Letters B 772 (2017) 865–869

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Physics Letters B

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Constraining spacetime nonmetricity with neutron spin rotation 4 in liquid He ∗ Ralf Lehnert a,b, , W.M. Snow a,c,d, Zhi Xiao a,e, Rui Xu c a Indiana University Center for Spacetime Symmetries, Bloomington, IN 47405, USA b Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany c Physics Department, Indiana University, Bloomington, IN 47405, USA d Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USA e Department of and Physics, North China Electric Power University, Beijing 102206, China a r t i c l e i n f o a b s t r a c t

Article history: General spacetime nonmetricity coupled to neutrons is studied. In this context, it is shown that certain Received 3 June 2017 nonmetricity components can generate a rotation of the neutron’s spin. Available data on this effect Received in revised form 19 July 2017 obtained from slow-neutron propagation in liquid helium are used to constrain isotropic nonmetricity Accepted 27 July 2017 − (6) components at the level of 10 22 GeV. These results represent the first limit on the nonmetricity ζ S Available online 1 August 2017 2 000 Editor: A. Ringwald parameter as well as the first measurement of nonmetricity inside matter. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction The specialized situation in which the nonmetricity van- ishes Nαβγ = 0 and only torsion is nonzero represents the widely The idea that spacetime geometry represents a dynamical phys- known Einstein–Cartan theory [3]. In that context, torsion has ical entity has been remarkably successful in the description of been the subject of various investigations during the last four classical gravitational phenomena. For example, , decades [4]. Considering the question of the presence of torsion which is based on , has recently passed a fur- in nature as an experimental one has spawned numerous phe- ther experimental test: the theory predicts gravitational waves, and nomenological studies of torsion [5–15] yielding bounds on various these have indeed been observed by the LIGO Scientific Collabora- torsion couplings. tion and the Virgo Collaboration [1]. An analogous phenomenological investigation of nonmetricity At the same time, anumber of observational as well as the- has been instigated last year [16]. Paralleling the torsion case, that oretical issues motivate the construction and study of alterna- analysis treats the question regarding the presence of nonmetricity tive gravity theories. Most of these efforts recognize the elegance as an experimental one, and the nonmetricity field Nαβγ is taken and success of a geometric underpinning for gravitational phe- as a large-scale background extending across the solar system. The nomena and therefore retain this feature in model building. One particular physical situation considered in Ref. [16] lends itself to popular approach in this context, known as metric-affine grav- an effective-field-theory description in which Nαβγ represents a ity [2], employs an underlying geometry more general than that prescribed external field selecting preferred spacetime directions. of a Riemannian . The basic idea behind this approach Thus, such a set-up embodies in essence a Lorentz-violating sce- can be summarized as relaxing the metric-compatibility condition nario amenable to theoretical treatment via the Standard-Model = Dα gβγ 0 and the symmetry condition on the coeffi- Extension (SME) framework [17]. For example, sidereal and annual cients α −α = 0. In general, this idea introduces two tensor βγ γ β variations of physical observables resulting from the motion of fields an Earth-based laboratory through this solar-system nonmetricity α α α Nαβγ ≡−Dα gβγ , T βγ ≡  βγ −  γ β , (1) background represent a class of characteristic experimental signals in that context [18]. relative to the Riemannian case known as nonmetricity and tor- sion, respectively. The present work employs a similar idea to obtain further, com- plementary constraints on nonmetricity. The specific set-up we have in mind consists of liquid 4He as the nonmetricity source. * Corresponding author. Polarized neutrons generated at the slow-neutron beamline at the E-mail address: [email protected] (R. Lehnert). National Institute of Standards and Technology (NIST) Center for http://dx.doi.org/10.1016/j.physletb.2017.07.059 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 866 R. Lehnert et al. / Physics Letters B 772 (2017) 865–869   + 1 + + Neutron Research traverse the helium and serve as the nonmetric- 9 (N2)μ ηαβ (N2)α ηβμ (N2)β ημα , ity probe. It is apparent that our set-up involves an Earth-based   M ≡ 1 2N − N − N nonmetricity probe. Thus, the key difference between our study μαβ 3 μαβ αβμ βμα  and that in Ref. [16] is that we examine the situation of non- + 1 − − 9 2(N1)μ ηαβ (N1)α ηβμ (N1)β ηαμ metricity sourced locally in a terrestrial laboratory by the 4He.   − 1 2(N ) η − (N ) η − (N ) η . (3) This implies that the presumed nonmetricity in our case is co- 9 2 μ αβ 2 α βμ 2 β αμ moving with the laboratory, and thus the neutron probe, which With these pieces, the nonmetricity tensor can be reconstructed as precludes certain experimental signatures, such as sidereal and an- follows [16]: nual variations. Instead, we utilize the prediction presented below  that certain components of N lead to neutron spin rotation in N = 1 − 5(N ) η + (N ) η + (N ) η αβγ μαβ 18 1 μ αβ 1 α βμ 1 β μα this system. + 2(N ) η − 4(N ) η − 4(N ) η The outline of this paper is as follows. Section 2 reviews the 2 μ αβ 2 α βμ 2 β μα basic ideas behind the effective-field-theory description of a back- + Sμαβ + Mμαβ . (4) ground Nαβγ in flat Minkowski space and derives the resulting The sign changes in Eqs. (3), (4), and some subsequent equations spin motion for nonrelativistic neutrons. This effect provides the relative to the corresponding equations in Ref. [16] arise due to dif- for our limits on nonmetricity. The details of the measure- fering conventions for the metric signature and for the sign of the ment of neutron spin rotation in liquid 4He including the exper- Levi-Civita symbol. We also remark that although Eqs. (3) and (4) imental set-up are discussed in Sec. 3. Abrief summary is con- employ a notation similar to that for the irreducible components tained in Sec. 4. Throughout, we adopt natural units c = h¯ = 1. Our of torsion T α [14], the nonmetricity and torsion pieces are un- conventions for the metric signature and the Levi-Civita symbol are βγ related. ημν = diag(+, −, −, −) and 0123 =+1, respectively. With this decomposition, the following Lagrangian contribu- tions can be constructed [16]: 2. Theory (4) (4) μ (4) μ L = ζ (N1)μ ψγ ψ + ζ (N1)μ ψγ5γ ψ Our analysis is based on the approach to nonmetricity couplings N 1 2 + (4) μ + (4) μ taken in Ref. [16], so we begin with a brief review of that ap- ζ3 (N2)μ ψγ ψ ζ4 (N2)μ ψγ5γ ψ, ↔ ↔ proach. The basic idea is to follow the usual reasoning that the (5) (5) (5) L =−1 iζ (N )μ ψ∂ ψ − 1 ζ (N )μ ψγ ∂ ψ construction of an effective Lagrangian should include all terms N 2 1 1 μ 2 2 1 5 μ ↔ ↔ compatible with the symmetries of the model. In the present − 1 (5) μ − 1 (5) μ 2 iζ3 (N2) ψ∂μψ 2 ζ4 (N2) ψγ5∂μψ context, possible couplings between the background nonmetric- ↔ − 1 (5) ρ μν ity Nαβγ and the polarized-neutron probe need to be classified. 4 iζ5 Mμν ψσ ∂ρψ ↔ Since we are interested in a low-energy experiment, we may disre- + 1 (5) κ λρ μν 8 iζ6 κλμν M ψσ ∂ρψ gard the neutron’s internal structure and model it as a point Dirac ↔ ↔ ↔ (5) (5) L = 1 μ − + 1 iζ (N ) ψσ μν∂ ψ + 1 iζ (N ) ψσ μν∂ ψ fermion with free Lagrangian 0 2 ψ γ i∂μψ m ψψ, where m 2 7 1 μ ν 2 8 2 μ ν ↔ denotes the neutron mass. Conventional gravitational effects are (5) − 1 iζ λμνρ(N ) ψσ ∂ ψ negligible, so that the flat-spacetime Minkowski limit gμν → μν 4 9 1 λ μν ρ η ↔ suffices for our present purposes. − 1 (5) λμνρ 4 iζ10 (N2)λ ψσμν∂ρψ, The next step is to enumerate possible couplings of ψ to the (6) (6) L ⊃−1 S μν λ + h c background nonmetricity Nαβγ . This yields a hierarchy of possi- N 4 ζ1 λ ψγ ∂μ∂νψ . . (n) ble Lagrangian terms L labeled by the mass n of the − 1 (6) μν λ + N 4 ζ2 Sλ ψγ5γ ∂μ∂νψ h.c. (5) corresponding field operator: (n) Here, the real-valued couplings ζ are taken as free parameters; L = L + L(4) + L(5) + L(6) + l N 0 N N N ... . (2) they can in principle be fixed by specifying a