Can we live in the bulk without a brane?

0.13 MB·English

by  Kurt Hinterbichler

Can we live in the bulk without a brane?
Kurt Hinterbichler1, Janna Levin2,3, and Claire Zukowski2,4
1Center for Particle Cosmology, University of Pennsylvania, Philadelphia, PA 19104
2Institute for Strings, Cosmology and Astroparticle Physics,
Physics Department Columbia University, New York, NY 10027
3Department of Physics and Astronomy, Barnard College, New York, NY 10027 and
4Department of Physics, University of California, Berkeley, CA 94720
We suggest a braneless scenario that still hides large-volume extra dimensions. Ordinarily the
strength of bulkgauge interactions would bediluted overthelarge internalvolume, makingall the
four dimensional forces weak. We use the fact that if the gauge fields result from the dimensional
reduction of pure higher-dimensional gravity, then the strengths of the four dimensional gauge
interactions are related to the sizes of corresponding cycles averaged over the compact internal
1 manifold. