Lorentz and CPT Tests with Clock-Comparison Experiments

PHYSICAL REVIEW D 98, 036003 (2018)

Lorentz and CPT tests with clock-comparison experiments

V. Alan Kostelecký1 and Arnaldo J. Vargas2 1Physics Department, Indiana University, Bloomington, Indiana 47405, USA 2Physics Department, Loyola University, New Orleans, Louisiana 70118, USA

(Received 20 May 2018; published 6 August 2018)

Clock-comparison experiments are among the sharpest existing tests of Lorentz symmetry in matter. We characterize signals in these experiments arising from modifications to electron or nucleon propagators and involving Lorentz- and CPT-violating operators of arbitrary mass dimension. The spectral frequencies of the atoms or ions used as clocks exhibit perturbative shifts that can depend on the constituent-particle properties and can display sidereal and annual variations in time. Adopting an independent-particle model for the electronic structure and the Schmidt model for the nucleus, we determine observables for a variety of clock-comparison experiments involving fountain clocks, comagnetometers, ion traps, lattice clocks, entangled states, and antimatter. The treatment demonstrates the complementarity of sensitivities to Lorentz and CPT violation among these different experimental techniques. It also permits the interpretation of some prior results in terms of bounds on nonminimal coefficients for Lorentz violation, including first constraints on nonminimal coefficients in the neutron sector. Estimates of attainable sensitivities in future analyses are provided. Two technical appendices collect relationships between spherical and Cartesian coefficients for Lorentz violation and provide explicit transformations converting Cartesian coefficients in a laboratory frame to the canonical Sun-centered frame.

DOI: 10.1103/PhysRevD.98.036003

I. INTRODUCTION physics, numerous searches for Lorentz violation have now reached sensitivities to physical effects originating at the Among the best laboratory tests of rotation invariance are Planck scale M ≃ 1019 GeV [3]. Since the three boost experiments measuring the ticking rate of a clock as its P generators of the Lorentz group close under commutation orientation changes, often as it rotates with the Earth. into the three rotation generators, any deviations from A spatial anisotropy in the laws of nature would be revealed Lorentz symmetry in nature must necessarily come with if the clock frequency varies in time at harmonics of the violations of rotation invariance. Searches for rotation rotation frequency. Detecting any time variation requires a violations therefore offer crucial tests of Lorentz symmetry. reference clock that either is insensitive to the anisotropy or In the present work, we pursue this line of reasoning by responds differently to it. Typically, the two clock frequen- developing and applying a theoretical treatment for the cies in these experiments are transition frequencies in atoms analysis of clock-comparison experiments searching for or ions, and the spatial orientation of a clock is the Lorentz violation. quantization axis established by an applied magnetic field. To date, no compelling experimental evidence for Lorentz These clock-comparison experiments can attain impressive violation has been adduced. Even if Lorentz violation does sensitivities to rotation violations, as originally shown by occur in nature, identifying the correct realistic model among Hughes et al. and Drever [1]. a plethora of options in the absence of positive experimental Rotation invariance is a key component of Lorentz guidance seems a daunting and improbable prospect. An symmetry, the foundation of relativity. Tests of this symmetry alternative is instead to adopt a general theoretical framework have experienced a revival in recent years, stimulated by the for Lorentz violation that encompasses specific models and possibility that minuscule violations could arise from a permits a comprehensive study of possible effects. Since any unification of quantum physics with gravity such as string Lorentz violation is expected to be small, it is reasonable to theory [2]. Using techniques from different subfields of use effective field theory [4] for this purpose. A realistic treatment then starts from well-established physics, which can be taken as the action formed by coupling general Published by the American Physical Society under the terms of relativity to the Standard Model of particle physics, and adds the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to all possible Lorentz-violating operators to yield the frame- the author(s) and the published article’s title, journal citation, work known as the Standard-Model Extension (SME) [5,6]. and DOI. Funded by SCOAP3. Each Lorentz-violating operator in the SME is contracted

2470-0010=2018=98(3)=036003(37) 036003-1 Published by the American Physical Society V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) with a coefficient that determines the magnitude of its Lorentz violation in the electron and proton sectors have physical effects while preserving coordinate independence been derived from laboratory experiments to date [35,43– of the theory. The operators can be classified according to 48], while the neutron sector is unexplored in the literature. their mass dimension d, with larger values of d associated Here, we seek to improve this situation by developing with greater suppression at low energies. The limiting case techniques for analyzing clock-comparison experiments with d ≤ 4 is power-counting renormalizable in Minkowski and identifying potential signals from nonminimal spacetime and is called the minimal SME. Since CPT Lorentz and CPT violation. Several clock modalities are violation in effective field theory is concurrent with considered in this context for the first time, including ones Lorentz violation [5,7], the SME also characterizes general yielding measurements of nonminimal SME coefficients effects from CPT violation. Experimental constraints on the from atomic fountains, comagnetometers, ion traps, lattice parameters of any Lorentz-violating model that is consistent clocks, and antimatter spectroscopy. We adopt existing with realistic effective field theory can be found by identify- results to deduce numerous first constraints on nonminimal ing the model parameters with specific SME coefficients and coefficients in the neutron sector, and we estimate sensi- their known constraints [3,8]. tivities to electron, proton, and neutron nonminimal coef- Signals arising from Lorentz and CPT violation are ficients that are attainable in future analyses. predicted by the minimal SME to appear in clock- The organization of this work is as follows. In Sec. II,we comparison experiments with atoms or ions [9]. The signals present the theoretical techniques that enable a perturbative include observable modifications of the spectra that can treatment of the effects of Lorentz and CPT violation on exhibit time variations and that depend on the electron and the spectra of atoms and ions. A description of the nucleon composition of the species used as clocks. Null perturbation induced by Lorentz- and CPT-violating oper- results from early clock-comparison experiments [10–13] ators of arbitrary mass dimension d is provided in Sec. II A. can be reinterpreted as bounds on coefficients for Lorentz The perturbative shifts in energy levels are discussed in violation in the minimal SME [9]. Many minimal-SME Sec. II B along with generic features of the resulting coefficients have been directly constrained in recent experi- spectra, and some useful formulas for subsequent calcu- ments, including clock comparisons performed using a lations are derived. In Sec. II C, we consider methods for 133 87 hydrogen maser [14,15], Cs and Rb fountain clocks determining expectation values of electronic states, with 199 [16], trapped ultracold neutrons and Hg atoms [17], emphasis on an independent-particle model. The corre- 3He-K and 21Ne-Rb-K comagnetometers [18,19], 133Cs and sponding techniques for nucleon states are presented in 199Hg magnetometers [20], transitions in 162Dy and 164Dy Sec. II D, primarily in the context of a comparatively simple atoms [21], 129Xe and 3He atoms [22–25], and entangled nuclear model. We then turn to evaluating the time states of 40Caþ ions [26]. The results represent competitive variations in the spectrum due to the noninertial nature tests of Lorentz and CPT symmetry [3,27], and additional of the laboratory frame, first examining effects induced by constraints on minimal-SME coefficients have been the rotation of the Earth about its axis in Sec. II E and next extracted by detailed theoretical analyses [28–34]. discussing ones induced by the revolution of the Earth In this work, we extend the existing theoretical treatment about the Sun in Sec. II F. The latter section also considers of Lorentz and CPT violation in clock-comparison experi- related issues associated with space-based missions. ments to include SME operators of nonminimal mass Applications of these theoretical results in the context of dimension d>4 that modify the Dirac propagators of various clock-comparison experiments are addressed in the constituent electrons, protons, and neutrons in atoms Sec. III. Searches for Lorentz and CPT violation using and ions. At an arbitrary given value of d, all Lorentz- and fountain clocks are considered in Sec. III A, and estimates CPT-violating operators affecting the propagation have for attainable sensitivities are obtained. Studies with been identified and classified [35], which in the present comagnetometers are investigated in Sec. III B, and first context permits a perturbative analysis of the effects of sensitivities to many nonminimal coefficients for Lorentz general Lorentz and CPT violation on the spectra of the violation are deduced from existing data. Optical transi- atoms or ions used in clock-comparison experiments. tions in ion-trap and lattice clocks are discussed in Sec. III Nonminimal SME operators are of direct interest in various C, and potential sensitivities in available systems are theoretical contexts associated with Lorentz-violating considered. Some comments about prospects for antimatter quantum field theories including, for instance, formal experiments are offered in Sec. III D, where the first SME studies of the underlying Riemann-Finsler geometry [36] constraints from antihydrogen spectroscopy are presented. or of causality and stability [37] and phenomenological A summary of the work is provided in Sec. IV.Two investigations of supersymmetric models [38] or noncom- appendices are also included. Appendix A describes the mutative quantum field theories [39,40]. They are also of general relationship between spherical and Cartesian coef- interest in experimental searches for geometric forces, such ficients for Lorentz violation and tabulates explicit expres- as torsion [41] and nonmetricity [42]. Only a comparatively sions for 3 ≤ d ≤ 8. Appendix B presents techniques for few constraints on nonminimal SME coefficients for transforming Cartesian coefficients for Lorentz violation

036003-2 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) from the laboratory frame to the Sun-centered frame jpj=mw, so the corrections to the propagation of the 3 ≤ ≤ 8 0 1 and collects explicit results for d . In this work, particles at order ðjpj=mwÞ and ðjpj=mwÞ and at leading we use conventions and notation matching those of order in Lorentz violation dominate any effects due to Ref. [35] except as indicated. Note that natural units with Lorentz-violating operators coupled to the interactions c ¼ ℏ ¼ 1 are adopted throughout. between the fermions. With these considerations in mind, it is reasonable to II. THEORY proceed under the usual assumption that the dominant Lorentz-violating shifts of the spectrum of the atom or ion This section discusses the theoretical techniques for the arise from corrections to the propagation of the constituent perturbative treatment of Lorentz and CPT violation in particles. For most purposes, these particles can be taken as atoms and ions. The perturbation is described using a electrons e, protons p, and neutrons n, so that w takes the framework that encompasses all Lorentz-violating quantum values e, p, and n. Applications to exotic atoms or ions can operators affecting the motion of the component particles in be accommodated by extending appropriately the values of the atom. Generic restrictions on the induced energy shifts w. The relevant Lorentz-violating terms in the Lagrange arising from symmetries of the system are considered. The density are then quadratic in the fermion fields for the perturbative calculation of the energy shift is formulated, constituent particles. All terms of this type have been and expressions useful for application to experiments are classified and enumerated [35], and applications to hydro- obtained. Simple models are selected for the electronic and gen and hydrogen-like systems have been established [43]. nuclear structure so that derivations of the relevant expect- For convenience, we reproduce in this section the key ation values can be performed for a broad range of atomic results relevant in the present context. species used in experiments. The conversion from the For a Dirac fermion ψ w of flavor w and mass mw, all laboratory frame to the Sun-centered frame is provided, quadratic terms in the Lagrange density L can be expressed accounting both for the rotation of the Earth about its axis as [35] and for the revolution of the Earth about the Sun at first order in the boost parameter. 1 L ⊃ ψ γμ ∂ − Qˆ ψ 2 wð i μ mw þ wÞ w þ H:c: ð1Þ A. Description of the perturbation Here, Qˆ is a spinor matrix describing modifications of The experiments of interest here involve comparisons of w the standard fermion propagator, including all Lorentz- transitions in atoms or ions, seeking shifts in energy levels invariant and Lorentz-violating contributions obtained by due to Lorentz and CPT violation. All possible shifts are contracting SME coefficients with operators formed from controlled by SME coefficients, which can be viewed as a ∂ Qˆ set of background fields in the vacuum. The energy-level derivatives i μ. The matrix w can be decomposed in a shifts arise from the coupling of these background fields to basis of Dirac matrices and can be converted to momentum the elementary particles and interactions comprising the space with the identification i∂μ → pμ. Individual oper- atom or ion. An exact theoretical treatment of the shifts is ators with definite mass dimension d in the Lagrange prohibitive. However, since any Lorentz and CPT violation density incorporate d − 3 momentum factors, and the is small, a perturbative analysis is feasible and sufficient to corresponding SME coefficients have dimension 4 − d. establish the dominant effects. The Lagrange density (1) has been extended to include From the perspective of perturbation theory, the inter- operators at arbitrary d in the photon sector [45,49,50]. action between the electrons and the protons inside an atom Analogous constructions exist for the neutrino [51] and or ion has some common features with the interaction gravity [52] sectors. between the nucleons inside the nucleus. In both cases, the For present purposes, the SME coefficients can be magnitude jpj of the momentum p of a fermion of flavor w assumed uniform and time independent within the solar in the zero-momentum frame is smaller than its rest mass system [5] and so can be taken as constants when specified mw. One consequence is that the dominant contribution due in the canonical Sun-centered frame [53]. Using standard to a perturbation added to the Hamiltonian of the system procedures, an effective nonrelativistic one-particle can be obtained by expanding the perturbation in terms of Hamiltonian that includes the leading-order correction the ratio jpj=mw, keeping only leading terms in the power due to Lorentz- and CPT-violation to the propagation of series. In most cases, it suffices to treat the system as a fermion of flavor w can be derived from the Lagrange effectively nonrelativistic. Another feature of interest is that density (1). This Hamiltonian can be separated into the the energy per particle due to the interaction between the conventional Hamiltonian for a free nonrelativistic fermion NR nucleons and the interaction between the electrically and a perturbation term δhw containing the Lorentz- and NR charged particles in the bound states is comparable to CPT-violating contributions. The perturbation δhw is thus the nonrelativistic kinetic energy of the particles. The a 2 × 2 matrix, with each component being a function of the nonrelativistic kinetic energy is second order in the ratio momentum operator and independent of the position. It can

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V NR NR − NR be expanded in terms of the identity matrix and the vector wkjm ¼ cwkjm awkjm; σ ¼ðσ1; σ2; σ3Þ of Pauli matrices. For convenience, this T NRðqPÞ NRðqPÞ − NRðqPÞ expansion can be performed using a helicity basis instead wkjm ¼ gwkjm Hwkjm ; ð5Þ of a Cartesian one. The corresponding three basis vectors pffiffiffi ϵˆ pˆ ≡ p p ϵˆ θˆ ϕˆ 2 where the a- and g-type coefficients are contracted with can be taken as r ¼ =j j and ¼ð i Þ= , θˆ ϕˆ CPT-odd operators and the c- and H-type coefficients with where and are the usual unit vectors for the polar angle CPT-even ones. The notation here parallels the standard θ and azimuthal angle ϕ in momentum space, with pˆ θ ϕ θ ϕ θ assignments in the minimal SME [5]. Each nonrelativistic ¼ðsin cos ; sin sin ; cos Þ. In this helicity basis, coefficient on the right-hand side of this equation can be δ NR the perturbation hw takes the form [35] expressed as a sum of SME coefficients in the Lagrange NR density, suitably weighted by powers of m . The explicit δh ¼ h 0 þ h σ · ϵˆ þ h σ · ϵˆ− þ h −σ · ϵˆ ; ð2Þ w w w wr r wþ w þ expressions for these sums are given in Eqs. (111) and where hw0 contains spin-independent effects and the (112) of Ref. [35]. The allowed ranges of values for the remaining terms describe spin-dependent ones. indices k, j, m and the numbers of independent components Many experiments searching for Lorentz and CPT for each coefficient are listed in Table IVof Ref. [35]. Note violation focus on testing the rotation subgroup of the that in the present work we follow the convention of Lorentz group. To facilitate the analysis of rotation proper- Ref. [43] and adopt the subscript index k instead of n,to avoid confusion with the principal quantum number of the ties, it is useful to express the components hw0, hwr, hw of the perturbation in spherical coordinates. It is opportune to atom or ion. δ NR express the spherical decomposition of the operators in the Given the perturbation hw affecting the propagation of NR each fermion in an atom or ion, we can formally express the perturbation δhw in terms of spin-weighted spherical ˆ perturbation δhatom of the system as a whole as harmonics sYjmðpÞ of spin weight s, as these harmonics capture in a comparatively elegant form the essential X XNw properties of the perturbation under rotations. The usual δ atom δ NR h ¼ ð hw Þa; ð6Þ spherical harmonics are spin-weighted harmonics with 1 0 θ ϕ ≡ θ ϕ w a¼ spin-weight s ¼ , Yjmð ; Þ 0Yjmð ; Þ. Definitions and some useful features of spin-weighted spherical har- where a ¼ 1; …;Nw labels the fermions of given flavor w monics are presented in Appendix A of Ref. [50]. in the atom or ion. The Lorentz-violating operators con- In terms of the spherical decomposition, the components sidered in this work are functions of the momentum and the δ NR δ NR p σ of the perturbation (2) can be expanded as spin of the particle, so ð hw Þa ¼ hw ð a; aÞ depends on X the momentum operator pa and the spin operator σa for the k NR h 0 ¼ − jpj 0Y ðpˆÞV ð3Þ δ NR w jm wkjm ath fermion. Each term ð hw Þa is understood to be the kjm tensor product of the perturbation (2) acting on the states of for the spin-independent terms, and the ath fermion of flavor w with the identity operator acting X on the Hilbert space of all other fermions. Note that the − p k pˆ T NRð0BÞ index a is tied to the momentum and spin, whereas the hwr ¼ j j 0Yjmð Þ wkjm ; kjm index w controlling the flavor of the particle is contained in X the coefficient for Lorentz violation. Note also that the p k pˆ T NRð1EÞ T NRð1BÞ hw ¼ j j 1Yjmð Þði wkjm wkjm Þð4Þ perturbation (6) can be separated according to operator kjm flavor as for the spin-dependent ones. The coefficients V NR , wkjm δhatom ¼ δhatom þ δhatom þ δhatom; ð7Þ T NRðqPÞ 0 1 1 e p n wkjm , where qP takes values B, B,or E, are atom nonrelativistic spherical coefficients for Lorentz violation. where δhw is the sum of all operators of flavor w that These effective coefficients are linear combinations of SME δ atom δ atom contribute to h . ForP example, the expression for he coefficients for Lorentz violation that emerge naturally in δ atom Ne δ NR is given by he ¼ a¼1 ð he Þa, where a ranges over the nonrelativistic limit of the one-particle Hamiltonian the N electrons in the atom. obtained from the Lagrange density (1). e For applications, it is useful to perform a further decomposition of the components of the perturbation B. Energy shifts Hamiltonian according to their CPT handedness. In par- The corrections to the spectrum of the atom or ion due to ticular, each nonrelativistic spherical coefficient can be Lorentz and CPT violation can be obtained from the δ separated into two pieces characterized by the CPT perturbation hatom using the Raleigh-Schrödinger pertur- handedness of the corresponding operator. This decom- bation theory. At first order, the shift of an energy level is position can be expressed as [35] obtained from the matrix elements of the perturbation

036003-4 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) evaluated in the subspace spanned by the degenerate numbers including F that together with mF forms a unperturbed energy eigenstates, as usual. In typical appli- complete set of quantum numbers. Using the Wigner- cations of relevance here, the degeneracy in the energy Eckart theorem, the energy shift due to the propagation levels is lifted by an external field such as an applied of the fermions in the atom or ion can then be written as magnetic field. In this scenario, the first-order shift of an X 0 atom 0 energy level is obtained from the expectation value of the δϵ ¼hα mFjδh jα mFi¼ Aj0hFmFj0jFmFi; ð8Þ perturbation with respect to the unperturbed state. Since the j exact unperturbed energy states for multielectron atoms or ions are typically unknown, approximations to these states where hj2m2j3m3jj1m1i denote Clebsch-Gordan coeffi- 0 must be used to obtain the first-order Lorentz-violating cients. The factors Aj0 ¼ Aj0ðα Þ are independent of mF. shift of the spectrum. However, the symmetries of the The sum over j in Eq. (8) involves the index j labeling the unperturbed system place restrictions on the expectation coefficients for Lorentz violation in δhatom, in parallel with values of the perturbation. In this section, we describe some the sums over j in Eqs. (3) and (4). Note that the Clebsch- of these constraints and establish the general form of the Gordan coefficient hFmFj0jFmFi vanishes when j>2F, perturbative energy shifts. implying that no operator with j>2F contributes to the Assuming that the degeneracy of the energy levels is energy shift. broken by an applied magnetic field, parity is a symmetry To find useful expressions for the factors Aj0, we make of the system and so the states of the atom or ion must be some additional assumptions that are broadly valid for the parity eigenstates. As a result, the expectation values of systems considered in this work. Except where stated parity-odd operators with respect to parity eigenstates must otherwise, we suppose that both the magnitude J of the vanish, so only parity-even operators can affect the spec- total angular momentum J of the electrons and the δ trum. This prevents some terms in the perturbation hatom magnitude I of the nuclear spin I are good quantum from contributing to the energy shift. Each operator in the numbers for the system. We also assume that the states spherical decomposition (3) and (4) of the perturbation jα0m i can be expressed as a tensor product jΨðα0Þi ⊗ δ F hatom is either odd or even under parity, with handedness jFmFi, where determined by the indices j and k of the corresponding X coefficient for Lorentz violation. The coefficients a NR wkjm jFmFi¼ hImIJmJjFmFijImIi ⊗ jJmJi: ð9Þ NR and cwkjm can contribute to energy shift at first-order in mJmI perturbation theory only for even values of j and k, while NRðqBÞ NRðqBÞ Here, the kets jFm i, jJm i, and jIm i are associated with the coefficients g and H can contribute only F J I wkjm wkjm the angular momenta F, J, and I, respectively. These states if j is odd and k even. also depend on other quantum numbers that are suppressed Another constraint arises from time-reversal invariance in the notation. For example, the ket jFmFi depends on J, and the Wigner-Eckart theorem [54], and it concerns the on I, and also on other quantum numbers established by the expectation value in any angular-momentum eigenstate of couplings of the orbital angular momenta and spins of the Lorentz-violating operators controlled by spin-dependent component particles to form F. For later use, it is also coefficients with P ¼ E. It can be shown that this expect- 0 0 convenient to introduce the notation jα mJi¼jΨðα Þi ⊗ ation value must vanish when the Lorentz-violating oper- jJm i and jα0m i¼jΨðα0Þi ⊗ jIm i. Under these assump- ators transform as spherical operators under rotations J I I tions, we can expand the factors A 0 appearing in Eq. (8) in generated by the angular-momentum operator [43], which j the form is the case for the perturbation δhatom of interest here. As a result, none of the spin-dependent coefficients with P ¼ E e e p p n n A 0 ¼ C W þ C W þ C W ; ð10Þ contribute to the perturbative shift of the spectrum for any j j j0 j j0 j j0 values of k and j. This result applies for all atoms and ions w w α0 considered in the present work. where Cj ðFJIÞ are weights for the quantities Wj0ð Þ δ atom In the absence of Lorentz violation, the total angular containing the expectation values of hw with respect to α0 α0 momentum F of the atom or ion commutes with the the states j mJi and j mIi. w Hamiltonian of the system. When a magnetic field B ¼ The analytical expressions for the factors Cj ðFJIÞ in terms of Clebsch-Gordan coefficients are BBˆ is applied, the rotational symmetry is broken. If the perturbative shift due to the magnetic field is smaller than X 2 hIm Jm jFFi the hyperfine structure, both the quantum number F Cn ¼ Cp ¼ I J hIm j0jIm i; F j j hFFj0jFFi I I corresponding to and the quantum number mF corre- mJmI F Bˆ X 2 sponding to · can be approximated as good quantum hIm Jm jFFi α0 Ce I J Jm j0 Jm : 11 numbers. Suppose the states j mFi represent a basis of j ¼ 0 h J j Ji ð Þ 0 hFFj jFFi eigenstates of the Hamiltonian, where α is a set of quantum mJmI

