Constraining spacetime nonmetricity with neutron spin rotation in liquid 4He

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Physics Letters B

www.elsevier.com/locate/physletb

Constraining spacetime nonmetricity with neutron spin rotation in liquid ⁴ He

Ralf Lehnert

^a,b,∗

, W.M. Snow

^a,c,d

, Zhi Xiao

^a,e

, Rui Xu

^c

aIndianaUniversityCenterforSpacetimeSymmetries,Bloomington,IN47405,USA bLeibnizUniversitätHannover,Welfengarten1,30167Hannover,Germany cPhysicsDepartment,IndianaUniversity,Bloomington,IN47405,USA

dCenterforExplorationofEnergyandMatter,IndianaUniversity,Bloomington,IN47408,USA eDepartmentofMathematicsandPhysics,NorthChinaElectricPowerUniversity,Beijing102206,China

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received3June2017

Receivedinrevisedform19July2017 Accepted27July2017

Availableonline1August2017 Editor:A.Ringwald

Generalspacetimenonmetricitycoupledtoneutronsisstudied.Inthiscontext,itisshownthatcertain nonmetricity components can generatea rotationofthe neutron’s spin.Available data on thiseffect obtainedfromslow-neutron propagationinliquidheliumare usedtoconstrain isotropicnonmetricity componentsatthelevelof10⁻²²GeV.Theseresultsrepresentthefirstlimitonthenonmetricityζ₂⁽⁶⁾S000 parameteraswellasthefirstmeasurementofnonmetricityinsidematter.

©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Theideathatspacetimegeometryrepresentsadynamicalphys- ical entity has been remarkably successful in the description of classicalgravitationalphenomena.Forexample,GeneralRelativity, whichisbasedonRiemanniangeometry,hasrecentlypassedafur- therexperimentaltest:thetheorypredictsgravitationalwaves,and thesehaveindeedbeenobservedbytheLIGOScientificCollabora- tionandtheVirgoCollaboration[1].

At the same time, a number of observational as well asthe- oretical issues motivate the construction and study of alterna- tivegravity theories.Mostoftheseeffortsrecognizetheelegance and success of a geometric underpinning for gravitational phe- nomenaand thereforeretain this featurein model building. One popular approach in this context, known as metric-affine grav- ity[2], employs an underlying geometry more general than that of a Riemannian manifold. The basic idea behind this approach canbesummarizedasrelaxingthemetric-compatibilitycondition Dαg_βγ=^{0 andthesymmetrycondition ontheconnectioncoeffi-} cients^αβγ−^α_γβ=^{0.Ingeneral,thisideaintroducestwotensor} fields

N_αβγ

≡ −

^Dαg_βγ

,

T^α_βγ

≡

^α_βγ

−

^α_γβ

,

(1) relative to the Riemannian case known asnonmetricity and tor- sion,respectively.

*

Correspondingauthor.

E-mailaddress:ralehner@indiana.edu(R. Lehnert).

Thespecializedsituationinwhichthenonmetricitytensorvan- ishesNαβγ=^{0 andonlytorsionisnonzerorepresentsthewidely} known Einstein–Cartan theory [3]. In that context, torsion has been the subject of various investigations during the last four decades [4]. Considering the question of the presence of torsion in nature as an experimental one has spawned numerous phe- nomenologicalstudiesoftorsion[5–15]yieldingboundsonvarious torsioncouplings.

An analogous phenomenological investigation of nonmetricity hasbeeninstigatedlastyear[16].Parallelingthetorsioncase,that analysistreatsthequestionregardingthepresenceofnonmetricity asanexperimentalone,andthenonmetricityfield Nαβγ istaken asalarge-scalebackgroundextendingacrossthesolarsystem.The particularphysicalsituationconsidered inRef. [16]lendsitselfto an effective-field-theory description in which Nαβγ represents a prescribed external field selecting preferredspacetime directions.

Thus, such a set-up embodies inessence a Lorentz-violatingsce- nario amenable to theoretical treatment via the Standard-Model Extension(SME)framework[17].Forexample,siderealandannual variations of physical observables resulting from the motion of an Earth-based laboratory throughthis solar-systemnonmetricity backgroundrepresentaclassofcharacteristicexperimentalsignals inthatcontext[18].

Thepresentworkemploysasimilarideatoobtainfurther,com- plementary constraints on nonmetricity. The specific set-up we have in mind consistsof liquid ⁴He asthe nonmetricity source.

Polarizedneutronsgeneratedattheslow-neutronbeamlineatthe National Institute of Standards andTechnology (NIST) Center for http://dx.doi.org/10.1016/j.physletb.2017.07.059

0370-2693/©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

NeutronResearchtraversetheheliumandserveasthenonmetric- ityprobe. It is apparent that our set-up involves an Earth-based nonmetricity probe. Thus, the key difference between our study and that in Ref. [16] is that we examine the situation of non- metricity sourced locally in a terrestrial laboratory by the ⁴He.

This implies that the presumed nonmetricity in our case is co- moving withthe laboratory, and thus the neutron probe, which precludescertainexperimentalsignatures,suchassiderealandan- nualvariations.Instead,weutilizethepredictionpresentedbelow thatcertaincomponentsof Nαβγ leadtoneutronspin rotationin thissystem.

The outline of thispaper is asfollows. Section 2 reviewsthe basicideas behindtheeffective-field-theorydescriptionofaback- ground Nαβγ in flat Minkowski space and derives the resulting spin motion for nonrelativisticneutrons. Thiseffect provides the basis forour limitson nonmetricity. The details ofthe measure- ment ofneutronspin rotation in liquid⁴He including theexper- imental set-up are discussed in Sec. 3. A brief summary is con- tainedinSec.4.Throughout,weadoptnaturalunitsc= ¯^h=^1.Our conventionsforthemetricsignatureandtheLevi-Civitasymbolare

η

^μν=diag(+,−,−,−)and

⁰¹²³= +1,respectively.

2. Theory

Ouranalysisisbasedontheapproachtononmetricitycouplings taken in Ref. [16], so we begin with a brief review of that ap- proach. The basic idea is to follow the usual reasoning that the construction of an effective Lagrangian should include all terms compatible with the symmetries of the model. In the present context, possible couplings between the background nonmetric- ity Nαβγ andthe polarized-neutron probe need to be classified.

Sinceweareinterestedinalow-energyexperiment,wemaydisre- gardtheneutron’sinternalstructureandmodelitasapointDirac fermionwithfreeLagrangianL0=¹₂ψ

γ

^μi∂^↔_μψ−^mψψ,wherem denotes the neutron mass. Conventional gravitational effects are negligible,sothat theflat-spacetimeMinkowskilimit gμν→

η

^μν

sufficesforourpresentpurposes.

The next step isto enumeratepossible couplings ofψ tothe background nonmetricity Nαβγ. This yields a hierarchyof possi- bleLagrangiantermsL⁽Nⁿ⁾ labeledbythemassdimensionnofthe correspondingfieldoperator:

LN

=

L0

+

L⁽_N⁴⁾

+

L⁽_N⁵⁾

+

L⁽_N⁶⁾

+ . . . .

(2) Forthe experimental set-up we havein mind,nonmetricity cou- plingsaffectingthepropagationofneutronsarethemostrelevant ones. Moreover, Nαβγ must be small on observational grounds.

Wetherefore focusoncontributions toL⁽_N^{n) that arequadratic in} ψ andlinear in Nαβγ. General arguments in effective field the- orysuggestthatLagrangiantermsoflowermassdimensionnmay bemoredominant.Capturingtheleadingeffectsofallnonmetric- itycomponentsthen requiresinclusionofLagrangiantermsupto massdimensionn=^6[16].

Theconstructionoftheexplicitformofeachindividual contri- butionL⁽_N^{n) ismosteasilyachievedby}decomposing Nαβγ intoits Lorentz-irreducible pieces. These are givenby two vectors (N1)_μ and(N2)_μ,a totallysymmetricrank-threetensorSμαβ,andarank- threetensorMμαβ withmixedsymmetry[16]:

(

N1

)

_μ

≡ − η

^αβN_μαβ

, (

N₂

)

_μ

≡ − η

^αβN_αμβ

,

S_μαβ

≡

¹₃

N_μαβ

+

^Nαβμ

+

^Nβμα

+

₁₈¹

(

N1

)

_μ

η

_αβ

₊ (

N1

)

_α

η

_βμ

₊ (

N1

)

_β

η

_μα

+

¹₉

(

N2

)

_μ

η

_αβ

+ (

N2

)

_α

η

βμ

+ (

N2

)

β

η

_μα

,

M_μαβ

≡

¹₃

2N_μαβ

−

N_αβμ

−

Nβμα

+

¹₉

2

(

N1

)

_μ

η

_αβ

− (

N1

)

_α

η

βμ

− (

N1

)

β

η

_αμ

−

¹₉

2

(

N2

)

_μ

η

_αβ

− (

N2

)

_α

η

βμ

− (

N2

)

β

η

_αμ

.