definite underly- (6) ing nonmetricity model. For the mass-dimension six term L , we For the experimental set-up we have in mind, nonmetricity cou- N have only listed those contributions that contain the S irre- plings affecting the propagation of neutrons are the most relevant μαβ ducible piece; all other components of N are already present ones. Moreover, N must be small on observational grounds. αβγ αβγ (4) (5) (n) in the terms L or L of lower mass dimension. We therefore focus on contributions to L that are quadratic in N N N Equations (2), (3), and (5) determine the low-energy neutron ψ and linear in Nαβγ . General arguments in effective field the- effective Lagrangian in the presence of general background non- ory suggest that Lagrangian terms of lower mass dimension n may metricity relevant for the experimental situation we have in mind. be more dominant. Capturing the leading effects of all nonmetric- We note, however, that the terms (5) would generally be viewed ity components then requires inclusion of Lagrangian terms up to as part of a more complete Lagrangian L ⊃ L that also treats mass dimension n = 6 [16]. N N as a dynamical variable. The nonmetricity field equations The construction of the explicit form of each individual contri- αβγ (n) then contain ∂L/∂ Nαβγ , and thus neutron source terms. This idea bution L is most easily achieved by decomposing Nαβγ into its N provides the justification for taking the 4He nucleus as a non- Lorentz-irreducible pieces. These are given by two vectors (N1)μ metricity source in the experimental set-up discussed below. The and (N2)μ, atotally symmetric rank-three tensor Sμαβ , and a rank- protons and electrons of the 4He atom may produce additional three tensor Mμαβ with mixed symmetry [16]: nonmetricity contributions if these particles exhibit nonmetricity αβ couplings analogous to those in Eq. (5). In what follows, we make (N1)μ ≡−η Nμαβ , no assumptions regarding the dynamics of Nαβγ or additional αβ 4 (N2)μ ≡−η Nαμβ , nonmetricity–matter couplings; we simply presume that the He   1 generates some nonzero nonmetricity. Sμαβ ≡ Nμαβ + Nαβμ + Nβμα 3   A model refinement can be achieved by focusing on the lead- + 1 + + = 18 (N1)μ ηαβ (N1)α ηβμ (N1)β ημα ing contribution to Nαβγ . Note that Nαβγ Nαβγ (x) must exhibit R. Lehnert et al. / Physics Letters B 772 (2017) 865–869 867

  α β j a nontrivial spacetime dependence determined by the interatomic 1 (5) ˜ 3 (5) (6) p p σ + ζ M jαβ + ζ M jαβ + m ζ S jαβ distance and the velocity of the 4He atoms. However, the ran- 2 5 2 6 2 m dom nature of these two quantities suggests that the leading non- pα pβ p · + 1 (6) σ metricity effects are actually governed by the spacetime average ζ S0αβ . (7) 2 2 m Nαβγ (x). For this reason, we may take Nαβγ = const. in what fol- lows. The nonmetricity contributions (5) then form a subset of the This expression contains the leading contribution in the nonrel- | | flat-space SME Lagrangian, afact that permits us to employ the full ativistic order p /m for each nonmetricity component. In the ˜ ≡ μν μ = repertoire of theoretical tools developed for the SME framework. above equation, we have set Mαβγ αβ Mμνγ . Moreover, p 0 = 0 j j One such SME result relevant for the present situation con- (p , p) (p , p ) denotes the neutron’s 4-momentum, and σ cerns the observability of constant background fields [17,19]. For are the usual Pauli matrices. Note that nonmetricity effects corre- (6) example, it is known that contributions associated with the cou- sponding to ζ1 only produce spin-independent effects. They are (4) (4) (5) (5) (5) (5) (5) (5) therefore absent from δhs and cannot be determined by observa- plings ζ1 , ζ3 , ζ1 , ζ2 , ζ3 , ζ4 , ζ7 , and ζ8 can be removed from the Lagrangian—at least at linear order—via judiciously cho- tions of neutron spin rotation. sen field redefinitions. We may therefore disregard these terms in what follows. Their measurement would require situations involv- 3. Experimental analysis ing nonconstant Nαβγ , the presence of gravity, or the consideration of higher-order effects. To extract experimental signatures resulting from the non- metricity correction (7), we analyze the aforementioned experi- An additional simplification arises