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Their numerical values can be obtained for any given XNw hα0Ijjp jk Y ðpˆ Þσ · pˆ jα0Ii Λ ð0BÞ − a 0 j0 a a a allowed values of F, J, and I. Some of the properties of w ¼ ; kj II j0II w a¼1 h j i Cj ðFJIÞ are induced by features of the Clebsch-Gordan w XNw α0 p k pˆ σ ϵˆ ϵˆ α0 coefficients. For example, C0 ¼ 1 for any values of F, J, 1 h Ijj aj 1Y 0ð aÞ a · ð þa þ −aÞj Ii Λ ð BÞ þ j ; and I because hKm 00jKm i¼1 for K equal to F, J, wkj ¼ 0 PK K 1 hIIjj IIi or I and because Im Jm FF 2 1. As another a¼ mJmI h I Jj i ¼ p XNw α0 p k pˆ α0 Ce FJI 0 j>2J C FJI ð0EÞ h Ijj aj 0Yj0ð aÞj Ii example, j ð Þ¼ whenever and j ð Þ¼ Λ ¼ − ; ð15Þ n 0 2 0 wkj II j0II Cj ðFJIÞ¼ whenever j> I because hKmKj jKmKi¼ a¼1 h j i 0 if j>2K. The explicit relationships between the expectation values δ atom w α0 of the perturbations hw and the Wj0ð Þ in Eq. (10) can be written as where the sum on a ranges over all particles with ∈ X flavors w fp; ng. α0 δ atom α0 e 0 Explicit determination of the nonvanishing expectation h mJj he j mJi¼ Wj0hJmJj jJmJi; j values in Eqs. (14) and (15) requires models for the X electronic states and for the nuclear states, as discussed α0 δ atom α0 p 0 h mIj hp j mIi¼ Wj0hImIj jImIi; below in Secs. II C and II D, respectively. However, certain j Λ ðqPÞ X components of wkj vanish. We saw above that only α0 δ atom α0 n 0 h mIj hn j mIi¼ Wj0hImIj jImIi: ð12Þ coefficients with even values of k can contribute due to j Λ ðqPÞ α0 parity invariance. This implies that wkj ð Þ vanishes if k Λ ð0BÞ Λ ð1BÞ These expressions are based on using the Wigner-Eckart is even. For wkj and wkj it follows that j must be odd, Λ ð0EÞ theorem, which is valid because the single-particle oper- while for wkj we find j must be even. These results are a atom ators in the spherical decomposition of δh transform as consequence of the relationships between the indices k and spherical operators with respect to rotations generated by I j of the nonrelativistic coefficients, as listed in Table IV J w and . The Wj0 are combinations of coefficients for Lorentz of Ref. [35]. violation with expectation values of the one-particle oper- Collecting the results discussed in this section yields a ators in Eqs. (3) and (4). The combinations take the form set of constraints determining which coefficients for Lorentz violation can affect the shift of an energy level X∞ 0 0 1 1 in an atom or ion. Table I compiles information about the w T NRð BÞΛ ð BÞ T NRð BÞΛ ð BÞ Wj0 ¼ ð wkj0 wkj þ wkj0 wkj Þ nonrelativistic spherical coefficients that can contribute to −1 k¼j spectral shifts. The first column of the table lists the X∞ 0 coefficients, which we denote generically by K NR . The þ V NRΛ ð EÞ; ð13Þ wkjm wkj0 wkj flavor of the operator associated to the coefficient is k¼j specified in the second column. The third column gives the angular momenta K that restrict the values of the j index where the indicated restrictions of the values of k in the on the coefficient according to the constraint 2K ≥ j.For sums originate in the properties of the nonrelativistic electron coefficients these angular momenta are the total coefficients provided in Table IV of Ref. [35]. Generic angular momentum F of the system and the total angular Λ ðqPÞ expressions for the quantities wkj can be found in terms momentum J of the electronic shells, while for nucleon α0 α0 of expectation values of the states j mJi and j mIi.For coefficients they are F and the nuclear spin I. The next two the electron operators, we have columns provide conditions on the values of j for the cases of integer K and of half-integer K. The value of j must be XNe 0 k ˆ ˆ 0 0 hα Jjjpaj 0Y 0ðpaÞσa · pajα Ji even for coefficients in the first two rows and odd for other Λ ð BÞ ¼ − j ; ekj hJJjj0JJi coefficients. This can constrain the maximum allowed a¼1 value of j. For example, for even j and half-integer K XNe α0 p k pˆ σ ϵˆ ϵˆ α0 1 h Jjj aj 1Y 0ð aÞ a · ð þa þ −aÞj Ji the equality in the condition 2K ≥ j cannot be satisfied Λ ð BÞ þ j ; ekj ¼ 0 because 2K is odd, so the maximum allowed value of j is 1 hJJjj JJi a¼ 2K − 1. The final column in the table displays the allowed XNe 0 k ˆ 0 0 hα Jjjpaj 0Y 0ðpaÞjα Ji values of k. Note that the appearance of a coefficient in the Λ ð EÞ − j ; ekj ¼ 0 ð14Þ table is necessary but not sufficient for it to contribute to a 1 hJJjj JJi a¼ Λ ðqPÞ theoretical energy shift because some wkj may vanish where the sum on a ranges over the electrons in the atom. for other reasons when a particular model is used to For the nucleon operators, we find compute the expectation values.

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TABLE I. Contributing nonrelativistic spherical coefficients.

K NR wkj0 wKjvalues, integer Kjvalues, half-integer Kkvalues V NR NR NR 2 ≥ ≥ 0 2 − 1 ≥ ≥ 0 ≥ wkj0, awkj0, cwkj0 eF, J even, K j even, K j even, k j V NR NR NR 2 ≥ ≥ 0 2 − 1 ≥ ≥ 0 ≥ wkj0, awkj0, cwkj0 n, pF, I even, K j even, K j even, k j T NRð0BÞ NRð0BÞ NRð0BÞ eF, J odd, 2K − 1 ≥ j ≥ 1 odd, 2K ≥ j ≥ 1 even, k ≥ j − 1 wkj0 , gwkj0 , Hwkj0 T NRð0BÞ NRð0BÞ NRð0BÞ n, pF, I odd, 2K − 1 ≥ j ≥ 1 odd, 2K ≥ j ≥ 1 even, k ≥ j − 1 wkj0 , gwkj0 , Hwkj0 T NRð1BÞ NRð1BÞ NRð1BÞ eF, J odd, 2K − 1 ≥ j ≥ 1 odd, 2K ≥ j ≥ 1 even, k ≥ j − 1 wkj0 , gwkj0 , Hwkj0 T NRð1BÞ NRð1BÞ NRð1BÞ n, pF, I odd, 2K − 1 ≥ j ≥ 1 odd, 2K ≥ j ≥ 1 even, k ≥ j − 1 wkj0 , gwkj0 , Hwkj0

XNe C. Electron expectation values 0 1 Λ ð EÞ ¼ − pffiffiffiffiffiffi hjp jki; ð16Þ In this section, the calculation of the electronic ek0 4π a a 1 expectation values (14) is discussed. The situation where ¼ F or J vanish is considered first. We then outline an where the sum on a ranges over all the electrons in the atom. approach to more complicated cases that is general enough p k The quantities hj aj i are the expectation values of powers of to cover systems of interest here while yielding a sufficient the momentum magnitude. approximation to the effects of Lorentz and CPT violation. Denote the contribution to the∘ energy shift due to the This involves modeling the electromagnetic interaction w electron isotropic coefficients by δϵe. Recalling that C0 ¼ 1 between the electrons and the nucleus via a central for any value of F, J, and I, it follows from Eqs. (8) and Coulomb potential and treating the repulsion between the ∘ ∘ (10) that δϵ takes the simple form δϵ We . Using electrons using a mean-field approximation. The approach e e ¼ 00 0 Eqs. (13) and (16) then yields an expression for the energy provides enough information about the states jα mJi to permit a reasonable estimation of the perturbative energy shift due to the electron isotropic coefficients, shift due to Lorentz- and CPT-violating effects on the N ∘ X ∘ NR Xe electron propagators. δϵ − V p 2q e ¼ e;2q hj aj i; ð17Þ q a¼1 1. Case F = 0 or J = 0 The ground states of many atoms and ions considered in where the index q is related to the index k of the coefficients 2 this work have quantum numbers F ¼ 0 or J ¼ 0.For for Lorentz violation by q ¼ k. This enforces the con- example, this holds for the ground state of any noble gas dition that only even values of k can contribute to the and any IIB transition metal such as Hg. It also holds for the energy shift. ground states of many ions of interest, including 27Alþ, 113 115 171 199 2. One open subshell with one electron Cdþ, Inþ, Ybþ, and Hgþ. The excited states of some systems of interest also have these quantum numbers, For atoms or ions with all electronic subshells closed 27 þ 115 þ such as the P0 state in Al or In . except for a single subshell occupied by one electron, we If either of the quantum numbers F or J vanishes, then the can find closed-form expressions for the expectation values electron coefficients for Lorentz violation that can contribute Λ ðqPÞ under suitable simplifying approximations. Treating 0 ekj to the energy shift must have j ¼ . These coefficients the electrons in the closed subshells as forming states with control isotropic Lorentz- and CPT-violating effects. The zero total angular momentum, the value of J for the whole discussion in Sec. II B reveals that the only relevant isotropic system can be identified with the total angular momentum V NR coefficients for electrons are ek00. These special coefficients of the electron in the open subshell. It follows that the only ∘ NR ∘ NR pffiffiffiffiffiffi V V ≡ V NR 4π Λ ðqPÞ ≠ 0 are commonly denoted as e;k , where e;k ek00= . contribution to ekj with j can arise from the valence ∘ NR Since only Ve;k can affect the energy shift, the quantities electron. Contributions from isotropic coefficients with 0 1 0 Λ ð BÞ and Λ ð BÞ cannot contribute to Eq. (13) and so j ¼ are given by Eq. (17). ekj ekj The closed shells produce a spherically symmetric become irrelevant. Moreover, when only the isotropic electronic distribution. For present purposes, the effective coefficients for electrons can provide nonvanishing contri- potential acting on the valence electron due to the repulsion Λ ð0EÞ butions, we can simplify the expression (14) for ek0 .The from the closed-shell electrons can be approximated as values of the relevant Clebsch-Gordan coefficient and central. One consequence of this is that the magnitude L of spherical harmonic are hJJ00jJJi¼1 and Y00 ¼ L pffiffiffiffiffiffi 0 the orbital angular momentum of the valence electron is a 1= 4π. This yields good quantum number for the system. It then becomes

036003-7 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) feasible to obtain explicit expressions for the quantities The results (17) and (18) involve expectation values of ðqPÞ powers of the magnitude of the electron momentum. An Λe defined in Eq. (14). We find kj analytical evaluation of these expectation values is imprac- J ð0EÞ ðj − 1Þ!! Mj tical, even for comparatively simple cases such as the Λ ¼ ij hjpjki; p k ekj j!! ΛJ expectation values hj j i for a valence electron. Numerical j methods can be adopted to resolve this issue, in conjunction 0 j!! Λ ð BÞ ¼ ij−1 MJΛJhjpjki; with techniques such as a self-consistent mean-field ekj ðj − 1Þ!! j j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approximation. However, the principle goal of this work is to serve as a guide to search for Lorentz and CPT 1 2J þ 1 j!!ðj − 2Þ!! Λ ð BÞ ij−1 MJΛJ p k : violation. In this context, a precise determination of these ekj ¼ − j j 2 1 !! − 1 !!hj j i L J ðj þ Þ ðj Þ expectation values is often inessential. For example, some ð18Þ transitions studied here are hyperfine or Zeeman transitions. These involve two levels with similar momentum In this equation, p is the momentum of the valence electron, expectation values, and the difference leaves unaffected the and we define qualitative form of experimental signals for Lorentz and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CPT violation. For these and many other transitions, 2j þ 1 estimates of the expectation values of the electron momen- MJ ¼ ; j 4πð2J þ 1Þ tum suffice as a guide to the sensitivity of experiments sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi across a broad range of systems. An accurate determination ð2J þ jÞ!!ð2J − j − 1Þ!! of the expectation values relevant to a given experimental ΛJ ¼ : ð19Þ j ð2J − jÞ!!ð2J þ j þ 1Þ!! setup may become useful once enough data are collected and a detailed analysis is being performed to extract the Note that the spin-independent operators in Eq. (3) coefficients for Lorentz violation. For a few transitions used transform as spherical operators with respect to rotations in experiments, estimates may be inadequate even as a generated by L, which suffices to exclude contributions guide to the sensitivity. For example, for optical transitions V NR 2 from ekjm to the energy shift when j> L. However, this the difference between the expectation values in the two requirement is already implied in the present context by the states can be significant and must be included in the condition 2J − 1 ≥ j presented in Table I, because the treatment, as described in Sec. III C below. lowest value of L for a given J is L ¼ J − 1=2. For atoms or ions with more than one electron in an open The results (18) can be applied to alkali-metal atoms and subshell, it is typically infeasible to obtain closed-form to singly ionized alkaline-earth ions. In both cases, the expressions like Eqs. (17) and (18). These systems can have electrons in the closed subshells belong to closed shells, so substantial many-body effects, and their treatment requires the approximations made above are comparatively good. a more sophisticated and individualized approach. This can be illustrated by comparing our results with Investigations of such systems are likely also to be of interest detailed calculations for specific states of particular sys- in searches for Lorentz and CPT violation, but a discussion tems. For example, consider the numerical results presented along these lines lies beyond the scope of this work. þ in Table 1 of Ref. [55] for the D3=2 and D5=2 states in Ca , Baþ, and Ybþ. The table provides the reduced matrix D. Nucleon expectation values elements of the operator rffiffiffiffiffiffiffiffi Next, we turn to the evaluation of the nucleon expectation 16π values (15). The simplest situation arises when F or I ð2Þ − p 2 pˆ T0 ¼ 5 j j 0Y20ð Þ; ð20Þ vanishes, for which a compact expression for the energy shift can be presented. For more complicated scenarios, a calculated using several many-body techniques and defined model accounting for the strong nuclear interactions is in terms of Wigner 3-j symbols instead of Clebsch-Gordan required. The central effective potential and mean-field coefficients. The ratio of the reduced matrix elements for approximation used above for the electronic structure are þ the two states D3=2 and D5=2 is 0.77 for Ca and 0.79 for inappropriate to describe the nucleon interactions. Instead, we Baþ. Converting the notation appropriately, we find that adopt here a simple nuclear shell model that permits analytical Eq. (18) predicts a ratio of 0.76 for both systems, in ðqPÞ evaluation of the quantities Λw . This enables an evaluation reasonable agreement with the many-body calculations. kj of the effects of Lorentz and CPT violation from nucleon However, for Ybþ the results obtained in Ref. [55] give a propagators on spectral shifts in a broad range of systems. ratio of 0.82, revealing a greater deviation from our prediction. This is unsurprising because in this ion some 1. Case F = 0 or I = 0 electrons in a closed subshell lie outside the closed shells, so the accuracy of our approximation is expected to be A number of atoms or ions have either vanishing total reduced. angular momentum F ¼ 0 or vanishing nuclear spin I ¼ 0.

036003-8 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The latter situation arises in nuclei with an even number of 2j þ 1 neutrons and an even number of protons. In these cases, MI ¼ ; j 4πð2I þ 1Þ independently of the nuclear model adopted, the energy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δϵ shift w due to a nucleon of flavor w ¼ p or n receives 2I j !! 2I − j − 1 !! ΛI ð þ Þ ð Þ contributions only from isotropic coefficients for Lorentz j ¼ 2 − !! 2 1 !!: ð23Þ violation. The arguments here parallel those in Sec. II C 1. ð I jÞ ð I þ j þ Þ ∘ NR Introducing the special isotropic coefficients Vw;k≡ pffiffiffiffiffiffi ∘ The primary advantage of the Schmidt model in the V NR 4π δϵ wk00= , the expression for w is found to be present context is its application to a broad range of systems Λ ðqPÞ for which the quantities wkj can be evaluated using N Eq. (22). The model has previously been used to determine ∘ X ∘ NR Xw δϵ − V p 2q signals arising from Lorentz- and CPT-violating operators w ¼ w;2q hj aj i; ð21Þ q¼0 a¼1 in the minimal SME for numerous experiments comparing atomic or ionic transitions [9]. A significant limitation of the Schmidt model in this respect is that only one flavor of where the sum over a spans the Nw nucleons of flavor w in nucleon is assumed to contribute to transitions in any given the nucleus. Like the electron case, this isotropic shift can atom or ion, which implies the corresponding experiment is also affect other energy levels having F ≠ 0 or I ≠ 0 sensitive only to coefficients for Lorentz violation in that through its contribution to Eq. (13). flavor sector. A more realistic treatment can be expected to reveal dependence on coefficients for both values of w. 2. Schmidt model for one unpaired nucleon This was illustrated in Ref. [9] using more detailed wave functions for the nuclei of 7Li and 9Be. Recently, The Schmidt model [56,57] offers a comparatively calculations using semiempirical models [32] and many- simple description of a broad range of nuclei. The model body methods [58] have obtained improved values for the assumes a shell structure for the nucleus in which any pair Λ ðqPÞ Λ ð0EÞ of nucleons of a given flavor combines to form states with coefficients wkj , particularly for the coefficient w22 . total angular momentum equal to zero. If only one unpaired These improved values emphasize the disadvantages of nucleon exists in the nucleus, then it is treated as a single- using a single-valence model to study Lorentz- and CPT- particle state with total angular momentum equal to the spin violating effects involving the nucleus. Nonetheless, to I of the nucleus. The magnitude L of the orbital angular maintain generality in this work and to permit the dis- momentum of the unpaired nucleon is treated as a good cussion of a broad range of atoms and ions, we adopt the quantum number. The model can be expected to deviate Schmidt model throughout, commenting where appropriate significantly from observation for nuclei lying away from a on the likely consequences of using improved nuclear magic number. modeling. We remark also that it suffices to estimate the Mathematically, the Schmidt model represents a setup expectation values of the magnitude of the linear nucleon equivalent to the one described in Sec. II C 2 for a valence momentum for all experiments considered here because no electron outside closed subshells. The contribution to the nuclear transitions are involved. perturbative energy shift involving isotropic coefficients is obtained from Eq. (21). When j>0, the expressions for E. Energy shift at zeroth boost order Λ ðqPÞ In any Cartesian inertial frame within the solar system, the quantities wkj can be calculated in closed form and are given by the coefficients for Lorentz violation can reasonably be taken as constant in both time and space [5,6]. However, the energy shift (8) is calculated in a laboratory frame. I 0 ðj − 1Þ!! M Laboratories on the surface of the Earth or on orbiting Λ ð EÞ ¼ ij j hjpjki; wkj j!! ΛI satellites generically correspond to noninertial frames, so j most coefficients appearing in Eq. (8) vary with time due 0 j!! Λ ð BÞ ¼ ij−1 MIΛIhjpjki; to the laboratory rotation and boost [59]. Moreover, the wkj ðj − 1Þ!! j j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi explicit forms of the coefficients for Lorentz violation differ in distinct inertial frames. To permit meaningful compari- 1 2I þ 1 j!!ðj − 2Þ!! Λ ð BÞ ij−1 MIΛI p k ; son of different experiments, experimental coefficient wkj ¼ − j j 2 1 !! − 1 !!hj j i L I ðj þ Þ ðj Þ values must therefore be reported in a canonical inertial ð22Þ frame. The standard frame adopted in the literature for this purpose is the Sun-centered celestial-equatorial frame [53], with Cartesian coordinates denoted ðT;X;Y;ZÞ. In this where p is the linear momentum of the unpaired nucleon of frame, the origin of the time coordinate T is defined as the I ΛI flavor w. The factors Mj and j are defined as vernal equinox 2000. The X axis points from the location of

036003-9 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) the Earth at this equinox to the Sun, the Z axis is aligned where φ is the angle between the X axis and the projection with the Earth’s rotation axis, and the X, Y, Z axes form a of the magnetic field on the XY plane at T⊕ ¼ 0. Note that right-handed coordinate system. The Sun-centered frame is both TL and T⊕ are offset from the standard time T in the appropriate for reporting measurements of coefficients Sun-centered frame by an amount that depends on the because it is inertial to an excellent approximation over longitude of the laboratory, given explicitly for T⊕ in the time scale of typical experiments. Eq. (43) of Ref. [45]. Λμ θ β ≡ lab The observer Lorentz transformation νð ; Þ between The factors Aj0 Aj0 appearing in Eq. (8) are defined in the laboratory frame and the Sun-centered frame can be the laboratory frame. They transform in the same way expressed as the composition of an observer rotation under rotations as the nonrelativistic coefficients for μ μ R νðθÞ with an observer boost B νðβÞ, Lorentz violation, so we can convert them to factors Sun Aj0 defined in the Sun-centered frame via the relation μ μ α Λ νðθ; βÞ¼R αðθÞB νðβÞ: ð24Þ X lab imω⊕T⊕ j −ϑ Sun Aj0 ¼ e d0mð ÞAjm : ð27Þ m The boost parameter β is the velocity of the laboratory frame with respect to the Sun-centered frame, while the The energy shift (8) can therefore be expressed in the Sun- rotation parameter θ fixes the relative orientation between centered frame as the laboratory frame and the frame obtained via the boost. X The magnitude β of β is small compared to the speed of δϵ ðjÞ −ϑ 0 ¼ d0jmjð ÞhFmFj jFmFi light. For example, the speed of the Earth in the Sun- jm β ≃ 10−4 centered frame in natural units is . At zeroth order Sun × ½ReA cos ðjmjω⊕T⊕Þ in β the boost transformation is simply the identity map, so jjmj the Lorentz transformation between the two frames − Sun ω Imjjmj sin ðjmj ⊕T⊕Þ; ð28Þ becomes a pure rotation. In this section, we consider the energy shift (8) at zeroth boost order. Effects at linear boost thereby explicitly demonstrating the time variation of the order are discussed in Sec. II F. spectrum of the atom or ion at harmonics of the sidereal In the laboratory frame, only the nonrelativistic coef- frequency ω⊕. ficients K NR;lab for Lorentz and CPT violation with m ¼ 0 wkjm For any m, a given factor ASun contains coefficients for contribute to the energy shift. At zeroth boost order and jjmj for a laboratory on the Earth, these coefficients can be Lorentz violation labeled with the same index j. However, converted to coefficients K NR;Sun in the Sun-centered as summarized in Table I, only restricted values of j for wkjm nonrelativistic coefficients can contribute to a specific frame via energy shift. Since the highest harmonic that can contribute X to the sidereal variation is determined by the maximum ω j K NR;lab im ⊕TL −ϑ K NR;Sun wkj0 ¼ e d0mð Þ wkjm : ð25Þ value of jmj, which in turn is fixed by the largest allowed m value of the index j, we can use the information in Table I ϑ to deduce constraints on the possible harmonics contrib- Here, is the angle between the applied magnetic field and uting to the time variation of any particular energy level. ’ j the Earth s rotation axis Z, and the quantities dmm0 are the Table II summarizes these constraints for various condi- little Wigner matrices given in Eq. (136) of Ref. [50]. The tions on the quantum numbers F, J, and I. The first column conversion (25) reveals the time variations of the labora- of the table lists the conditions, while the second column tory-frame coefficients, which occur at harmonics of the displays the range of allowed values of jmj, which ’ ω ≃ 2π 23 56 Earth s sidereal frequency ⊕ =ð h minÞ. The corresponds to the possible harmonics of ω⊕ that can local sidereal time TL is a convenient local Earth sidereal appear. For example, the first row of the table shows that time with origin chosen as the time when the magnetic field the time variation of an energy level with quantum numbers lies in the XZ plane in the Sun-centered frame. This choice F ¼ 3, I ¼ 7=2, and J ¼ 3=2 can in principle contain up to yields the comparatively simple expression (25). For some the sixth harmonic of ω⊕. Note, however, that special applications below it is preferable instead to adopt a different local sidereal time T⊕, which is associated with ω the laboratory frame introduced in Ref. [53] and has as TABLE II. Possible harmonics of ⊕ at zeroth boost order. origin the time at which the tangential velocity of the Conditions on F, J, I Possible harmonics laboratory frame points along the Y axis. The relationship F ≤ J or F ≤ I or both 2F ≥ jmj ≥ 0 between TL and T⊕ is F ≥ J, F ≥ I, J ≥ I 2J ≥ jmj ≥ 0 F ≥ J, F ≥ I, I ≥ J 2I ≥ jmj ≥ 0 ω⊕T⊕ ¼ ω⊕TL − φ; ð26Þ

