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Constraining Spacetime Nonmetricity with Neutron Spin Rotation in Liquid

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Constraining Spacetime Nonmetricity with Neutron Spin Rotation in Liquid Physics Letters B 772 (2017) 865–869 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 Mathematics 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 tensor 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, General Relativity, decades [4]. Considering the question of the presence of torsion which is based on Riemannian geometry, 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 manifold. 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 D g = 0 and the symmetry condition on the connection coeffi- α βγ 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(N2)μηαβ − 4(N2)αηβμ − 4(N2)β ημα The outline of this paper is as follows. Section 2 reviews the 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- basis 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 ↔ ↔ (5) (5) compatible with the symmetries of the model. In the present − 1 μ − 1 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 ↔ (5) suffices for our present purposes. − 1 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) (6) ble Lagrangian terms L labeled by the mass dimension n of the − 1 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 Sμαβ irre- plings affecting the propagation of neutrons are the most relevant ducible piece; all other components of Nαβγ are already present ones.
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