Constraining interactions mediated by axion-like particles with ultracold neutrons

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Constraining

interactions

mediated

by

axion-like

particles

with

ultracold

neutrons

S. Afach

a

,

b

,

c

,

G. Ban

d

,

G. Bison

b

,

K. Bodek

e

,

M. Burghoff

f

,

M. Daum

b

,

M. Fertl

a

,

b

,

1

,

B. Franke

a

,

b

,∗,

2

,

Z.D. Gruji ´c

g

,

V. Hélaine

b

,

d

,

M. Kasprzak

g

,

Y. Kermaïdic

h

,

K. Kirch

a

,

b

,

P. Knowles

g

,

3

,

H.-C. Koch

g

,

i

,

S. Komposch

a

,

b

,

A. Kozela

j

,

J. Krempel

a

,

B. Lauss

b

,

∗

,

T. Lefort

d

,

Y. Lemière

d

,

A. Mtchedlishvili

b

,

O. Naviliat-Cuncic

d

,

4

,

F.M. Piegsa

a

,

G. Pignol

h

,

P.N. Prashanth

k

,

G. Quéméner

d

,

D. Rebreyend

h

,

D. Ries

a

,

b

,

S. Roccia

l

,∗

,

P. Schmidt-Wellenburg

b

,

A. Schnabel

f

,

N. Severijns

k

,

J. Voigt

f

,

A. Weis

g

,

G. Wyszynski

a

,

e

,

J. Zejma

e

,

J. Zenner

a

,

G. Zsigmond

b

a_{ETHZürich,InstituteforParticlePhysics,CH-8093Zürich,Switzerland} b_{PaulScherrerInstitute(PSI),CH-5232Villigen-PSI,Switzerland}

c_{HansBergerDepartmentofNeurology,JenaUniversityHospital,D-07747Jena,Germany} d_{LPCCaen,ENSICAEN,UniversitédeCaen,CNRS/IN2P3,Caen,France}

e_{MarianSmoluchowskiInstituteofPhysics,JagiellonianUniversity,30-059Cracow,Poland} f_{PhysikalischTechnischeBundesanstalt,D-10587Berlin,Germany}

g_{PhysicsDepartment,UniversityofFribourg,CH-1700Fribourg,Switzerland} h_{LPSC,UniversitéGrenobleAlpes,CNRS/IN2P3,Grenoble,France}

i_{InstitutfürPhysik,Johannes–Gutenberg–Universität,D-55128Mainz,Germany} j_{HenrykNiedwodnicza´nskiInstituteforNuclearPhysics,31-342Cracow,Poland} k_{InstituutvoorKern–enStralingsfysica,UniversityofLeuven,B-3001Leuven,Belgium} l_{CSNSM,UniversitéParisSud,CNRS/IN2P3,OrsayCampus,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received9December2014

Receivedinrevisedform30March2015 Accepted14April2015

Availableonline20April2015 Editor:M.Doser

Keywords: CPviolation

Shortrangespin-dependentinteraction Axion

Axion-likeparticle Ultracoldneutrons

Neutronelectricdipolemoment

We report a new limit on a possible short range spin-dependent interaction from the precise measurement of the ratio of Larmor precession frequencies of stored ultracold neutrons and 199Hg atoms confined inthe same volume. The measurement was performed ina ∼1μT vertical magnetic holding fieldwiththe apparatussearching forapermanentelectricdipole momentofthe neutronat the PaulScherrerInstitute.Apossible couplingbetweenfreelyprecessingpolarized neutronspinsand unpolarized nucleons ofthe wall materialcan be investigated by searching for atiny change of the precessionfrequenciesofneutronand mercuryspins.Suchafrequencychangecanbeinterpretedasa consequenceofashortrangespin-dependentinteractionthatcouldpossiblybemediatedbyaxionsor axion-like particles.The interactionstrengthis proportional tothe CPviolatingproductofscalarand pseudoscalarcouplingconstantsgSgP.Ourresultconfirmslimitsfromcomplementaryexperimentswith

spin-polarizednucleiinamodel-independentway.Limitsfromotherneutronexperimentsareimproved byuptotwoordersofmagnitudeintheinteractionrangeof10−6_{< λ<10}−4_m.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

*

Correspondingauthor.

E-mailaddresses:[email protected](B. Franke),[email protected] (B. Lauss),[email protected](S. Roccia).

