Physics Le e s B 582 (2004) 15–20
www.else ie .com/loca e/physle b
P obing addi ional dimensions in he uni e se
wi h neu on expe imen s
Alejand o F anka,b, Pie Van Isacke c, Joaquín Gómez-Camachod
aIns i u o de Ciencias Nuclea es, Uni e sidad Nacional Au ónoma de México, Apa ado Pos al 70-543, 04510 México, D.F., Mexico
bCen o de Ciencias Físicas, Uni e sidad Nacional Au ónoma de México, Apa ado Pos al 139-B, 62251 Cue na aca, Mo elos, Mexico
cGANIL, B.P. 55027, F-14076 Caen Cedex 5, F ance
dDepa amen o de Física A ómica, Nuclea y Molecula , Facul ad de Física, Uni e sidad de Se illa, Se illa, Spain
Recei ed 12 May 2003; ecei ed in e ised o m 17 No embe 2003; accep ed 9 Decembe 2003
Edi o : W. Hax on
Abs ac
We ca y ou a simple analysis o (n +3)-dimensional g a i y in he con ex o ecen wo k on ‘la ge’ supplemen a y
dimensions and deduce a o mula o he expec ed compac i ica ion adius o he naddi ional dimensions in he uni e se, as
a unc ion o he Planck and he elec o-weak scales. We a gue ha he co espondingly modi ied g a i a ional o ce gi es ise
o e ec s ha migh be wi hin he de ec ion ange o dedica ed neu on expe imen s. A sca e ing analysis o he co esponding
modi ied g a i a ional o ces sugges s ha slow neu on sca e ing o a omic nuclei wi h null spin may p o ide an expe imen al
es o hese ideas.
2004 Published by Else ie B.V.
The s udy o g a i y a sho ange has ecen ly
been he subjec o nume ous heo e ical and expe -
imen al in es iga ions, spa ked by he p oposal by
A kani-Hamed, Dimoupoulos, and D ali (ADD) [1]
ha g a i y may depa om New on’s in e se squa e
law a scales which could be as la ge as a millime-
e , a scena io ha was subsequen ly shown o be
consis en wi h s ing heo y [2]. Di e se expe imen-
al g oups ha e buil e ined e sions o o sion bal-
ance expe imen s and o he ingenious designs o es
g a i y a submillime e anges [3]. On he heo e i-
cal on , hund eds o pape s ha e been w i en, ang-
ing om al e na i e o mula ions o he la ge ex a-
E-mail add ess: isacke @ganil. (P. Van Isacke ).
dimensions amewo k (LED) [4], o he s udy o as-
ophysical cons ain s and he expec ed expe imen-
al consequences in u u e high-ene gy collide expe -
imen s o black-hole p oduc ion and he e ec o LED
on undamen al symme ies. Bu pe haps he mos e-
ma kable esul o da e is he ac ha no known phys-
ical cons ain s ha e as ye alsi ied he LED heo ies.
F om he expe imen alpoin o iew he new measu e-
men s ha e con i med he alidi y o New on’s law
o abou 0.1 mm. Al hough such e o s a e o g ea
alue, i can ha dly be expec ed ha his kind o ex-
pe imen s can lowe his limi signi ican ly.
We show in his Le e ha a simple analysis o N-
dimensional (ND) g a i y leads o compac i ica ion
leng hsin close ag eemen wi h he mo e sophis ica ed
calcula ions o gene al ela i i y and s ing heo y [5].
0370-2693 2004 Published by Else ie B.V.
doi:10.1016/j.physle b.2003.12.026
Open access unde CC BY license.
Open access unde CC BY license.
16 A. F ank e al. / Physics Le e s B 582 (2004) 15–20
Following his cue, we hen p esen calcula ions ha
sugges ha he e ec s o ND o ces, al hough e y
small, migh o pa icula alues o N all wi hin he
de ec ion ange o dedica ed neu on expe imen s.
In hei pape s, ADD conjec u e he exis ence o
wo o mo e addi ional dimensions in which g a -
i y, bu no he s ong o he elec o-weak o ces,
migh be ac ing, dilu ing i sel by sp eading i s lines
o o ce in o hese ex a dimensions. Essen ially, his
would explain i s appa en weakness [1]. The p o-
posal a ises om a bold modi ica ion o p e-exis ing
many-dimensional s ing heo ies and he mo e ecen
M- heo ies (M o memb ane) which encompass he
o me [6]. In hese heo ies only g a i ons a e able
o a e se he ex a dimensions, whe eas o he pa i-
cles a e ixed o ou obse able 3D wo ld, since he
o me a e desc ibed as closed s ings ee o wande
while he la e a e open s ings wi h hei ends ixed o
ou ‘b ane’.Addi ionaldimensionsa e‘compac i ied’,
i.e., hey a e closed on hemsel es wi h a cha ac e is ic
adius o compac i ica ion Rc(which o simplici y is
assumed o be he same o all addi ional dimensions).
