this respect, dimension-3 and -4 operators are less natural in an EFT for LV, where one assumes that . Additional symmetry arguments are then needed to prevent the LV physics at high energy from resulting in large dimension-3 and -4 LV operators at a low-energy scale such as . To evade the strong experimental limits on LV, these symmetry arguments should forbid the appearance of the dimension-3 and -4 operators, or at least make them scale like and respectively, where is e.g. the scale of supersymmetry breaking, . Remarkably, for LV in the minimal supersymmetric standard model, the lowest dimension for LV operators is 5 Gro05 ; Bol05 ; Mat08 , so that LV is suppressed by at least one power of the high-energy scale. A similar suppression of dimension-3 and -4 operators occurs when we construct the effective chiral Lagrangian that is induced by dimension-5 operators in the LV QCD Lagrangian.
In Ref. dim5lagrangian all dimension-5 operators were classified that can be built out of SM fields and are restricted by a set of “UV-safety” conditions that protect the operators from transmuting into lower-dimensional operators by quantum effects. In this paper, we will restrict ourselves to a subset of the quark and gluon operators listed in Ref. dim5lagrangian . The operators we choose are, for our exploratory purpose, the most interesting ones from the point of view of PT. Other LV QCD operators can be treated in the same way, but we leave this for future work.
At a scale of 1 GeV there is a limited set of protected dimension-5 operators in the quark sector. They are summarized in Eq. (18) of Ref. dim5lagrangian . Of this set, we consider the only two that explicitly contain the gluon field strength (, , where are the Gell-Mann matrices, are the generators of the color group). They are given by the Lagrangian density
Both operators also violate CPT invariance. Our naming of the LV tensors differs from Ref. dim5lagrangian , wherin is called , while is . Although these operators should be considered as part of a theory where the and bosons are already integrated out, one should keep in mind that and contain (different) contributions from the same high-energy operators, because of mixing due to - and -boson loops. Lacking a renormalization-group analysis for such operators, we will here consider and to be independent. For isospin considerations we split the Lagrangian density in Eq. (1) in two parts,
with , for , and the third Pauli matrix.
Operators similar to those in Eq. (1) exist that contain the photon field strength instead of the gluon field strength. Some phenomenological effects of these operators are considered in Refs. Bol08 ; Sta14 . Since we are interested in observables for non-strange baryons, we have focused on operators with up and down quarks only. Our analysis can be extended to include the strange quark and observables with kaons and hyperons.
In addition to the quark operators we consider the only dimension-5 pure-gauge term that satisfies the UV-safety conditions of Ref. dim5lagrangian . It is given by the Lagrangian density
where is the dual tensor of . In Ref. dim5lagrangian is called .
The real tensor components and describe LV in the quark-gluon interactions, whereas parametrizes the LV of gluonic interactions. All the LV tensor components have mass-dimension . Constraints on the symmetry of the components are derived from UV-safety considerations in Ref. dim5lagrangian : , with , while is fully symmetric in all its Lorentz indices. Additionally, all traces of the LV tensors vanish. Due to these symmetries there are 16 independent components of , while the observable parts of and , i.e. , each also have 16 independent components. The transformation properties of the LV operators under the discrete-symmetry transformations , , and are summarized in Table 1.
ii.2 Operators with nucleons and pions
At momenta of order of the pion mass GeV, the above operators induce interactions among the relevant low-energy degrees of freedom, pions (), nucleons (), and photons (). To derive these interactions we employ PT Wei79 ; Gas84 . The standard PT Lagrangian contains all interactions allowed by the QCD symmetries. In the limit of zero up- and down-quark masses and charges, the QCD Lagrangian has an chiral symmetry. Chiral symmetry is spontaneously broken to its isospin subgroup, resulting in a triplet of (almost) massless Goldstone bosons, the pions. In this limit, pions only interact via space-time derivatives, allowing for the calculation of hadronic observables in perturbation theory, with expansion parameter , where is the typical momentum of the process under consideration. The pion fields can be parametrized in infinitely many ways. We use stereographic coordinates Wei96 , as reviewed briefly in Appendix A, but different choices give identical results. For a generalization to the standard formalism reviewed in Ref. Sch12 would be indicated.
Although the operator form of the effective hadronic interactions is dictated by symmetry considerations, each interaction is multiplied by a low-energy constant (LEC) that parametrizes the nonperturbative dynamics. The values of these LECs do not follow from symmetry arguments alone. In principle these LECs can be calculated with lattice QCD, but for the LV cases discussed here this has not been done. Alternatively, if a Lorentz- or CPT-violating signal would be detected, the LV LECs can be fitted to the experimental data. In the absence of such LV signals, we resort to naive dimensional analysis (NDA) Man84 , cf. Appendix A, to estimate the LECs at the order-of-magnitude level.
The chiral (and gauge) symmetries are incorporated with covariant derivatives for the pion,
and for the nucleon,
where , is the proton charge, are the Pauli isospin matrices, and are isospin indices. The low-energy effective Lagrangian involves an infinite number of interactions ordered by the expected size of their contributions to physical processes. Each effective interaction is associated with a chiral index Wei79 ; Gas84
where counts the number of (covariant) derivatives and the number of nucleon fields appearing in the interaction. Because is not a small number, time derivatives acting on nucleon fields are not suppressed. However, the combination is still small and increases by one. The leading terms in the chiral-symmetric Lagrangian (that is, with the lowest chiral index ) are then given by
in terms of the nucleon mass and the axial-vector coupling .
Chiral symmetry is broken by the masses of the up and down quarks, but, being small, these can be incorporated in the expansion by letting increase by two for each quark-mass insertion. The most important consequence is that the pion acquires a small mass through
In a similar fashion we can construct the hadronic interactions induced by the LV operators in Eqs. (2) and (3). We assume that there arises no additional Lorentz or CPT violation from the QCD phase transition itself, such that the symmetry properties of the LV coefficients remain intact when going from the quark-gluon to the PT Lagrangian. Although all operators in Eqs. (2) and (3) break Lorentz symmetry, they transform differently under chiral symmetry. The operator in Eq. (3) and the first two terms in Eq. (2) are invariant under global chiral transformations, and therefore induce low-energy interactions that are chiral invariant as well. The interactions, however, have different symmetrization properties of the Lorentz indices as well as different properties under the individual discrete-symmetry transformations , , and (see Table 1). Therefore, they lead to different chiral-invariant interactions at lower energies.
In contrast, the last two terms in Eq. (2) break chiral symmetry explicitly and thus induce chiral-breaking hadronic interactions. In particular, they give rise to operators that involve pion fields without the spacetime derivatives that are necessary for chiral-invariant interactions Wei96 . The chiral operators resulting from the and terms can be easily constructed by noticing that the corresponding operators transform as, respectively, the and components of the antisymmetric tensor
As we discuss below, the strongest experimental constraints result from LV two-point interactions for the nucleon. These two-point interactions are induced by the LV tensors and . At the level of pions and nucleons the former give rise to the operators
where H.c. means hermitian conjugate. We denote the LV LECs at the hadronic level with a tilde. The LV components are related to by , where is a strong-interaction matrix element estimated with NDA Man84 . We introduce a factor of for each covariant nucleon derivative to keep the time derivatives from spuriously lowering the chiral index of the operators, given by for the dominant terms in Eqs. (10).
Chiral symmetry relates the nucleon-nucleon () operators to pion-nucleon () interactions. However, the strongest constraints result from the terms without pions. Operators of different form exist at this order, but in Appendix B we show that these are redundant. In all hadronic interactions we also omit terms with additional nucleon covariant derivatives, because by using the equations of motion such terms can be reduced to operators of the same form plus higher-order terms. The form of the free-nucleon operators in Eqs. (10) agrees with the effective operator for obtained in Ref. dim5lagrangian .
Similar to , the operator induces only contributions to the nucleon two-point function at this order, viz.
with the LV LEC , where is a strong-interaction matrix element estimated by NDA. Redundant terms are again discussed in Appendix B.
In contrast to the and tensors, the tensors do not lead to a nucleon two-point function at any chiral order. In fact, at lowest order () only contributes. The relevant Lagrangian is given by
The LV LEC is again defined by , with the strong-interaction matrix element according to NDA. Because and are components of the same tensor, chiral symmetry gives the relation . Additional redundant operators are discussed in Appendix B. The leading terms for , with chiral index , read
with given by
and the LV LECs defined as with . The metric tensor in the first term of contracts two covariant derivatives. Since at lowest order , we see that it represents the simple operator .
The operators in Eqs. (12) and (13) will induce loop corrections to the nucleon Lagrangian. We will see an example of this in Section II.4. However, since nucleon two-point functions are not allowed by the symmetries of the original operators, they will also not be induced by quantum effects at first order in LV. It turns out that the dominant observable effects of the loop corrections are represented by nucleon two-point functions coupled to the electromagnetic field strength. At leading order, such operators have to take the form (see Appendix B)
where is an isospin matrix analogous to Eq. (14) and the dots represent interactions that chiral symmetry relates to the displayed operator. The tensor is built from the LV components , , the metric tensor, and low-energy constants of order one. This results in many inequivalent contributions to , each of which has its own LEC. It goes beyond the scope of this work to list them all, but two relevant examples are
The tensor in Eq. (16a) is the simplest contribution to , with and , together with LV LECs of order . ( contributes to both and , due to isospin-breaking from the quark charges.) Eq. (16b) is interesting because this operator gets a contribution from loop corrections due to the dominant -dependent interaction given in Eq. (12), which is therefore enhanced by a chiral logarithm, cf. Eq. (25) below.
ii.3 Heavy-baryon formalism
Loop calculations in a relativistic meson-nucleon field theory performed with dimensional regularization receive contributions from loop momenta of order . Since , this upsets the assumed power counting. (When more complicated regularization schemes are adopted the power counting can be made consistent, for a review see Ref. Bernard .) In heavy-baryon PT (HBPT) Jen91 , this problem is overcome by introducing heavy-nucleon fields with fixed velocity , defined by
where , with a small residual momentum. Derivatives acting on the heavy fields give, instead of the large nucleon mass, the small residual momenta. Because the propagator of a heavy-nucleon field does not contain the nucleon mass, the results of loop integrals scale with powers of and , where is of order or the external momentum and . In HBPT the Dirac matrices are eliminated in favor of the simpler nucleon velocity and the covariant spin vector with and in the nucleon rest frame, where .
All coupling constants of these interactions scale as or and thus suffer a suppression of order compared to LECs appearing in standard PT, if is identified with the Planck scale. In the heavy-baryon limit, the tensors and lead to an identical leading-order operator. However, because the symmetrization properties of the tensors are different they can, in principle, still be distinguished.
In HBPT, the subleading operators in Eqs. (13) give
while the terms parametrizing the -dependent nucleon coupling to the photon field in Eq. (15) give
The nucleon operators in Eqs. (18) and (20) can be used directly as the LV perturbation of the proton or neutron Hamiltonian. As shown in Section IV, the Hamiltonian can be used to determine LV contributions to observables such as the nucleon spin-precession frequency and transition frequencies in clock-comparison experiments. Taking , we see that, in the nucleon rest-frame, Eq. (18) gives exactly the result obtained later on in Eq. (31). In addition, the heavy-baryon framework greatly simplifies loop calculations, as discussed in the next section. On the other hand, at leading order in the heavy-baryon expansion we neglect terms of order , such that the results only apply in the limit. Terms of higher order in , which can become relevant in, for example, storage-ring experiments, can be explicitly calculated in HBPT by including subleading terms in the heavy-baryon expansion. However, when such terms are needed below in Sect. III we find it more convenient to derive a relativistic expression for the Hamiltonian.
ii.4 Pion-loop diagrams
In contrast to the and components, the LV tensors give no contribution to free nucleons at tree level, since we cannot write down a two-point function that does not vanish on-shell. Pion-loop corrections, however, can induce a LV contribution to the electromagnetic form factor via the loop diagrams shown in Fig. 1. The squares represent a LV vertex from Eq. (18). We assign the external momenta , , and to the incoming nucleon, the outgoing nucleon, and the photon, respectively. In leading order in the heavy-baryon expansion, we have .
The LV current that follows from the loop calculation has the form
where and . The loop contributions to the isovector form factors and turn out to vanish, while
in terms of the two functions
and , where is the number of spacetime dimensions and is the Euler-Mascheroni constant.
The terms proportional to in Eq. (22) resemble that of the anapole Zel58 form factor Mae00 , where the role of the nucleon spin is taken over by a LV absolute direction that depends on . Although it is potentially relevant for e.g. electron-nucleon scattering, it does not contribute for on-shell photons and we will neglect this term from now on.
The terms proportional to do contribute for on-shell photons. In that case, the isoscalar form factor can be written as
which contains a logarithmic divergence. This divergence needs to be compensated by a counterterm that naturally appears at this order in the chiral expansion, as seen in Eq. (21b). The chiral power counting indicates that the long-range contribution from the pion loop and the short-range term in Eq. (21b) are of similar size. However, the long-range part is somewhat enhanced by the chiral logarithm, as mentioned below Eq. (16). In any case, a cancellation is unlikely considering the non-analytic dependence of the loop contributions on . The isovector piece in Eq. (21b), proportional to , is not needed for renormalization purposes, but there is no reason to assume it is very small either. Absorbing and the associated dependence into the short-range terms and taking as the renormalization scale, we obtain for the form factors
where the bar on indicates that this is a renormalized quantity.
In the following sections, we study the phenomenological consequences of the relativistic LV chiral Lagrangians, obtained in Section II.2. The isocalar LV form factor in Eq. (II.4) gives a slightly enhanced contribution to the operators summarized in Eq. (15), which are studied in Section IV.3. We mention already here that, in the rest-frame, the operators that follow from the present loop calculation do not couple to the magnetic field, which is most easily seen from Eq. (21b). This is important when considering experimental methods to limit the LV coefficients.
ii.5 Nucleon-nucleon interactions from one-pion exchange
Our analysis can be extended to systems with multiple interacting nucleons, and in particular to the few-nucleon sector, where PT is often called EFT. EFT allows the derivation of the structure and hierarchy of multi-nucleon interactions (for reviews see e.g. Refs. Bedaque:2002mn ; Epelbaum:2008ga ). EFT is usually formulated for nonrelativistic nucleons, which fits naturally with the heavy-baryon framework discussed above. We briefly discuss here the LV interaction arising from one-pion exchange with the vertices from Eqs. (18) and (19). Although the tensors and give contributions to interactions at the same chiral order as , we omit them here, because, in contrast to , there exist nucleon two-point functions for and that already provide very strict limits (see below, in Section IV).
In combination with the standard leading-order vertex multiplied by , we obtain the LV potential
where are the spin (isospin) operators of the interacting nucleons and the momentum transfer flows from nucleon to nucleon ; and are the relative momenta of the incoming and outgoing nucleon pair in the center-of-mass frame. The Latin indices denote spatial directions. At the same order as the second term in Eq. (27), there exist contributions from LV contact interactions, which we ignore here.
We postpone a detailed study of this potential and its consequences to future work, but point out that the interactions between nucleons can lead to measurable LV in clock-comparison experiments on nuclei or in the spin precession of nuclei in storage rings. This is especially relevant because the effects could be considerably larger than for nucleons where, in case of the operators, a coupling to an electromagnetic field is required. As discussed in Secs. IV.2 and IV.3, this greatly weakens the constraints on the LV tensors. A study of the effects of Eq. (27) on, for example, the spin precession of deuterons in storage rings would therefore be very interesting.
Iii Hamiltonian with Lorentz violation
Having obtained the low-energy chiral Lagrangian, we now obtain the limits that are set by existing experimental constraints, from which we deduce which parts of the parameter space have room for improvement. As mentioned, the strictest limits are on the nucleon two-point functions and come from clock-comparison experiments datatables . For the analysis of clock-comparison experiments, the block-diagonalized form of the Hamiltonian has proven to be convenient clockcomparison . In this diagonal form the Dirac equation for the particle and the antiparticle are decoupled. The diagonalization is achieved by performing a unitary Foldy-Wouthuysen transformation of the fields Foldwout . A comparable particle-antiparticle decoupling is obtained in HBPT.
The heavy-baryon approach employs a nonrelativistic expansion in , which implies that observer Lorentz invariance can only be restored perturbatively rpi . As for the Foldy-Wouthuysen transformation, for some Hamiltonians an exact diagonalization can be achieved Foldwout ; exactfw . In most cases, however, the transformation is done such that the off-diagonal parts of the Hamiltonian can be made of arbitrary order in some small quantity. Often, is chosen as the small parameter, which results in a nonrelativistic expansion of the Hamiltonian, comparable to the heavy-baryon approach. Here, we adopt the approach of Ref. kostmuon , where the relevant Hamiltonian is obtained with a Foldy-Wouthuysen transformation on the relativistic muon Hamiltonian that follows from the mSME sme . The small quantities in which the off-diagonal parts of the Hamiltonian are expanded are the LV tensor components and the electromagnetic fields. This results in a relativistic expression for the relevant parts of the Hamiltonian for free nucleons (at least when restricting to frames where the LV coefficients and the EM fields are small with respect to the nucleon mass).
where denotes proton or neutron and
with , , and where for the proton and neutron is given by and . We added the term for the anomalous magnetic moment of the nucleon, where is the nuclear magneton.
