Ultra cold neutrons: determination of the electric dipole moment and gravitational corrections via matter wave interferometry

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by  E. Abdalla

Ultra cold neutrons: determination of the electric dipole moment and gravitational
corrections via matter wave interferometry
∗
E. Abdalla and M. S. Hussein
Instituto de F´ısica, Universidade de S˜ao Paulo
C. P. 66318, 05315-970 S˜ao Paulo, Brasil
(Dated: 1 defevereiro de2008)
We propose experiments using ultra cold neutrons which can be used to determine the electric
dipole moment of the neutron itself, a well as to test corrections to gravity as they are foreseen by
string theories and Kaluza-Klein mechanisms.
Matter Wave Interferometry Devices (MWID) are part of physics or are just mathematically well defined
aimed at testing one of the important pilars of quan- objects beyond physical reality.
5 tum mechanics, namely wave-like interference of matter. In a recent publication [14], an electric field-induced
0 These devices use the idea that quantum mechanical in- phase was measured with great precision for atomic
0
terference results in diffraction patterns of the intensity, waves. We propose that the same idea may be used to
2
whichcriticallydependontherelativephaseoftheinter- extract the dipole moment of the neutron, and test the

n fering waves. This allows the extraction of useful infor- behaviour of gravity for very short distances.
a
mationabout the phase generating,refractive-diffractive The first idea is to use Ultra-Cold Neutrons (UCN) of
J
7 medium through which the wave propagates. a velocity of vn ≃ 5m/s, which pass through a splitter
There are two main questions we are concerned with (crystal) before they actually interact with the electric

4 here, which appear in the context of grand unification field of an appropriate capacitor. The idea of using ul-
v and superstring theories. The first is that within grand tracold neutrons in order to probe properties of elemen-
1 unified theories with supersymmetry there are several tary particles has already been advocated [13]. When
1 plausible contributions to the electric dipole moments of a neutron moves in a potential, U(x), the phase of its
0 eletrons and neutrons [1, 2], a problem known as the wave function can be written as (considering this poten-
5
SUSY CP problem. CP violation is very little known, tial much smaller than its kinetic energy)
0
andthe violatingparameters,especiallythe neutronand
4 1
0 the eletron electric dipole moments should be tested. ∆φ U(x)dx , (2)
≈−~v
h/ In fact, the Standard Model (SM) is an extremely ef- n Z
p ficient theory in what concerns experience today, but it where the limits of the integral are determined by the
- is widely believed that there exists new physics beyond length of the phase generating apparatus. If U(x) is a
p it and any extension of the SM, especially in the case constant, then
e
it contains supersymmetry (SSM), which is also wide-
h
ly believed to be certain, foresees CP violation, since it 1
: ∆φ UL . (3)
v is necessary to explain, from the observational point of ≈−ℏv
n
i
X view, the baryon asymmetry of the Universe [3].
Therefore, the interference of the two split neutron
r In the case of the neutron electric dipole moment the waves, one of which affected by (3), would behave as
a relevant CP-violating terms are flavour blind. The con-
tribution from the Kobayashi-Maskawa matrix arises at In A+Bcos(φ0+∆φcosθ) , (4)
three loop level, and the experimental bound is [4] ≈
where θ is the angle between the parallel to the
d 6.3 10−26e cm . (1) propagation-in-mediumwave, and that of the ¨screen¨.
n
≤ × × Astheneutronwavetraversestheregionofstrongelec-
tric field it feels an interaction which has the form
The secondquestionwedealwithhereconcernsaspe-
cific point of string theories, which is the fact that they v α ε2
requiretheexistenceoffurtherdimensions[5,6,7]which U =d~n ~ε µn nε n , (5)
· − c − 2
canbebothofinfinite[8,9,10]orfinite[11,12]size. This (cid:16) (cid:17)
leadstocorrectionsofNewton’sgravitylawatverysmall wheretheterm vnε istheinducedmagneticfieldow-
− c
distances. ingtothemotionoftheparticlethroughtheelectricfield
(cid:0) (cid:1)
An experimentalconfirmation(or denial) of the above ε. Thethirdtermcorrespondstotheneutronstarkeffect
facts lies in the heartof the questionwhether granduni- interactionwhere the electric polarizabilityαn entersin-
fied theorieswith supersymmetryandstring theoriesare to the picture. Of course, at the very slow velocities we
are considering,gravityeffects are relevant. These, how-
ever canbe eliminated by a proper calibrationof the ex-
perimentthroughameasurementthematterinterference
∗Electronicaddress: [email protected],[email protected] when the electric field is switched off.