Fo anges smalle han Rc, we hus expec a modi ied
g a i a ional o ce. The basic ques ion is hen: why
should Rcbe la ge compa ed wi h M- heo y’s o igi-
nal Planck scale o ∼10−35 m? The answe es s on
he empi ical ac ha g a i a ion has ne e been mea-
su ed below abou 10−4m and mo e signi ican ly, on
he p o ound heo e ical implica ions ha ‘la ge’ ex a
dimensions would ha e on he solu ion o he hie a -
chy p oblem whose o igin can be aced o he huge
di e ence in s eng h obse ed be ween g a i y on he
one hand and he o he o ces desc ibed by he s an-
da d model [7]. In his scena io, ins ead o ca ching
up wi h he o he o ces a Planck’s leng h scale, he
ND g a i a ional o ce ac ually joins he o he in e -
ac ions a a dis ance abou 1016 imes la ge , namely,
a he elec o-weak uni ica ion scale o ∼10−19 m. As
will be shown below, his conjec u e ixes he s eng h
o he ND g a i a ional o ce and he alue o Rc.The
mos impo an consequence o he LED hypo hesis is
he possible ansi ion om he en i ely Pla onic, inac-
cessible si ua ion posed by he Planck-scale compac i-
ica ion, o one whe e i is concei able ha expe imen-
al measu emen s may ac ually es hese ideas. The
expe imen s en isioned o da e a e p edominan ly o
wo kinds. The ones in ol ing submillime e o sion-
balance expe imen s [3] men ioned abo e, and high-
ene gy collide expe imen s in he TeV ene gy egion,
whe e di e se heo e ical p edic ionsexis o he indi-
ec obse a ion o addi ional dimensions, such as he
occu ence o missing ene gy ca ied away by unde-
ec ed g a i ons [8]. The ques ion a ises as o whe he
o he expe imen s can be designed o p obe g a i y a
e y sho anges. I has been sugges ed, o example,
ha he inc easing p ecision o expe imen al es s o
he Casimi e ec may be used o p obe new o ces
a mic ome e dis ances [9]. He e we explo e he pos-
sible e ec s o ND g a i y in expe imen s wi h neu-
ons.
We s a in he spi i o ADD [1] by ca ying ou a
classical analysis o g a i y in N=n+3 dimensions.
I space would ha e n+3 (ex ended) dimensions,
Gauss’ law implies ha he o ce o g a i y would be
o he o m
(1)Fn=−m1m2Gn
n+2,
whe e Gnis a cons an which educes o New on’s
g a i a ional cons an G o n=0. As explained
abo e, we shall ollow M- heo y and assume ha he
masses, as well as all o he o ces emain in 3D space
and only he g a i a ional o ce ield leaks in o he
addi ional dimensions. E en a e compac i ica ion o
hese addi ional dimensions, o mula (1) should be
co ec o ⩽Rc.
In gene al, oneshould assume a so ansi ion om
ND g a i y o 3D g a i y. We shall ins ead ollow a
simple p ocedu e and conside a sudden ansi ion
om ND o 3D using Eq. (1). We i s impose he
equali y o o ces a he compac i ica ion leng h Rc,
Fn(Rc)=F0(Rc), which implies ha
(2)Fn=−m1m2GRn
c
n+2.
We hen implemen he ADD conjec u e ha he ND
o ce a he elec o-weak leng h Reis as s ong as he
3D o ce a he Planck leng h RP,Fn(Re)=F0(RP).
This leads o he desi ed equa ion
(3)
Rc
Ren
=Re
RP2
,
which exp esses he compac i ica ion leng h Rcas a
unc ion o he wo undamen al scales Reand RP.