1 _{NowatUniversityofWashington,SeattleWA,USA.}

2 _{NowatMax-Planck-InstituteofQuantumOptics,Garching,Germany.} 3 _{NowatLogrusData,Vienna,Austria.}

4 _{NowatMichiganStateUniversity,East-Lansing,USA.}

1. Introduction

Wepresentaninterpretationofourrecentmeasurementofthe ratio

γ

n

/

γ

Hg of theneutron and199Hg magnetic moments [1]in

terms of the strength of a possible short range spin-dependent neutron–nucleon interaction.Thisratiowas inferred froma com-parisonofthesimultaneouslyrecordedLarmorprecession frequen-ciesofthetwospeciescontainedinthesamestoragevolume.The http://dx.doi.org/10.1016/j.physletb.2015.04.024

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

measurementwasperformedusingtheapparatusdedicatedtothe searchfortheneutronelectricdipole moment(nEDM)[2]bythe nEDMCollaboration atthesourceforultracoldneutrons[3]ofthe PaulScherrerInstitute,Switzerland.

Inthecentralstoragevesselofthe apparatus,thespinsofthe neutronsandmercuryatomsaremadeto precesssimultaneously inthesamevolume.Theratio

R

=

fn fHg

(1) constitutes a sensitive tool for the control of systematic effects during the measurement ofthe nEDM. By correcting R properly

forknowndifferencesoftheLarmorprecessionofthetwospecies neutrons and199_{Hg, respectively, theratio ofmagnetic moments}

γ

n

/

γ

Hg can be extracted. A dataset of R taken in 2012was

in-dependentlyanalysedin[1]andin[4],whereweadditionally ex-amineditssensitivity tohypotheticalshortrange spin-dependent interactions.Possibleforcemediatorsareaxions,oraxion-like par-ticlesandtheinteractionstrengthisproportionaltotheproductof scalarandpseudoscalarcouplingconstants gSgP.Ithasbeen pro-posedin[5,6] to usean nEDMapparatusforthe investigationof suchaforce.

A motivation to search for an interaction involving gSgP is giveninSection 2.Theinfluenceofashortrangespin-dependent interactionon the observable R isexplained andderived in Sec-tion3, whereadditionallysome related details aboutthe experi-mentareshown.Ourresultiscomparedtoothercurrentlimitson theproduct gSgP inSection4.

2. Motivation

The investigation of CP violating processes is a major line of research in particle physics. In contrast to the weak interaction, thereissofarnoevidencethat thestronginteractionviolatesCP symmetry.The non-observationofan nEDMatcurrentsensitivity levelsconstrains theCPviolating term(

θ

-term) intheLagrangian ofthe stronginteraction to be nine orders of magnitudesmaller than naturally expected [7]. This fact is known as thestrongCP problem andasolutiontoitwasproposedin[8],wherethe spon-taneouslybrokenPeccei–Quinnsymmetrywasintroduced.

A new pseudoscalar boson emerges from this symmetry, the axion [9,10]. Anintrinsic feature ofthe Peccei–Quinn modelis a fixedrelationbetweenmassandinteractionstrengthoftheaxion. Theoriginallyassumedsymmetrybreakingscale(correspondingto theelectroweak scale)was ruled out, leavingonly higherenergy scalespossible. Fortheaxion one thusexpectsa smallmassand afeebleinteractionwithotherparticles.Thepossiblemassofthe axionisconstrainedbycosmologyandastro-particlephysics mea-surementstotheso-calledaxionwindow[11].

Ashortrangespin-dependent interaction whichcould be me-diated by an axion was proposed in [12]. There, three classes of interactions were presented, involving either g2

S-, gSgP-, or g2_P-couplings,whereasgSgP-couplingsareconsideredofparticular interest,sincetheyviolateCPsymmetry.A gSgP-couplingdiagram isshowninFig. 1(a)andtakesplacebetweenanunpolarized par-ticle



(whereunpolarized meansrandomlypolarizedwithrespect to any quantizationaxis) anda polarized particle



. The sym-bol  is used to denote properties of the particle interacting at thepseudoscalar vertexwitha strength proportional to the cou-plingconstant g_P oftheparticle



.Thepotentialcausedbysuch a gSgP-couplingbetweenanunpolarizedparticleandapolarized particlewithmassmandspin

σ

isderivedas[12]:

V

(

r

)

=

gSgP

(

hc

¯

)

2 8

π

mc2

(

σ

ˆ



_{· ˆ}

_r

₎



1 r

λ

+

1 r2



e−r/λ

,

(2)

Fig. 1. (a)Interactiondiagramofascalar-pseudoscalarcouplingbetweenparticles and.isunpolarizedandinteractsatthescalarvertexwiththecoupling con-stantgS,whereasispolarizedandinteractsatthepseudoscalarvertexwiththe couplingconstantgP.Thetotalinteractionstrengthisproportionaltotheproduct gSgP.(b)Apolarizedneutronnwithspinσ interactswithanunpolarizednucleon Natdistancer withinbulkmattershapedasaplateofthicknessd.Aviewofthe –z planeinacylindricalcoordinatesystem(,φ,z)isshown.

where

σ

ˆ

 istheunitvectorofthespin,r is

ˆ

theunitvectoralong thedistancer betweentheparticles,and

λ

theinteractionrange. Theproduct

(

σ

ˆ



· ˆ

r

)

alsoviolatesparityPandtimereversal sym-metryT.

gSgP-couplings can also be mediated by other hypothetical spin-zeroparticleswhichare generictotheaxion andusually re-ferredtoasaxion-likeparticles.However,forthesegenericbosons no relation between mass and interaction strength is given, as compared to the genuine axion. The originof such particles can besymmetriesotherthanPeccei–Quinnsymmetry,whichare bro-kenatveryhighenergiesandoftenpostulatedintheoriesbeyond theStandardModelofparticlephysics,suchase.g. StringTheory. Thus,bothaxionsandaxion-likeparticles,areintriguingdark mat-tercandidatesandbeyondStandardModelphysicsprobes[13–15]. However,duetothenon-observationofthenEDMashortrange spin-dependent interaction mediated by an axion is constrained to gSgP

<

10−40

. . .

10−34 [16].Onthe otherhand,ifthe force is mediated by an axion-like particle, gS and gP are not related to aspecificsymmetrybreakingscale.Thus,nosignificantconstraint (i.e. comparable to experimental sensitivity ranges) on gSgP can bededucedfromcurrentEDMlimits[16]forthecaseofageneric bosonbeingtheinteractionmediator.

Our measurement withultracold neutrons is particularly sen-sitive to axion-likeparticles witha mass intherange ofroughly 10 meVto100 meVcouplingtofermions.Italsomatchesthemass rangetargetedby helioscopessuch asCAST[17] whichwouldbe sensitivetoaxion-likeparticlescouplingtophotons.

3. ThemeasurementwiththenEDMapparatus

The experiment is performedby confining ultracold neutrons (UCN) ofenergies below 160 neV [18,19] in a cylindrical storage chamber withvertical axisat thecenter ofthe nEDMapparatus. Thedimensionsofthestoragechamberandsomespecificfeatures areshowninFig. 2.A homogeneousverticalmagneticholdingfield of

∼

1μT is appliedwith a cos

θ

-coilwound around the horizon-tally cylindricalvacuumtank. This vacuumtank isenclosed by a four-layermagneticshielding[20]andanactivemagneticfield sta-bilisationsystemfortheexternalmagneticfield[21].

Spin-polarizedUCNarefilledintothestoragechamber approx-imatelyevery340 swheretheyprecessfreelyfor180 sduring the described measurements.Theprecessionfrequencyisinferred us-ingRamsey’smethod[2].Thespinsofpolarized199Hg atoms pre-cess simultaneously in the same volume allowing to correctthe Larmor precession frequency of the neutrons for potential, small magnetic field fluctuations whichcan occur inside thefour-layer magneticshieldingmadeofμ-metal.

We searchfor asignature ofa spin-dependent interaction be-tween polarizedparticles insidethestoragechamberandthe un-polarizedwall of thischamber. This interactioncan be described