2
Table: fundamental constants of the electric dipole moment, (6) is overcome (with the
Neutron mass mn =1.67×10−27Kg cancellationof the velocity dependent term for back and
Magnetic dipole moment −1.91× e~ forth travels) and we have a result with a 10% confi-
2mpc2 dence. In fact angular shifts of the order of 10−3 have
Neutron Lifetime τ =(889.1±2.1)s
alreadybeenmeasuredinanotherquantalwavemechani-
Electric polarizability α=(0.98±0.20)×10−42cm3 calprobleminvolvingMott-type oscillationinthe elastic
Charge (−0.4±1.1)×10−21e scattering of two slowleads ions [14]. The purpose of
Electric Dipole moment dn ≤6.3×10−26e×cm these measurements was the verification of the possible
Newton constant G=6.67×10−11m3/Kg s2 existence of long-range multigluon exchange interaction
(colourvan der Waals force)whichwere predictedto ex-
ist in several theories [15]. Therefore an enhancement
of one order of magnitude in the neutron electric dipole
From the numbers shown in the table we anticipate moment limit can be obtained.
that the third term in Eq. (5) is much smaller than the We should emphasize that, in ∆φe, we have na very
firsttwo,evenfor the reasonablystrongfieldconsidered, small number, d ( 10−25ecm) multiplying a very large
n
namely eε 100KeV. In so far as the competition be- number,N L 1≃. This latter canbe made largerwith
∼ cm a ~vn
tween the electric and magnetic interaction of the neu- a suitable changes in the macroscopic variables that de-
(cid:0) (cid:1)
tron is concerned, we have evaluated both terms using fine it. For a time of flight approximately equal to the
the values of µ and d given in the table, a neutron neutron lifetime, N can be as large as 5000.
n n
velocity of 5m/s and the above cited value of ε. We find Wenowpasstothesecondproposal. Asmentioned,ac-
cording to string theory, world is multidimensional. The
dn −4 question of how do the extra dimensions behave is how-
10 . (6)
µnvcn ≈ ever not fixed, at least from the theoretical side, since
there are several possibilities, namely the extra dimen-
It is clear that the measurement of the phase shift of sions might be of the size of the Planck lenght [5], they
thetwobeamsofneutronswillsimplybesensitivetoµn. can have a finite submilimetric size in order to unify the
In order to remove the effect of the magnetic interac- PlanckandStandardModelscales[11]ortheycanbein-
tion to first order we suggestusing an externalmagnetic finite with a nontrivial warp in the extra dimensions [9].
field that is opposite to the induced one, It is an important experimental problem to prove these
extradimensionsbyproposingnewsituationswherethey
~v
B~ = n ~ε . (7) canleaveanexperimentalimprint[10,16]. Arecentpro-
ext
c × posal in this direction was given in [17].
We thus propose to use the slow neutrons to measure
This field is quite weak and can be easily attained. In
thenewcontributionstoNewtonspotential. Wesuppose
fact,duetothesmallnessof(6),thereisnopracticalway
thatsincegravitationalfieldsareweak,themetric,either
of cancelling such a term with the corresponding one in
3+1-dimensional,or 4+1-dimensional is well described
(5) to the desired accuracy, since the velocity v itself is
defined to an accuracy of a few percent. Howevner, if we in terms of a potential as g00 =−1+ 2cU2g, with Ug given
allowforseveraltotalreflections,N,oftheneutronsfrom bythesolutionofthePoissonequation. Weproposethat
mirrorsplacedatthe ends ofthe capacitorthis wouldon the neutrons are colimated into a small hollow cilinder.
theonehandenhancetheelectricinteractionasthephase In accordance with 3 + 1 dimensional Newton law the
becomes potential due to a line of matter with linear density λ,
located at the z-axis from z =0 to z =L is
NUL N L
∆φe =− ~v = ~v dn(eV)a , (8) (L z)+ r2+(L z)2
n n U (x)=Gλln − − , (10)
g z+√r2+z2
p
with a still verysmall(almost cancelled)magnetic inter- −
action which reverses sign for a neutron traversing the where r is the distance to the z-axis. We shall suppose
cylinder back and forth. In eq. (8), eV is the voltage thatforahollowcilindes thepotentialinside thecilinder
across the capacitor and a is the distance between the is given by the above with r = r0 being the internal
plattes. Taking L = 104 (L = 1m, a = 1mm) and using radius.