These leng hs a e de ined as
RP=¯
hG/c3,R
e=GF/¯
hc,
A. F ank e al. / Physics Le e s B 582 (2004) 15–20 17
whe e Gand GFa e New on’s and Fe mi’s cons an s,
espec i ely, he la e de ined by [7]
GF=√2¯
h2g2
8c2M2
W=(1.166×10−5GeV−2)(¯
hc)3,
whe e MWis he mass o he W±bosons and g×
sinθW=e, wi h θW he Weinbe g angle. We use cgs
uni s h oughou . We hen ind ha o mula (3) can be
w i en in he o m:
(4)
Rc
Ren
=c2
¯
h2
GF
G.
No e ha his exp ession ela es he numbe o ad-
di ional dimensions n, he adius o compac i ica ion
Rc, and he elec o-weak scale Re, o he a io o wo
undamen alnumbe s in na u e:Fe mi’s and New on’s
cons an s, wi h c2/¯
h2as a p opo ionali y cons an .
To ha e an indica ion o whe he o mula (4) is
obus , we ema k ha a simila esul can be ob ained
using a di e en a gumen . We equi e ha a he
elec oweak leng h scale he ND o ce be compa able
o he elec omagne ic in e ac ion. We may assume
ha a dis ances o o de Remos pa icles a e ul a-
ela i is ic and hus ha hei masses should be o
o de ¯
h/Rec. Equa ing he elec omagne ic in e ac ion
o he g a i a ional o ce be ween wo masses o his
o de we a i e a
(5)
¯
h2
R2
ec2
GRn
c
Rn+2
e=e2
R2
e
.
This leads o he ela ion
(6)
Rc
Ren
=αc2
¯
h2
GF
G=αRe
RP2
,
whe e αis he ine-s uc u e cons an which is abou
1/128 a he elec oweak scale Re. Values o Rc
ob ained wi h exp essions (4) and (6) a e compa ed in
Table 1.
The p esen model can be in e p e ed as ollows:
he Fe mi cons an is ela ed in he s anda d elec o-
weak heo y o he alue o he scala ield ha p o-
duces he Higgs mechanism o spon aneoussymme y
b eaking. The Higgs boson, which is ye o be dis-
co e ed, co esponds o he exci a ion mode o his
scala ield. Howe e , in his model GFalso ep e-
sen s he in ensi y o he g a i a ional ield in Ndi-
mensions. Thus, his model sugges s he exci ing pos-
sibili y ha he scala ield which is needed in he s an-
da d model may jus be he g a i a ional ield which is
w apped ou in o he addi ional dimensions [10]. The
e y small New on cons an Ggi es he esidual alue
o he g a i a ional ield which spills ou in o he usual
h ee dimensions, beyond he adius o compac i ica-
ion Rc.
Ca ying u he his line o hinking, we can
conside ha , o dis ances la ge han Rc, he gauge
o ces li e in he 3-dimensional b ane, and so does
he no mal, e y weak g a i y. Be ween Rcand Re,
he gauge o ces s ill li e in h ee dimensions, while
g a i y li es in Ndimensions, inc easing i s s eng h
as he dis ance dec eases, so ha o =Reis
becomes compa able o he elec o-weak o ce. A his
poin , and due o a ye unknown mechanism, g a i y
gene a es he scala ield ha couples o he gauge ield
gi ing ise o he Higgs mechanism.
In Table 1 we display bo h alues o Rcas a
unc ion o n, which u n ou o be close o he ones
e alua ed by o he means. No e ha n=1 can be
eadily disca ded since i leads o a alue o Rcla ge
Table 1
Es ima es o a ious leng hs (in m), ene gies (in MeV), and phase shi s (in ad) as a unc ion o he numbe o ex a dimensions n
n1234567
Rc(Eq. (3)) 1.2×1030 2.8×1013 8.1×1071.4×1053.0×1032.3×10238
Rc(Eq. (6)) 9.3×1027 2.5×1012 1.6×1074.2×1041.1×1031.0×10219
Vn( =30)1.6×10−72.4×10−12 4.0×10−17 7.1×10−22 1.3×10−26 2.6×10−31 5.1×10−36
R=18 25 32 39 47 54 62
Vn(R=)4.4×10−74.1×10−12 3.1×10−17 1.9×10−22 9.0×10−28 4.2×10−33 1.5×10−38
Emin –– 2.3×10−13 7.9×10−81.7×10−42.7×10−21.1
Eop 4.62.31.50.98 0.68 0.51 0.39
φop 7.1×10−81.2×10−12 1.6×10−17 2.1×10−22 4.9×10−28 5.6×10−33 1.2×10−38
18 A. F ank e al. / Physics Le e s B 582 (2004) 15–20
han he size o he sola sys em and hence o uns able
plane a y o bi s. Fo n=2we indRc∼mm o cm.
De ia ions om New on’s law a his scale seem o
be disca ded by expe imen . Ne e heles, i should
be no ed ha ou analysis can only be expec ed o
gi e ough es ima es wi h conside able unce ain ies
in he p edic ion o Rc. I is he ange below 1 mm,
which is e y di icul o explo e wi h mac oscopic
g a i y expe imen s, ha we would like o in es iga e
by means o neu ons. We should no e ha e en i
he LED hypo hesis u ns ou o be w ong, i is s ill
an in e es ing ques ion o analyze whe he neu on
expe imen s can un eil de ia ions om New on’s law
a sho dis ances.
The physics o slow neu ons has unde gone sig-
ni ican e olu ion in he las decades. Neu ons ha e
become a s anda d p obe o nuclea physics expe i-
men s as well as o o he a eas including he s udy
and dynamics o condensed ma e [11]. Pulsed neu-
on beams can cu en ly be gene a ed wi h e y p e-
cise ene gies and pola iza ions and neu on de ec o s
achie e e y high e iciencies. Delica e expe imen s
wi h he mal neu ons ha e ecen ly demons a ed he
quan iza ion o hei ene gies when subjec o Ea h’s
g a i a ional ield [12]. In o de o a emp neu on
ND g a i y expe imen s a sho anges we ace wo
p oblems om he ou se . The mo e ob ious one is
he s ong nuclea o ce, p esen a ange scales o
he o de o 10−15 m. A second, less ob ious p ob-
lem is ha e en o a spin-ze o a ge nucleus, a neu-
on app oaching i wi h speed eels a magne ic ield

B=1
2c
E× in i s es ame, whe e 
Eis he nuclea
elec ic ield due o i s cha ge Z. A long- ange elec-
omagne ic in e ac ion ( he Schwinge e ec [13]) o
s eng h µ·
B esul s, whe e µis he neu on magne ic
momen . Neu ons ha e been p oposed p ima ily o
a oid di ec compe i ion wi h he much s onge elec-
omagne ic in e ac ion. We see ha he e is a esidual,
ela i is ic e ec which needs o be deal wi h. To min-
imize his po en ially compe ing in e ac ion slow neu-
ons a e equi ed, possibly pola ized in he inciden
di ec ion. (No e ha he e ec a e ages ou o ze o o
unpola ized p ojec iles.) Ve y slow neu ons will su -
e essen ially pu e s-wa e nuclea sca e ing, while
he longe ange ND g a i a ional o ce can in p in-
ciple p oduce sca e ing o highe l-wa es. The main
ques ion is whe he in e e ence e ec s be ween nu-
clea and g a i a ional sca e ing can be de ec ed, in
a ashion simila o he obse ed in e e ence e ec s
be ween nuclea and elec omagne ic o ces in hea y-
ion eac ions which gi e ise o ‘ ainbow’ sca e ing
and o he such phenomena [14].
The po en ial ha p oduces he modi ied g a i a-
ional o ce o dis ances below Rccan be w i en
as
(7)Vn( ) =m1m2GRn
c
(n +1) n+1.
Le us i s conside he in e ac ion o a neu on beam
wi h a hea y nucleus such as 208Pb. The cons an
m1m2Gis ex emely small in his case gi ing ise o
e y small alues o he po en ial ene gy a a ypical
dis ance o =30 m (see Table 1). We will discuss
wha a e he op imal expe imen al condi ions which
could allow obse a ion o his iny e ec in neu on-
sca e ing expe imen s and will conside wha is he
adequa e ene gy and angula momen um so ha he
phase shi due o he g a i a ional o ce is as la ge as
possible.
The nuclea po en ial can be pa ame ized wi h a
Woods–Saxon shape, so ha
(8)Vnucl( ) =V0
1+exp( −R)/a .