036003-10 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) circumstances might contrive to lower the maximum ð4Þð1BÞ ˜ð4Þj3j g210 is proportional to the combination geff of harmonic affecting a given transition frequency. For exam- Cartesian coefficients in the laboratory frame. This combi- Sun 4 μνα ple, a factor Ajjmj might vanish identically, or the two ˜ð Þ nation can be converted to Cartesian coefficients geff;rot in energy levels involved in the transition might have identical the rotated frame as contributions at a particular harmonic so that the transition frequency is unaffected. Note also that time variations at ˜ð4Þj3j Rj R3 Rj ˜ð4ÞKLM R3 ˜ð4ÞJKJ ˜ð4ÞJKJ ˆ K higher harmonics than those displayed in Table II can geff ¼ K L Mgeff;rot ¼ Kgeff;rot ¼ geff;rot B ; become allowed when effects at linear or higher order in the ð30Þ boost are incorporated, but any such variations are suppressed by powers of the boost. where the second equality follows from the iden- j j tity R KR L ¼ δKL. F. Energy shift at linear boost order The above discussion shows that the number of factors of ˆ J Since the magnitude of the boost between the laboratory B appearing in a given term contributing to the energy frame and the Sun-centered frame is small, it is reasonable shift at linear boost order is determined by the index μ structure of the corresponding coefficient for Lorentz to expand the boost transformation B νðβÞ of Eq. (24) in powers of the relative speed β. In this section, we consider violation. To keep the explicit tables appearing in contributions to the energy shift that appear at linear order Sec. III of reasonable size, we limit attention below to in the boost. At this order, the components of the observer Lorentz- and CPT-violating operators of mass dimension μ d ≤ 8. Expressions relating all the corresponding Cartesian Lorentz transformation Λ νðθ; βÞ take the form coefficients in the laboratory frame to those in the Sun- 0 0 centered frame at linear boost order are given in Λ ¼ 1; Λ ¼ −βJ; Λj ¼ −Rj βJ; Λj ¼ Rj ; T J T J J J Appendix B. Inspection of these results reveals that the ð29Þ form of the shift δν in a transition frequency for an atom or ion takes the generic form where lowercase and uppercase indices represent spatial Cartesian coordinates in the laboratory frame and in the X8 X5 … ˆ J1 ˆ J Sun-centered frame, respectively. δν ¼ VðdÞJJ1 Js B …B s βJ ð31Þ Given an expression for the energy shift in the laboratory d¼3 s¼0 frame in terms of spherical coefficients for Lorentz viola- d JJ1…J tion, converting to the Sun-centered frame at linear boost at linear boost order, where the quantities Vð Þ s are order can be performed in two steps. First, the spherical linear combinations of Cartesian coefficients for Lorentz coefficients in the laboratory frame can be rewritten in violation in the Sun-centered frame. The explicit forms of β ˆ terms of Cartesian coefficients in the same frame. The and B in this equation depend on the choice of laboratory transformation (29) can then be applied to express the frame. We consider here in turn two types of laboratory, one Cartesian coefficients in the laboratory frame in terms of located on the surface of the Earth and another on a Cartesian coefficients in the Sun-centered frame. spacecraft orbiting the Earth. Explicit expressions relating spherical coefficients to For a laboratory on the surface of the Earth, the boost Cartesian coefficients in any inertial frame are given in velocity β in Eq. (31) can be taken as Appendix A. To implement the conversion to the Sun- centered frame, note that only spherical coefficients for β ¼ β⊕ þ βL; ð32Þ Lorentz violation with m ¼ 0 contribute to the energy shift (8) in the laboratory frame. This implies that all uncon- where β⊕ is the instantaneous Earth orbital velocity in the tracted spatial indices on the corresponding Cartesian Sun-centered frame and βL is the instantaneous tangential coefficients are in the x3 direction. The relevant part of velocity of the laboratory relative to the Earth’s rotation j the rotation matrix R J converting the Cartesian compo- axis. Approximating the Earth’s orbit as circular, the nents between the laboratory frame and the Sun-centered velocity β⊕ can be written as 3 frame therefore involves the row R J. This row can be 3 ˆ ˆ ˆ viewed as the components of a unit vector along x . Since β⊕ ¼ β⊕ sinΩ⊕TX − β⊕ cosΩ⊕TðcosηY þ sinηZÞ; ð33Þ by construction this is the quantization axis of the atom or ˆ −4 ion, which points in the direction B of the applied magnetic where β⊕ ≃ 10 is the Earth’s orbital speed, Ω⊕ ≃ 3 ˆ J 2π 365 26 ’ field, it follows that R J ¼ B . =ð . dÞ is the Earth s orbital angular frequency, To illustrate this idea with an example, consider the and η ≃ 23.4° is the angle between the XY plane and the ð4Þð1BÞ spherical coefficient g210 given in the laboratory frame. Earth’s orbital plane. Also, treating the Earth as spherical, From Appendix A we see that the spherical coefficient the tangential velocity βL takes the form

036003-11 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) 0 1 ˆ ˆ β ¼ −β sin ω⊕T⊕X þ β cos ω⊕T⊕Y; ð34Þ −cosαcosωsTs þ cosζ sinαsinωsTs L L L B C ˆ xs ¼ @ −sinαcosωsTs − cosαcosζ sinωsTs A; where βL ≈ r⊕ω⊕ sin χ is determined by the colatitude χ of −sinζ sinωsTs the laboratory the radius r⊕ of the Earth, and the sidereal 0 1 ω Bˆ sinαsinζ frequency ⊕. The unit vector in Eq. (31) can conven- β xˆ B C ˆ s × s iently be expressed in an instantaneous Earth-centered ys ≡ ¼ @−cosαsinζ A; βs coordinate system with axes parallel to those of the Sun- cosζ centered frame, β zˆ ¼ s : ð38Þ ˆ ˆ s β B ¼ sin ϑ cos ðω⊕T⊕ þ φÞX s ˆ ˆ þ sin ϑ sin ðω⊕T⊕ þ φÞY þ cos ϑZ; ð35Þ Using this basis, the direction Bˆ of the magnetic field in the space-based experiment can be expressed as where ϑ and φ are the polar and azimuthal angles of the ⊕ 0 ˆ ˆ ˆ ˆ magnetic field at T ¼ . B ¼ sin θs cos ϕsxs þ sin θs sin ϕsys þ cos θszs; ð39Þ Next, consider a laboratory located on a space-based platform orbiting the Earth. Examples include experiments ˆ ˆ ˆ where cos θs ¼ βs · B=βs and cos ϕs ¼ xs · B. on board the International Space Station (ISS) such as the Atomic Clock Ensemble in Space (ACES) [60] and the III. APPLICATIONS Quantum Test of the Equivalence Principle and Space Time (QTEST) [61], or dedicated missions searching for Lorentz In this section, we comment on some applications of the violation such as the Space-Time Explorer and Quantum formulas derived above. Many existing searches for Equivalence Space Test (STE-QUEST) [62] and the Boost Lorentz and CPT violation are based on the study of Symmetry Test (BOOST) [63]. We adopt the coordinates transitions in fountain clocks, in comagnetometers, and in depicted in Fig. 2 of Ref. [64]. Assuming for definiteness a trapped ions or lattice clocks. Each of these experimental trajectory with negligible eccentricity, the parameters for the approaches is considered in turn. We present expressions relevant to the analysis of data from a variety of experi- orbit are the mean orbital radius rs, the mean orbital angular ments, and we estimate the attainable sensitivities to speed ωs, the angle ζ between the satellite orbital axis and the Earth’s rotation axis, and the azimuthal angle α between the coefficients for Lorentz violation along with actual con- Earth and satellite orbital planes. In this scenario, the boost straints from existing data where possible. velocity β in Eq. (31) can be written as the vector sum A. Fountain clocks β β β 133 ¼ ⊕ þ s ð36Þ Fountain clocks using Cs atoms have been widely adopted as primary time and frequency standards. The ’ β of the Earth s orbital velocity ⊕ in the Sun-centered frame standard transition in these clocks, jF ¼ 3;mF ¼ 0i ↔ β and the satellitevelocity s relative to an instantaneous Earth- jF ¼ 4;mF ¼ 0i, is insensitive to the Lorentz- and CPT- centered frame. Explicitly, the components of the satellite violating spectral shifts discussed above. This implies that 133 velocity βs in the Sun-centered frame take the form the Cs frequency standard can be used as a reference in 0 1 experimental studies searching for Lorentz violation, in −β α ω − β ζ α ω s cos sin sTs s cos sin cos sTs parallel with the hydrogen-maser standard [43]. Searches B C 133 βs ¼ @ −βs sin α sin ωsTs þ βs cos α cos ζ cos ωsTs A; for violations of Lorentz and CPT symmetry using a Cs β ζ ω fountain clock can instead be performed by studying the s sin cos sTs ν 3 ↔ 4 frequencies mF for transitions jF ¼ ;mFi jF ¼ ;mFi ð37Þ with mF ≠ 0. These transitions are individually sensitive to the linear Zeeman shift, and hence their precision is limited β ω where s ¼ rs s and the local satellite time Ts has origin by systematic effects. However, the systematics can be fixed as the satellite crosses the equatorial plane on an significantly reduced by measuring the observable νc ¼ ν ν − 2ν ascending orbit. Obtaining also an explicit expression for the þ3 þ −3 0 [16,65]. unit vector Bˆ in Eq. (31) requires a further specification of the The total electronic angular momentum for the states orientation of the space-based laboratory relative to the Earth. with mF ¼1 is J ¼ 1=2. Consulting Table I reveals that For example, when this orientation is fixed then an instanta- only electron operators with j ≤ 1 can in principle shift the ν ν neous satellite frame can be defined with x axis pointing frequencies 3. However, the observable c remains radially towards the Earth and z axis aligned along βs.The unaffected by these shifts. To evaluate the nucleon con- ˆ ˆ ν components of the corresponding unit spatial vectors xs, ys, tributions to c we adopt the Schmidt model as discussed ˆ 133 zs can be expressed in the Sun-centered frame as above, in which the nuclear spin I ¼ 7=2 of Cs is

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NR;Sun assigned to the unpaired proton. With this assumption, V 4 → 0; p jm rffiffiffiffiffiffi the Lorentz-violating shift δνc of the observable νc is 1 4π 4 given by V NR;Sun → − c˜ ð Þ; p220 3m2 5 pQ prffiffiffiffiffiffi rffiffiffi 1 2π NR;Sun ˜ ð4Þ 3 5 ReVp221 → − cp ; 2πδν ¼ − ðhjpj2iV NR þhjpj4iV NR Þ m2 15 Y c 14 π p220 p420 rp ffiffiffiffiffiffi 45 1 2π 4 4 NR V NR;Sun → ˜ ð Þ pffiffiffi p V ; 40 Im p221 2 cp ; þ 77 π hj j i p440 ð Þ m 15 X p rffiffiffiffiffiffi 1 2π V NR;Sun → ˜ ð4Þ Re p222 2 15cp− ; p mp where is the momentum of the valence proton. Note that rffiffiffiffiffiffi the results presented in Refs. [16,65] based on the minimal 1 2π 4 ImV NR;Sun → − c˜ ð Þ ð43Þ SME analysis in Ref. [9] can be recovered from the above p222 m2 15 pZ expression by excluding contributions from nonminimal p Lorentz-violating operators. In practice, this corresponds to making the replacements in the entries in the fourth column of Table III. Next, consider the contribution to the frequency shift δν rffiffiffiffiffiffi c;1 at linear order in the boost. Applying the trans- 1 4π formation (29) to Eq. (40) and writing the result in the form V NR → 3 ð4Þ − ð4Þ p220 ð cpzz cpjj Þ; (31) yields 3mp 5 V NR → 0 V NR → 0 p420 ; p440 ð41Þ X p 2 2πδν hj j i ðdÞJβJ ðdÞJKL ˆ K ˆ LβJ c;1 ¼ 5−d ðVCs;2 þ VCs;2 B B Þ d mp in Eq. (40). X p 4 hj j i ðdÞJβJ ðdÞJKL ˆ K ˆ LβJ To convert the above expression to the Sun-centered þ 7−d ðVCs;4 þ VCs;4 B B Þ d mp frame, consider first the frequency shift δνc;0 at zeroth X 4 boost order. Applying the transformation rule (25) for hjpj i þ VðdÞJKLMNBˆ KBˆ LBˆ MBˆ NβJ: ð44Þ nonrelativistic coefficients to the result (40) yields a 7−d Cs;4 d mp somewhat lengthy form for δνc;0 in the Sun-centered frame. The result is presented in tabular form in ðdÞJJ1…Js Table III. In each row of the table, the first entry contains Expressions for the quantities VCs;k in terms of the the harmonic dependence on the sidereal frequency ω⊕ effective Cartesian coefficients can be deduced from the and the local sidereal time TL. The second entry results presented in Appendix B and are displayed in describes the dependence on the orientation of the Table IV for mass dimensions 5 ≤ d ≤ 8. In each row of … magnetic field in the laboratory frame. The third entry ðdÞJJ1 Js this table, the first entry lists a specific quantity VCs;k , provides the relevant expectation value of the proton while the second entry gives its expression as a combina- p momentum magnitude j j, while the fourth entry con- tion of effective Cartesian coefficients in the Sun- tains the numerical factor and the coefficient for Lorentz centered frame. δν violation. To obtain the frequency shift c;0, it suffices Note that the minimal-SME limit of the result (44) can be to multiply the columns and add the rows in the table. ðdÞJJ1…Js obtained by setting to zero all the quantities VCs;k For example, the contributions to δνc;0 from the first and second rows are except for

4 2 4 4 6 4 rffiffiffi rffiffiffi Vð ÞJ ¼ − c ð ÞTJ;Vð ÞJKL ¼ c ð ÞTKδJL; ð45Þ 3 5 9 5 Cs;2 7 peff Cs;2 7 peff − p 2 V NR;Sun − 2ϑ p 2 V NR;Sun 56 πhj j i p220 56 π cos hj j i p220 : ð4Þμν ð42Þ where the coefficients ceff are defined as the symmetric 1 ð4Þμν ð4Þνμ combination 2 ðc þ c Þ, as in Ref. [35]. In contrast to the minimal-SME case, the nonminimal terms introduce Note that the corresponding expression for δνc;0 in sidereal variations incorporating the third, fourth, and fifth the minimal SME can be obtained by making the harmonics of the sidereal frequency. For example, the replacements contribution to δνc;1 from the fifth harmonic is given by

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TABLE III. Frequency shift δνc;0 at zeroth boost order.

k ω⊕T ϑ p Coefficient L hj j i qffiffi p 2 − 3 5V NR;Sun 11hj j i 56 qπffiffi p220 2ϑ p 2 − 9 5V NR;Sun 1 cos hj j i qffiffi 56 π p220 3 5 405 p 4 − V NR;Sun pffiffi V NR;Sun 11hj j i 56 qπffiffi p420 þ 4928 π p440 2ϑ 4 − 9 5V NR;Sun 225 ffiffi V NR;Sun 1 cos hjpj i 56 π p420 þ 1232pπ p440 4ϑ p 4 225 ffiffi V NR;Sun 1 cos hj j i 704pπ p440 qffiffiffiffi 2 sin ω⊕TL sin 2ϑ hjpj i 3 15 NR;Sun 14 2πImVp221 qffiffiffiffi qffiffi 4 sin ω⊕TL sin 2ϑ hjpj i 3 15 NR;Sun 45 5 NR;Sun 14 2πImVp421 − 616 πImVp441 qffiffi ω 4ϑ p 4 − 45 5 V NR;Sun sin ⊕TL sin hj j i 176qffiffiffiffiπIm p441 ω 2ϑ p 2 − 3 15 V NR;Sun cos ⊕TL sin hj j i qffiffiffiffi 14 2πRe p221qffiffi 4 3 15 NR;Sun 45 5 NR;Sun cos ω⊕TL sin 2ϑ hjpj i − 14 2πReVp421 þ 616 πReVp441 qffiffi 4 cos ω⊕TL sin 4ϑ hjpj i 45 5 NR;Sun 176 πReVp441 qffiffiffiffi 2 sin 2ω⊕TL 1 hjpj i 3 15 NR;Sun 28 2πImVp222 qffiffiffiffi 2 3 15 NR;Sun sin 2ω⊕TL cos 2ϑ hjpj i − 28 2πImVp222 qffiffiffiffi qffiffiffiffi 4 sin 2ω⊕TL 1 hjpj i 3 15 NR;Sun 135 5 NR;Sun 28 2πImVp422 − 1232 2πImVp442 qffiffiffiffi qffiffiffiffi 4 3 15 NR;Sun 45 5 NR;Sun sin 2ω⊕TL cos 2ϑ hjpj i − 28 2πImVp422 − 308 2πImVp442 qffiffiffiffi 4 sin 2ω⊕TL cos 4ϑ hjpj i 45 5 NR;Sun 176 2πImVp442 qffiffiffiffi 2 3 15 NR;Sun cos 2ω⊕TL 1 hjpj i − 28 2πReVp222 qffiffiffiffi 2 cos 2ω⊕TL cos 2ϑ hjpj i 3 15 NR;Sun 28 2πReVp222 qffiffiffiffi qffiffiffiffi 4 3 15 NR;Sun 135 5 NR;Sun cos 2ω⊕TL 1 hjpj i − 28 2πReVp422 þ 1232 2πReVp442 qffiffiffiffi qffiffiffiffi 4 cos 2ω⊕TL cos 2ϑ hjpj i 3 15 NR;Sun 45 5 NR;Sun 28 2πReV 422 þ 308 2πReV 442 p qffiffiffiffi p 2ω 4ϑ p 4 − 45 5 V NR;Sun cos ⊕TL cos hj j i 176qffiffiffiffi2πRe p442 4 45 5 NR;Sun sin 3ω⊕TL sin 2ϑ hjpj i − 88 7πImVp443 qffiffiffiffi 4 sin 3ω⊕TL sin 4ϑ hjpj i 45 5 NR;Sun 176 7πImVp443 qffiffiffiffi 4 cos 3ω⊕TL sin 2ϑ hjpj i 45 5 NR;Sun 88 7πReVp443 qffiffiffiffi 3ω 4ϑ p 4 − 45 5 V NR;Sun cos ⊕TL sin hj j i 176qffiffiffiffiffiffi7πRe p443 4 135 5 NR;Sun sin 4ω⊕TL 1 hjpj i − 352 14πImVp444 qffiffiffiffiffiffi 4 sin 4ω⊕TL cos 2ϑ hjpj i 45 5 NR;Sun 88 14πImVp444 qffiffiffiffiffiffi 4 45 5 NR;Sun sin 4ω⊕TL cos 4ϑ hjpj i − 352 14πImVp444 qffiffiffiffiffiffi 4 cos 4ω⊕TL 1 hjpj i 135 5 NR;Sun 352 14πReV 444 qffiffiffiffiffiffi p 4 45 5 NR;Sun cos 4ω⊕TL cos 2ϑ hjpj i − 88 14πReVp444 qffiffiffiffiffiffi 4 cos 4ω⊕TL cos 4ϑ hjpj i 45 5 NR;Sun 352 14πReVp442