Fig. 2. VerticalcutviewoftheUCNstoragechamberwithimportantparts schemat-ically indicated(not toscale). The biggrey arrowindicates the main magnetic holdingfieldB0.Theblueshadedregionistheinsideofthechamberandindicates theinhomogeneousUCNdensitydistribution.TheredshapesdepictCs magnetome-ters[22]whicharemountedaboveandbelowthestoragechamber:fourontopand sevenatthebottom.TheseconsistofaCs-vapourfilledglassbulbinsideashielded housingandservetomeasuretheverticalmainmagneticfieldB0andalsovertical magneticfieldgradientsG.Themeasuresarethethicknessd ofthetopandbottom platesofthevessel (coatedwithdiamond-likecarbonforimprovedUCNstorage properties[18]),distanceD betweentopandbottomCsmagnetometers,heightH ofthevessel,andradiusC ofthevessel.UCNarefilledthroughaguidefrom be-low.Thelateralwallofthevesselisapolystyreneringwithadeuteratedcoating forimprovedUCNstorageproperties[19].Thevacuumtank(VT)oftheapparatus isenclosedbyafour-layermagneticshielding(MS)made fromμ-metal [20]and asurroundingfieldcompensation(SFC)whichactivelystabilisestheexternal mag-neticfield[21].(Forinterpretationofthereferencestocolorinthisfigurelegend, thereaderisreferredtothewebversionofthisarticle.)

by thepotential ofEq.(2) andanexample isshownin Fig. 1(b). Integratingtheinteractionoverallthenucleonspresentinuniform bulkmatterresultsinaneffectivefieldnormaltothesurface.

Sincethepotentialisspin-dependent,itcanalsoberegardedas a pseudomagneticfield b which can affectthe Larmor frequency of precessing spins. For a symmetric setup withidentical mate-rialfor thebottomandtop ofthestorage vessel,thefield atthe vesselsurfacespointsinoppositedirections.Therefore,weexpect no shiftin theLarmor frequencyof the Hg atomswhich sample the volume homogeneously. However, UCNs havesuch low ener-giesthattheyaresignificantlyaffectedbygravityandtheirdensity increasestowardsthebottomofthestoragevessel.Thus,thevessel isinhomogeneously sampled andthe effectof apseudomagnetic fieldatthebottomofthechamberwillnotcanceloutcompletely. Dependingon thesign ofthe verticalmagnetic field,the preces-sion frequency ofthe UCN spins will be increasedor decreased. Consequently, R will beshifted by a constant withopposite sign fortheupwardordownwardorientedmagneticholdingfield B0: R↑↓

=

γ

n

γ

Hg



1

±

b B0



,

(3)

wherethe

±

signappliestotheupward/downwardoriented mag-neticholdingfield,respectively.

3.1. Derivationofthepseudomagneticfield

IntegratingthepotentialgiveninEq.(2)overbulkmatter,e.g. a plateofthicknessd,radiusC ,andnucleonnumberdensityN, re-sultsinatotalpotential Vtot atheight z abovethesurfaceofthe

plate(cylindricalcoordinatesr

= (



,

φ,

z

)

areused)[4]:

Vtot

(

z

)

=

−d



0 2π



0 C



0 N V

(

r

)



d



d

φ

dz (4)

=

gSgP

¯

h2N

λ

4m



e−λz

−

_e−z+λd

−

_e−  C 2+z2 λ

+

_e−  C 2+(z+d)2 λ



(5)

≈

gSgP

¯

h2N

λ

4m

1

−

e−d/λ

e−z/λ

.

(6)

TheapproximationfromEq.(5)to(6)neglectsthethirdandfourth terms intheroundbracketsandiswelljustifiedduetothe dimen-sionsofthechamber(d



C ),andtherangeofinterestof

λ

andz

(

λ,

z



C ).Italsocorresponds to simplifyingtheintegration over theradialcomponent



toaninfiniteplane.

Thepseudomagneticfieldb normaltothesurfacecanbe writ-tenasafunctionofheight z:

b

(

z

)

=

2Vtot

(

z

)

γ

h

¯

≈

gSg  P

¯

h

λ

N 2

γ

m

1

−

e−d/λ



e−z/λ



,

(7) where

γ

isthegyromagneticratioofthepolarizedparticle.This resultagreeswiththeonederivedin[5].

InordertocalculatethefieldpresentinthenEDMsetupbnedm,

boththebottomandthetopofthestoragevesselhavetobetaken intoaccount.Thevessel’sinnerhorizontalsurfacesare perpendic-ulartoz,withthebottomsurfaceatz

= −

H

/

2 andthetopsurface atz

= +

H

/

2,asshowninFig. 2.UsingEq.(7)onefinds

bnedm

(

z

)

=

bbottome− z+H/2

λ

−

_btope−− z+H/2

λ

_.