Reasonable pa ame e s a e V0=50 MeV, R=1.2×
A1/3 m, and a=0.6 m. This gi es, o dis ances
below =10 m, alues o he po en ial in he MeV
ange and any g a i a ional e ec a ha dis ance
would be d owned by he unce ain ies in he nuclea
po en ial. Ins ead, one mus p obe dis ances a which
he nuclea and g a i a ional po en ial a e o he same
o de . In Table1 we indica e he dis ances R=a which
he nuclea and g a i a ional po en ial a e equal, as
a unc ion o n. Fo hese calcula ions we ha e used
he es ima es o Rc om Eq. (4) al hough Eq. (6)
leads o simila esul s. Any sca e ing obse able
ha is a ec ed by dis ances smalle han R=will be
con amina ed by nuclea e ec s. We ind ha , i he
numbe o addi ional dimensions is la ge han n=6,
hen he g a i a ional o ce will be smalle han he
nuclea o ce o any alue o <R
c.Fo n⩽6 he e
exis s a ange R=< <R
cin which he g a i a ional
e ec s a e la ge han nuclea e ec s, and, a leas in
p inciple, migh be measu able al hough small.
The sca e ing obse ables a e also a ec ed by
he ac ha he neu on has bound s a es in he
A. F ank e al. / Physics Le e s B 582 (2004) 15–20 19
nuclea po en ial gene a ed by 208Pb. F om he shell
s uc u e o his nucleus one knows ha he single-
pa icle po en ial suppo s bound s a es up o angula
momen um L=7( he1j15/2o bi al). Thus, he
sca e ing o neu ons wi h L⩽7 is a ec ed by he
nuclea po en ial, e en i he sca e ing is e y small,
because he sca e ing wa e unc ions ha e o be
o hogonal o hebounds a es.Consequen ly, oob ain
sca e ing obse ables ee o nuclea con amina ion,
we need o conside L>7.
In o de o in es iga e g a i a ional e ec s, he
ene gy o he neu on canno be a bi a ily low. As one
is in es iga ing e ec s which occu a dis ances below
he compac i ica ion leng h Rc, he wa e leng h o he
neu on should be smalle han Rc. Mo e speci ically,
he momen um o he neu on should be such ha
(9)pnRc>(L
min +1/2)¯
h,
whe e Lmin =8 in he example o 208Pb. The co e-
sponding minimum ene gies o he neu on,
(10)Emin =p2
n
2mn=(Lmin +1/2)2¯
h2
2mnR2
c
,
a e gi en in Table 1.
Es ima es o he phase shi s due o he g a i a ional
in e ac ion can be ound in he eikonal app oxima ion
whe e hey a e ob ained o a gi en angula momen-
um in e ms o he ime in eg al o he po en ial along
a s aigh line ajec o y which has he same angula
momen um:
φ(L,E) ≃1
¯
hVb2+ 2 2d ,
(11)b=¯
h(L +1/2)
mn .
The ime in eg al can be es ima ed by aking he
po en ial a he poin o closes app oach =b,
mul iplied by he app oxima e cha ac e is ic ime τ=
b/ o in e ac ion. In his way, we ind
(12)φ(L,E,n) ≃m1m2GRn
c
(n +1)bn¯
h .
F om his exp ession we see ha , in o de o enhance
he sca e ing e ec s o he g a i a ional o ce, one
would need o ha e, in p inciple, small impac pa a-
me e and small eloci y which a e ela ed h ough
b mn=(L +1/2)¯
h. To maximize he phase shi
φ(L,E,n), he bes choice is o ake he minimum
angula momen um Lmin =8 and he minimum im-
pac pa ame e which co esponds o he dis ance R=
a which he nuclea o cemay s a o play a ole. This
gi es he ollowing op imal ene gy o sca e ing:
(13)Eop =(Lmin +1/2)2¯
h2
2mnR2
=
.
These ene gies a e shown in Table 1 and a e o he
o de o 1 MeV. A hese ene gies he op imal phase
shi s ( o L=Lmin), also shown in Table 1, can be
e alua ed o gi e
(14)φop ≃Vn(R=)
2Eop .