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… ðdÞJJ1 Js 5 ≤ ≤ 8 TABLE IV. The quantities VCs;k for d . extracting real constraints from data would be of definite interest. ðdÞJJ1…Js VCs;k Combination Further developments of these results are also possible. 5 5 5 Corrections at second boost order that are sensitive to Vð ÞJ 3 ða ð ÞJKK þ 2a ð ÞJTTÞ Cs;2 7 peff peff isotropic coefficients in the minimal SME are analyzed in 9 ð5ÞJJ1J2 ð5ÞTTJ2 ð5ÞJJ1J2 − a 2δJJ1 a V 2 7 ð peff þ peff Þ Ref. [65]. Generalizing this analysis to the nonminimal Cs; 6 6 ð6ÞJ 12 ð ÞJTKK ð ÞJTTT − 7 ðcp þ cp Þ sector is a worthwhile open project. Another line of VCs;2 eff eff 6 JJ1J2 36 6 JTJ1J2 6 TTTJ2 reasoning extending the above results would involve Vð Þ ðc ð Þ þ δJJ1 c ð Þ Þ Cs;2 7 peff peff replacing the Schmidt model with a more realistic nuclear ð7ÞJ 10 3 ð7ÞJTTKK 2 ð7ÞJTTTT 133 VCs;2 7 ð apeff þ apeff Þ model for the Cs nucleus. In the minimal SME, this 30 ð7ÞJTTJ1J2 ð7ÞTTTTJ2 ð7ÞJJ1J2 − 3 2δJJ1 replacement reveals that neutron Lorentz-violating oper- V 7 ð apeff þ apeff Þ Cs;2 ators with j ¼ 2 also contribute to the frequency shift, ð7ÞJ 60 ð7ÞJKKLL 4 ð7ÞJTTKK VCs;4 77 ðapeff þ apeff Þ thereby leading to constraints on coefficients for Lorentz 540 ð7ÞJTTJ1J2 ð7ÞTTJ1KK ð7ÞJJ1J2 JJ1 − 77 ðap þ δ ap Þ violation in the neutron sector [65]. For the nonrelativistic VCs;4 eff eff − 270 ð7ÞJJ1J2KK coefficients, the neutron-sector corrections can be incorpo- 77 apeff rated into the expressions given above via the replacement 7 JJ1J2J3J4 150 7 JJ1J2J3J4 7 TTJ2J3J4 ð Þ ð Þ 4δJJ1 ð Þ VCs;4 77 ðapeff þ apeff Þ 8 8 ð8ÞJ 30 ð ÞJTTTKK ð ÞJTTTTT NR NR NR − 7 ð2cp þ cp Þ V → V þ 0.021V 22 : ð47Þ VCs;2 eff eff p22m p22m n m 8 JJ1J2 8 JTTTJ1J2 8 TTTTTJ2 ð Þ 90 2 ð Þ δJJ1 ð Þ VCs;2 7 ð cpeff þ cpeff Þ 8 8 We can therefore estimate the attainable sensitivities to ð8ÞJ 360 ð ÞJTKKLL ð ÞJTTTKK − 77 ðcp þ 2cp Þ VCs;4 eff eff these neutron-sector coefficients by reducing by about 2 8 JJ1J2 1620 8 JTTTJ1J2 8 TTTJ2KK ð Þ ð Þ δJJ1 ð Þ VCs;4 77 ðcpeff þ cpeff Þ orders of magnitude the corresponding proton-sector esti- 8 þ 1620 c ð ÞJTJ1J2KK mates given in Table V. Note that neutron Lorentz-violating 77 peff 2 4 4 900 ð8ÞJTJ1J2J3J4 1800 ð7ÞTTTJ2J3J4 operators with j ¼ , k ¼ or with j ¼ may also affect ð8ÞJJ1J2J3J4 JJ1 − 77 cp − 77 δ cp VCs;4 eff eff the energy shift, but this possibility remains unexplored in the literature to date. Atomic clocks placed on orbiting satellites or other 1 X p 4 4 hj j i spacecraft offer qualitatively different experimental oppor- 2πδν 1 5ω ¼ β sin ϑ c; ; ⊕ 16 L 7−d d mp tunities for studying Lorentz and CPT symmetry. Since typical space missions involve different clock trajectories 5ω 10 ðdÞXXXYY × ½sin ⊕Tð VCs;4 than those relevant to Earth-based laboratories, they pro- ðdÞXXXXX ðdÞXYYYY vide access to different combinations of coefficients for − V 4 − 5V 4 Þ Cs; Cs; Lorentz violation [64]. For example, the orbital plane of 5ω 5 ðdÞXXXXY þ cos ⊕Tð VCs;4 space-based laboratories like the International Space − 10 ðdÞXXYYY ðdÞYYYYY VCs;4 þ VCs;4 Þ ð46Þ TABLE V. Potential sensitivities to coefficients in the Sun- centered frame from sidereal and annual variations in a 133Cs fountain clock. and is suppressed by the boost factor βL, in agreement with the discussion following Eq. (28). Coefficient Sensitivity NR NR −24 −1 Taken together, the above results permit estimates of the jap221j, jcp221j 10 GeV potential sensitivity to Lorentz and CPT violation that is ja NR j, jc NR j 10−22 GeV−3 133 p421 p421 attainable in experiments with Cs fountain clocks via −24 −1 ja NR j, jc NR j 10 GeV studies of sidereal and annual variations. Adopting as a p222 p222 NR NR 10−22 −3 benchmark the measurements of minimal-SME coefficients jap422j, jcp422j GeV reported in Ref. [65], it is reasonable to expect sensitivities ð4ÞTJ 10−20 jcpeff j in the Sun-centered frame of the orders of magnitude listed 5 5 −20 −1 ja ð ÞTTJj, ja ð ÞKKJj 10 GeV in Table V. The first four lines of this table provides peff peff ð6ÞTTTJ ð6ÞTKKJ 10−21 GeV−2 estimated sensitivities to the nonrelativistic coefficients jcpeff j, jcpeff j V NR ð7ÞTTTTJ ð7ÞTTKKJ 10−21 GeV−3 pk2m, while the remainder of the table concerns the japeff j, japeff j ðdÞJ ð7ÞKKLLJ 10−18 GeV−3 effective Cartesian coefficients VCs;k. For the entries japeff j involving the latter, the uncontracted Cartesian spatial ð8ÞTTTTTJ ð8ÞTTTKKJ 10−21 GeV−4 jcpeff j, jcpeff j index J represents any of the possible values X, Y, Z. ð8ÞTKKLLJ −19 −4 jc j 10 GeV These estimated attainable sensitivities are competitive, so peff

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ω ω Station (ISS) is inclined relative to the equatorial plane and compared the angular frequencies Xe and He of precesses, thereby sampling orientation-dependent effects Larmor transitions in the ground states of 129Xe and 3He in a unique way. The satellite orbital velocity can also atoms by measuring the observable exceed the rotational velocity of the Earth, which γ can enhance some signals for Lorentz violation and can ω ω − He ω ¼ He γ Xe; ð49Þ permit faster data accumulation. For instance, the period of Xe the ISS is approximately 92 minutes, so over 15 orbits are completed during a sidereal day. γ 133 which is insensitive to the linear Zeeman shift. Here, Xe is A Cs cold-atom clock is a component of the ACES 129 γ the gyromagnetic ratio for the ground state of Xe and He platform on the ISS [60]. The proposed STE-QUEST is that for the ground state of 3He. mission [62] may also involve a 133Cs clock. Frequency 133 Since the total electronic angular momentum of the noble data obtained from operating a Cs clock in the spin- gases in the ground state is J ¼ 0, the Larmor transitions polarized mode can be converted to the Sun-centered frame are unaffected by electron coefficients for Lorentz viola- – using Eqs. (36) (39) or similar expressions, leading to tion. The contributions from the nucleon coefficients can be bounds on combinations of coefficients for Lorentz viola- estimated using the Schmidt model, in which the nuclear tion that are inaccessible to Earth-based experiments. For spin I ¼ 1=2 of each species is assigned to the unpaired V NR example, Table III shows that the coefficients pkj0 neutron. The analysis in Sec. II then yields the Lorentz- produce no sidereal effects in an Earth-based laboratory, violating shift δω of the observable ω as but they can be measured on a space platform. 87 Fountain clocks using Rb atoms have been considered as 1 X2 γ interesting alternatives for a primary frequency standard δω ¼ − pffiffiffiffiffiffi hjpj2qi − He hjpj2qi 3π He γ Xe [66,67] and for studying Lorentz symmetry via the proposed q¼0 Xe 133 87 space-based mission QTEST [61]. A double Cs and Rb 0 1 × ðT NRð BÞ þ 2T NRð BÞÞ; ð50Þ fountain clock has been used to search for Lorentz and CPT nð2qÞ10 nð2qÞ10 violation [16,65]. Like its 133Cs and H analogues, the 87Rb clock transition jF ¼ 1;mF ¼ 0i ↔ jF ¼ 2;mF ¼ 0i is evaluated in the laboratory frame. In this expression, p k p k insensitive to Lorentz and CPT violation and can thus be hj j iHe and hj j iXe are the expectation values of the used as a reference in experiments. However, the frequencies Schmidt neutron in 3He and 129Xe. These quantities can νRb 1 ↔ 2 mF associated with the transitions jF¼ ;mFi jF¼ ;mFi reasonably be taken as roughly the same order of magni- ≠ 0 p k ∼ p k δω with mF do experience Lorentz- and CPT-violating tude, hj j iHe hj j iXe, so the shift can be written as shifts. The systematics associated with the linear Zeeman shift can in this case be reduced by considering the X2 2q γ hjpj i NR 0B NR 1B observable νRb ¼ νRb þ νRb − 2νRb. Coefficients in the δω ¼ He − 1 pffiffiffiffiffiffi ðT ð Þ þ 2T ð ÞÞ: c þ1 −1 0 γ 3π nð2qÞ10 nð2qÞ10 Rb q¼0 Xe electron sector leave νc unaffected. In the context of the Schmidt model the valence nucleon is a proton with ð51Þ Rb Rb spin I ¼ 3=2, so the shift δνc in the observable νc depends on coefficients for Lorentz violation in the proton sector. We This result reduces to the minimal-SME expressions find presented in Refs. [22–25] based on the theoretical treat- 1 ment of Ref. [9], by taking the limit 2πδνRb ¼ − pffiffiffiffiffiffi ðhjpj2iV NR þhjpj4iV NR Þ: ð48Þ c 5π p220 p420 pffiffiffiffiffiffi NRð0BÞ NRð1BÞ ˜ n T n010 þ 2T n010 → 2 3πb3; 87 133 Since the nuclear spin of Rb is smaller than that of Cs, NR 0B NR 1B T ð Þ þ 2T ð Þ → 0; fewer coefficients appear in Eq. (48) than in Eq. (40).All n210 n210 133 NRð0BÞ NRð1BÞ results for Cs fountains discussed in the present section can T n410 þ 2T n410 → 0; ð52Þ be transcribed to results for 87Rb fountains by matching the changes between Eqs. (40) and (48). as expected. Conversion of Eq. (51) to the Sun-centered frame reveals B. Comagnetometers the time variations in the observable ω. At zeroth boost Comagnetometers form another category of sensitive order, the nonminimal terms produce time variations at the tools used for studies of Lorentz and CPT symmetry. High- first harmonic of the sidereal frequency, which can be sensitivity searches for Lorentz and CPT violation in both explicitly obtained using Eq. (25). We can then translate ˜ n sidereal and annual variations have been achieved using existing bounds on the minimal SME coefficients bX and 129 3 – ˜ n Xe- He comagnetometers [22 25]. The experiments bY obtained from studies of this harmonic to constraints on

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TABLE VI. Constraints on the moduli of the real and imaginary at the one sigma level. Following the standard procedure parts of neutron nonrelativistic coefficients determined from in the literature of taking one coefficient to be nonzero 129 3 Xe- He comparisons using Eq. (54). at a time [3], we find the maximal sensitivities to nonrelativistic coefficients shown in Table VI. These are Coefficient K Constraint on jReKj; jImKj the first constraints on neutron nonrelativistic coefficients NRð0BÞ;Sun NRð0BÞ;Sun <4 × 10−33 GeV Hn011 , gn011 in the literature. They correspond to substantially greater NRð1BÞ;Sun NRð1BÞ;Sun <2 × 10−33 GeV sensitivities to Lorentz and CPT violation than those attained Hn011 , gn011 NRð0BÞ;Sun NRð0BÞ;Sun <4 × 10−31 GeV−1 to date on electron or proton nonrelativistic coefficients, and Hn211 , gn211 they exceed even the comparatively tight constraints NRð1BÞ;Sun NRð1BÞ;Sun <2 × 10−31 GeV−1 Hn211 , gn211 on muon nonminimal coefficients obtained from laboratory NRð0BÞ;Sun NRð0BÞ;Sun <4 × 10−29 GeV−3 Hn411 , gn411 measurements of the muon anomalous magnetic moment NRð1BÞ;Sun NRð1BÞ;Sun −29 −3 [68,69]. H , g <2 × 10 GeV n411 n411 At linear boost order in the Sun-centered frame, the Lorentz-violating shift δω1 in ω follows the generic structure (31) and can be written as nonminimal coefficients for Lorentz violation. For this purpose, it suffices to implement the identifications X8 X2 γ p 2q He hj j i ðdÞJK ˆ J K δω1 ¼ − 1 T B β : ð55Þ X2 γ 3þ2q−d HeXe;ð2qÞ 1 0 1 d¼3 q¼0 Xe mn b˜ n → − pffiffiffiffiffiffi hjpj2qiRe½T NRð BÞ þ 2T NRð BÞ; X 6π nð2qÞ11 nð2qÞ11 q¼0 The quantities TðdÞJK are the linear combinations X2 HeXe;k 1 0 1 b˜ n → pffiffiffiffiffiffi hjpj2qiIm½T NRð BÞ þ 2T NRð BÞð53Þ of effective Cartesian coefficients displayed in Table VII. Y 6π nð2qÞ11 nð2qÞ11 q¼0 In this table, parentheses around indices are understood to represent symmetrization with a suitable factor, ˜ ð4ÞJðTKÞ ˜ ð4ÞJTK ˜ ð4ÞJKT 2! on existing minimal-SME limits. For example, the bound e.g., gneff ¼ðgneff þ gneff Þ= . Also, repeated ˜ n on the coefficient b⊥ reported in Ref. [25] then yields the dummy indices are understood to be summed, constraint ˜ ð5ÞTJTJ ˜ ð5ÞTXTX ˜ ð5ÞTYTY ˜ ð5ÞTZTZ e.g., Hneff ¼ Hneff þ Hneff þ Hneff . X2 The explicit form of the result (55) can be displayed 0 1 – p 2q T NRð BÞ;Sun 2T NRð BÞ;Sun by substituting Eqs. (32) (35) given in Sec. II F for hj j ið nð2qÞ11 þ nð2qÞ11 Þ q¼0 the boost velocity of the laboratory and for the direction of the magnetic field. This ensuing expression takes 3 7 10−33 < . × GeV ð54Þ the form

ðdÞJK 3 ≤ ≤ 8 TABLE VII. The quantities THeXe;k for d .

ðdÞJK THeXe;k Combination 3 ð3ÞJK ˜ ð ÞJK −2Hn THeXe;0 eff ð4ÞJK 4˜ ð4ÞJðTKÞ THeXe;0 gneff 5 ð5ÞJK ˜ ð ÞJðTTKÞ −6Hn THeXe;0 eff ð6ÞJK 8˜ ð6ÞJðTTTKÞ THeXe;0 gneff 7 ð7ÞJK ˜ ð ÞJðTTTTKÞ −10Hn THeXe;0 eff ð8ÞJK 12˜ ð8ÞJðTTTTTKÞ THeXe;0 gneff 5 5 5 ð5ÞJK 4 ˜ ð ÞTLTL JK ˜ ð ÞJðTTKÞ ˜ ð ÞJðLLKÞ − 3 Hn δ − 4Hn − 2Hn THeXe;2 eff eff eff ð6ÞJK 2˜ ð6ÞTLTTLδJK 8˜ ð6ÞJðTTTKÞ 8˜ ð6ÞJðTLLKÞ THeXe;2 gneff þ gneff þ gneff 7 7 7 ð7ÞJK 8 ˜ ð ÞTLTTTL JK 40 ˜ ð ÞJðTTTTKÞ ˜ ð ÞJðTTLLKÞ − 3 Hn δ − 3 Hn − 20Hn THeXe;2 eff eff eff ð8ÞJK 10 ˜ ð8ÞTLTTTTLδJK 20˜ ð8ÞJðTTTTTKÞ 40˜ ð8ÞJðTTTLLKÞ THeXe;2 3 gneff þ gneff þ gneff 7 7 7 ð7ÞJK 8 ˜ ð ÞTLTMML JK ˜ ð ÞJðTTLLKÞ ˜ ð ÞJðLLMMKÞ − 5 Hn δ − 8Hn − 2Hn THeXe;4 eff eff eff ð8ÞJK 4˜ ð8ÞTMTTLLMδJK 24˜ ð8ÞJðTTTLLKÞ 12˜ ð8ÞJðTLLMMKÞ THeXe;4 gneff þ gneff þ gneff

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λðdÞk ðdÞJK TABLE VIII. The quantities in terms of THeXe;k.

ðdÞk λ Combination λðdÞk ϕ ðdÞ½XY 1 ϕ ðdÞXX ðdÞYY ⊕ cos THeXe;ð2qÞ þ 2 sin ðTHeXe;k þ THeXe;kÞ λðdÞk ðdÞZX L THeXe;k d ZY d ZZ λðdÞk −ðcos ηTð Þ þ sin ηTð Þ Þ cΩ HeXe;k HeXe;k λðdÞk ðdÞZX sΩ THeXe;k λðdÞk ðdÞZY cω THeXe;k ðdÞk −TðdÞZX λsω HeXe;k λðdÞk ϕ ðdÞðXYÞ ϕ 1 ðdÞYY − ðdÞXX c2ω cos THeXe;k þ sin 2 ðTHeXe;k THeXe;kÞ ðdÞk ðdÞðXYÞ 1 ðdÞYY ðdÞXX λ − sin ϕT − cos ϕ 2 ðT − T Þ s2ω HeXe;k HeXe;k HeXe;k λðdÞk η ϕ ðdÞXY − ϕ ðdÞYY η ϕ ðdÞXZ − ϕ ðdÞYZ cΩsω cos ðsin THeXe;k cos THeXe;kÞþsin ðsin THeXe;k cos THeXe;kÞ d XZ d YZ d XY d YY λðdÞk − sin ηðcos ϕTð Þ þ sin ϕTð Þ Þ − cos ηðcos ϕTð Þ þ sin ϕTð Þ Þ cΩcω HeXe;k HeXe;k HeXe;k HeXe;k λðdÞk ϕ ðdÞYX − ϕ ðdÞXX sΩsω cos THeXe;k sin THeXe;k λðdÞk ϕ ðdÞYX ϕ ðdÞXX sΩcω sin THeXe;k þ cos THeXe;k

δω1 ¼ β⊕ sin ϑλ⊕ þ βL cos ϑλL þ β⊕ cos ϑ cos ðΩ⊕TÞλcΩ þ β⊕ cos ϑ sin ðΩ⊕TÞλsΩ þ βL cos ϑ cos ðω⊕T⊕Þλcω

þ βL cos ϑ sin ðω⊕T⊕Þλsω þ β⊕ sin ϑ cos ðω⊕T⊕Þ cos ðΩ⊕TÞλcωcΩ þ β⊕ sin ϑ sin ðω⊕T⊕Þ cos ðΩ⊕TÞλsωcΩ

þ β⊕ sin ϑ cos ðω⊕T⊕Þ sin ðΩ⊕TÞλcωsΩ þ β⊕ sin ϑ sin ðω⊕T⊕Þ sin ðΩ⊕TÞλsωsΩ þ βL sin ϑ cos ð2ω⊕T⊕Þλc2ω

þ βL sin ϑ sin ð2ω⊕T⊕Þλs2ω; ð56Þ

λ −27 where the twelve quantities with subscripts ranging λcωcΩ ¼ð−3.9 3.5Þ × 10 GeV; over the values ⊕, L, cΩ, sΩ, cω, sω, cωcΩ, sωcΩ, cωsΩ, −27 λ ω Ω 0.7 6.3 × 10 GeV; sωsΩ, c2ω;s2ω can be decomposed in terms of quantities c s ¼ð Þ −27 ðdÞk λ ω Ω −6 3 6 7 10 ; λ with fixed values of k and d via the relation s s ¼ð . . Þ × GeV −27 λsωcΩ ¼ð−3.9 2.8Þ × 10 GeV: ð58Þ X8 X2 2q γ hjpj i 2 λ ¼ He − 1 λðdÞð qÞ: ð57Þ Note that the dependence of the quantities λ on the angle ϕ γ 3þ2q−d d¼3 q¼0 Xe mn means that these bounds hold only at ϕ ¼ 90°. Using the results in Tables VII and VIII and the bounds (58), we can extract maximal sensitivities to many nonminimal effective λðdÞkÞ ðdÞJK Expressions for the quantities in terms of THeXe;k are Cartesian coefficients for the neutron. These constraints are given in Table VIII. listed in Table IX. They are the first of their kind reported in An experiment using a dual 129Xe-3He maser to study the literature for neutrons. sidereal variations in the observable ω at different times of Improvements over the results in Table IX are within the year was performed at the Harvard-Smithsonian Center reach of existing experiments. The sensitivity recently for Astrophysics [23]. In this experiment, the magnetic attained in the Heidelberg apparatus described in field was oriented west to east, corresponding to ϑ ¼ 0° and Ref. [25] represents a gain of about 2 orders of magnitude, ϕ ¼ 90°. We can use the bounds on δω1 reported in so sufficient sidereal data accumulated at the annual Ref. [23] to determine limits on nonminimal effective frequency with this apparatus could in principle better Cartesian coefficients for the neutron. The published the constraints in Table IX by a similar factor. Moreover, analysis neglected contributions proportional to the labo- with the sidereal data already in hand, the time variations at ratory velocity βL in the Sun-centered frame, so we can the second harmonic of the sidereal frequency appearing in deduce the four bounds Eq. (56) could in principle be studied and would be

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TABLE IX. Constraints on the moduli of neutron effective Cartesian coefficients determined from 129Xe-3He comparisons using Eq. (58).