from the isotropy of the liq- mental situation, namely spin motion of a neutron as it passes uid helium. The 4He ground state has spin zero, so anisotropies through liquid 4He. As argued above, our Lagrangian (5) implies would have to be tied to excited states of 4He or arrangements of that neutrons, and hence 4He nuclei, can generate nonmetricity. the helium atoms involving preferred directions. However, the ab- The injected neutron beam would then be affected by this non- sence of polarization and the aforementioned random nature of metricity background. Moreover, our “in-matter” approach permits both position and velocity of individual 4He particles precludes us to search for short-ranged or non-propagating nonmetricity. sizeable, large-scale anisotropies. The leading background non- In particular, this encompasses situations analogous to minimally metricity contributions generated by the liquid-helium bath can coupled torsion, where the vanishes outside the therefore also be taken as isotropic in the helium’s center-of-mass spin-density source [4]. Such an approach rests on the premise frame. It follows that the present experimental set-up is only sen- that the probe penetrates the matter and that the effects of con- sitive to the rotationally invariant pieces of N . αβγ ventional Standard-Model (SM) physics are minimized. The 4He– To uncover the isotropic content of N , we may proceed by αβγ neutron system appears to be ideal in this respect for two reasons. inspecting its irreducible pieces (3). Clearly, components without First, the neutron mean free path inside liquid 4He is relatively spatial indices are rotation symmetric: (N ) , (N ) , and S . Note 1 0 2 0 000 long allowing for the accumulation of the predicted spin-rotation that M obeys the cyclic property αβγ effect. This is due to the small elastic and the essentially vanish- ing inelastic cross sections as well as rapidly decreasing neutron– M + M + M = 0 , (6) αβγ βγα γαβ phonon scattering as T → 0. Second, contamination of the non- metricity spin rotation by ordinary SM physics can be excluded on which implies M000 = 0. Further isotropic components in S and the grounds that these conventional effects lie below the current M with spatial indices must have spatial-index structure δ jk or detection sensitivity. This latter fact is explained in more detail be- jkl, where Latin indices run from 1 to 3. Since both S and M are symmetric in their last two indices, they cannot contain pieces low. The rotation of the spin of a transversely polarized slow- of jkl. This only leaves contributions involving δ jk. But these do not yield independent isotropic contributions because both S and neutron beam is called neutron optical activity. It is quantified by M are traceless. To see this, consider as an example a piece of the the rotary power dφPV/dL defined as the rotation angle φPV of the neutron spin about the neutron momentum p per traversed form S0 jk = s δ jk, where s is the isotropy parameter in question. αβ distance L. The nonmetricity correction (7) leads to the following But S is traceless, so that we have 0 = S0αβ η = S000 − S0 jk δ jk = S − s δ δ . It follows that 3s = S does not represent an expression for the rotary power: 000 jk jk 000     additional independent isotropic contribution to S. An analogous dφPV = (4) − (5) + (4) − (5) reasoning applies to M, so that (N ) , (N ) , and S are indeed 2 ζ m ζ (N1)0 2 ζ m ζ (N2)0 1 0 2 0 000 dL 2 9 4 10 the only isotropic nonmetricity components. + 2 (6) The model determined by Eqs. (2) and (5) permits a fully rel- m ζ2 S000 , (8) ativistic description of all dominant nonmetricity effects on the where we have implemented the isotropic limit. The neutron ro- propagation of both neutrons and antineutrons in the present tary power is amenable to high-precision experimental studies and context. Since our current goal is an analysis of the spin mo- can therefore be employed to measure or constrain the combina- tion of slow neutrons, we may disregard all antineutron physics, tion of nonmetricity components appearing on the right-hand side and focus entirely on the 2 × 2nonrelativistic neutron Hamilto- of Eq. (8). = + + nian h h0 δh δhs resulting from our model Lagrangian (5). The