Fo n>2 he phase shi is ex emely small and
dec eases as ng ows. Howe e , we belie e ha
he case o n=2 may be wi hin each o cu en
dedica ed sca e ing expe imen s. Fo his case, he
elas ic sca e ing ampli ude p oduced by he modi ied
g a i a ional o ce is gi en by he exp ession
(15)Ag(E, θ) =i
k
L
(2L+1)φ(L, E, 2)PL(cosθ),
whe e φ(L,E,2)is gi en by Eq. (12). This ampli ude
u ns ou o be ene gy-independen and can be w i en
in he closed o m
Ag(E, θ) =iAg g(θ), Ag=2¯
hc(mnmTGF)2
3(mn+mT),
(16) g(θ) =
L
(L +1/2)−1PL(cosθ),
whe e we ha e made use o Eq. (4). The modi ied
g a i a ional ampli ude can now be e alua ed o he
sca e ing o neu ons on 208Pb, using Eq. (16) and
we ind Ag=0.298 ×10−8 m, o be compa ed wi h
he ypical sca e ing ampli udes o he nuclea o ce,
which o a ange o ene gies o a ew MeV, a e
o he o de o An=7 m, al hough he ampli ude
can s ongly luc ua e wi h ene gy as esonances a e
c ossed. I would seem ha i is impossible o ob-
se e such a iny g a i a ional e ec , being so small
compa ed o he nuclea ampli ude. Howe e , he an-
gula dependence o hese ampli udes is qui e di e -
en . In con as o he nuclea pa , which is essen ially
independen o he sca e ing angle o θ1/Lmin,
he g a i a ional ampli ude in ol es he con ibu ion
o a signi ican numbe o angula momen a. This in

20 A. F ank e al. / Physics Le e s B 582 (2004) 15–20
u n implies ha he g a i a ional ampli ude di e ges
o small sca e ing angles. I is he combina ion o
hese cha ac e is ics which may open a window o ob-
se ean in e e encee ec .Mo e speci ically,weha e
de i ed ha g(θ) =K0(θ/2)+δg(θ),whe eK0is
he Bessel unc ion, which di e ges loga i hmically as
θ→0, and δg(θ) is a smoo h unc ion o he angle.
Ou analysis shows ha a e y small sca e ing
angles, he neu on–nucleus di e en ial c oss sec ion
is gi en by
(17)
dσ
dΩ =An(E, θ) +Ag(E, θ)
2
≃An(E, 0)
2
+2An(E, 0)Ag
(18)×sina gAn(E, 0)K0(θ/2).
No e ha while |An(E, 0)|2is abou 5 b/s , 2|An(E,
0)|Agis abou 0.4 nb/s . Disce ning such a ain whis-
pe in he mids o he nuclea backg ound oa can be
a o midable ask. Bu his ea may be accomplished
by ca e ully moni o ing bo h he angula and ene gy
dependence o he c oss sec ion. As he phase o he
nuclea ampli ude changes as esonances a e c ossed,
he diminu i e in e e ence be ween nuclea and g a -
i a ional ampli udes changes om cons uc i e o de-
s uc i e in e e ence.Ou p oposal hus equi es mea-
su emen s in ol ing wo de ec o s, one a he smalles
possible angles and a second a sligh ly la ge ones.
Fo pu ely nuclea sca e ing he a io o he c oss sec-
ions ( he a io o de ec ed neu ons) should emain
cons an as ene gy is changed. The p esence o a g a -
i a ional e ec o he kind discussed he e would be
signaled by small luc ua ions associa ed o he in e -
e ence be ween he wo in e ac ions. Mo eo e , hese
luc ua ions would no be andom, bu should co e-
la e wi h he magni ude o he c oss sec ion. A ca e-
ul analysis o hese luc ua ionsmay isola e a g a i a-
ional signa u e.
Conside ing he p og ess ha has been achie ed
ecen ly in neu on physics, as shown, o example,
in ime-o - ligh expe imen smeasu ing he p ecession
o pola ized neu ons h ough a gas [15], hese ideas
could be es ed in he nea u u e.
Al hough we ha e concen a ed he e on he e ec s
o ex a-dimensional g a i y on neu on sca e ing,
an en i ely di e en app oach can be a emp ed [16].
The quan um e ec s o ea h’s g a i y on neu ons
ha e been obse ed in expe imen s by Nes izhe sky
e al. [12]. The in e e ence pa e n o he neu on
densi y, which is o µm scale, could be a ec ed by
de ia ions om New on’s law a µm scale, p oduced
by he modi ied g a i y o he pla es. Fo n=2 he
es ima e o Rcis in he cm ange, so ND g a i y, a he
µm ange, is 108 imes la ge han no mal g a i y. The
cu en expe imen al se up [12] could disce n e ec s
o o ces 1010 imes la ge han no mal g a i y [16].
Bo h kinds o dedica ed neu on expe imen s could
shed ligh on he ques o addi ional dimensions in
he uni e se.
Acknowledgemen s
We wish o hank H.G. Bö ne , R.F. Cas en, V.
Nes izhe sky and A. Villa i o encou aging discus-
sions. A.F. is suppo ed by CONACyT, Mexico, and
J.G.C. by he Spanish DGICyT p ojec FPA2002-
04181-C04-04.
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