(3) Withthesepieces,thenonmetricity tensorcanbereconstructedas follows[16]:

N_μαβ

=

₁₈¹

−

5

(

N1

)

_μ

η

_αβ

+ (

N1

)

_α

η

_βμ

+ (

N1

)

_β

η

_μα

+

2

(

N2

)

_μ

η

_αβ

−

4

(

N2

)

_α

η

βμ

−

4

(

N2

)

β

η

_μα

+

^Sμαβ

+

^Mμαβ

.

(4) The signchangesinEqs. (3),(4),andsome subsequentequations relativetothecorrespondingequationsinRef.[16]ariseduetodif- feringconventionsforthemetricsignatureandforthesignofthe Levi-Civita symbol.Wealsoremarkthat althoughEqs.(3)and(4) employ a notation similarto that fortheirreducible components oftorsion Tα

βγ [14],thenonmetricity and torsionpiecesare un- related.

With this decomposition, the following Lagrangian contribu- tionscanbeconstructed[16]:

L⁽_N⁴⁾

= ζ

₁⁽⁴⁾

(

N₁

)

μ

ψ γ

^μ

ψ + ζ

₂⁽⁴⁾

(

N₁

)

μ

ψ γ

5

γ

^μ

ψ + ζ

₃⁽⁴⁾

(

N₂

)

μ

ψ γ

^μ

ψ + ζ

₄⁽⁴⁾

(

N₂

)

μ

ψ γ

5

γ

^μ

ψ ,

L⁽_N⁵⁾

= −

¹₂i

ζ

₁⁽⁵⁾

(

N1

)

^μ

ψ ∂

^↔_μ

ψ −

¹₂

ζ

₂⁽⁵⁾

(

N1

)

^μ

ψ γ

5

∂

↔_μ

ψ

−

¹₂ⁱ

ζ

₃⁽⁵⁾

(

N₂

)

^μ

ψ ∂

^↔μ

ψ −

¹₂

ζ

₄⁽⁵⁾

(

N₂

)

^μ

ψ γ

5

∂

↔μ

ψ

−

¹₄i

ζ

₅⁽⁵⁾M_μν^ρ

ψ σ

^μν

∂

^↔_ρ

ψ +

¹₈ⁱ

ζ

₆⁽⁵⁾ κλμνM^κλρ

ψ σ

^μν

∂

^↔ρ

ψ

+

¹₂ⁱ

ζ

₇⁽⁵⁾

(

N1

)

_μ

ψ σ

^μν

∂

^↔_ν

ψ +

¹₂ⁱ

ζ

₈⁽⁵⁾

(

N2

)

_μ

ψ σ

^μν

∂

^↔_ν

ψ

−

¹₄ⁱ

ζ

₉^{(5) λμνρ}

(

N₁

)

λ

ψ σ

μν

∂

↔ρ

ψ

−

¹₄ⁱ

ζ

₁₀^{(5) λμνρ}

(

N2

)

_λ

ψ σ

_μν

∂

^↔_ρ

ψ ,

L⁽_N⁶⁾

⊃ −

¹₄

ζ

₁⁽⁶⁾S_λ^μν

ψ γ

^λ

∂

μ

∂

ν

ψ +

^h

.

c

.

−

¹₄

ζ

₂⁽⁶⁾Sλμν

ψ γ

5

γ

^λ

∂

_μ

∂

_ν

ψ +

h

.

c

.

(5) Here,the real-valuedcouplingsζ_l⁽ⁿ⁾are takenasfreeparameters;

they can in principle be fixed by specifying a definite underly- ingnonmetricitymodel.Forthemass-dimensionsixtermL⁽N⁶⁾,we have only listed those contributions that contain the Sμαβ irre- ducible piece; all other components of Nαβγ are alreadypresent inthetermsL⁽_N⁴⁾orL⁽_N⁵⁾oflowermassdimension.

Equations (2), (3), and (5) determine the low-energy neutron effective Lagrangian in the presence of general background non- metricityrelevantfortheexperimentalsituationwehaveinmind.

We note, however,that theterms (5)wouldgenerallybe viewed as part of a more complete Lagrangian L⊃LN that also treats Nαβγ as a dynamical variable. The nonmetricity field equations thencontain∂L/∂^Nαβγ,andthusneutronsourceterms.Thisidea provides the justification for taking the ⁴He nucleus as a non- metricitysource intheexperimental set-up discussedbelow. The protons and electrons of the ⁴He atom may produce additional nonmetricity contributions if theseparticles exhibit nonmetricity couplingsanalogoustothoseinEq.(5).Inwhatfollows,wemake no assumptions regarding the dynamics of Nαβγ or additional nonmetricity–matter couplings; we simply presume that the⁴He generatessomenonzerononmetricity.

A modelrefinementcan beachieved byfocusing onthe lead- ingcontributiontoNαβγ.Notethat Nαβγ=^Nαβγ(x)mustexhibit

anontrivialspacetimedependencedeterminedbytheinteratomic distance and the velocity of the ⁴He atoms. However, the ran- domnatureofthesetwoquantitiessuggeststhattheleadingnon- metricityeffects are actually governed by the spacetime average ^Nαβγ(x)^{.Forthisreason,wemaytakeN}αβγ=^{const.inwhatfol-} lows.Thenonmetricitycontributions(5)thenformasubsetofthe flat-spaceSMELagrangian,a factthatpermitsustoemploythefull repertoireoftheoreticaltoolsdevelopedfortheSMEframework.

One such SME result relevant for the present situation con- cerns theobservability of constant backgroundfields [17,19]. For example,itis knownthat contributions associated withthecou- plingsζ₁⁽⁴⁾,ζ₃⁽⁴⁾,ζ₁⁽⁵⁾,ζ₂⁽⁵⁾,ζ₃⁽⁵⁾,ζ₄⁽⁵⁾,ζ₇⁽⁵⁾,andζ₈⁽⁵⁾canberemoved fromthe Lagrangian—atleastatlinearorder—viajudiciouslycho- senfield redefinitions.Wemaythereforedisregardthesetermsin whatfollows.Theirmeasurementwouldrequiresituationsinvolv- ingnonconstantNαβγ,thepresenceofgravity,ortheconsideration ofhigher-ordereffects.

Anadditionalsimplificationarisesfromtheisotropyoftheliq- uid helium.The ⁴He ground state hasspin zero, so anisotropies wouldhavetobetiedtoexcited statesof⁴Heorarrangementsof theheliumatomsinvolvingpreferreddirections.However,theab- sence of polarization and the aforementioned random nature of both position and velocity of individual ⁴He particles precludes sizeable, large-scale anisotropies. The leading background non- metricity contributions generated by the liquid-helium bath can thereforealsobetakenasisotropicinthehelium’scenter-of-mass frame.Itfollowsthatthepresentexperimentalset-upisonlysen- sitivetotherotationallyinvariantpiecesofNαβγ.

Touncover theisotropiccontentof Nαβγ,we mayproceed by inspecting its irreducible pieces (3).Clearly, components without spatialindicesarerotationsymmetric:(N₁)0,(N₂)0,andS₀₀₀.Note thatMαβγ obeysthecyclicproperty

M_αβγ

+

^Mβγ α

+

^Mγ αβ

=

⁰

,

(6) which implies M000=^{0. Further isotropic components in S and} M with spatial indices must have spatial-index structure δjk or jkl, where Latin indices run from 1 to 3. Since both S and M aresymmetricintheirlasttwoindices,theycannotcontainpieces of _jkl. Thisonly leavescontributions involving δ_jk.But thesedo notyieldindependentisotropic contributionsbecauseboth S and Maretraceless.Toseethis,considerasanexampleapieceofthe form S_0jk=^sδjk,where s isthe isotropy parameter in question.

ButSistraceless,sothatwehave0=^S0αβ

η

^αβ=^S000−^S0jkδjk= S000−^sδ_jkδ_jk. It follows that 3s=^S000 does not represent an additionalindependent isotropic contribution to S. An analogous reasoningappliesto M,sothat (N1)0,(N2)0,andS000areindeed theonlyisotropicnonmetricitycomponents.

Themodeldetermined by Eqs.(2)and(5)permitsa fullyrel- ativistic description of all dominant nonmetricity effects on the propagation of both neutrons and antineutrons in the present context. Since our current goal is an analysis of the spin mo- tionof slow neutrons, we maydisregard all antineutron physics, andfocus entirely on the 2×2 nonrelativistic neutron Hamilto- nian h=^h0+δh+δhs resulting from our modelLagrangian (5).

Here,h0istheordinarynonrelativisticpiece.Thespin-independent nonmetricity contributionδhisirrelevant forthiswork.Thespin- dependent correction δhs resulting from Eq. (5) can be gleaned frompreviouslyestablished SME studies[20].The resultforboth isotropicaswellasanisotropiccontributionreads

δ

h_s

=

ζ

₂⁽⁴⁾

−

^m

ζ

₉⁽⁵⁾

(

N₁

)

_j

+

ζ

₄⁽⁴⁾

−

^m

ζ

₁₀⁽⁵⁾

(

N₂

)

_j

σ

^j

+

ζ

₂⁽⁴⁾

−

m

ζ

₉⁽⁵⁾

(

N1

)

0

+

ζ

₄⁽⁴⁾

−

m

ζ

₁₀⁽⁵⁾

(

N2

)

0

_p

· σ

m

+

¹₂

ζ

₅⁽⁵⁾M

˜

_jαβ

+

³₂

ζ

₆⁽⁵⁾M_jαβ

+

^m

ζ

₂⁽⁶⁾S_jαβ

_p^α_p^β

σ

^j

m

+

¹₂

ζ

₂⁽⁶⁾S_0αβ p^αp^βp

· σ

m

.

(7)

This expression contains the leading contribution in the nonrel- ativistic order | ^p|/^{m for each} nonmetricity component. In the aboveequation,wehavesetM˜αβγ≡ αβμνMμνγ.Moreover,pμ= (p⁰,p)=(p⁰,p^j) denotes the neutron’s 4-momentum, and

σ

^j

aretheusual Paulimatrices.Notethatnonmetricityeffectscorre- spondingto ζ₁⁽⁶⁾ onlyproducespin-independent effects.Theyare thereforeabsent fromδhs andcannot be determinedby observa- tionsofneutronspinrotation.

3. Experimentalanalysis

To extract experimental signatures resulting from the non- metricity correction (7), we analyze the aforementioned experi- mental situation, namely spin motion of a neutron as it passes through liquid ⁴He. As argued above, our Lagrangian (5) implies that neutrons, and hence ⁴He nuclei, can generate nonmetricity.

The injected neutron beamwould then be affected by thisnon- metricitybackground.Moreover,our“in-matter”approachpermits us to search for short-ranged or non-propagating nonmetricity.

In particular,thisencompasses situationsanalogousto minimally coupled torsion, where the torsion tensor vanishes outside the spin-density source [4]. Such an approach rests on the premise that theprobepenetrates thematter andthat theeffects ofcon- ventional Standard-Model(SM) physics are minimized. The ⁴He–

neutronsystemappearstobeidealinthisrespectfortworeasons.

First, the neutron mean free path inside liquid ⁴He is relatively long allowingforthe accumulationofthepredictedspin-rotation effect.Thisis duetothesmallelastic andtheessentially vanish- inginelasticcrosssectionsaswellasrapidlydecreasingneutron–

phonon scatteringas T →^{0. Second,} contamination of the non- metricityspinrotationbyordinarySMphysicscanbeexcludedon thegrounds that theseconventionaleffects liebelow thecurrent detectionsensitivity.Thislatterfactisexplainedinmoredetailbe- low.

The rotation of the spin of a transversely polarized slow- neutronbeamiscalledneutronopticalactivity.Itisquantifiedby the rotary power dφP V/dL definedas the rotation angle φP V of the neutron spin aboutthe neutron momentum p per traversed distance L.The nonmetricitycorrection(7)leads tothe following expressionfortherotarypower:

d

φ

P V

dL

=

2

ζ

₂⁽⁴⁾

−

m

ζ

₉⁽⁵⁾

(

N1

)

0

+

2

ζ

₄⁽⁴⁾

−

m

ζ

₁₀⁽⁵⁾

(

N2

)

0

+

^m2

ζ

₂⁽⁶⁾S₀₀₀

,

(8) wherewe have implementedthe isotropiclimit. The neutronro- tarypowerisamenabletohigh-precisionexperimentalstudiesand cantherefore beemployed tomeasureorconstrain thecombina- tionofnonmetricitycomponentsappearingontheright-handside ofEq.(8).

The experiment described in detail below measured the neu- tronrotarypowertobe

d

φ

P V

dL

= +

¹

.