₍₈₎

Becauseoftheirinhomogeneousdensitydistribution

ρ

(

z

)

,theUCN experienceaneffectivefieldgivenby

bUCN

=

H 2



−H 2 bnedm

(

z

)

ρ

(

z

)

dz

.

(9)

ThefirstorderestimatefortheUCNdensitydistributionis

ρ

(

z

)

=

1 H



1

+

12h H2z



,

(10)

whereh

= −

2

.

35

(

5

)

mm isthemeasuredcenter-of-massoffset be-tweenUCNandHgatomdistributions(seealsoSection3.2).

Since the pseudomagnetic field isexpected to be of short in-teraction range

λ

given by the limits which have already been imposed on gSgP, the UCNdensity distribution can be approxi-mated witha constantvalue within adistance

∼λ

to thebottom andtopsurfaces.Asaconsequence,Eq.(9)canbesimplifiedinthe followingway: bUCN

≈

H 2



−H 2

ρ

bottombbottome− z+H/2 λ

−

ρ

_topbtope−− z+H/2 λ

dz

,

(11) where

ρ

bottom

=

ρ

(

−

H

/

2

)

and

ρ

top

=

ρ

(

+

H

/

2

)

.Thusthe integral

canbesolvedanalyticallyandthescalarandpseudoscalarcoupling constantscanbeisolatedasfollows:

gSgP

=

bUCN H2

γ

m 6hhN

¯

λ

2

1

−

e−H/λ

₋1

1

−

e−d/λ

₋1

.

(12)

The bottom and top of theUCN storage vessel are madeof alu-minumandhaveathicknessofd

=

2

.

5cm each.WeuseN

≡

NAl

=

1

.

62

·

1024_cm−3 _and

_γ



_≡

_γ

_n

₌

₂

_π

_·

₂₉

_.

_1646943MHz

_/

_T[23].The

surface ofthe aluminum platesis coatedwith diamond-like car-bon in order to improve neutron storage properties.The coating thickness is below3 μm and the densityof diamond-like carbon

issimilartothatofaluminum.Thus thecoatingisdisregardedin thecalculation butforthefact thatwerestrictthevalidityofour derivedlimitto

λ

>

1μm.

The center-of-mass offset h contributes to the denominator of Eq. (12) and depends on the energy spectrum of the UCN. Hence, a changeinthe energy spectrum,or alsoa different ves-sel height H ,will influence the sensitivityof our apparatusto a pseudomagneticfield.

3.2.DetailsonthemeasurementoftheratioR andthecenter-of-mass offseth

In [1], the measurement of R and its experimental setup are thoroughlydescribed.Herewebrieflyrecapitulateselectedaspects of the experiment and related systematics relevant to our mea-surement.

Taking into account systematicshifts on the UCN or mercury spinprecessionfrequency,theratioR canberewrittenas: R

=

fn

fHg

=

γ

n

γ

Hg



1

+ δ

Grav

+ δ

Trans

+ δ

Light

+ δ

Earth



.

(13)

Alinear, G-dependent fit of the form x

+

G y, where G

= ∂

B

/∂

z

istheverticalmagneticfield gradient,wasperformedtothedata listedinTable 2 of[1].Thefirst twoterms inEq.(13)constitute thetermsofthefit.Theothertermsshifttheconstantfit parame-terandaddtothesystematicerror.Thedifferentcontributionsto Eq.(13)arediscussedinthefollowing.

Firstlywewanttofocusontheshifts

δ

Gravand

δ

Transwhichare

bothrelatedtotheverylowkineticenergiesoftheUCN.

The gravitation-induced systematic effect is described at first orderby

δ

↑↓_Grav

= ±

h B0

G

,

(14)

where the plus-sign refers to B0↑ and the minus-sign to B0↓. Eq.(14)is derived assuming a UCNdensitydistribution asgiven in Eq. (10) and thus the common slope in both equations cor-responds to the center-of-mass offset h of the UCN. The linear gradient dependenceis assumed to hold for small G. Measuring

R asafunctionofvaryingverticalmagneticfieldgradientsenables usextractthecenter-of-massoffseth fromtheslopeofR vs. G.

