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Searching for New Spin-Velocity Dependent Interactions by Spin Relaxation of

Polarized 3He Gas

H. Yan,1, ∗ G.A. Sun,1 S.M. Peng,2 Y. Zhang,1, 3 C. Fu,4 H. Guo,5 and B.Q. Liu1

1Key Laboratory of Neutron Physics, Institute of Nuclear Physics and Chemistry, CAEP, Mianyang, Sichuan, 621900,China2Institute of Nuclear Physics and Chemistry,CAEP, Mianyang, Sichuan, 621900,China

3School of Nuclear Science and Technology, University of Science and Technology of China, Hefei, 230026, China4Department of Physics, Shanghai Jiaotong University, Shanghai, 200240, China

5Department of Physics, Southeast University, Nanjing, 211189, China

(Dated: February 10, 2018)

We have constrained possible new interactions which produce nonrelativistic potentials betweenpolarized neutrons and unpolarized matter proportional to α~σ · ~v where ~σ is the neutron spin and~v is the relative velocity. We use existing data from laboratory measurements on the very long T1

and T2 spin relaxation times of polarized 3He gas in glass cells. Using the best available measuredT2 of polarized 3He gas atoms as the polarized source and the earth as an unpolarized source,we obtain constraints on two new interactions. We present a new experimental upper bound onpossible vector-axial-vector(VV A) type interactions for ranges between 1 ∼ 108m. In combinationwith previous results, we set the most stringent experiment limits on gV gA ranging from ∼ µm to∼ 108m. We also report what is to our knowledge the first experimental upper limit on the possibletorsion fields induced by the earth on its surface. Dedicated experiments could further improve thesebounds by a factor of ∼ 100. Our method of analysis also makes it possible to probe many velocitydependent interactions which depend on the spins of both neutrons and other particles which havenever been searched for before experimentally.

PACS numbers: 13.88.+e, 13.75.Cs, 14.20.Dh, 14.70.Pw

In recent years, various models of new physics beyondthe Standard Model have been studied in which new mas-sive particles such as the axion, familon and majoron, etc.were theoretically introduced [1]. New macroscopic inter-actions meditated by WISPs (weakly-interacting sub-eVparticles) have also been theoretically proposed. The in-teraction ranges of these new forces range from nanome-ters to astronomical distance scales. The fact that thedark energy density is on the order of (1 meV)4 corre-sponding to a length scale of 100 µm also encouragespeople to search for new physical phenomena around thisscale [2]. Various experiments have been performed orproposed recently to search for a subset of these newinteractions which could couple to the spin of the neu-tron/electron. Polarized neutron beams were used inthe experiments presented in Refs.[3, 4]. Polarized no-ble gases were used in Refs.[5–8]. Atomic magnetome-ters were applied in Refs.[9, 10]. Experimental schemesusing polarized atom beams have been recently proposedin Ref.[11].

The idea that exotic new interactions might bespin/velocity dependent is quite fascinating. For ex-ample, the theoretically proposed photinos interact withnuclei only through spin-dependent forces [12]. Ref.[13]analyzed the possible nonrelativistic potentials betweenspin-1/2 fermions from spin 0 and spin 1 boson exchangeand found 16 possible new interactions, 10 of which de-pend both on the spin states and the relative velocity be-

∗Corresponding author: hyan@caep.cn

tween particles. Torsion, a twisting of spacetime coupledto intrinsic spin, which has been included in many modelswhich extend general relativity [14, 15], can also inducespin-velocity dependent interactions between an unpo-larized source and the spin. Hari-Dass interaction [16]is another example of spin-velocity dependent interac-tions. These possible new interactions have eluded de-tection so far. It might be that the new interactions areeither too weak to be detected, or the spin/velocity de-pendence causes difficulties for the conduct of sensitiveexperiments.Either for the torsion-induced or the vector-axial-

vector (VV A) type interaction, the spin of polarized noblegases like 3He can interact with an unpolarized sourcethrough a potential proportional to