7

±

⁹

.

1(stat.)

±

¹

.

4(sys)

×

^10−7rad

/

m (9) atthe1-

σ

level.ConversiontonaturalunitstogetherwithEq.(8) yieldsthefollowingnonmetricitymeasurement:

2

ζ

₂⁽⁴⁾

−

m

ζ

₉⁽⁵⁾

(

N1

)

0

+

2

ζ

₄⁽⁴⁾

−

m

ζ

₁₀⁽⁵⁾

(

N2

)

0

+

m²

ζ

₂⁽⁶⁾S000

= (

3

.

4

±

¹⁸

.

2

) ×

^10−23GeV

.

(10)

Weinterpretthisresultasthe2-

σ

constraint

²

ζ

₂⁽⁴⁾

−

^m

ζ

₉⁽⁵⁾

(

N₁

)

₀

+

²

ζ

₄⁽⁴⁾

−

^m

ζ

₁₀⁽⁵⁾

(

N₂

)

₀

+

^m2

ζ

₂⁽⁶⁾S₀₀₀

<

3

.

6

×

^10−22GeV

.

(11)

Disregarding the possibilityof extremelyfine-tuned cancellations betweenthevariousnonmetricitycouplingsintheconstraint(11), wecanestimatethefollowingindividualbounds:

| ζ

₂⁽⁴⁾

(

N1

)

0

| <

10⁻²²GeV

, | ζ

₄⁽⁴⁾

(

N2

)

0

| <

10⁻²²GeV

,

| ζ

₉⁽⁵⁾

(

N1

)

0

| <

10⁻²²

, | ζ

₁₀⁽⁵⁾

(

N2

)

0

| <

10⁻²²

,

| ζ

₂⁽⁶⁾S000

| <

10⁻²²GeV⁻¹

.

(12) Theabovelimitsrepresenttheprimaryresultofthiswork.Toour knowledge,theyprovidethefirstmeasurementofζ₂⁽⁶⁾S000aswell as the first measurement of any nonmetricity component inside matter.

The measurement (9) performed at the NG-6 slow-neutron beamline at NIST’s Center for Neutron Research has already ap- pearedintheliterature[21].Neutronswithtransversespin polar- ization traversed 1 meter of liquid⁴He that was kept at a tem- perature of 4 K in a magnetically shielded cryogenic target. The neutron beam’s energy distribution was well approximated by a Maxwellianwithamaximumcloseto3 meV.Parallelingtheusual light-opticsset-upofacrossedpolarizer–analyzerpair,theexperi- mentsearchedforanonzerorotationintheneutrons’polarization.

FurtherdetailsofthismeasurementcanbefoundinRefs.[22–27].

Theresultquoted intheabove Eq.(9)representstheupperlimit on the parity-oddneutron-spin rotation angle per unit length in liquidheliumat4 Kextractedfromthemeasureddata.

Theusual SMincorporatesknownparity-violating physicsthat canalsoleadtoneutronspinrotation,forinstanceviainteractions with electrons or nucleons. In fact, this phenomenon has been measured inheavy nuclei[28–30].A convincing interpretation of the abovenonmetricity constrainttherefore requiresa discussion ofthis SM background. From a theoretical perspective, parityvi- olationinneutron–electronphysicsintheSM iswell understood.

Inparticular, it issuppressed relative to theparity-odd neutron–

nucleoninteractionbytheweak charge(1−^4sin2θW)≈⁰.1.The neutron–nucleon parity violation, on the other hand, is induced by quark–quark weak interactions. This system also involves the strong-couplinglimit ofQCD, which still evades solid theoretical tractability.Nevertheless,nucleon–nucleonweak-interactionampli- tudes have been argued to be six to seven orders of magnitude belowstrong-interaction amplitudes atneutron energies relevant forourpresentpurposes[31].Althoughreliantonphenomenolog- icalinput inthe formofnuclearparity-violation datafolded into a specific model, the value dφP V/dL= −⁶.5±².2×^{10−7 rad/m} fortheSMspinrotationinthe⁴He–neutronsystemisregardedas the mostdecent theoretical estimate [32]. Our experimental up- per limiton nonmetricity(11)is larger thanthisSM-background estimate.Forthisreason, wedisregardtheremote possibilityofa cancellationbetween SM and nonmetricity contributions to neu- tronspinrotation.