α
for N 1001, (odd number of collisions guarantees that For a 4+1-dimensional world the expression is differ-
≃
theneutronscomeoutattheotherendofthecapacitor), ent, and we suppose that it is effective only for dis-
we find (for d 0.6 10−25ecm) tances smaller than a typical size of the extra dimen-
n
∼ ×
sion ξ 0.1mm. Therefore, inside the hollow cylinder
∆φe 2 10−3rad 0.1o , (9) we subt∼ract the 3+1-dimensional contribution of a slice
≃ × ≃
zǫ[z ξ,z+ξ] and add a 4+1-dimensionalcontribution
a measurableshift in the diffractionmaxima seenon the −
of a slice zǫ[z ξ,z+ξ], obtaining
screen. For N 1000andanexternalB fieldgivenwith −
∼
an accuracy of 1% the difficulty related to the smallness
3
for a cylinder of linear density λ 0.03Kg/m, and N
∼ ∼
1000,which is comparableto the value of ∆φ in eq. (9)
e
(L z)+ r2+(L z)2 2ξ for the electric dipole measurement. The above numbers
U (z) Gλ ln − − +
g ≃ ( z+p√r2+z2 r arecompatiblewithathincylinderofradius1mm. Fora
− cylinderof0.3mm,quitecomparabletothepresumedsize
ξ+ r2+ξ2 of the extra dimension, we still get a measurable result
ln (11)
− −ξ+p r2+ξ2) [n1e4u].troNnotliifceetitmhea.t N ∼ 5000 is still compatible with the
4z(L z) p2ξ ξ+ r2+ξ2
Gλ ln − + ln .
≃ ( r2 r − ξ+p r2+ξ2)
−
p
Theorderofmagnitudeoftheeffectonaslowneutron
is not negligible, that is
ACKNOWLEDGEMENT: This work was partially
U m L Gλm L
∆φ =N g n N n =6 10−3 =0.3o (12) supported by FAPESP and CNPQ, Brazil.
g ~v ≃ ~v ×
[1] Y.Y.KeumandOttoC.W.Kong,Phys.Rev.D63(2001) (1999) 4690-4693.
113012,hep-ph/0101113;Phys.Rev.Lett.86(2001)393- [10] ElcioAbdalla,AdenauerCasali,BerthaCuadros-Melgar,
396, hep-ph/0004110. Nucl. Phys. B644 (2002) 201-222, hep-th/0205203.
[2] Yosef Nir, 27th SLAC Summer Institute on Particle [11] Ignatios Antoniadis, Nima Arkani-Hamed, Savas Di-
Physics: CPViolationinandBeyondtheStandardMod- mopoulos and G.R. Dvali Phys. Lett. B436 (1998) 257-
el (SSI 99), Stanford, California, 7-16 Jul 1999, *Trieste 263, hep-ph/9804398.
1999, Particle physics* 165-243, hep-ph/9911321. [12] Lisa Randall and Raman Sundrum Phys. Rev. Lett. 83
[3] A. D. Sakharov JETP Lett. 5 (1967) 24; G. R. Farrar (1999)3370-3373.
and M. E. Shaposhnikov Phys. Rev. D50 (1994) 774; P. [13] O. Bertolami and F. M. Nunes Class. Quantum Grav.
Huet and E. SatherPhys. Rev. D51 (1995) 379. 20 (2003) L61-L66.
[4] P.G. Harris et al Phys. Rev. Lett. 82 (1999) 904. [14] A.C.C. Villari et al Phys. Rev. Lett. 71 (1993) 2551.
[5] MichaelB.Green,J.H.Schwarz,Edward Witten,Super- [15] M. S. Hussein,C. L. Lima, M. P.Pato and C. A.Bertu-
stringTheory,CambridgeMonographsOnMathematical lani Phys. Rev. Lett. 65 (1990) 839.
Physics, 1987. [16] M. S. Hussein. R. Lichtenth¨aler, M. P. Pato and C. A.
[6] J. Polchinski String Theory Cambridge Univ.Pr., 1998. Bertulani Braz. J. Phys. 27 (1997).
[7] J. Scherk Rev. Mod. Phys. 47 (1975) 123-164. [17] A. Frank, P. Van Isacker and J. Gomez-Camacho Phys.
[8] Petr Horava and Edward Witten, Nucl. Phys. B460 Lett. B582 (2004) 15-20.
(1996) 506-524, hep-th/9510209.
[9] Lisa Randall and Raman Sundrum Phys. Rev. Lett. 83