V = α~σ · ~v, (1)

where ~σ are the Pauli matrices and ~v is the relative ve-locity between the spin and the unpolarized source. Theparameter α has dimensions of momentum and dependson factors specific to the interaction such as the probe tosource distance, the source mass or the nucleon number,etc. Since it is known that the spin of the 3He nucleusis dominated by the spin of its unpaired neutron [17], inthis paper we interpret constraints on α in terms of neu-tron properties. To help develop intuition for the physicaleffects of V it is useful to imagine that V produces aneffective pseudo-magnetic field:

V = ~µ · ~B′, (2)

where ~µ is the magnetic moment, γ the gyromagnetic

ratio of the spin polarized particle, and ~B′ = 2α~v/~γ

2

the pseudo-magnetic field. This pseudo-magnetic fieldis along the ~v direction and has a strength of 2αv/~γ.Searching for the new physics from this potential is then

equivalent to detecting the pseudo-magnetic field ~B′

.Spin polarized neutron/atom beams are convenient to

probe these spin-velocity dependent interactions since alarge relative velocity between the probe and the sourcecan be easily realized. However, the number of the probeparticles is limited by the phase space density of thebeam. Larger phase space densities of polarized probeparticles can be obtained by using ensembles of polar-ized gases, but the polarized noble gas ensembles whichcan support sensitive NMR measurements of the spin dy-namics needed for this search are usually sealed in glasscells. It would be technically difficult to realize a largerelative velocity between the source mass and the probeparticles inside delicate glass cells.Though 〈~v〉 is zero for atoms of the glass sealed no-

ble gas, 〈v2〉 is not. The nonzero 〈v2〉 in the presence ofa ~σ · ~v type interaction will change the spin relaxationtimes of polarized noble gases. Although it is a secondorder effect, in this case there is no need for bulk mo-tion of either the polarized or unpolarized masses in theexperiments. Thus it is possible to detect or constrainthe new physics by the longitudinal spin relaxation time(T1) or the transverse relaxation time (T2) of polarizednoble gases. Here T1 refers to the mean time for a spinpolarized ensemble to return to its thermal equilibriumstate and T2 the mean time that polarized spins to losecoherence when processing along the longitudinal mainfield while the polarization is tipped to the transverse di-rection [18]. For the best available T1 [19] and T2 [20, 21]data, our research indicates that the constraint on α fromT2 is tighter than that from T1. In what follows, we willfirst describe how the α~σ · ~v interaction affects the spinrelaxation times of the polarized 3He gas, then we willconstrain α by using the best available T2 measured inthe experiment. Furthermore, by using this constraintof α and the earth as a source, we obtain new limitson two different types of new interactions, vector-axial-vector interaction (VV A) and a linear combination of thetime component of possible torsion fields from the earth.

I. CONSTRAINING α BY T1 AND T2 OF SPIN

POLARIZED 3HE GAS

Highly polarized, dense ensembles of polarized 3He gashave been developed over the last few decades for sci-entific applications in nuclear/particle physics, neutronspin filters and medical imaging [22–24]. There are al-ready some examples using the spin relaxation time toconstrain the scalar-pseudo-scalar type [25] interactionwhich is spin dependent. T1 is used in Ref.[19] while T2in Refs.[21, 26]. Assume the magnetic field is along z,then using the Redfield theory [21, 27], the longitudinaland transverse relaxation times of the polarized 3He gasdue to a randomly fluctuating magnetic field can be ex-

pressed as:

Γ1 =1

T1=γ2

2[SBx(ω0) + SBy(ω0)],

Γ2 =1

T2=γ2

4[SBx(ω0) + SBy(ω0) + 2SBz(0)], (3)

where

SBx(ω0) =

∫ +∞

−∞

〈Bx(t)Bx(t+ τ)〉e−iω0τdτ. (4)