Todetermine additional limitson in-matternonmetricity, one could also consider using data from other high-precision parity- violationexperiments.Oneexampleinthecontextofneutronsare measurementsofparity-breakingeffectsinatomsthatareaffected bythenuclearanapolemomentandarisefromparity-oddinterac- tionsbetweennucleons [33,34].Anidea forextractingnonmetric- ityconstraintsinvolvingelectronscould,forexample,bebasedon theconsistencybetweenthetheoreticalSM resultandtheexperi- mentalvalueoftheweakchargeofthe¹³³Csatom[35].

Additional nonmetricity components maybecome experimen- tally accessible with a set-up in which both the slow-neutron beamaswellasthenucleartarget arepolarized:thealignedtar- get spins wouldcoherently generatelarge-scale anisotropic com- ponents of Nαβγ, which were disregarded in ourabove analysis.

High-sensitivitystudiesofthistypehavereceivedconsiderableat- tention forquite some time [36]. Theneutron–nucleus scattering amplitude exhibitsasignificantpolarizationdependence,aneffect known asnuclear pseudomagnetic precession [37]: the neutron’s spinprecessesaboutthenuclearpolarizationvectorastheneutron traversesthepolarizedmedium.Inthepast,thismethodhasbeen employed to determine the spin dependence ofneutron–nucleus scattering cross sections for a number of nuclei [38]. However, the nuclear-pseudomagnetism spin-precession contributions from thestrongneutron–nucleusinteractiontosuchameasurementare substantial and currently evade theoretical treatment from first principles.Itisthereforeexpectedthattheexperimentalreachre- garding in-matter anisotropic Nαβγ components would be more modestthanthatinthisstudy.

Wefinallymentionthatahigh-precisiontransmission-asymme- trymeasurementutilizingtransverselypolarized5.9 MeVneutrons was performed ina nuclear spin-alignedtarget ofholmium [39].

This experiment explored the presence of P-invariant but T- violating interactions of the neutron. The measurement yielded A₅= ^σ_σ^P₀ = +⁸.6±⁷.7(stat.+^sys.)×^{10−6. Here, A}5 denotes the transmission asymmetry for neutrons polarized parallel and an- tiparallel to the normal of the plane spanned by the neutron momentum and the spin polarization of the holmium target. An open question iswhetheror not polarizednuclear mattergener- atesaneffectiveNαβγ thatdiffersfromthatofunpolarizednuclear matter, and how such a difference would manifest itself in this experiment.Thatsaid,theneutronenergyinthismeasurementre- mains nonrelativistic, soour above methodologyshould continue tobeapplicable.

4. Summary

In this work, we have considered the possibility of nontrivial nonmetricityinnature.Wehavearguedthatinthiscontextanef- fective nonmetricityfield could be generated inside a liquid⁴He target. We have shown that the spin of nonrelativistic neutrons traversing such a target would then precess. This prediction, to- gether with existingdata on neutronspin rotation inliquid ⁴He, implies the primary result of thiswork, namely the bound (11).

Toourknowledge,thisisthefirstexperimentallimitonin-matter nonmetricity.

Wehavefurtherconcludedthatitwouldbedifficulttoimprove ourbound viahigher-precisionspin-rotationdataduetothecon- ventionalSM backgroundarisingfromquark–quarkweak interac- tions.However,otheratomicandnuclearparity-violationtestsmay havethepotential toyieldcomplementarylimitsonnonmetricity interactions ofneutrons andelectrons. Moreover, polarizedslow- neutrontransmissionmeasurementsthroughpolarizednucleartar- gets could be studied with the approach presented in this work andmay give boundsonadditional in-matter Nαβγ components.

We encourageother researchers to perform further nonmetricity searchesusingthegeneralframeworkemployedinthisstudywith theaimtoturnnonmetricitytestsintoamorequantitativeexper- imentalscience.

Acknowledgements

This work was supported by the DOE under grant No. DE- SC0010120, by theNSF under grant No. PHY-1614545, by the IU CenterforSpacetimeSymmetries,bytheIUCollaborativeResearch

andCreativeActivity Fund ofthe Officeofthe VicePresidentfor Research,bytheIUCollaborativeResearchGrantsprogram,bythe NationalScience Foundation ofChina undergrant No. 11605056, andbytheChinese ScholarshipCouncil. RLacknowledgessupport fromthe Alexandervon HumboldtFoundation. ZXis gratefulfor thehospitality of theIUCSS, where mostof thiswork was com- pleted.

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