Twocorrectioncoils(woundassaddlecoilsontopandbottom ofthe vacuumtank) were powered in an anti-Helmholtz config-uration to superimpose a vertical magnetic field gradient to B0. A measurementofthemagnitudeofG isprovidedbycesium mag-netometers[22]whicharelocatedaboveandbelowtheUCN stor-agevessel(seeFig. 2).Asecondorderpolynomialparametrization ofthemagneticfieldisusedtoextractG fromthemagnetometers’ readingsandtheirpositions.

Thesystematiceffectduetopossibletransversecomponentsof

B0 reads

δ

↑↓_Trans

=

B_⊥2



↑↓ 2B02

,

(15)

where

B_⊥2



↑↓ is the storage chamber volume average of trans-versemagneticfieldcomponentsB_⊥forthecaseofthemain mag-neticfield B0beingorientedupwardsordownwards,respectively. Thiscorrectionresultsfromthefactthat,givenbytheirlow veloci-ties,theneutronsareintheadiabaticregime(Larmorfrequency

>

wallcollision rate)andtheir precessionfrequency isproportional totheaveragemagneticfieldmodulus fn

∝ |

B

|

.Whereasforthe mercuryatomswiththermalvelocities(Larmor frequency

<

wall collisionrate), fHg

∝ |



B

|

.

Table 1

Relativecontributionstotheoverallerrorlistedbytheeffectwhichcontributesto themagneticfield.

Effect B0↑ B0↓ Statistics ±0.5·10−6 _±0 .5·10−6 Gravitational shift (−8.9±2.3)·10−6 _{(−1.8±2.7)·10}−6 Transverse shift (3.7±0.8)·10−6 _{(3.0±1.2)·10}−6 Light shift (1.3±0.7)·10−6 _{(0.8±0.6)·10}−6 Earth rotation shift −5.3·10−6 _+5.3·10−6

A magnetic field mapping was performed inside the vacuum tank with a sophisticated non-magnetic robot using a custom-madehighsensitivitytriaxialfluxgate magnetometer.Thevolume oftheprecessionchamberhasbeenmapped.Thefollowingresults forthetwomagneticfieldorientationswereobtained:



B_⊥2



_↑

= (

2

.

1

±

0

.

5

)

nT2

;



B_⊥2



_↓

= (

1

.

7

±

0

.

7

)

nT2

.

(16) These numbers have to be regarded in relation to the absolute value of the magnetic field of

∼

1μT.Comparing the two values for

↑

and

↓

witheach other also demonstrates how well a po-laritychangeof B0 resemblesa perfectinversion ofthemagnetic

field.

δ

Lightresultsfromashiftof fHgduetolight-intensity-dependent

effectsduring theoptical read-outofthemercuryprecession fre-quencyasexplainedin[24].Thecorresponding correction factors aregiveninTable 1(fourthrow).

The remaining

δ

Earth is a consequence of the Earth being a

rotatingframeofreference.Inthiscontextitisofparticular inter-estthatneutronsandmercuryatomshavedifferentsignsoftheir gyromagnetic ratios,i.e. they precess in differentdirections with respectto themagneticfield. Thiscanbe correctedforby apply-ingthetermsinTable 1(fifthrow).

Allsystematicerrorcontributionsaresummarizedintheerror budget for the measurement of the ratio R in Table 1. We can derive two independent valuesfor R using the datafor B0↑ and B0↓,respectivelyandtheerrorcontributionsfromTable 1:

R↑

=

3

.

8424583

(

26

)

(17)

R↓

=

3

.

8424562

(

30

).

(18)

FromthedifferenceofR↑ andR↓(asgiveninEq.(3))andthesum

R↑

+

R↓

=

2

γ

n

/

γ

Hgwecanextract bUCN

=



R↑

−

R↓





R↑

+

R↓



B0

= (

0

.

28

±

0

.

53

)

pT

.

(19)

4. Resultandcomparisontootherexperiments

UsingEq.(12),themeasuredpseudomagneticfieldbUCNcanbe

convertedtoa95 %confidencelevellimitongSgP

gSgP

λ

2

<

2

.

2

·

10−27m2 (20)

for1μm

< λ

<

5mm.Attheupperendofthisrange,thelastfactor inEq.(12)departs from

∼

1 andthe relation gSgP

∝

1

/λ

2 isnot fulfilledanymore. Asa consequencethe sensitivityofour experi-mentto gSgP decreases whichresultsina flatteningofthelimit inthe gSgP vs.