Here 〈...〉 represents the ensemble average, Bx is the xcomponent of the fluctuating magnetic field as seen in therest frame of the diffusing polarized 3He atom, and ω0 =γB0 is the Larmor frequency. Using above formulas, itis easy to see that the pseudo-magnetic field induced bythe α~σ · ~v interaction will change T1 as follows:

Γ1 =1

T1=

4α2

~2

∫ +∞

−∞

〈vx(t)vx(t+ τ)〉e−iω0τdτ. (5)

When the Lamor frequency ω0 is much larger than 1/τDwith τD ∝ L2/D (L is the characteristic length of the gassealed cell in the dimensions transverse to the magneticfield), the autocorrelation function for the velocity is [27]:

〈vx(t)vx(t+ τ)〉 = 〈v2x〉e−

|τ|τc =

1

3〈v2〉e−

|τ|τc , (6)

where τc is the average collision time of the atoms.Plugging in the data given in Ref.[19], τc = 3×10−10s,

T rem1 = 2664h (1σ value), ω0 = 105s−1, we obtain an

upper limit to α as:

α ≤ 7.7× 10−37kg ·m · s−1. (7)

Similarly, the relaxation time T2 caused by the α~σ ·~v typeinteraction can be expressed as:

1

T2=

α2

~2

∫ +∞

−∞

[〈vx(t)vx(t+ τ) + vy(t)vy(t+ τ)〉e−iω0τ

+2〈vz(t)vz(t+ τ)〉]dτ. (8)

Here we need to calculate 〈vx(t)vx(t+ τ)〉 under the con-dition that the magnetic field is small such that Eqn.(6)cannot be applied anymore. According to Refs.[28, 29],we can derive

∫ +∞

−∞

〈vx(t)vx(t+ τ)〉e−iω0τdτ = ω20Sx(ω0), (9)

where Sx(ω0) is defined as the Fourier transformation of〈x(t)x(t + τ)〉. Now the problem of finding the velocityautocorrelation function reduces to calculating the posi-tion autocorrelation function. The latter can be solvedusing diffusion theory [21, 27]. For a spherical cell withradius R, it can be shown that:

Sx(ω0) = 4R2∑

n

1

x21n(x21n − 2)

x2

1nDR2

(x21nDR2 )2 + ω2

0

, (10)

3

where x1n are the zeros of the derivatives of sphericalBessel functions [30] andD is the diffusion constant of the3He gas. The relaxation time T2 can be finally expressedas

Γ2 =1

T2=α2

~2ω20 [Sx(ω0) + Sy(ω0)]. (11)

Now using the data given in Ref.[20], which used a radiusof the spherical cell R = 3cm with D = 470cm2 · s−1 andΓ2 = 8.0 × 10−7s−1 (1σ value [21]) into Eqn.(11), weobtain an upper limit on α as

α ≤ 6.9× 10−37kg ·m · s−1. (12)

This upper limit is slightly more stringent than that givenby Eqn.(7) and so we are going to use T2 to constrain thepossible spin-velocity dependent new physics. However,the T1 estimate is simple and direct, and in some morecomplicated cell geometries one can get a constraint moreeasily and quickly by using the T1 estimate.

II. CONSTRAINING NEW INTERACTIONS

USING THE EARTH AS A SOURCE

A. Vector-axial-vector interaction

Taking advantage of the fact that there are about 1042

polarized electrons in the earth, Ref.[10] used the po-larized electron spins of the earth as a source to con-strain several possible exotic long-range spin-spin inter-actions. Inspired by this spirit, we notice that there are1051 nucleons in the earth. It might be even more advan-tageous to use the earth as a source to constrain somespin-velocity dependent interactions. For example, forthe vector-axial-vector interaction VV A(r) from the La-grangian LX = ψ(gV γ

µ + gAγµγ5)ψXµ, this parity vio-

lating potential has the form:

VV A(r) =~gV gA2π

e−r/λ

r~σ · ~v, (13)

where λ = ~/mXc is the interaction range, mX the massof the new vector boson and gV ,gA the vector and axialvector couplings. VV A(r) is the Yukawa potential mul-tiplied by the ~σ · ~v factor. Using the earth as a source,the Vector-Axial(V-A) potential generated by the earthon its surface is[31]:

VV A = ~gV gAρNλ2[(1−

λ

R⊕

)e−∆y

λ

+(1 +λ

R⊕

)e−2R⊕

λ ]~σ · ~v, (14)

where ρN is the nucleon number density of the earth, R⊕

the earth radius, ∆y the probe to ground distance. Ifthe force ranges are close to or smaller than the typical∼ 1m distance between the cell and ground, the earth

source can be considered as semi-infinite and Eqn.(14)is still valid as we will see later. If the new interactionVV A(r) exists, it will induce the α~σ ·~v interaction on theearth surface and affect the relaxation times of the spinpolarized 3He gas. Plugging in ρN = 3.3× 1030m−3 andusing Eqn.(11), we get

gV gAλ2[(1−

λ

R⊕

)e−∆y

λ +(1+λ

R⊕

)e−2R⊕

λ ] ≤ 2.0×10−34m2.

(15)The derived constraint on gV gA is shown in FIG.1. For

Log(|

gVg

A|)

λ (m)

Mass (eV)

-50

-45

-40

-35

-30

-25

-20

10-6

10-4

10-2

100

102

104

106

108

1010

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

FIG. 1: (Color online)Constraint to the coupling constantproduct |gV gA| as a function of the interaction range λ(newvector boson mass). The bold solid line is the result of thiswork; The dashed line is the result of Ref.[4]; The blue and reddotted lines are the result of Ref.[33] which were derived bycombining gV of Refs.[34, 35] with gA of Ref.[9] from separateexperiments.The dark grey area is excluded by experimentsof previous work [4] and this work, both directly constrain|gV gA| in a single experiment.

λ ≤ ∆y much smaller than R⊕, the nonzero ∆y limitsthe practical force range. In this case Eqn.(14) can beapproximated as:

VV A = ~gV gAρNλ2e−

∆y

λ ~σ · ~v, (16)

which is the same as the V-A potential derived in Ref.[11]for a plane plate in which the thickness goes to infinity.In comparison with the previous result given by Ref.[4],

the constraint derived in this work has no improvementfor ranges below ∼ 1m. In the long distance limit withλ≫ R⊕, Eqn.(15) becomes:

gV gA ≤ 7.4× 10−47. (17)

If the constraint derived from the neutron spin rotationexperiment is extended to the long range ∼ 108m, ourwork improves the existing direct experimental upperbound by as much as ∼ 16 orders of magnitude. Whencomparing with the results of Ref.[33], which were de-rived by combining gV in Refs.[34, 35] and gA in Ref.[9],we get ∼ 3 orders of magnitude improvement. The

4

present work gives the best known constraint on |gV gA|for ranges between 10−6m to 108m. We emphasize thatthe limits on the vector-axial-vector interaction presentedhere are derived directly from a single laboratory exper-iment.

B. Torsion induced by the earth on its surface

Using neutron spin rotation in liquid helium, Refs.[36,37] constrained possible “in-matter ”torsion for the firsttime. For torsion mentioned here, we refer to exactly thesame type as in Refs.[36, 37]. These works showed that,for nonrelativistic motion of a spin in a torsion field, sometime components of the torsion field couple to the spinthrough the form [36] :