experiment described in detail below measured the neu- Here, h0 is the ordinary nonrelativistic piece. The spin-independent tron rotary power to be nonmetricity contribution δh is irrelevant for this work. The spin- dφPV − dependent correction δhs resulting from Eq. (5) can be gleaned =+1.7 ± 9.1(stat.) ± 1.4(sys) × 10 7 rad/m(9) from previously established SME studies [20]. The result for both dL isotropic as well as anisotropic contribution reads at the 1-σ level. Conversion to natural units together with Eq. (8)      yields the following nonmetricity measurement: = (4) − (5) + (4) − (5) j     δhs ζ2 m ζ9 (N1) j ζ4 m ζ10 (N2) j σ (4) (5) (4) (5) (6) 2 ζ − m ζ (N ) + 2 ζ − m ζ (N ) + m2 ζ S      2 9 1 0 4 10 2 0 2 000 p · σ (4) (5) (4) (5) −23 + ζ − m ζ (N1)0 + ζ − m ζ (N2)0 = (3.4 ± 18.2) × 10 GeV . (10) 2 9 4 10 m 868 R. Lehnert et al. / Physics Letters B 772 (2017) 865–869

We interpret this result as the 2-σ constraint Additional nonmetricity components may become experimen-         tally accessible with a set-up in which both the slow-neutron  (4) − (5) + (4) − (5) + 2 (6)  beam as well as the nuclear target are polarized: the aligned tar- 2 ζ2 m ζ9 (N1)0 2 ζ4 m ζ10 (N2)0 m ζ2 S000 get spins would coherently generate large-scale anisotropic com- −22 < 3.6 × 10 GeV . (11) ponents of Nαβγ , which were disregarded in our above analysis. High-sensitivity studies of this type have received considerable at- Disregarding the possibility of extremely fine-tuned cancellations tention for quite some time [36]. The neutron–nucleus scattering between the various nonmetricity couplings in the constraint (11), amplitude exhibits a significant polarization dependence, an effect we can estimate the following individual bounds: known as nuclear pseudomagnetic precession [37]: the neutron’s spin precesses about the nuclear polarization vector as the neutron | (4) | −22 | (4) | −22 ζ2 (N1)0 < 10 GeV , ζ4 (N2)0 < 10 GeV , traverses the polarized medium. In the past, this method has been (5) −22 (5) −22 employed to determine the spin dependence of neutron–nucleus |ζ (N1)0| < 10 , |ζ (N2)0| < 10 , 9 10 scattering cross sections for a number of nuclei [38]. However, | (6) | −22 −1 ζ2 S000 < 10 GeV . (12) the nuclear-pseudomagnetism spin-precession contributions from the strong neutron–nucleus interaction to such a measurement are The above limits represent the primary result of this work. To our substantial and currently evade theoretical treatment from first (6) knowledge, they provide the first measurement of ζ2 S000 as well principles. It is therefore expected that the experimental reach re- as the first measurement of any nonmetricity component inside garding in-matter anisotropic Nαβγ components would be more matter. modest than that in this study. The measurement (9) performed at the NG-6 slow-neutron We finally mention that a high-precision transmission-asymme- beamline at NIST’s Center for Neutron Research has already ap- try measurement utilizing transversely polarized 5.9 MeV neutrons peared in the literature [21]. Neutrons with transverse spin polar- was performed in a nuclear spin-aligned target of holmium [39]. 4 ization traversed 1meter of liquid He that was kept at a tem- This experiment explored the presence of P-invariant but T- perature of 4K in a magnetically shielded cryogenic target. The violating interactions of the neutron. The measurement yielded σP −6 neutron beam’s energy distribution was well approximated by a A5 = =+8.6 ± 7.7(stat. + sys.) × 10 . Here, A5 denotes the σ0 Maxwellian with a maximum close to 3meV. Paralleling the usual transmission asymmetry for neutrons polarized parallel and an- light-optics set-up of a crossed polarizer–analyzer pair, the experi- tiparallel to the normal of the plane spanned by the neutron ment searched for a nonzero rotation in the neutrons’ polarization. momentum and the spin polarization of the holmium target. An Further details of this measurement can be found in Refs. [22–27]. open question is whether or not polarized nuclear matter gener- The result quoted in the above Eq. (9) represents the upper limit ates an effective Nαβγ that differs from that of unpolarized nuclear on the parity-odd neutron-spin rotation angle per unit length in