Coefficient Constraint Coefficient Constraint ˜ ð5ÞXðTXTÞ <1 × 10−27 GeV−1 ˜ ð6ÞXðTXTTÞ <9 × 10−28 GeV−2 Hneff gneff ˜ ð5ÞXðTYTÞ <8 × 10−28 GeV−1 ˜ ð6ÞXðTYTTÞ <7 × 10−28 GeV−2 Hneff gneff ˜ ð5ÞXðTZTÞ <2 × 10−27 GeV−1 ˜ ð6ÞXðTZTTÞ <2 × 10−27 GeV−2 Hneff gneff ˜ ð5ÞYðTXTÞ <8 × 10−28 GeV−1 ˜ ð6ÞYðTXTTÞ <6 × 10−28 GeV−2 Hneff gneff ˜ ð5ÞYðTYTÞ <8 × 10−28 GeV−1 ˜ ð6ÞYðTYTTÞ <7 × 10−28 GeV−2 Hneff gneff ˜ ð5ÞYðTZTÞ <2 × 10−27 GeV−1 ˜ ð6ÞYðTZTTÞ <2 × 10−27 GeV−2 Hneff gneff ˜ ð5ÞXðJXJÞ <4 × 10−25 GeV−1 ˜ ð6ÞXðJXJTÞ <9 × 10−26 GeV−2 Hneff gneff ˜ ð5ÞXðJYJÞ <3 × 10−25 GeV−1 ˜ ð6ÞXðJYJTÞ <7 × 10−26 GeV−2 Hneff gneff ˜ ð5ÞXðJZJÞ <6 × 10−25 GeV−1 ˜ ð6ÞXðJZJTÞ <2 × 10−25 GeV−2 Hneff gneff ˜ ð5ÞYðJXJÞ <2 × 10−25 GeV−1 ˜ ð6ÞYðJXJTÞ <2 × 10−25 GeV−2 Hneff gneff ˜ ð5ÞYðJYJÞ <3 × 10−25 GeV−1 ˜ ð6ÞYðJYJTÞ <7 × 10−26 GeV−2 Hneff gneff ˜ ð5ÞYðJZJÞ <6 × 10−25 GeV−1 ˜ ð6ÞYðJZJTÞ <2 × 10−25 GeV−2 Hneff gneff ˜ ð5ÞTJTJ <6 × 10−25 GeV−1 ˜ ð6ÞTJTJT <5 × 10−25 GeV−2 Hneff gneff ˜ ð7ÞXðTXTTTÞ <8 × 10−28 GeV−3 ˜ ð8ÞXðTXTTTTÞ <7 × 10−28 GeV−4 Hneff gneff ˜ ð7ÞXðTYTTTÞ <6 × 10−28 GeV−3 ˜ ð8ÞXðTYTTTTÞ <5 × 10−28 GeV−4 Hneff gneff ˜ ð7ÞXðTZTTTÞ <2 × 10−27 GeV−3 ˜ ð8ÞXðTZTTTTÞ <1 × 10−27 GeV−4 Hneff gneff ˜ ð7ÞYðTXTTTÞ <6 × 10−28 GeV−3 ˜ ð8ÞYðTXTTTTÞ <5 × 10−28 GeV−4 Hneff gneff ˜ ð7ÞYðTYTTTÞ <6 × 10−28 GeV−3 ˜ ð8ÞYðTYTTTTÞ <5 × 10−28 GeV−4 Hneff gneff ˜ ð7ÞYðTZTTTÞ <2 × 10−27 GeV−3 ˜ ð8ÞYðTZTTTTÞ <1 × 10−27 GeV−4 Hneff gneff ˜ ð7ÞXðJXJTTÞ <4 × 10−26 GeV−3 ˜ ð8ÞXðJXJTTTÞ <2 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞXðJYJTTÞ <3 × 10−26 GeV−3 ˜ ð8ÞXðJYJTTTÞ <1 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞXðJZJTTÞ <7 × 10−26 GeV−3 ˜ ð8ÞXðJZJTTTÞ <4 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞYðJXJTTÞ <3 × 10−26 GeV−3 ˜ ð8ÞYðJXJTTTÞ <1 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞYðJYJTTÞ <3 × 10−26 GeV−3 ˜ ð8ÞYðJYJTTTÞ <1 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞYðJZJTTÞ <7 × 10−26 GeV−3 ˜ ð8ÞYðJZJTTTÞ <3 × 10−26 GeV−4 Hneff gneff ˜ ð7ÞTJTJTT <2 × 10−25 GeV−3 ˜ ð8ÞTJTJTTT <4 × 10−25 GeV−4 Hneff gneff ˜ ð7ÞXðJXJKKÞ <4 × 10−23 GeV−3 ˜ ð8ÞXðJXJTKKÞ <7 × 10−24 GeV−4 Hneff gneff ˜ ð7ÞXðJYJKKÞ <3 × 10−23 GeV−3 ˜ ð8ÞXðJYJTKKÞ <5 × 10−24 GeV−4 Hneff gneff ˜ ð7ÞXðJZJKKÞ <7 × 10−23 GeV−3 ˜ ð8ÞXðJZJTKKÞ <1 × 10−23 GeV−4 Hneff gneff ˜ ð7ÞYðJXJKKÞ <3 × 10−23 GeV−3 ˜ ð8ÞYðJXJTKKKÞ <5 × 10−24 GeV−4 Hneff gneff ˜ ð7ÞYðJYJKKÞ <3 × 10−23 GeV−3 ˜ ð8ÞYðJYJTKKÞ <5 × 10−24 GeV−4 Hneff gneff ˜ ð7ÞYðJZJKKÞ <7 × 10−23 GeV−3 ˜ ð8ÞYðJZJTKKÞ <1 × 10−23 GeV−4 Hneff gneff ˜ ð7ÞTJTJKK <6 × 10−23 GeV−3 ˜ ð8ÞTJTJTKK <2 × 10−23 GeV−4 Hneff gneff expected to yield additional measurements of interest. coefficients, as sidereal variations are insensitive to the Although this signal is suppressed by about 2 orders of combinations λcΩ and λsΩ even when monitored throughout magnitude compared to annual-variation effects, the greater the year. sensitivity of the Heidelberg apparatus suggests constraints Another avenue offering potential improvements is the of the same order of magnitude as those in Table IX could adoption of better nuclear models beyond the Schmidt be obtained. Note also that direct measurements of the model. These techniques have already been used to show annual modulation would lead to new constraints on SME that contributions from proton coefficients to Eq. (51) are

036003-19 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) significant, being suppressed only by a factor of about 5 C. Trapped ions and lattice clocks 0 for coefficients with k ¼ [30]. If a similar relationship The stability and accuracy of optical frequency standards 2 for coefficients with k ¼ , 4 can be demonstrated, then currently exceeds the performance of fountain clocks. It is the constraints on the neutron coefficients listed in thus natural to consider the prospects for testing Lorentz Tables VI and IX could be extended to bounds on the and CPT symmetry using optical transitions. However, corresponding proton coefficients by multiplying by a sensitivities to many coefficients for Lorentz violation factor of 5. This would represent a striking gain in depend on the absolute uncertainty of the frequency sensitivity to the proton nonrelativistic coefficients com- measurement rather than on its relative precision. As the pared to the existing results obtained using data from a absolute uncertainties of fountain clocks still surpass those hydrogen maser [43]. of optical clocks, the advantages of the latter lie primarily in Other comagnetometers can also be used to test Lorentz their ability to access distinct Lorentz- and CPT-violating and CPT symmetry and may offer sensitivities to addi- effects. In particular, optical clocks offer sensitivities to tional coefficients. One potential example is the coefficients for Lorentz violation in the electron sector that 21 Ne-Rb-K comagnetometer described in Ref. [19], which are unattainable in other clock-comparison experiments. In 3 is designed to extend the reach achieved earlier by a He-K this section, we study sensitivities to electron coefficients in self-compensating comagnetometer [18]. The addition of trapped-ion and lattice optical clocks. Since any signals 21Ne to the system is of particular interest here because the from proton and neutron coefficients are better accessed via nuclear spin of 21Ne is I ¼ 3=2 and so this comagne- other techniques, we disregard nucleon contributions in tometer can access more coefficients for Lorentz violation. what follows. 1 3 A glance at Table I reveals that there are prospects for The transition S0- P0 is commonly used in optical measuring the coefficients with j ¼ 2 and j ¼ 3. The frequency standards. It has been studied with trapped ions, underlying physics of this comagnetometer system differs including 27Alþ [70–73] and 115Inþ [74–76], and also in the significantly from that of the other systems discussed in context of optical lattice clocks based on 87Sr [77–82], this work, so the results obtained in Sec. II cannot be 171Yb [83–86], and 199Hg [87–89]. For this transition, the directly applied to estimate sensitivities. However, some of total electronic angular momentum of the two states the bounds presented in Ref. [19] can be converted to involved is J ¼ 0, so only isotropic electron Lorentz- constraints on nonrelativistic coefficients for the neutron violating operators can contribute. The Lorentz-violating by applying the relationship (43) between the nonrelativ- shift δν in the transition frequency ν therefore involves only ð4Þ 00 istic coefficients and the coefficients cμν . Table X lists coefficients with jm ¼ . In the independent-particle the corresponding maximal sensitivities achieved, model discussed in Sec. II C, the shift in the laboratory which are the first of this kind in the literature. As before, frame is given by these results can be expected to extend to constraints 1 ffiffiffiffiffiffi 2 NR 4 NR on nonrelativistic coefficients for the proton because 2πδν ¼ − p ðΔp Ve200 þ Δp Ve400Þ; ð59Þ nuclear models beyond the Schmidt model are known 4π to allow contributions from proton operators to Lorentz- Δ k p k violating expectation values with j ¼ 2 [33]. It is also where p is the difference in the expectation values hj j i plausible that a similar situation holds for coefficients with of the energy levels involved in the transition. Some optical frequency standards involve transitions j ¼ 3. All these interesting issues are open for future between energy levels with J ≠ 0. For example, the investigation. 2 2 transition S1=2- D5=2 is used as a frequency standard in ion-trap clocks based on 40Caþ [90–93] and 88Srþ [94–96]. TABLE X. Constraints on neutron nonrelativistic coefficients 21 For these systems, certain systematic effects can be mini- determined using data from the Ne-Rb-K comagnetometer. mized by measuring transitions involving different Zeeman Coefficient Constraint sublevels. These techniques typically also eliminate sensitivity to some Lorentz-violating effects, as is to be expected NR NR −ð3.3 3.0Þ × 10−29 GeV−1 Rean221,Recn221 given that the coefficients for Lorentz violation behave in NR NR − 1 9 2 3 10−29 −1 Iman221,Imcn221 ð . . Þ × GeV many ways as effective external fields. NR NR −29 −1 Rean222,Recn222 ð1.0 1.2Þ × 10 GeV One common technique to remove the linear Zeeman NR NR 0 83 0 96 10−29 −1 shift of the clock transition is averaging over the Zeeman Iman222,Imcn222 ð . . Þ × GeV 2 2 2 2 pair S1 2 1 2 − D5 2 and S1 2 −1 2 − D5 2 − . NR NR − 3 7 3 4 10−27 −3 = ; = = ;mJ = ; = = ; mJ Rean421,Recn421 ð . . Þ × GeV Similarly, the electric quadrupole shift can be removed NR NR − 2 2 2 6 10−27 −3 Iman421,Imcn421 ð . . Þ × GeV by averaging over three different Zeeman pairs. NR NR −27 −3 Rean422,Recn422 ð1.1 1.3Þ × 10 GeV Implementing this process eliminates any contributions −27 −3 from Lorentz-violating operators with j ≠ 0 at linear order Ima NR ,Imc NR ð0.9 1.1Þ × 10 GeV n422 n422 in perturbation theory, due to the identity

036003-20 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) X5=2 5 5 the electric quadrupole shift. It is conceivable that coef- 0 6δ 4 2 mJj j 2 mJ ¼ j0: ð60Þ ficients with j ¼ contribute to the frequency shift, but −5 2 mj¼ = establishing this lies outside our present scope. The coefficients in the above expressions are in the As a result, the Lorentz- and CPT-violating frequency shift laboratory frame and hence may vary with time. In δν measurable in these systems is still given by Eq. (59), converting to the Sun-centered frame, the isotropic fre- despite the nonzero value of J. quency shift (59) receives contributions that depend on the Another technique to remove the quadrupole shift uses boost velocity of the laboratory frame. At linear boost instead two Zeeman pairs to interpolate the value of the order, we find that the shift δν1 is given by 2 35 12 frequency at mJ ¼ = , which corresponds to zero X k quadrupole shift because the shift is proportional to Δp ðdÞX 2 2πδν − ffiffiffiffiffiffi β Ω 35 12 − 1 ¼ p ½ ⊕ sin ⊕TVe;k = mJ. This method eliminates contributions involv- 4π ing coefficients for Lorentz violation with j ¼ 2, but it d;k 4 − β Ω η ðdÞY η ðdÞZ retains contributions with j ¼ . In this scenario, the ⊕ cos ⊕Tðcos Ve;k þ sin Ve;k Þ Lorentz- and CPT-violating shift (59) is replaced by the β ω ðdÞY − ω ðdÞX expression þ Lðcos ⊕T⊕Ve;k sin ⊕T⊕Ve;k Þ; ð63Þ

1 ðdÞJ 2 NR 4 NR where expressions for the quantities Ve;k in terms of 2πδν ¼ − pffiffiffiffiffiffi ðΔp V 200 þ Δp V 400Þ 4π e e effective Cartesian coefficients are given in Table XI. 7 The result (63) predicts annual and sidereal variations of ffiffiffi p 4 V NR þ 27pπ hj j i e440; ð61Þ the transition frequency, which can in principle be detected by comparison to a reference. Since optical clocks can where the expectation value hjpj4i is evaluated in the outperform other frequency standards, an effective way to 2 search for the effects predicted by Eq. (63) is to compare state D5=2. 2 2 two optical clocks and search for a sidereal or annual The clock transition S1 2ðF ¼ 0Þ- D3 2ðF ¼ 2Þ with = = modulation of their frequency difference. For systems with Δm ¼ 0 in 171Ybþ has also been used as a frequency F long-term stability, studying annual variations is preferable standard [97,98]. In the context of the independent-particle because the speed β⊕ is typically about 2 orders of model described in Sec. II C, the Lorentz- and CPT- magnitude bigger than β . Note also that the two clocks violating frequency shift δν for this system is given in L can be located in different laboratories. Using Eq. (63),we the laboratory frame by see that the annual and sidereal modulations of the 1 frequency difference between clocks A and B are given by 2πδν ¼ − pffiffiffiffiffiffi ðΔp2V NR þ Δp4V NR Þ 4π e200 e400 X Δpk − Δpk 2πδν B ffiffiffiffiffiffi A β Ω ðdÞX 1 AB ¼ p ½ ⊕ sin ⊕TVe;k þ pffiffiffiffiffiffi ðhjpj2iV NR þhjpj4iV NR Þ; ð62Þ 4π 2 5π e420 e420 d;k − β Ω η ðdÞY η ðdÞZ ⊕ cos ⊕Tðcos Ve;k þ sin Ve;k Þ p 4 where the expectation value hj j i is evaluated in the state X Δ k β − Δ k β 2 pB L;B pA L;A ðdÞY D3 2. However, to suppress the contribution from the þ pffiffiffiffiffiffi ðcos ω⊕T⊕V = 4π e;k electric quadrupole shift, an averaging of the frequency d;k over three orthogonal directions of the magnetic field is ðdÞX − sin ω⊕T⊕V Þ; ð64Þ performed. This procedure suppresses the contribution e;k from coefficients with j ¼ 2. As a result, in the limit that the three directions are exactly orthogonal, the shift (62) ðdÞJ 5 ≤ ≤ 8 TABLE XI. The quantities Ve;k for d . reduces to the expression (59). Other frequency standards are provided by the electric ðdÞJ Ve;k Combination 171 þ 199 þ octopole transitions in Yb [98,99] and Hg ð5ÞJTT ð5ÞJKK ð5ÞJ −2a − a [100,101]. The clock transition used in these systems is Ve;2 eff eff Δ 0 ð6ÞJ 4 ð6ÞJTTT 4 ð6ÞJTKK the transition mF ¼ , which is insensitive to B-type Ve;2 ceff þ ceff 7 7 ð7ÞJ 10 ð ÞJTTTT ð ÞJTTKK coefficients for Lorentz violation. As before, the contribu- − 3 ð2a þ 3a Þ Ve;2 eff eff tion to the Lorentz- and CPT-violating frequency shift δν 8 8 8 Vð ÞJ 10cð ÞJTTTTT 20cð ÞJTTTKK arising from coefficients with j ¼ 0 is given by Eq. (59). e;2 eff þ eff 7 7 The contribution from coefficients with j 2 is again ð7ÞJ −að ÞJKKLL − 4að ÞJTTKK ¼ Ve;4 eff eff eliminated by the averaging procedure over three different ð8ÞJ ð8ÞJTKKLL ð8ÞJTTTKK V 6c þ 12c directions of the magnetic field, which is designed to cancel e;4 eff eff

036003-21 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) where Δpk is the expectation value p k for the transitions 12 11 A hj j i V NR → − V NR ; V NR → V NR : 66 β pk2m ek2m pk4m ek4m ð Þ in clock A, L;A is the speed of the laboratory containing 5 45 clock A, and Δpk , β are defined similarly for clock B. B L;B δ¯ Several laboratories have the potential to compare two At linear boost order, the contribution f1 is clocks at the same location, searching for the effects X p 2 ¯ hj j i ðdÞJ J ðdÞJKL ˆ K ˆ L J predicted in Eq. (64) in a scenario with βL; ¼ βL; .For 2πδ β β A B f1 ¼ 5−d ðVCa;2 þ VCa;2 B B Þ example, scanning the literature cited above suggests that d mp 87 171 X 4 comparisons of any two lattice clocks based on Sr, Yb, hjpj i 199 þ ðVðdÞJ βJ þ VðdÞJKLBˆ KBˆ LβJÞ or Hg could in principle be performed at Rikagaku m7−d Ca;4 Ca;4 Kenkyūsho (RIKEN) in Japan. Similarly, 87Sr and 199Hg d p X p 4 lattice clocks can be compared at the Système de Référence hj j i ðdÞJKLMN ˆ K ˆ L ˆ M ˆ NβJ 87 þ 7−d VCa;4 B B B B ; ð67Þ Temps-Espace (SYRTE) in France, ones based on Sr and d mp 171Yb can be compared at the National Metrology Institute 27 þ … of Japan (NMIJ), and the Al ion clock could be where expressions for the quantities VðdÞJJ1 Js in terms of 171 Ca;k compared to the Yb lattice clock at the National effective Cartesian coefficients are displayed in Table XII. Institute of Standards and Technology (NIST) in the For the nonminimal terms considered in this work, the United States. Many individual comparisons between result (67) incorporates time variation at the first five clocks located at different institutions are also possible harmonics of the sidereal frequency along with annual β ≠ β in principle, by using Eq. (64) with L;A L;B. Moreover, variations. At the sidereal frequency, the dominant con- some Lorentz- and CPT-violating effects that are absent in tributions to the variations in the first four harmonics are Eq. (64) and hence cannot be studied with any of these given by Table III with the substitutions (66). The variation clock combinations might become accessible given at the fifth harmonic is suppressed by βL, and it is given by suitable care for the treatment of systematics and its d J…K d J…K Eq. (46) with the replacement Vð Þ → Vð Þ . Using implication for cancellations of signals. Some examples Cs;k Ca;k 40 þ of such experiments with clocks at a single location these results, we can estimate the sensitivities of the Ca might include comparison of the 88Srþ and 171Ybþ ion experiment [26] to the nonrelativistic coefficients. clocks at the National Physical Laboratory (NPL) in Table XIII displays these sensitivities. In deriving them, p 2 ∼ 10−11 2 p 4 ∼ 10−22 4 England, the 87Sr lattice clock and the 171Ybþ ion clock we take hj j i GeV and hj j i GeV .We at the Physikalisch-Technische Bundesanstalt (PTB) in also suppose the experimental reach is 0.03 Hz. With 27 199 sufficient stability and data collection over a long time Germany, or the Alþ and Hgþ ion clocks at NIST. A qualitatively different approach to testing Lorentz and … ðdÞJJ1 Js 5 ≤ ≤ 8 CPT symmetry is to create an entangled state and monitor its TABLE XII. The quantities VCa;k for d . time evolution. In Ref. [26], the entangled state combinesffiffiffi the p ðdÞJJ1…Js states ðj5=2ij∓ 5=2iþj1=2ij∓ 1=2iÞ= 2 of two VCa;k Combination 40 5 5 þ ð5ÞJ 36 ð ÞJKK ð ÞJTT Ca ions, where the kets jmFi represent the mF Zeeman − 35 ðae þ 2ae Þ 2 VCa;2 eff eff 5 5 5 level of the energy state D5=2. The experimental observable ð ÞJJ1J2 108 ð ÞJJ1J2 JJ1 ð ÞTTJ2 V 2 ða þ 2δ a Þ f¯ is obtained by averaging the energy difference between the Ca; 35 eeff eeff ð6ÞJ 144 ð6ÞJTKK ð6ÞJTTT product states j5=2ij∓ 5=2i and j1=2ij∓ 1=2i. VCa;2 35 ðceeff þ ceeff Þ 432 ð6ÞJTJ1J2 ð6ÞTTTJ2 Following the approach in Sec. II C, we assign angular ð6ÞJJ1J2 − c δJJ1 c V 2 35 ð eeff þ eeff Þ 5 2 2 Ca; 7 7 momenta J ¼ = and L ¼ to the valence electron. The ð7ÞJ − 24 3 ð ÞJTTKK 2 ð ÞJTTTT V 7 ð aeeff þ aeeff Þ Lorentz- and CPT-violating shift δf of the observable f¯ in Ca;2 7 JKL 7 JTTJ1J2 7 TTTTJ2 ð Þ 72 3 ð Þ 2δJJ1 ð Þ the laboratory frame is found to be VCa;2 7 ð apeff þ apeff Þ 7 7 ð7ÞJ − 10 a ð ÞJKKLL 4a ð ÞJTTKK 18 V 4 7 ð eeff þ eeff Þ ¯ 2 NR 4 NR Ca; 2πδ ffiffiffiffiffiffi p V p V 7 JJ1J2 7 JTTJ1J2 7 TTJ2KK 7 JJ1J2KK f ¼ p ðhj j i e220 þhj j i e420Þ ð Þ 8 ð Þ δJJ1 ð Þ 4 ð Þ 7 5π VCa;4 ðaeeff þ aeeff Þþ aeeff

7 JJ1J2J3J4 10 7 JJ1J2J3J4 7 TTJ2J3J4 1 ð Þ ð Þ 4δJJ1 ð Þ ffiffiffi p 4 V NR VCa;4 21 ðaeeff þ aeeff Þ þ 7pπ hj j i p440: ð65Þ ð8ÞJ 72 2 ð8ÞJTTTKK ð8ÞJTTTTT VCa;2 7 ð ceeff þ ceeff Þ 216 ð8ÞJTTTJ1J2 ð8ÞTTTTTJ2 This expression has a structure similar to that of the ð8ÞJJ1J2 − 2 δJJ1 V 7 ð ceeff þ ceeff Þ frequency shift (40) in fountain clocks, so we can adapt Ca;2 ð8ÞJ 60 ð8ÞJTKKLL 2 ð8ÞJTTTKK the results presented in Sec. III A to convert the expression VCa;4 7 ðceeff þ ceeff Þ ð8ÞJTTTJ1J2 ð8ÞTTTJ2KK ð8ÞJTJ1J2KK ð8ÞJJ1J2 JJ1 (65) to the Sun-centered frame. The expression for the −24ðce þδ ce Þ−24ce VCa;4 eff eff eff ¯ 20 ð8ÞJTJ1J2J3J4 ð7ÞTTTJ2J3J4 shift δf0 at zeroth boost order is therefore given by Table III ð8ÞJJ1J2J3J4 − − 2δJJ1 V 7 ðceeff ceeff Þ with the replacements Ca;4

036003-22 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018)