V = ζ~σ · ~v, (18)

which has exactly the same form as Eqn.(1). ζ includesall other factors such as the distance, the source mass,etc. As in Refs.[36–38], here we only consider the lead-ing order of the torsion background which is a constant.The torsion effect is considered to be very small andit is extremely difficult to be detected. Constraints onEarth-sourced torsion had been discussed in Ref.[38],where rotating the apparatus or comparing the behav-ior of particles and antiparticles were proposed to detecttorsion.These authors showed that the extensive bodyof sensitive experiments which have been conducted tosearch for CPT/Lorentz violation could be reinterpretedto constrain many components of a possible torsion ten-sor field in vacuum. We have found that relaxation timeof the polarized 3He gas can constrain ζ without movingthe apparatus. Using the earth as a source and applyingEqn.(12), it is easy to show that in natural units

ζ = α ≤ 1.3× 10−18GeV. (19)

It might be more convenient to rewrite ζ as ζ = mScξ,wheremS is the source mass. Now ξ is dimensionless andits dependence on the source mass is isolated. Pluggingin the earth mass m⊕ = 5.97× 1024kg, we get

ξ =α

mSc≤ 1.6× 10−70. (20)

III. OTHER APPLICATIONS AND PROPOSED

EXPERIMENTS

Applying the method to the axial-axial-vector interac-

tion — It is possible to apply this analysis to other possi-ble spin-velocity interactions. We have tried to constrainthe axial-axial-vector interaction(VAA) which also comesfrom LX . We find no improvement for ranges above∼ 1min comparison with Ref.[9].Using the cell wall as a source– For VV A or VAA and

ranges around ∼ 1mm, one might think to use the cell

wall as a source, as done in Refs.[19, 21]. In this caseα is not a constant anymore and cannot be moved outof the ensemble average. We would therefore need tocalculate more complicated correlation functions such as〈α[x(t), y(t), z(t)]α[x(t+ τ), y(t+ τ), z(t+ τ)]vx(t)vx(t+τ)〉. The situation is simpler for T1. The correlationtime of velocities is ∼ 10−10s, and on this time scale theatom can only move a distance

√

〈v2〉τ ∼ 10−7m. In thisshort distance for the force ranges under consideration,α(t) and α(t+ τ) can be treated as constants and movedout of the ensemble average. We have verified this ap-proximation by Monte Carlo simulations. Using T1, weobtain constraints for |gV gA| ∼ 3 orders less stringentthan Ref.[4] and for gAgA more than ∼ 1 order less strin-gent than Ref.[3]. For T2, the situation is much morecomplicated. If we approximate the spherical cell as acube, and consider each wall as an infinite plate as inRef.[21], we could get a constraint for |gAgA|. The resultis also more than one order less sensitive than the exist-ing constraint. This is not surprising since the relaxationtimes depend on the interactions through second orderprocesses. At short distances, with a small amount of thesource mass as the cell wall, it is hard for the relaxationtime method to compete with the first order methods asin Refs.[3–6, 8].

Possibilities of applying the method to spin-spin-

velocity dependent interactions— It is also possible toapply this method to search for many other types of spin-spin-velocity interactions. There are 6 different types ofthese interactions out of the 16 total derived in Ref.[13].For these interactions, not only a relative velocity be-tween the source and probe is required, but also both theprobe and source have to be spin polarized. Thus theyare even more difficult to search for experimentally. Tothe best of our knowledge, most of these interactions havenever been experimentally searched for. The plannedexperiments and projected constraints on such interac-tions were recently presented in Ref.[41]. However, asin Ref.[10], if we use polarized electrons in the earth asa source and the relaxation time method presented inthis work, it would be possible to constrain these veloc-ity dependent electron-spin-neutron-spin interactions forthe first time. Though the relaxation time method isbased on second order processes, it has advantages of us-ing the earth as a large source and no need to move theapparatus. The method presented in this work thereforeopens a new path for probing various new interactionswhich are spin-spin-velocity dependent. The more com-plicated calculations needed to derive these constraintswould require a more detailed consideration of the col-lisional spin dynamics in ensembles of polarized gas [42]and will be the subject of future work.