λ

-plane(seeFig. 3).Thelowerendofthisrangeis constrainedby thewavelengthofultracold neutrons andaffected bysurfacepropertiessuchascoating,roughness,etc.

Since we investigatedan interactionbetween unpolarized nu-cleons andpolarized neutrons, we can state that we probed the scalarcouplingconstantgenerallyvalidfornucleons gS

≡

gN_S and the pseudoscalar coupling constant specific to the neutron g_P

≡

gn

Fig. 3. Overviewofcurrentlimitsontheproductofscalarandpseudoscalar cou-plingconstantsgSgP asfunctionoftheinteractionrangeλofashortrange spin-dependentforceat95 %confidencelevel.Onthetop,thecorrespondingmassrange ofthemediatingparticle,i.e. axionoraxion-likeparticle,isshown.Theshaded re-gionisexcludedbydifferentexperiments.Solidlinelimitswereobtainedusingcold orultracoldneutrons.Dashedlinelimitswereobtainedusing3_He,129_Xe,or131_Xe precessionexperiments.A[30];B[31],assuminganattractiveinteraction;C[32]; D[6];E[29];F[26];G[27];andH (redinthewebversion)thiswork.ThelineI (dotted)depictstheachievablelimitbyasimplemodificationofourapparatus(see text).

Fig. 3comparesourlimitongSgP toresultsfromother exper-iments.Itcoverstheinteractionrangeof1μm

< λ

<

0

.

1m,which isnotyetstronglyexcludedbyastrophysicalorcosmological con-straints.

Experiments using free neutrons are depicted by solid lines. Experiments with precessing atoms, such as e.g. 3He, 129Xe, or

131_{Xe, are depicted by dashed lines. According to the Schmidt}

Model [25], polarized atoms ofodd isotopes (with one unpaired nucleon) can roughly be considered asa probe forthe magnetic properties ofthis unpaired nucleon, regardless ofthe other con-stituents of the nucleus. Under this assumption, both types of experimentsprobegN

SgnP.Whiletheseapproachesare complemen-tary, the direct neutron measurements are model independent. The moststringentlimitsfor

λ

>

10−4m havebeenimposed re-centlyin[26]and[27] (curveslabelledFandG inFig. 3, respec-tively) which improved the recent limits from [28]. For shorter interactionranges,the moststringentlimitwas givenin[29] (E), whererelaxationofspinpolarized3He gas was investigated.This limit has been model independently confirmed and slightly im-proved by the measurements presented in this work (H). Lim-its derived from experiments with neutrons (A,B,C,D) [30–32,6]

were improved by one order ofmagnitude for

λ

<

10−5 _andby

two orders of magnitude for

λ

>

10−5. In [33] a stronger but indirectlimit on gSgP was imposed by combininglaboratory re-sults with stellar energy loss arguments. Such limits might be reached with dedicated future laboratory searches e.g. proposed in[34].

Already our present result constitutes a new direct limit on

gSgP.Replacing inourexperimentthe centralvesselbottom and top with copper, a material with higher density and good UCN reflecting surface properties,would resultina sensitivity gain of

∼

3, corresponding to the density ratio betweencopper and alu-minum. Replacing either only the bottom ortop would createa trueasymmetric potentialandincrease the sensitivityby one or-der ofmagnitude[4].The consequentlyachievable limit depicted bythedottedcurve (I)inFig. 3wouldbean important contribu-tion toreduce the allowed parameter spaces ofbeyondStandard Modeltheories.

Acknowledgements

We are grateful to the PSI staff (the accelerator operating team and the BSQ group) forproviding excellent running condi-tions andacknowledge theoutstanding support ofM. Meier and F. Burri.SupportbytheSwissNationalScienceFoundationProjects 200020-144473(PSI),200021-126562(PSI), 200020-149211(ETH) and200020-140421(Fribourg)isgratefullyacknowledged.TheLPC Caen and the LPSC Grenoble acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-09–BLAN-0046.ThePolishcollaboratorsacknowledgethe Na-tional Science center, Poland, forthe grant No. UMO-2012/04/M/ ST2/00556 andthe support by the Foundation for PolishScience – MPD program, co-financed by the European Union within the European Regional DevelopmentFund. Thisworkwas partly sup-ported by the Fund for Scientific Research Flanders (FWO), and ProjectGOA/2010/10oftheKULeuven.Theoriginalapparatuswas fundedbygrantsfromtheUK’sPPARC.

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