matter, and how such a difference would manifest itself in this liquid helium at 4K extracted from the measured data. experiment. That said, the neutron energy in this measurement re- The usual SM incorporates known parity-violating physics that mains nonrelativistic, so our above methodology should continue can also lead to neutron spin rotation, for instance via interactions to be applicable. with electrons or nucleons. In fact, this phenomenon has been measured in heavy nuclei [28–30]. A convincing interpretation of 4. Summary the above nonmetricity constraint therefore requires a discussion of this SM background. From a theoretical perspective, parity vi- In this work, we have considered the possibility of nontrivial olation in neutron–electron physics in the SM is well understood. nonmetricity in nature. We have argued that in this context an ef- In particular, it is suppressed relative to the parity-odd neutron– fective nonmetricity field could be generated inside a liquid 4He − 2 ≈ nucleon interaction by the weak charge (1 4 sin θW ) 0.1. The target. We have shown that the spin of nonrelativistic neutrons neutron–nucleon parity violation, on the other hand, is induced traversing such a target would then precess. This prediction, to- by quark–quark weak interactions. This system also involves the gether with existing data on neutron spin rotation in liquid 4He, strong-coupling limit of QCD, which still evades solid theoretical implies the primary result of this work, namely the bound (11). tractability. Nevertheless, nucleon–nucleon weak-interaction ampli- To our knowledge, this is the first experimental limit on in-matter tudes have been argued to be six to seven orders of magnitude nonmetricity. below strong-interaction amplitudes at neutron energies relevant We have further concluded that it would be difficult to improve for our present purposes [31]. Although reliant on phenomenolog- our bound via higher-precision spin-rotation data due to the con- ical input in the form of nuclear parity-violation data folded into ventional SM background arising from quark–quark weak interac- −7 a specific model, the value dφPV/dL =−6.5 ± 2.2 × 10 rad/m tions. However, other atomic and nuclear parity-violation tests may for the SM spin rotation in the 4He–neutron system is regarded as have the potential to yield complementary limits on nonmetricity the most decent theoretical estimate [32]. Our experimental up- interactions of neutrons and electrons. Moreover, polarized slow- per limit on nonmetricity (11) is larger than this SM-background neutron transmission measurements through polarized nuclear tar- estimate. For this reason, we disregard the remote possibility of a gets could be studied with the approach presented in this work cancellation between SM and nonmetricity contributions to neu- and may give bounds on additional in-matter Nαβγ components. tron spin rotation. We encourage other researchers to perform further nonmetricity To determine additional limits on in-matter nonmetricity, one searches using the general framework employed in this study with could also consider using data from other high-precision parity- the aim to turn nonmetricity tests into a more quantitative exper- violation experiments. One example in the context of neutrons are imental science. measurements of parity-breaking effects in atoms that are affected by the nuclear anapole moment and arise from parity-odd interac- Acknowledgements tions between nucleons [33,34]. An idea for extracting nonmetric- ity constraints involving electrons could, for example, be based on This work was supported by the DOE under grant No. DE- the consistency between the theoretical SM result and the experi- SC0010120, by the NSF under grant No. PHY-1614545, by the IU mental value of the weak charge of the 133Cs atom [35]. Center for Spacetime Symmetries, by the IU Collaborative Research R. Lehnert et al. / Physics Letters B 772 (2017) 865–869 869 and Creative Activity Fund of the Office of the Vice President for H. Yan et al., Phys. Rev. Lett. 115 (2015) 182001; Research, by the IU Collaborative Research Grants program, by the A.N. Ivanov, W.M. Snow, Phys. Lett. B 764 (2017) 186. 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