Possible improvement in sensitivity from T1 — We cantry to further improve the sensitivity by dedicated exper-iments. Either for T1 or T2, the key is to constrain α bet-ter. For T1, in the limit that 1/τD ≪ ω0 ≪ 1/τc we find

that α ∝√

n/T1 after optimizing parameters. Since themeasured T1 is already as long as several thousand hours

5

for polarized 3He with cell pressure of ∼ 1 bar, it seemsthat the most promising way to improve the sensitivity isto reduce the gas number density. If n is reduced to ∼ 1mbar, and the conditions for T1 and ω0 remain the sameas before, the sensitivity could be improved by a factorof ∼ 30 if the same long T1 is observed. Furthermore,we notice that in Ref.[21], by subtracting the relaxationcontributed from the magnetic field gradient, the resid-ual T2 from other sources was improved by about oneorder of magnitude from ∼ 60h to ∼ 500h. Although atwice better T2 was reported in Ref.[39], no improvementin sensitivity can be made without knowing the magneticfield gradients. For T1, according to our best knowledge,the contribution from the magnetic field gradient has notbeen subtracted yet. For T1 measured as long as a fewthousand hours, the relative magnetic field gradient, forexample ∂xB/B0 is usually in order of ∼ 10−4cm−1. Atthis level, the gradient contributed relaxation is also inorder of ∼ 1000h. There could be improvement for the T1method by precisely mapping the holding magnetic field.By reducing the gas pressure and mapping the field pre-cisely, we might get an improvement in sensitivity by afactor of ∼ 100.Possible improvement in sensitivity from T2—A similar

result is obtained using T2: α ∝√

n/T2 after optimizingparameters. Since the cell pressure is already as low asa few mbars [20], it is hard to imagine that sensitivitycan be increased substantially by further reducing thepressure. Also the diffusion theory will break down for amuch lower cell pressure, though the Redfield theory isstill valid [29, 40]. On the other hand, lower cell pressuremeans fewer probing particles, a factor which will eventu-ally dominate. The optimum gas pressure in this case is atheoretically interesting question. One might find the an-swer by using the theory presented in Refs.[29, 40]. Fora ∼mbar cell, if a thousand-hour-long T2 could be ob-served the sensitivity could be improved by one order ofmagnitude. In this case, since ∼ 10−4cm−1 field gradientis already realized for a holding field as low as ∼ 100nT,it seems difficult though not impossible to improve the

transverse relaxation time substantially.

IV. CONCLUSION

By using the spin relaxation times of polarized 3Hegas measured in previous experiments and the earth asa source, we have constrained two types of possible newinteractions which are neutron spin-velocity dependent.We found that the best available T2 relaxation times giveslightly better constraints. We derived new experimen-tal limits on possible Vector-Axial type interactions withranges from ∼ 1m to ∼ 108m. At the distance of ∼ 108m,the limit is improved by∼ 16 orders in magnitude in com-parison with the previous result of the neutron spin rota-tion experiment. In combination with the previous result[4] which is more sensitive at short distances, we presentthe most stringent constraint derived directly from ex-periments on gV gA ranging from ∼ 10−6m to ∼ 108m(FIG.1). The methods presented in this work open upnew possibilities to search for or constrain many possiblespin-spin-velocity dependent interactions. By dedicatedexperiments, an improvement in sensitivity by a factorof ∼ 100 might be achieved using these ideas.

Acknowledgments

We acknowledge support from the National NaturalScience Foundation of China, under grant 11105128,91126001, 11204032 and 51231002. We thank Dr. AlanKostelecky, W. M. Snow, H.Gao, P.-H. Chu and Y.N.Pokotilovski for valuable discussions. We thank theanonymous referees for providing useful additional ref-erences. C.F. and H.Y. acknowledge partial supportfrom the Indiana University Center for Spacetime Sym-metries. H.G. acknowledges support from grant No.SBK201241926.

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