The operator is not a standard Hamiltonian because it contains extra terms with time derivatives. This is a well-known problem when dealing with LV. In the mSME it can be solved by applying a spinor redefinition that removes the extra time derivatives Blu97 . However, since we have time-derivative terms of higher order, we use the approach of Ref. Kos09 , where one first diagonalizes the Hamiltonian and then substitutes for the fermion and for the antifermion in the LV terms. Contributions that we miss in this way are higher order in the LV components, and hence negligible.
The Foldy-Wouthuysen transformation used to diagonalize is given by , with . We assume that all the electromagnetic fields are homogeneous and small and we neglect all contributions that are quadratic in these fields as well as products of LV and an electromagnetic field. This results in a Hamiltonian with off-diagonal blocks that are first order in the LV components or the - and -fields. We neglect these small off-diagonal contributions and take the upper left block () as the Hamiltonian for the particle and the lower right block () as the Hamiltonian for the antiparticle. We find that the resulting Hamiltonian is given by
where is the conventional particle or antiparticle Hamiltonian, while the LV perturbation is given by
where is the (anti)particle velocity, is the relativistic boost factor, and
with . We thus conclude that the part of that is symmetric in and is the only observable combination of the and tensors in experiments with free nucleons. This is consistent with Eq. (18), where the same combination of and appears. It confirms that the heavy-baryon and the Foldy-Wouthuysen approach are equivalent for nucleons at rest. The tensor is completely traceless and its observable part therefore has 32 independent components, while the observable parts of and both contain 16 independent components. In the following, we will derive bounds on a subset of these.
Iv Experimental constraints
iv.1 Clock-comparison experiments
The most restrictive limits on Lorentz and CPT violation for protons and neutrons come from clock-comparison experiments datatables ; clockcomparison . In these experiments transition frequencies of two colocated samples of atoms or ions are compared. The variation of these frequencies, as the clocks rotate with Earth, gives a limit on rotational noninvariance and hence on LV. In Ref. clockcomparison the combinations of mSME tensor components that are observable in clock-comparison experiments are determined by calculating expectation values of the particle Hamiltonian that is linear in LV. For an atom or ion , it is given by
where is the LV Hamiltonian for the th particle of species , the second sum runs over all particles of species that are present in the atom or ion , and the first sum runs over all species. In the present case, for protons and neutrons is given by in Eq. (31), while it is zero for electrons.
We take the laboratory axis as the axis of quantization. The LV corrections to the transition frequencies follow from the expectation value , where is the state corresponding to the atom or ion with total relevant angular momentum and projection . The LV shift in a frequency corresponding to a transition is then given by
Depending on the rotational transformation properties of the different parts of the Hamiltonian, LV will give rise to different multipole contributions to the transition frequencies. The LV shift can be written as
where the constants (), given by ratios of Clebsch-Gordan coefficients, read
Furthermore, , , and originate from spherical tensors of rank and , respectively, which require a total angular momentum of at least , , and to be nonvanishing. Following Ref. clockcomparison , we call , , and the dipole, quadrupole, and octupole contributions, respectively. We calculate these contributions in the nonrelativistic limit, keeping only terms up to first order in . We find
where we used that is completely traceless and defined the symmetrized parts of as and . In contrast to the mSME case in Ref. clockcomparison , we find an octupole contribution, which originates from the totally-symmetric gluon tensor . Due to the antisymmetry of the term in Eq. (32), the contributions of contain no component of or that is not already present in . It does contain different linear combinations of the tensor components, however.
The matrix elements to are sums of expectation values of spherical-tensor operators in the special “stretched” state . The relation between the expectation values in this special state and a state with general follows from the Wigner-Eckart theorem rose , which allows to separate the matrix elements of spherical tensors in a Clebsch-Gordan coefficient and a reduced matrix element. The ratios of these Clebsch-Gordan coefficients for the expectation values in the states and in the states are given by the factors in Eqs. (36). The relevant expectation values in the state with are given by