TABLE XIII. Potential sensitivities to coefficients in the Sun- Collaboration [112], the ALPHA Collaboration [113], centered frame from sidereal variations in entangled 40Caþ ions. and the Gravitational Behavior of Antihydrogen at Rest (GBAR) Collaboration [114], and the corresponding tech- Coefficient Sensitivity niques may also enhance future spectroscopic studies of NR NR 10−14 −1 jae22mj, jce22mj GeV antihydrogen. NR NR −3 −3 jae42 j, jce42 j 10 GeV One signal for nonzero CPT violation would be a m m Δν¯ ≡ ν − ν¯ NR NR 10−3 −3 measured difference 1S2S 1S2S 1S2S between the jae44mj, jce44mj GeV resonance frequency ν1S2S of the 1S-2S transition in hydrogen and the analogous resonance frequency ν¯1S2S period, constraints could also be placed on coefficients for in antihydrogen. Performing a general analysis in the Lorentz violation associated with the annual variation context of effective field theory [43] reveals that signal predicted by Eq. (67). CPT-violating effects contributing to a nonzero value of Related experiments have been proposed using Zeeman Δν¯1S2S can be classified as spin independent or spin þ transitions of the F7=2 state in Yb [55]. Experiments using a dependent and as isotropic or anisotropic, and they can dynamical decoupling technique have also been proposed for exhibit time variations induced by the noninertial nature of a broad class of trapped ions and lattice clocks [102]. All are the experimental laboratory. It turns out that the spin- expected to permit significant improvements over existing dependent effects are more readily studied using ground- ð4Þ constraints on the coefficients cμν . The energy shift produced state hyperfine transitions, while the time variations are ν¯ by these minimal-SME coefficients can be generalized to better explored by directly studying modulations of 1S2S. Δν¯ incorporate also the contributions from the nonminimal However, the difference 1S2S is particularly sensitive to NR isotropic, spin-independent, and time-constant CPT viola- coefficients V 22 in the nonrelativistic limit. The proposed e m ∘ NR ∘ NR ∘ NR ∘ NR experiments can therefore be expected to yield substantial tion controlled by the coefficients ae;2 , ae;4 , ap;2, ap;4.An improvements over the estimated sensitivities to the coef- explicit expression for Δν¯1S2S in terms of these coefficients NR NR 8 → 16 ficients ae22m and ce22m given in Table XIII. is given by Eq. (86) of Ref. [43], with the correction in the denominator. Note that these nonrelativistic coefficients incorporate effects from CPT-violating operators D. Antimatter clocks of arbitrary mass dimension [35]. In this final section on applications, we offer some Based on an analysis that assumes no spin-, geometry-, or comments about the prospects for spectroscopic experiments time-dependent CPT violation, the ALPHA Collaboration using antimatter. Comparisons of the properties of matter and reported agreement between the 1S-2S resonance frequen- antimatter are of particular interest for testing the CPT cies of hydrogen and antihydrogen at a precision of 2 × 10−12 symmetry of quantum field theory. The line of reasoning [108]. We can therefore deduce the constraint outlined in the Introduction reveals that effective field theory also provides the general model-independent framework for ∘ ∘ 67 ∘ ∘ NR NR α 2 NR NR 1 10−9 −1 analyzing antimatter systems, so the results of experiments ae;2 þ ap;2 þ 12 ð mrÞ ðae;4 þ ap;4Þ < × GeV ; testing CPT symmetry can be expressed in a model- independent way as constraints on SME coefficients. ð68Þ A diverse set of such constraints has already been α obtained via precision spectroscopy of positrons and where is the fine-structure constant and mr is the reduced antiprotons confined in a Penning trap [45–47,103–105]. mass of hydrogen. The result (68) represents the first Studying antimatter instead of antiparticles offers advan- constraint on SME coefficients extracted from antihydrogen tages in searches for CPT violation [43,106], and several spectroscopy. Table XIV lists the corresponding maximal collaborations are developing techniques for the precision sensitivities obtained by taking each coefficient to be nonzero spectroscopy of antihydrogen. Recently, the Antihydrogen in turn, following the standard procedure in the literature [3]. Laser Physics Apparatus (ALPHA) Collaboration has Note that several factors currently limit the precision of the ν¯ measured the antihydrogen ground-state hyperfine transi- measurement of 1S2S, including the comparatively smaller tions [107] and the 1S-2S transition [108], heralding an era of precision antimatter spectroscopy. Other collab- TABLE XIV. Constraints on electron and proton nonrelativistic orations pursuing this goal include the Atomic Spectro- coefficients determined from 1S-2S hydrogen and antihydrogen scopy and Collisions Using Slow Antiprotons (ASACUSA) spectroscopy. Collaboration [109,110], and the Antihydrogen Trap (ATRAP) Collaboration [111]. Experiments investigating Coefficient Constraint ∘ ∘ NR NR −9 −1 the gravitational response of antihydrogen are also jae;2 j, jap;2j 1 × 10 GeV ∘ ∘ being developed, including the Antihydrogen Experi- NR NR −3 ja 4 j, ja 4j 14 GeV ment: Gravity, Interferometry, Spectroscopy (AEGIS) e; p;

036003-23 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) number and higher temperature of atoms in antihydrogen can be obtained from the corresponding expression for the δν experiments relative to hydrogen ones. However, there is shift D in deuterium given as Eq. (103) of Ref. [43], every reason to expect improvements in the future. One yielding the result proposal along these lines is to trap the ultracold antiatoms from the GBAR antihydrogen beam in an optical lattice m¯ 1 2πδν ffiffiffir ε − ε V¯ NR V¯ NR V¯ NR [115], which could enable measurements of the 1S-2S D¯ ¼ pπ ð n0 nÞ e200 þ 4 ð p200 þ n200Þ transition in antihydrogen at a level approaching the pre- 4 2 10−15 p2 V¯ NR V¯ NR cision of . × already obtained with hydrogen [116]. þh p¯ d¯ ið p400 þ n400Þ Other signals for CPT violation can appear in compar- 2 ¯ 2 8 0 8 isons of the hyperfine structure of hydrogen and antihy- mr 2 n 2 n − pffiffiffi ε 0 − 3 − ε − 3 drogen [43,106]. High-precision measurements of the π n 2L0 þ 1 n 2L þ 1 hyperfine transition of hydrogen can be obtained using a 1 V¯ NR V¯ NR V¯ NR hydrogen maser [117], but these methods are impractical × e400 þ 16 ð p400 þ n400Þ ; ð70Þ for antihydrogen due, for example, to collisions with the walls in the maser bulb. One different approach already V¯ NR NR NR ¯ realized by the ALPHA Collaboration is to perform hyper- where wkjm ¼ cwkjm þ awkjm, mr is the reduced mass of ε ≡ −α2 ¯ 2 2 p2 ≃ 104 2 fine spectroscopy on trapped antihydrogen [107].An antideuterium, n mr= n , and h p¯ d¯ i MeV . alternative option being pursued by the ASACUSA Collaboration involves using instead an antihydrogen beam IV. SUMMARY [110]. Testing the latter method with hydrogen has demonstrated a precision only 3 orders of magnitude below that This work studies Lorentz and CPT violation in clock- achieved via the hydrogen maser. The prospects are comparison experiments by incorporating effects on elec- excellent for further substantial improvements in hyperfine tron and nucleon propagators arising from SME operators spectroscopy using advanced tools such as ultracold anti- of arbitrary mass dimension d. It begins with a discussion hydrogen beams, and perhaps ultimately adapting tech- of theoretical issues in Sec. II. The general Lagrange niques similar to those used for atomic fountain clocks. density (1) for a fermion propagating in the presence of In the longer term, antiatom spectroscopy could con- arbitrary Lorentz and CPT violation implies the perturba- ceivably evolve to include also experiments with heavier tive result (2) for the corresponding nonrelativistic one- antiatoms. The simplest candidate system is antideuterium, particle Hamiltonian. Combining the expressions for the which has the antideuteron as its nucleus. Unlike antipro- constituent particles yields the Hamiltonian (6) for an atom tonic deuterium, antideuterium is expected to be stable and or ion, which is the basis for our analysis of clock- is therefore in principle a candidate for precision spectros- comparison experiments. copy and hence for CPT tests. Deuterium spectroscopy is The experimental observables are transition frequencies in known to be many orders of magnitude more sensitive than atoms or ions. The Lorentz- and CPT-violating signals in hydrogen spectroscopy to certain kinds of Lorentz and these frequencies can be calculated from the perturbative CPT violation [43], and the same arguments hold for the shifts (8) in energy levels. These shifts involve products of comparative sensitivities of antideuterium and antihydro- Clebsch-Gordan coefficients with expectation values of the gen spectroscopy. The production of a single heavier perturbative Hamiltonian. The symmetries of the system antiion is also of real interest, as it could in principle be imply that contributions to the energy shifts can arise only confined in an ion trap and repeatedly interrogated to from specific nonrelativistic spherical coefficients for Lorentz perform high-precision spectroscopy. violation, as listed in Table I. Explicit computation of the Whatever the future of antimatter experiments with expectation values requires modeling the electronic and heavier systems than antihydrogen, the theoretical treat- nuclear states. Our approach for electrons adopts the inde- ments presented in Sec. II and in Ref. [43] can readily be pendent-particle model described in Sec. II C, while for the adapted to antiatoms and antiions. In particular, the nucleus we use the Schmidt model as discussed in Sec. II D. expression for the shift in an energy level of an antiatom A laboratory on the surface of the Earth or on an orbiting or antiion can be obtained from the corresponding expres- satellite typically represents a noninertial frame. As a result, sion for an atom or ion by implementing the substitutions most SME coefficients measurable in the laboratory acquire a dependence on time due to the laboratory rotation and NR → − NR NR → NR awjkm awjkm;cwjkm cwjkm; boost relative to the canonical Sun-centered frame, which is NRðsBÞ NRðsBÞ NRðsBÞ NRðsBÞ an approximately inertial frame over the experimental Hw → −Hw ;gw → gw ð69Þ jkm jkm jkm jkm timescale. Determining the time dependence induced by for the SME coefficients. For example, an expression for the rotation of the Earth is the subject of Sec. II E. This δν 0 0 the frequency shift D¯ of the nL-n L transition in treatment is extended in Sec. II F to include effects at linear antideuterium due to isotropic Lorentz and CPT violation order in the Earth’s boost as it orbits the Sun. The time

036003-24 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018) dependence in a space-based laboratory arising from the theoretically both in the minimal SME [106] and allowing for orbital motion of a satellite is also discussed. nonminimal terms of arbitrary mass dimension [43].These The application of our results to the analysis of clock- treatments are combined with recent spectroscopic measure- comparison experiments is described in Sec. III. We first ments of the 1S-2S transition in antihydrogen to extract first consider fountain clocks, deriving expressions for the constraints on SME coefficients from this system, summa- frequency shift in terms of coefficients expressed in the rized in Table XIV. We also propose that in the long term it Sun-centered frame. At zeroth boost order the frequency may become feasible to perform experiments with heavier shift is given in Table III, while at linear boost order it is antiatoms and antiions, with options possibly including the given by Eq. (44) and the entries in Table IV. Estimates for precision spectroscopy of antideuterium or of trapped anti- attainable sensitivities to SME coefficients using existing ions. A technique is presented to convert theoretical results devices are provided in Table V. The discussion covers both for frequency shifts in atoms or ions to the corresponding 133Cs and 87Rb fountain clocks, and it is also applicable to ones in antiatoms or antiions. clocks located on a space-based platform. The primary The two appendices following the present summary collect sensitivity in these systems is to coefficients for Lorentz some results that are useful in handling coefficients for violation in the proton sector. Lorentz violation. Appendix A includes relations connecting We next consider the prospects for using comagnetom- spherical and Cartesian coefficients and provides explicit 3 ≤ ≤ 8 eters to search for nonminimal violations of Lorentz and expressions between them for the cases d . CPT symmetry. The methodology developed in Sec. II is Appendix B discusses the transformation between the labo- well suited for application to investigations using 129Xe and ratory frame and the Sun-centered frame and tabulates explicit 3 results connecting Cartesian coefficients in the two frames for He atoms as comagnetometers. Within the nuclear model 3 ≤ ≤ 8 adopted here, the Lorentz- and CPT-violating signals are the cases d . The results in these appendices are affected predominantly by SME coefficients in the neutron generally applicable and so have implications extending sector. In Sec. III B, we determine the shift in the outside the analysis of clock-comparison experiments. experimental frequency observable at zeroth boost order Throughout this work, we have noted possibilities for and extract the bound (54) by extending to arbitrary d the pursuing investigations that go beyond our present scope while remaining within the context of Lorentz- and CPT- known results for the minimal SME. This leads to the violating corrections to the propagators of the constituents of constraints on neutron nonrelativistic coefficients listed in atoms and ions. In principle, our scope could also be Table VI. We also establish the Lorentz- and CPT-violating extended by incorporating effects arising from other SME shift at linear boost order, using existing data to place sectors. For instance, the Maxwell equations acquire mod- constraints on neutron effective Cartesian coefficients in ifications due to Lorentz and CPT violation in the pure- Table IX. Other comagnetometers can also place competi- photon sector. Including these might further enhance the tive limits on the neutron sector of the SME. We derive a reach of clock-comparison experiments, though in practice partial map from known minimal-SME bounds to non- most relevant photon-sector coefficients are already tightly minimal coefficients, which permits using data from a 21 bounded from analyses of other systems [3,118,119]. Effects Ne-Rb-K comagnetometer to place the additional con- involving U(1)-covariant Lorentz- and CPT-violating cou- straints on the neutron sector given in Table X. All these plings between photons and fermions are of interest as well, constraints on neutron coefficients for Lorentz violation are with only a few SME coefficients currently constrained by the first of their kind reported in the literature. experiment [3,45,120]. One could also envisage the inclusion As another application, we consider the attainable reach in of SME effects arising in the strong, electroweak, or clock-comparison experiments using trapped ions and lattice gravitational sectors, although some of these are expected clocks. In this case, interesting sensitivities are in principle either to be suppressed or to be more readily studied by other attainable to coefficients for Lorentz violation in the electron means. An exception might be countershaded Lorentz and sector. Various transitions are considered for a range of atoms CPT violation [121], for which unexpectedly large effects and ions. The expression (64) is found to describe the annual can appear in the context of special measurements. For and sidereal modulations of the frequency difference example, sensitivity to countershaded coefficients has been between two clocks, including ones located in distinct demonstrated using atom interferometry, which can be laboratories. In this section, we also consider tests of interpreted in terms of clock comparisons [122]. Lorentz and CPT symmetry based on studying the time Overall, the content of this paper provides a broad evolution of an entangled state. The shift in the experimental methodology for exploring Lorentz and CPT symmetry frequency observable is determined at both zeroth and first using clock-comparison experiments. While our treatment boost order and is used to estimate attainable sensitivities to has yielded many first constraints, numerous coefficients electron nonrelativistic coefficients, as listed in Table XIII. for Lorentz violation are unmeasured to date. The striking Our final application considers the prospects for experi- potential sensitivities attainable either from reanalysis of ments using antimatter. Signals for Lorentz and CPT existing data or in future searches suggest that further work violation in antihydrogen have previously been investigated with clock-comparison experiments remains one of the

036003-25 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) most interesting prospects for uncovering these novel Using Eq. (A2), we can extract explicit expressions for physical effects in nature. the spin-independent spherical coefficients in terms of effective Cartesian coefficients. Table XV contains the ACKNOWLEDGMENTS results for spherical coefficients with 3 ≤ d ≤ 8, m ¼ 0, even values of j in the range 0 ≤ j ≤ k, and even values of k This work was supported in part by the United States in the range 0 ≤ k ≤ d − 2. The table consists of two pairs Department of Energy under Grant No. DE-SC0010120 of columns. In each pair, the first entry in a given row lists a and by the Indiana University Center for Spacetime spherical coefficient, while the second entry provides its Symmetries. equivalent as a linear combination of effective Cartesian coefficients. APPENDIX A: RELATION BETWEEN Next, we consider the spin-dependent part of the single- SPHERICAL AND CARTESIAN COEFFICIENTS particle Hamiltonian. For the component involving only the T ðdÞð0BÞ coefficients kjm , the relation between the Cartesian and This Appendix presents relationships between spherical spherical terms is coefficients and effective Cartesian coefficients and tabu- ≤ 8 μ lates explicit results for d . We focus on spherical ˆ˜ ðdÞ t ˜ ðdÞμtα1α2…αd−3 T pμ ¼ T pμpα pα …pα coefficients for Lorentz violation with even k and eff Xeff 1 2 d−3 0 d−3−k kþ1 ˆ ðdÞð0BÞ m ¼ , which are centrally relevant to analyses of clock- ¼ E0 jpj ðk þ 1ÞYjmðpÞT : ðA3Þ 1 kjm T ðdÞð EÞ kjm comparison experiments. The coefficients kjm controlling spin-dependent operators of the E type are disregarded Using orthonormality of the spherical harmonics then here as they leave unaffected the energy shifts. As else- where in this work, we follow Ref. [35] in using the symbol yields V Z with appropriate subscripts and superscripts to indicate Xd−3 ˜ μ 0 the difference of c- and a-type coefficients and T to Ω pˆ Tˆ ðdÞ t d−3−k p kþ1 1 T ðdÞð BÞ d Yjmð Þ eff pμ ¼ E0 j j ðk þ Þ kjm indicate the difference of g- and H-type coefficients. k¼j−1 VðdÞ VðdÞ For instance, kjm represents the difference kjm ¼ ðA4Þ ðdÞ − ðdÞ ckjm akjm. The spherical coefficients are assigned indices kj0, while t, x, y, z are used for specific index values on the between effective Cartesian coefficients and spherical effective Cartesian coefficients in a chosen frame. Dummy coefficients. This result permits the extraction of explicit spatial Cartesian indices are represented by l, m, n, and expressions for the spin-dependent spherical coefficients 0 ð4Þll T ðdÞð BÞ as linear combinations of effective Cartesian repeated Cartesian indices are summed. For example, ceff kjm ð4Þll ð4Þxx ð4Þyy ð4Þzz coefficients. Table XVI contains these expressions for represents the sum ceff ¼ ceff þ ceff þ ceff . T ðdÞð0BÞ 3 ≤ ≤ 8 0 The single-particle Hamiltonian can be decomposed in spherical coefficients kjm with d , m ¼ , either the spherical or the Cartesian bases. The connection odd values of j in the range 0 ≤ j ≤ k þ 1, and even between these decompositions is presented in Sec. IV of values of k in the range 0 ≤ k ≤ d − 3. The structure of this Ref. [35]. Consider first the spin-independent component table parallels that of Table XV. T ðdÞð1BÞ of the Hamiltonian. The corresponding match between the Determining the spherical coefficients kjm in terms Cartesian and spherical bases is fixed by of effective Cartesian coefficients requires more work 1 μ μα α …α ðdÞð BÞ ˆ ðdÞ ðdÞ 1 2 d−3 because the relation containing T also incorporates V pμ ¼ V pμpα pα …pα kjm eff eff 1 2 d−3 0 X the coefficients T ðdÞð BÞ. We find d−2−k p k pˆ VðdÞ kjm ¼ E0 j j Yjmð Þ kjm; ðA1Þ kjm Z ν ˆ˜ ðdÞj j dΩT ϵˆ pν1Y ðpˆÞ where pμ ¼ðE0; −pÞ. Using the orthogonality of the eff þ jm rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spherical harmonics, the connection between the Xd−2 jðj þ 1Þ 0 Cartesian and spherical terms can be written as Ed−2−k p k T ðdÞð BÞ ¼ 0 j j 2 kjm Z −1 Xd−2 k¼j ˆ ðdÞμ d−2−k k ðdÞ dΩY ðpˆÞV pμ ¼ E jpj V ; ðA2Þ jm eff 0 kjm T ðdÞð1BÞ T ðdÞð1EÞ k¼j þ kjm þ i kjm ; ðA5Þ Ω where d is the differential element of solid angle in pffiffiffi ϵˆ θˆ ϕˆ 2 momentum space. The upper and lower bounds for the where ¼ð i Þ= . This result links three types of summation index k are determined by the spherical-index spherical coefficients with the effective Cartesian coeffi- relations listed in Table III of Ref. [35]. cients. It can be disentangled first by eliminating the

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VðdÞ 3 ≤ ≤ 8 TABLE XV. Relations between spherical coefficients kj0 and effective Cartesian coefficients for d . Spherical Cartesian Spherical Cartesian pffiffiffiffiffi pffiffiffiffiffi ð3Þ 4π ð3Þt ð6Þ 1 4π ð6Þllmm a000 aeff c400 5 ceff pffiffiffiffiffi pffiffi ð5Þ 4π ð5Þttt ð6Þ 4 π 3 ð6Þttzz − ð6Þttmm a000 aeff c220 5ð ceff ceff Þ ffiffiffiffiffi pffiffi ð5Þ p ð5Þtll ð6Þ 4 π ð6Þllzz ð6Þllmm a200 4πa c420 7 5ð3c − c Þ qffiffiffiffi eff ffiffiffiffi eff eff ffiffiffiffi 5 6 p 6 6 p 6 að Þ 4π 3 ð5Þtzz − ð5Þtll cð Þ 4π ð7cð Þzzzz − 6cð ÞzzllÞþ 4π cð Þmmll 220 5 ð aeff aeff Þ 440 21 eff eff 35 eff pffiffiffiffiffi pffiffiffiffiffi ð7Þ 4π ð7Þttttt ð8Þ 4π ð8Þtttttt a000 aeff c000 ceff pffiffiffiffiffi pffiffiffiffiffi ð7Þ 10 4π ð7Þtttll ð8Þ 5 4π ð8Þttttll a200 3 aeff c200 ceff pffiffiffiffiffi pffiffiffiffiffi ð7Þ 4π ð7Þtmmll ð8Þ 3 4π ð8Þttmmll a400 aeff c400 ceff pffiffiffiffiffi pffiffiffiffiffi ð7Þ 4 5π 3 ð7Þtttzz − ð7Þtttmm ð8Þ 1 4π ð8Þnnllmm a220 3 ð aeff aeff Þ c600 7 ceff pffiffiffiffiffi pffiffiffiffiffi ð7Þ 4 5π 3 ð7Þtllzz − ð7Þtllmm ð8Þ 2 5π 3 ð8Þttttzz − ð8Þttttmm a420 7 ð aeff aeff Þ c220 ð ceff ceff Þ pffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffi ð7Þ 100π 7 ð7Þtzzzz − 6 ð7Þtzzll 4π ð7Þtmmll ð8Þ 12 5π 3 ð8Þttllzz − ð8Þttllmm a440 21 ð aeff aeff Þþ 7 aeff c420 7 ð ceff ceff Þ pffiffiffiffiffi pffiffiffiffiffi ð4Þ 4π ð4Þtt ð8Þ 2 5π 3 ð8Þnnllzz − ð8Þnnllmm c000 ceff c620 21 ð ceff ceff Þ pffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffi ð4Þ 1 4π ð4Þll ð8Þ 100π 7 ð8Þttzzzz − 6 ð8Þttzzll 3 4π ð8Þttmmll c200 3 ceff c440 7 ð ceff ceff Þþ 7 ceff pffiffi pffiffiffiffiffiffiffi pffiffiffiffi ð4Þ 2 π 3 ð4Þzz − ð4Þll ð8Þ 100π 7 ð8Þmmzzzz − 6 ð8Þmmzzll 4π 3 ð8Þnnmmll c220 3 5ð ceff ceff Þ c640 77 ð ceff ceff Þþ 77 ceff pffiffiffiffiffi pffiffiffiffi ð6Þ 4π ð6Þtttt ð8Þ 4π 231 ð8Þzzzzzz − 5 ð8Þnnmmll c000 ceff c660 231 ð ceff ceff Þ pffiffiffiffiffi pffiffiffiffiffiffiffi ð6Þ 2 4π ð6Þttll 100π ð8Þzzmmll − 3 ð8Þzzzzmm c200 ceff þ 11 ðceff ceff Þ

T ðdÞð0BÞ 3 ≤ ≤ 8 TABLE XVI. Relations between spherical coefficients kj0 and effective Cartesian coefficients for d . Spherical Cartesian Spherical Cartesian qffiffiffiffi qffiffiffiffi 3 0 4 0 Hð Þð BÞ 4π ˜ ð3Þtz gð Þð BÞ 4π ˜ð4Þtzt 010 3 Heff 010 3 geff qffiffiffiffi qffiffiffiffi 5 0 6 0 Hð Þð BÞ 4π ˜ ð5Þtztt gð Þð BÞ 4π ˜ð6Þtzttt 010 3 Heff 010 3 geff qffiffiffiffi qffiffiffiffi 5 0 6 0 Hð Þð BÞ 1 4π 2 ˜ ð5Þtllz ˜ ð5Þtzll gð Þð BÞ 1 4π 2˜ð6Þtltlz ˜ð6Þtztll 210 15 3 ð Heff þ Heff Þ 210 5 3 ð geff þ geff Þ pffiffi pffiffi pffiffi pffiffi ð5Þð0BÞ 2 π ˜ ð5Þtzzz ˜ ð5Þtzll 4 π ˜ ð5Þtllz ð6Þð0BÞ 2 π ˜ð6Þtztzz ˜ð6Þtztll 4 π ˜ð6Þtltlz H230 15 7ð5H − H Þ − 15 7H g230 5 7ð5g − g Þ − 5 7g qffiffiffiffi eff eff eff qffiffiffiffi eff eff eff 7 0 8 0 Hð Þð BÞ 4π ˜ ð7Þtztttt gð Þð BÞ 4π ˜ð8Þtzttttt 010 3 Heff 010 3 geff qffiffiffiffi qffiffiffiffi 7 0 8 0 Hð Þð BÞ 2 4π 2 ˜ ð7Þtmttmz ˜ ð7Þtzttmm gð Þð BÞ 2 4π 2˜ð8Þtmtttmz ˜ð8Þtztttmm 210 5 3 ð Heff þ Heff Þ 210 3 3 ð geff þ geff Þ pffiffiffiffiffi pffiffiffiffiffi ð7Þð0BÞ 2 3π 4 ˜ ð7Þtmmllz ˜ ð7Þtzllmm ð8Þð0BÞ 2 3π 4˜ð8Þtmtmllz ˜ð8Þtztllmm H410 175 ð Heff þ Heff Þ g410 35 ð geff þ geff Þ pffiffi pffiffi pffiffi pffiffi ð7Þð0BÞ 4 π 5 ˜ ð7Þtzttzz − ˜ ð7Þtzttmm − 8 π ˜ ð7Þtmttmz ð8Þð0BÞ 4 π 5˜ð8Þtztttzz − ˜ð8Þtztttmm − 8 π ˜ð8Þtmtttmz H230 5 7ð Heff Heff Þ 5 7Heff g230 3 7ð geff geff Þ 3 7geff 7 0 pffiffi 7 7 8 0 pffiffi 8 8 Hð Þð BÞ 4 πð3H˜ ð Þtzmmzz þ 2H˜ ð ÞtmmzzzÞ gð Þð BÞ 4 πð3g˜ð Þtztmmzz þ 2g˜ð ÞtmtmzzzÞ 430 45 7 pffiffieff eff 430 9 7 peffffiffi eff − 4 π 4 ˜ ð7Þtmmllz ˜ ð7Þtzmmll − 4 π 4˜ð8Þtmtmllz ˜ð8Þtztmmll 75 7ð Heff þ Heff Þ 15 7ð geff þ geff Þ pffiffiffiffi pffiffiffiffi ð7Þð0BÞ 2 π 3 ˜ ð7Þtzzzzz − 2 ˜ ð7Þtzmmzz ð8Þð0BÞ 2 π 3˜ð8Þtztzzzz − 2˜ð8Þtztmmzz H450 15 11ð Heff Heff Þ g450 3 11ð geff geff Þ pffiffiffiffi 7 7 pffiffiffiffi 8 8 þ 2 π ð4H˜ ð Þtmmllz þ H˜ ð ÞtzmmllÞ þ 2 π ð4g˜ð Þtmtmllz þ g˜ð ÞtztmmllÞ 105pffiffiffiffi11 eff eff 21pffiffiffiffi11 eff eff − 8 π ˜ ð7Þtmmzzz − 8 π ˜ð8Þtrtrzzz 45 11Heff 9 11geff

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T ðdÞð1BÞ 3 ≤ ≤ 8 TABLE XVII. Relations between spherical coefficients kj0 and effective Cartesian coefficients for d . Spherical Cartesian Spherical Cartesian pffiffi pffiffi pffiffi pffiffi ð5Þð1BÞ 2 π 3 ˜ ð5Þnztn − ˜ ð5Þtnnz 2 π ˜ ð5Þtznn ð8Þð1BÞ 5 π 4˜ð8Þtztttmm − 4˜ð8Þtmtttmz 5 π ˜ð8Þmzttttm H210 3 3ð Heff Heff Þþ3 3Heff g210 3 3ð geff geff Þþ 3geff pffiffi pffiffi pffiffiffiffiffi pffiffiffiffiffi ð7Þð1BÞ 4 π ˜ ð7Þtzttnn − ˜ ð7Þtnttnz 4 π ˜ ð7Þnztttn ð8Þð1BÞ 4 3π ˜ð8Þtztllmm − ˜ð8Þtmtmllz 2 3π˜ð8Þmzttmll H210 3ðHeff Heff Þþ 3Heff g410 5 ðgeff geff Þþ geff ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi ð7Þð1BÞ 4 p ˜ ð7Þtzllnn ˜ ð7Þtnnllz 4 p ˜ ð7Þnztnll ð8Þð1BÞ 1 p ð8Þmzmnnll H410 25 3πðH − H Þþ5 3πH g610 7 3πg˜ qffiffiffiffi eff eff eff qffiffiffiffi eff 7 1 8 1 ð Þð BÞ 2 6π ˜ ð7Þnztnzz ˜ ð7Þtznnzz ð Þð BÞ 6π ð8Þmzttmzz ð8Þtztmmzz H430 ð5H þ H Þ g430 ð5g˜ þ 2g˜ Þ 5 7 qffiffiffiffieff eff 7qffiffiffiffieff eff 7 7 8 8 − 2 6πðH˜ ð Þtnnzzz þ H˜ ð ÞnztnllÞ − 6πð2g˜ð Þtmtmzzz þ g˜ð ÞmzttmllÞ 5 q7ffiffiffiffi eff eff q7 ffiffiffiffi eff eff 2 6π ˜ ð7Þtnnllz − ˜ ð7Þtznnll 2 6π ˜ð8Þtmtmllz − ˜ð8Þtztmmll þ 25 7 ðHeff Heff Þ þ 5 7 ðgeff geff Þ qffiffiffiffi 4 1 pffiffi 4 8 1 gð Þð BÞ πg˜ð Þnzn gð Þð BÞ 1 2π 5˜ð8Þmzmllzz − ˜ð8Þmzmllnn 210 3 eff 630 3 21ð geff geff Þ qffiffiffiffi qffiffiffiffi 6 1 ffiffiffiffiffi 8 1 ð Þð BÞ 4π ð6Þtztnn ð6Þtntnz p ð6Þnzttn ð Þð BÞ 1 5π ð8Þmzmzzzz ð8Þmzmllzz g210 ðg˜ − g˜ Þþ 3πg˜ g650 ð3g˜ − 2g˜ Þ 3 eff eff eff 3 33 qeffffiffiffiffi eff 1 5π ˜ð8Þmzmllnn þ 21 33geff pffiffiffiffiffi ð6Þð1BÞ 1 3π˜ð6Þnznmm g410 5 qffiffiffiffi geff 6 1 gð Þð BÞ 1 3π 5˜ð6Þlzlzz − ˜ð6Þlzlnn 430 5 14ð geff geff Þ

ðdÞð0BÞ 4 ≤ ≤ 8 0 T via Eq. (A4) and then by grouping the remaining coefficients with d , m ¼ , odd values of j in the kjm 0 ≤ ≤ − 1 terms according to powers of the momentum magnitude. range j k , and even values of k in the range 0 ≤ k ≤ d − 2 The point is that the E-type and B-type coefficients are . The structure of this table again follows that of Table XV. proportional to distinct powers of the momentum when j is fixed. For example, if j is odd then the terms involving B-type and E-type coefficients can only contain even and APPENDIX B: TRANSFORMATIONS TO THE SUN-CENTERED FRAME odd powers of the momentum magnitude, respectively. For the particular case with m ¼ 0, the spherical coef- Constraints on the coefficients for Lorentz violation are ficients and the spin-weighted harmonics are all real commonly reported in the Sun-centered frame [3].This numbers. It is therefore useful to separate the real and Appendix describes the conversion of coefficients for imaginary parts of Eq. (A5). The real part is Lorentz violation in a laboratory frame into combinations Z of coefficients in the Sun-centered frame, including effects at ν ˆ˜ ðdÞj ˆj zeroth and linear boost order. The primary focus here is on dΩT θ pν Y ðpˆÞ eff 1 j0 effective Cartesian coefficients, which are better suited for Xd−2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi boost analyses. We use Greek indices to denote spacetime d−2−k p k 1 T ðdÞð0BÞ 2T ðdÞð1BÞ ¼ E0 j j ð jðj þ Þ kj0 þ kj0 Þ; indices and Latin indices to represent spatial components. k¼j−1 Generic indices in the laboratory frame are represented by lowercase letters, while indices in the Sun-centered frame are ðA6Þ represented by uppercase ones. For definiteness, we label and it contains only B-type coefficients. The imaginary part Cartesian components in the laboratory frame by 0,1,2,3 and assume that the Lorentz transformation is given by Eq. (29) of Eq. (A5) is given by 3 ˆ J with R J ¼ B . Cartesian components in the Sun-centered Z ffiffiffi Xd−2 frame are denoted by T, X, Y, Z, and contractions of spatial ˜ ν p 1 ΩTˆ ðdÞj ϕˆ j pˆ 2 d−2−k p kT ðdÞð EÞ uppercase indices imply summation over components in the d eff pν1Yj0ð Þ¼ E0 j j kj0 k¼j Sun-centered frame. Consider first the effective Cartesian coefficients asso- A7 ð Þ ciated with spin-independent Lorentz and CPT violation. The expressions for these effective Cartesian coefficients in and contains only E-type coefficients. By combining the laboratory frame in terms of effective Cartesian coef- Eqs. (A2) and (A6), we can extract explicit expressions ficients in the Sun-centered frame can be reconstructed at T ðdÞð1BÞ for the coefficients kjm in terms of effective Cartesian linear boost order from the information contained in components. Table XVII contains the results for spherical Table XVIII. The table limits attention to coefficients in

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TABLE XVIII. Relations between spin-independent Cartesian coefficients in laboratory and Sun-centered frames for 3 ≤ d ≤ 8.

Laboratory Factor Sun-centered Laboratory Factor Sun-centered ð3Þ0 1 ð3ÞT ð6Þ0000 1 ð6ÞTTTT aeff aeff ceff ceff −βJ ð3ÞJ −4βJ ð6ÞTTTJ aeff ceff ð5Þ000 1 ð5ÞTTT ð6Þ00jj 1 ð6ÞTTJJ aeff aeff ceff ceff −3βJ ð5ÞTTJ −2βJ ð6ÞTTTJ aeff ceff ð5Þ0jj 1 ð5ÞTKK −2βJ ð6ÞTKKJ aeff aeff ceff −2βJ ð5ÞTTJ ð6Þ0033 Bˆ J1 Bˆ J2 ð6ÞTTJ1J2 aeff ceff ceff 5 ˆ J1 ˆ J2 6 −βJ ð ÞKKJ −2B B βJ ð ÞTJ1J2J aeff ceff 5 033 5 ˆ J1 ˆ 6 ð Þ Bˆ J1 Bˆ J2 ð ÞTJ1J2 −2B B · β ð ÞTTTJ1 aeff aeff ceff ˆ J1 ˆ J2 5 6 6 −B B βJ ð ÞJ1J2J ð Þjjkk 1 ð ÞKKLL aeff ceff ceff ˆ J1 ˆ 5 6 −2B B · β ð ÞTTJ1 −4βJ ð ÞTKKJ aeff ceff ð7Þ00000 1 ð7ÞTTTTT ð6Þ33jj Bˆ J1 Bˆ J2 ð6ÞKKJ1J2 aeff aeff ceff ceff 7 ˆ J1 ˆ J2 6 −5βJ ð ÞTTTTJ −2B B βJ ð ÞTJ1J2J aeff ceff 7 000 7 ˆ J1 ˆ 6 ð Þ jj 1 ð ÞTTTKK −2B B · β ð ÞTKKJ1 aeff aeff ceff −2βJ ð7ÞTTTTJ ð6Þ3333 Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 ð6ÞJ1J2J3J4 aeff ceff ceff 7 ˆ J1 ˆ J2 ˆ J3 ˆ 6 −3βJ ð ÞTTKKJ −4B B B B · β ð ÞTJ1J2J3 aeff ceff ð7Þ00033 Bˆ J1 Bˆ J2 ð7ÞTTTJ1J2 ð8Þ000000 1 ð8ÞTTTTTT aeff aeff ceff ceff ˆ J1 ˆ J2 7 8 −3B B βJ ð ÞTTJ1J2J −6βJ ð ÞTTTTTJ aeff ceff ˆ J1 ˆ 7 8 0000 8 −2B B · β ð ÞTTTTJ1 ð Þ jj 1 ð ÞTTTTKK aeff ceff ceff ð7Þ0jjkk 1 ð7ÞTKKLL −2βJ ð8ÞTTTTTJ aeff aeff ceff −4βJ ð7ÞTTKKJ −4βJ ð8ÞTTTKKJ aeff ceff −βJ ð7ÞKKLLJ ð8Þ000033 Bˆ J1 Bˆ J2 ð8ÞTTTTJ1J2 aeff ceff ceff 7 0 33 7 ˆ J1 ˆ J2 8 ð Þ jj Bˆ J1 Bˆ J2 ð ÞTKKJ1J2 −4βJB B ð ÞTTTJJ1J2 aeff aeff ceff ˆ J1 ˆ J2 7 ˆ J1 ˆ 8 −2B B βJ ð ÞTTJ1J2J −2B B · β ð ÞTTTTTJ1 aeff ceff ˆ J1 ˆ 7 8 00 8 −2B B · β ð ÞTTKKJ1 ð Þ jjkk 1 ð ÞTTKKLL aeff ceff ceff ˆ J1 ˆ J2 7 8 −B B βJ3 ð ÞKKJ1J2J3 −4βJ ð ÞTTTKKJ aeff ceff ð7Þ03333 Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 ð7ÞTJ1J2J3J4 −2βJ ð8ÞTKKLLJ aeff aeff ceff ˆ J1 ˆ J2 ˆ J3 ˆ J4 7 8 00 33 8 −B B B B βJ ð ÞJ1J2J3J4J ð Þ jj Bˆ J1 Bˆ J2 ð ÞTTKKJ1J2 aeff ceff ceff ˆ J1 ˆ J2 ˆ J3 ˆ 7 ˆ J1 ˆ J2 8 −4B B B B · β ð ÞTTJ1J2J3 −2βJB B ð ÞTTTJJ1J2 aeff ceff 4 00 4 ˆ J1 ˆ J2 8 ð Þ 1 ð ÞTT −2βJB B ð ÞTKKJ1J2J ceff ceff ceff 4 ˆ J1 ˆ 8 −2βJ ð ÞTJ −2B B · β ð ÞTTTKKJ1 ceff ceff ð4Þjj 1 ð4ÞKK ð8Þ003333 Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 ð8ÞTTJ1J2J3J4 ceff ceff ceff ceff 4 ˆ J1 ˆ J2 ˆ J3 ˆ J4 8 −2βJ ð ÞTJ −2B B B B βJ ð ÞTJ1J2J3J4J ceff ceff 4 33 4 ˆ J1 ˆ J2 ˆ J3 ˆ 8 ð Þ Bˆ J1 Bˆ J2 ð ÞJ1J2 −4B B B B · β ð ÞTKKJ1J2J3 ceff ceff ceff ˆ J1 ˆ 4 −2B B · β ð ÞTJ1 ceff

036003-29 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018) the laboratory frame that contribute to the spherical ∂KNR KNR ≈ KNR kjm βJ coefficients with 3 ≤ d ≤ 8 discussed in Appendix A, kjm kjmjβJ¼0 þ J : ðB2Þ ∂β βJ 0 which are the ones relevant to the clock-comparison ¼ experiments analyzed in this work. The table contains For all coefficients, the zeroth-order term is given by Eq. (25). two triplets of columns. In each triplet, the first column lists At linear boost order, we are interested in the contribu- the Cartesian components of interest in the laboratory tion VNR O β to the nonrelativistic spin-independent frame. Entries in the second column are factors involving kj0ð ð ÞÞ coefficients with m ¼ 0. Decomposing this contribution the boost −βJ and the direction Bˆ J of the magnetic field. as a polynomial in the unit vector Bˆ J along the magnetic The third column lists the relevant Cartesian components in field yields the result the Sun-centered frame. The expression converting a given coefficient from the laboratory frame to the Sun-centered ∂VNR frame is obtained by multiplying the entries in the second VNR O β kj0 βJ kj0ð ð ÞÞ ¼ ∂βJ and third columns and adding the associated rows. For βJ¼0 example, the first two rows of the table generate the sffiffiffiffiffiffiffiffiffiffiffiffiffi X8 ð3Þt ð3ÞT ð3ÞJ 4π equation a ¼ a − βJa . Using the contents of this ¼ − md−k−3 VðdÞJβJ eff eff eff 2j 1 kj table and the results in Appendix A, it is straightforward to þ d¼3 convert spin-independent spherical coefficients in the X8 ðdÞJJ1J2 βJ ˆ J1 ˆ J2 laboratory frame to effective Cartesian coefficients in the þ Vkj B B Sun-centered frame at linear boost order. d¼5 To analyze the experiments discussed in this work, X8 ðdÞJJ1J2J3J4 βJ ˆ J1 ˆ J2 ˆ J3 ˆ J4 it is useful to find analogous expressions converting þ Vkj B B B B : ðB3Þ the nonrelativistic coefficients for Lorentz violation to the d¼7 Sun-centered frame. The nonrelativistic coefficients are … combinations of spherical coefficients for Lorentz violation ðdÞJJ1 Jn − 3 − 0 The quantities Vkj with d k< vanish. For of arbitrary mass dimension, as illustrated in Eqs. (111) and 3 ≤ d ≤ 8, Table XIX provides explicit expressions for (112) of Ref. [35]. All the spherical coefficients contrib- … VðdÞJJ1 Jn uting to a particular nonrelativistic coefficient behave the many nonvanishing kj in terms of combinations of same way under rotations, so at zeroth boost order the effective Cartesian coefficients in the Sun-centered frame. conversion between frames is given by the comparatively simple result (25). However, the spherical coefficients ðdÞJK…M 5 ≤ ≤ 8 TABLE XIX. The quantities Vkj for d . transform differently under boosts, so converting nonrelativistic coefficients at linear boost order becomes ðdÞJK…M Vkj Combination involved. In contrast, the effective Cartesian coefficients 5 5 ð5ÞJ −2að ÞTTJ − að ÞKKJ have comparatively simple transformations under boosts V20 eff eff ð6ÞJ 4 ð6ÞTTTJ 4 ð6ÞTKKJ and so are better suited for studying boost effects. V20 ceff þ ceff 7 7 To circumvent this issue, we limit attention here ð7ÞJ − 10 2að ÞTTTTJ 3að ÞTTKKJ V20 3 ð eff þ eff Þ to terms involving effective Cartesian coefficients that 8 8 8 Vð ÞJ 10cð ÞTTTTTJ 20cð ÞTTTKKJ contribute at zeroth order in jpj=m , which yields the 20 eff þ eff w 7 7 ð7ÞJ −að ÞLLKKJ − 4að ÞTTKKJ dominant contributions at linear boost order and suffices V40 eff eff for the experimental analyses of interest. With this ð8ÞJ ð8ÞTLLKKJ ð8ÞTTTKKJ V 6c þ 12c assumption, the spin-independent nonrelativistic coefficients 40 eff eff ð5ÞJJ1J2 ð5ÞTTJ2 ð5ÞJJ1J2 −3 − 6δJJ1 VNR V aeff aeff kjm in the laboratory frame are expressed in terms of 22 6 JJ1J2 6 TJJ1J2 6 TTTJ2 ð Þ 12 ð Þ 12δJJ1 ð Þ spherical coefficients as V22 ceff þ ceff ð7ÞTTJJ1J2 ð7ÞTTTTJ2 ð7ÞJJ1J2 −30a − 20δJJ1 a V22 eff eff X 8 8 8 −3− ðdÞ ð ÞJJ1J2 ð ÞTTTJJ1J2 JJ1 ð ÞTTTTTJ2 VNR ≈ d kV V22 60c þ 30δ c kjm mψ kjm: ðB1Þ eff eff 60 ð7ÞTTJJ1J2 ð7ÞTTLLJ2 d ð7ÞJJ1J2 − a δJJ1 a V42 7 ð eff þ eff Þ 30 ð7ÞLLJJ1J2 − 7 a The spherical coefficients can then be translated into effective ð8ÞJJ1J2 180 ð8ÞTTTJJ1J2 ð8ÞTKKJJ1J2 V42 ðc þ c Þ Cartesian coefficients in the laboratory frame using the 7 eff eff 180 8 TTTKKJ2 δJJ1 ð Þ results in Appendix A. To perform the conversion between þ 7 ceff ð7ÞTTJ2J3J4 ð7ÞJJ1J2J3J4 the laboratory frame and the Sun-centered frame, we note ð7ÞJJ1J2J3J4 −5 δJJ1 V ð aeff þ aeff Þ KNR 44 that any nonrelativistic coefficient can be expanded to 8 JJ1J2J3J4 8 TTTJ2J3J4 8 TJJ1J2J3J4 kjm Vð Þ 60δJJ1 cð Þ þ 30cð Þ linear boost order as 44 eff eff

036003-30 LORENTZ AND CPT TESTS WITH CLOCK-COMPARISON … PHYS. REV. D 98, 036003 (2018)

TABLE XX. Relations between spin-dependent Cartesian coefficients in laboratory and Sun-centered frames for 3 ≤ d ≤ 6.

Laboratory Factor Sun-centered Laboratory Factor Sun-centered ˜ ð3Þ03 Bˆ J1 ˜ ð3ÞTJ1 ˜ð6Þ03000 Bˆ J1 ˜ð6ÞTJ1TTT Heff Heff geff geff Bˆ J1 βJ ˜ ð3ÞJ1J 4Bˆ J1 βJ ˜ð6ÞJ1ðTTTJÞ Heff geff ˜ ð5Þ0300 Bˆ J1 ˜ ð5ÞTJ1TT ˜ð6Þ0j0j3 Bˆ J1 ˜ð6ÞTKTKJ1 Heff Heff geff geff 3Bˆ J1 βJ ˜ ð5ÞJ1ðTTJÞ Bˆ J1 βJ ˜ð6ÞJTTTJ1 Heff geff ˜ ð5Þj30j Bˆ J1 ˜ ð5ÞKJ1TK 2Bˆ J1 βJ ˜ð6ÞKðJTKÞJ1 Heff Heff geff 2Bˆ J1 βJ ˜ ð5ÞJ1ðJTÞT Bˆ · β ˜ð6ÞKTTTK Heff geff Bˆ J1 βJ ˜ ð5ÞJ1KKJ ˜ð6Þ030jj Bˆ J1 ˜ð6ÞTJ1TKK Heff geff geff Bˆ · β ˜ ð5ÞTKTK 2Bˆ J1 βJ ˜ð6ÞJ1TTTJ Heff geff ˜ ð5Þ0jj3 Bˆ J1 ˜ ð5ÞTKKJ1 2Bˆ J1 βJ ˜ð6ÞJ1ðTJÞKK Heff Heff geff Bˆ J1 βJ ˜ ð5ÞJTTJ1 ˜ð6Þj300j Bˆ J1 ˜ð6ÞKJ1TTK Heff geff geff Bˆ J1 βJ ˜ ð5ÞKJKJ1 2Bˆ J1 βJ ˜ð6ÞJ1KTJK Heff geff Bˆ · β ˜ ð5ÞKTKT 2Bˆ J1 βJ ˜ð6ÞJ1ðJTÞTT Heff geff ˜ ð5Þ03jj Bˆ J1 ˜ ð5ÞTJ1KK Bˆ · β ˜ð6ÞTKTTK Heff Heff geff 2Bˆ J1 βJ ˜ ð5ÞJ1TTJ ˜ð6Þj3jkk Bˆ J1 ˜ð6ÞJJ1JKK Heff geff geff Bˆ J1 βJ ˜ ð5ÞJ1JKK 4Bˆ J1 βJ ˜ð6ÞJ1ðKTKJÞ Heff geff ˜ ð5Þ0333 Bˆ J1 Bˆ J2 Bˆ J3 ˜ ð5ÞTJ1J2J3 Bˆ · β ˜ð6ÞTKKLL Heff Heff geff 2Bˆ J1 Bˆ J2 Bˆ · β ˜ ð5ÞJ1TTJ2 ˜ð6Þ03033 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð6ÞTJ1TJ2J3 Heff geff geff Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð5ÞJ1JJ2J1 2Bˆ J1 Bˆ J2 Bˆ · β ˜ð6ÞJ1TTTJ2 Heff geff ˜ð4Þ030 Bˆ J1 ˜ð4ÞTJ1T 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð6ÞJ1ðTJÞJ2J3 geff geff geff 2Bˆ J1 βJ ˜ð4ÞJ1ðTJÞ ˜ð6jl3j33 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð6ÞKJ1KJ2J geff geff geff 4 3 4 ˆ J1 ˆ J2 ˆ 6 ˜ð Þj j Bˆ J1 ˜ð ÞKJ1K −3B B B · β ˜ð ÞKðJ1TJ2ÞK geff geff geff 2Bˆ J1 βJ ˜ð4ÞJ1ðTJÞ 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð6ÞJ1ðTJÞJ2J3 geff geff Bˆ · β ˜ð4ÞTKK geff

The other quantities of relevance can be obtained from attention to coefficients with 3 ≤ d ≤ 6 and 7 ≤ d ≤ 8, entries in this table using the relations respectively, which are the ones of relevance to our analysis of clock-comparison experiments. Each table contains two 10 ðdÞJ ðdÞJ ðdÞJ ðdÞJ triplets of columns, and each triplet has the same structure V ¼ −V ;V¼ − V ; 22 20 42 7 40 as that of Table XVIII. Taking products of the second and 3 third entries in a row and summing over rows relevant to the ðdÞJ ðdÞJ ðdÞJJ1J2 − ðdÞJJ1J2 V44 ¼ 7 V40 ;V44 ¼ V42 : ðB4Þ chosen laboratory-frame coefficient yields the desired equation converting the effective Cartesian coefficients from the laboratory to the Sun-centered frame, as before. With the above results for spin-independent coefficients In parallel with the above discussion for spin-indepen- in hand, we next consider spin-dependent effects. The dent effects, the analysis of experiments is facilitated by relations connecting the sets of spin-dependent effective translating nonrelativistic coefficients for Lorentz violation Cartesian coefficients in the laboratory frame and the Sun- in the laboratory frame to expressions involving effective centered frame up to linear boost order can be found using Cartesian coefficients in the Sun-centered frame. Adopting the information in Tables XX and XXI. These tables restrict the assumptions leading to Eq. (B1), the spin-dependent

036003-31 V. ALAN KOSTELECKÝ and ARNALDO J. VARGAS PHYS. REV. D 98, 036003 (2018)

TABLE XXI. Relations between spin-dependent Cartesian coefficients in laboratory and Sun-centered frames for 7 ≤ d ≤ 8.

Laboratory Factor Sun-centered Laboratory Factor Sun-centered ˜ ð7Þ030000 Bˆ J1 ˜ ð7ÞTJ1TTTT ˜ð8Þ0300000 Bˆ J1 ˜ð8ÞTJ1TTTTT Heff Heff geff geff 5Bˆ J1 βJ ˜ ð7ÞJ1ðJTTTTÞ 6Bˆ J1 βJ ˜ð8ÞJ1ðTTTTTJÞ Heff geff ˜ ð7Þ0j00j3 Bˆ J1 ˜ ð7ÞTKTTKJ1 ˜ð8Þ0j000j3 Bˆ J1 ˜ð8ÞTKTTTKJ1 Heff Heff geff geff Bˆ J1 βJ ˜ ð7ÞJTTTTJ1 Bˆ · β ˜ð8ÞKTTTTTK Heff geff 3Bˆ J1 βJ ˜ ð7ÞKðJTTÞKJ1 Bˆ J1 βJ ˜ð8ÞJTTTTTJ1 Heff geff Bˆ · β ˜ ð7ÞKTTTTK 4Bˆ J1 βJ ˜ð8ÞKðJTTTÞJ1K Heff geff ˜ ð7Þ0300jj Bˆ J1 ˜ ð7ÞTJ1TTKK ˜ð8Þ03000jj Bˆ J1 ˜ð8ÞTJ1TTTKK Heff Heff geff geff 2Bˆ J1 βJ ˜ ð7ÞJ1TTTTJ 2Bˆ J1 βJ ˜ð8ÞJ1TTTTTJ Heff geff 3Bˆ J1 βJ ˜ ð7ÞJ1ðJTTÞKK 4Bˆ J1 βJ ˜ð8ÞJ1ðJTTTÞKK Heff geff ˜ ð7Þj3000j Bˆ J1 ˜ ð7ÞKJ1TTTK ˜ð8Þj30000j Bˆ J1 ˜ð8ÞKJ1TTTTK Heff Heff geff geff 2Bˆ J1 βJ ˜ ð7ÞJ1ðJTÞTTT 2Bˆ J1 βJ ˜ð8ÞJ1ðJTÞTTTT Heff geff 3Bˆ J1 βJ ˜ ð7ÞJ1KTTJK 4Bˆ J1 βJ ˜ð8ÞJ1KTTTKJ Heff geff Bˆ · β ˜ ð7ÞTKTTTK Bˆ · β ˜ð8ÞTKTTTTK Heff geff ˜ ð7Þj30jkk Bˆ J1 ˜ ð7ÞJJ1TJKK ˜ð8Þj300jkk Bˆ J1 ˜ð8ÞJJ1TTJKK Heff Heff geff geff 2Bˆ J1 βJ ˜ ð7ÞJ1ðJTÞTKK Bˆ · β ˜ð8ÞTKTTKLL Heff geff Bˆ J1 βJ ˜ ð7ÞJ1KJKLL 4Bˆ J1 βJ ˜ð8ÞJ1ðJTKKÞTT Heff geff 2Bˆ J1 βJ ˜ ð7ÞJ1KTTKJ 2Bˆ J1 βJ ˜ð8ÞJ1KTJKLL Heff geff Bˆ · β ˜ ð7ÞTKTKJJ ˜ð8Þ030jjkk Bˆ J1 ˜ð8ÞTJ1TKKJJ Heff geff geff ˜ ð7Þ03jjkk Bˆ J1 ˜ ð7ÞTJ1KKLL 4Bˆ J1 βJ ˜ð8ÞJ1TTTJKK Heff Heff geff Bˆ J1 βJ ˜ ð7ÞJ1JLLKK 2Bˆ J1 βJ ˜ð8ÞJ1ðJTÞLLKK Heff geff 4Bˆ J1 βJ ˜ ð7ÞJ1TTJKK ˜ð8Þ0j0jkk3 Bˆ J1 g˜ð8ÞTKTKLLJ1 Heff geff ˜ ð7Þ0jjkk3 Bˆ J1 ˜ ð7ÞTKKJJJ1 Bˆ · β ˜ð8ÞKTTTKLL Heff Heff geff Bˆ J1 βJ ˜ ð7ÞLJLKKJ1 2Bˆ J1 βJ ˜ð8ÞKðJTÞKLLJ1 Heff geff ˆ J1 7 ˆ J1 8 −3B βJ ˜ ð ÞTðKKJÞJ1T −3B βJ ˜ð ÞTðKKJÞTTJ1 Heff geff Bˆ · β ˜ ð7ÞKTTKLL ˜ð8Þj300j33 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð8ÞKJ1TTKJ2J3 Heff geff geff ˜ ð7Þ030033 Bˆ J1 Bˆ J2 Bˆ J3 ˜ ð7ÞTJ1TTJ2J3 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJ1KTJ2KJ3 Heff Heff geff 7 ˆ J1 ˆ J2 ˆ 8 3Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð ÞJ1ðJTTÞJ2J3 −3B B B · β ˜ð ÞKðJ1J2TÞTTK Heff geff 2Bˆ J1 Bˆ J2 Bˆ · β ˜ ð7ÞJ1TTTTJ2 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJ1ðJTÞTTJ2J3 Heff geff ˜ ð7Þj30j33 Bˆ J1 Bˆ J2 Bˆ J3 ˜ ð7ÞKJ1TKJ2J3 ˜ð8Þ0j0j333 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð8ÞTKTKJ1J2J3 Heff Heff geff geff 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞJ1ðJTÞTJ2J3 3Bˆ J1 Bˆ J2 Bˆ · β ˜ð8ÞKTTTKJ1J2 Heff geff Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞJ1KJKJ2J3 Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJTTTJ1J2J3 Heff geff ˆ J1 ˆ J2 ˆ 7 8 −3B B B · β ˜ ð ÞKðTJ1J2ÞTK 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð ÞKðJTÞKJ1J2J3 Heff geff

(Table continued)

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TABLE XXI. (Continued)

Laboratory Factor Sun-centered Laboratory Factor Sun-centered ˜ ð7Þ0jj333 Bˆ J1 Bˆ J2 Bˆ J3 ˜ ð7ÞTKKJ1J2J3 ˜ð8Þ030jj33 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð8ÞTJ1TKKJ2J3 Heff Heff geff geff 3Bˆ J1 Bˆ J2 Bˆ · β ˜ ð7ÞKTTKJ1J2 2Bˆ J1 Bˆ J2 Bˆ · β ˜ð8ÞJ1TTTJ2KK Heff geff Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞJTTJ1J2J3 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJ1TTTJ2J3J Heff geff Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞKJKJ1J2J3 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJ1ðJTÞJ2J3KK Heff geff ˜ ð7Þ03jj33 Bˆ J1 Bˆ J2 Bˆ J3 ˜ ð7ÞTJ1KKJ2J3 ˜ð8Þ0300033 Bˆ J1 Bˆ J2 Bˆ J3 ˜ð8ÞTJ1TTTJ2J3 Heff Heff geff geff 2Bˆ J1 Bˆ J2 Bˆ · β ˜ ð7ÞJ1TTJ2KK 4Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ð8ÞJ1ðTTTJÞJ2J3 Heff geff 2Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞJ1TTJJ2J3 2Bˆ J1 Bˆ J2 Bˆ · β ˜ð8ÞJ1TTTTTJ2 Heff geff Bˆ J1 Bˆ J2 Bˆ J3 βJ ˜ ð7ÞJ1JKKJ2J3 ˜ð8Þ0303333 Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 Bˆ J5 ˜ð8ÞTJ1TJ2J3J4J5 Heff geff geff 2Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 Bˆ J5 βJ ˜ð8ÞJ1ðJTÞJ2J3J4J5 geff 4Bˆ J1 Bˆ J2 Bˆ J3 Bˆ J4 Bˆ · β ˜ð8ÞJ1TTTJ2J3J4 geff

… nonrelativistic coefficients can be approximated in terms of ðdÞJJ1 Jn − 3 − 0 where the quantities TsB;kj with d k< vanish. spherical coefficients as … Explicit expressions for nonvanishing TðdÞJJ1 Jn in terms X 0B;kj T NRð0BÞ ≈ d−3−k 1 T ðdÞð0BÞ of effective Cartesian coefficients in the Sun-centered kjm mψ ðk þ Þ kjm ; d frame can be found in the first two columns of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Table XXII and by using the relations X 1 NRð1BÞ d−3−k ðdÞð1BÞ jðj þ Þ ðdÞð0BÞ T ≈ mψ T T : kjm kjm þ 2 kjm 14 d TðdÞJJ1 −TðdÞJJ1 ;TðdÞJJ1 − TðdÞJJ1 ; 0B;23 ¼ 0B;21 0B;43 ¼ 9 0B;41 ðB5Þ 5 ðdÞJJ1J2J3 − ðdÞJJ1J2J3 ðdÞJJ1 ðdÞJJ1 T0B;45 ¼ T0B;43 ;T0B;45 ¼ T0B;41 : ðB7Þ The spherical coefficients can then in turn be converted to 9 effective Cartesian coefficients using the results in ðdÞJJ1…Jn Appendix A. The conversion can be implemented to linear The nonvanishing quantities T1B;kj are compiled in the boost order via Eq. (B2), where the zeroth-order term is second pair of columns of Table XXII. again given by Eq. (25). In working with these results, the reader is cautioned that T NRð0BÞ T NRð1BÞ 1 At linear boost order, the relevant spin-dependent non- the coefficients kjm and kjm with j ¼ k þ are T NRðqBÞ O β 0 linearly dependent at zeroth order in p =m because the relativistic coefficients kj0 ð ð ÞÞ have m ¼ . j jw w ˆ J T ðdÞð1BÞ 1 Expanding them in powers of the unit vector B along spherical coefficients kjm vanish for j ¼ k þ .One the magnetic field, we obtain implication of this, for instance, is the existence of the relationships ∂T NRðqBÞ T NRðqBÞ O β kj0 βJ kj0 ð ð ÞÞ ¼ ∂βJ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βJ¼0 2 sffiffiffiffiffiffiffiffiffiffiffiffi ðdÞJJ1 k þ ðdÞJJ1 X8 T1 1 ¼ T0 1 ; 4π B;kðkþ Þ 2ðk þ 1Þ B;kðkþ Þ ¼ md−k−3 TðdÞJJ1 βJBˆ J1 2jþ 1 sB;kj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d¼3 2 ðdÞJJ1J2J3 k þ ðdÞJ1J2J3 X8 T ¼ T ; 1B;kðkþ1Þ 2 1 0B;kðkþ1Þ ðdÞJJ1J2J3 βJ ˆ J1 ˆ J2 ˆ J3 ðk þ Þ þ TsB;kj B B B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d¼5 2 ðdÞJJ1J2J3J4J5 k þ ðdÞJ1J2J3J4J5 X8 T T B8 1B;kðkþ1Þ ¼ 2 1 0B;kðkþ1Þ ð Þ ðdÞJJ1J2J3J4J5 βJ ˆ J1 ˆ J2 ˆ J3 ˆ J4 ˆ J5 ðk þ Þ þ TsB;kj B B B B B ; d¼7 ðB6Þ that link the quantities with subscripts 0B and 1B.

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ðdÞJK…M ðdÞJK…M 3 ≤ ≤ 8 TABLE XXII. The quantities T0B;kj and T1B;kj for d .

ðdÞJK…M ðdÞJK…M T0B;kj Combination T1B;kj Combination ð3ÞJ1J 1 ð5ÞJ1ðJTTÞ ð5ÞTðJ1JÞT ð3ÞJJ1 ˜ ð5ÞJJ1 ˜ ˜ −H − 5 ð15H þ 2H T0B;01 eff T1B;21 eff eff ð4ÞJJ1 2˜ð4ÞJ1ðJTÞ 6 ˜ ð5ÞJ1ðJKKÞ − 2 ˜ ð5ÞKðJ1JÞK T0B;01 geff þ Heff Heff 5 ˜ ð5ÞJ1ðJTTÞ 5 ð ÞJJ1 −3H 6 ˜ ð ÞTKTKδ T0B;01 eff þ Heff JJ1 Þ ð6ÞJJ1 4˜ð6ÞJ1ðJTTTÞ ð6ÞJJ1 1 60˜ð6ÞJ1ðJTTTÞ 12˜ð6ÞTðJ1JÞTT T0B;01 geff T1B;21 10 ð geff þ geff 7 ˜ ð7ÞJ1ðJTTTTÞ 6 6 ð ÞJJ1 −5H 48˜ð ÞJ1ðTJKKÞ − 18˜ð ÞKðJ1JTÞK T0B;01 eff þ geff geff ð8ÞJJ1 6˜ð8ÞJ1ðJTTTTTÞ 21˜ð6ÞTKTTKδ T0B;01 geff þ geff JJ1 Þ 2 ð7ÞJ1ðTTTTJÞ ð7ÞTðJ1JÞTTT ð5ÞJJ1 1 ð5ÞJðJ1KKÞ ð5ÞTðJ1JÞT ð7ÞJJ1 ˜ ˜ 3 ˜ 4 ˜ − 5 ð25H þ 6H T0B;21 5 ð Heff þ Heff T1B;21 eff eff 2 ˜ ð5ÞTKTKδ 30 ˜ ð7ÞJ1ðTTJKKÞ − 12 ˜ ð7ÞKðTTJJ1ÞK þ Heff JJ1 ÞþHeff Heff 3 ð6ÞJðJ1KKÞT ð6ÞTðJ1JÞTT ð6ÞJJ1 ð7ÞTKTTTK − 5 ð3g˜ þ 4g˜ 8 ˜ δ T0B;21 eff eff þ Heff JJ1 Þ 3˜ð6ÞTðJ1KKÞJ 2˜ð6ÞTKTTKδ ð8ÞJJ1 1 30˜ð8ÞJ1ðJTTTTTÞ 8˜ð8ÞTðJ1JÞTTTT þ geff þ geff JJ1 Þ T1B;21 2 ð geff þ geff ð7ÞJJ1 6 3 ˜ ð7ÞJðJ1KKÞTT 6 ˜ ð7ÞTðJ1KKÞTJ 48˜ð8ÞJ1ðTTTJKKÞ − 20˜ð8ÞKðJ1JTTTÞK T0B;21 5 ð Heff þ Heff þ geff geff 4 ˜ ð7ÞTðJ1JÞTTT 2 ˜ ð7ÞTKTTTKδ 9˜ð8ÞTKTTTTKδ þ Heff þ Heff JJ1 Þþgeff JJ1 Þ ð8ÞJðJ1KKÞTTT ð8ÞTðJ1KKÞTTJ 3 ð7ÞJ1ðJTTKKÞ ð7ÞKðJJ1ÞKLL ð8ÞJJ1 ð7ÞJJ1 ˜ ˜ −2ð3g˜ þ 9g˜ − 35 ð70H − 4H T0B;21 eff eff T1B;41 eff eff 4˜ð8ÞTðJ1JÞTTTT 2˜ð8ÞTKTTTTKδ 15 ˜ ð7ÞJ1ðJLLKKÞ 8 ˜ ð7ÞTðJJ1LLÞT þ geff þ geff JJ1 ÞþHeff þ Heff

7 JJ1 3 7 T J1JKK T 7 J J1KKLL 7 TKKLLT ð Þ 16 ˜ ð Þ ð Þ 5 ˜ ð Þ ð Þ 16 ˜ ð Þ δJJ1 T0B;41 35 ð Heff þ Heff þ Heff Þ 4 ˜ ð7ÞTKKLLT δ ð8ÞJJ1 3 18˜ð8ÞJ1ðJTKKLLÞ − 2˜ð8ÞKðJJ1TÞKLL þ Heff JJ1 Þ T1B;41 7 ð geff geff 3 ð8ÞTðJ1JKKÞTT ð8ÞJðJ1KKLLÞT ð8ÞJJ1 ˜ ˜ ð8ÞTðKKJ1JÞTT ð8ÞJ1ðJLLTTTÞ − 7 ð16g þ 5g 8˜ 42˜ T0B;41 eff eff þ geff þ geff

8 T J1KKLL J 8 TKKLLTT 8 TKKLLTT 5˜ð Þ ð Þ 4˜ð Þ δ 9˜ð Þ δJJ1 þ geff þ geff JJ1 Þþgeff Þ pffiffi 5 JJ1J2J3 5 JJ1J2J3 5 TJ2TJ3 7 JJ1 6 7 J1 JKKTT ð Þ ˜ ð Þ 2δJJ1 ˜ ð Þ ð Þ 15 ˜ ð Þ ð Þ T0B;23 Heff þ Heff T1B;43 15 ð Heff ð7ÞKðJJ1ÞKLL ð7ÞJ1ðJLLKKÞ 6 JJ1J2J3 6 J1 JT J2J3 6 TJ2J3TT ˜ ˜ ð Þ 6 ˜ð Þ ð Þ − δJJ1 ˜ð Þ −2H þ 5H T0B;23 ðgeff geff Þ eff eff 7 ˜ ð7ÞJ1ðJTTÞJ2J3 ˜ ð7ÞTJ2J3TTT ˜ ð7ÞTðJJ1LLÞT ˜ ð7ÞTKKLLT ð ÞJJ1J2J3 −18H þ 12δJJ1 H −4H þ 2H δJJ1 Þ T0B;23 eff eff eff eff 1 ð8ÞJ1ðJTKKLLÞ ð8ÞJJ1J2J3 40˜ð8ÞJ1ðJTTTÞJ2J3 ð8ÞJJ1 − pffiffi ð12g˜ T0B;23 geff T1B;43 6 eff ð8ÞTðJ2J3ÞTTTT −20δJJ1 g˜ 6˜ð8ÞKðJJ1TÞKLL − 8˜ð8ÞTðKKJ1JÞTT eff þ geff geff

7 JJ1J2J3 2 7 T JJ1J2J3 T 7 J J1J2J3KK 8 J1 JLLTTT 8 TKKLLTT ð Þ 8 ˜ ð Þ ð Þ 5 ˜ ð Þ ð Þ 18˜ð Þ ð Þ ˜ð Þ δJJ1 T0B;43 9 ð Heff þ Heff þ geff þ geff Þ pﬃﬃ 7 T J2J3KK T 7 JJ1J2J3 6 7 K JJ1J2J3 K 12δJJ1 ˜ ð Þ ð Þ ð Þ 12 ˜ ð Þ ð Þ þ Heff Þ T1B;43 9 ð Heff 10 ð8ÞTðJJ1J2J3ÞTT ð8ÞJJ1J2J3 ð7ÞJðJ1J2J3KKÞ ð7ÞJ1ðJTTÞJ2J3 − 9 ð8g˜ 5 ˜ − 27 ˜ T0B;43 eff þ Heff Heff 8 8 ˜ ð7ÞTðJJ1J2J3ÞT 5˜ð ÞJðJ1J2J3KKÞT 5˜ð ÞTðJ1J2J3KKÞJ −4H þ geff þ geff eff 7 T J2J3KK TT 7 T J2J3KK T 12δJJ1 ˜ð Þ ð Þ 12δJJ1 ˜ ð Þ ð Þ þ geff ÞþHeff

7 JJ1J2J3J4J5 7 JJ1J2J3J4J5 7 TJ2J3J4J5T 7 K J2J3TT K ð Þ ˜ ð Þ δJJ1 ˜ ð Þ 36δJJ1 ˜ ð Þ ð Þ T0B;45 Heff þ Heff þ Heff ) ð8ÞJJ1J2J3J4J5T ð8ÞTJ1J2J3J4J5J 5 ð8ÞKðJJ1J2J3TÞK ð8ÞJJ1J2J3J4J5 −5ðg˜ þ g˜ ð8ÞJJ1J2J3 − pﬃﬃ ð10g˜ T0B;45 eff eff T1B;43 3 6 eff ð8ÞTðJJ1J2J3ÞTT ð8ÞJ1ðJKKTÞJ2J3 8 TJ2J3J4J5TT 4δJJ1 ˜ð Þ −8g˜ − 24g˜ þ geff Þ eff eff −36˜ð8ÞJ1ðJTTTÞJ2J3 geff

8 T J2J3KK TT 24δJJ1 ˜ð Þ ð Þ þ geff

8 K J2J3TTT K 45δJJ1 ˜ð Þ ð Þ þ geff )

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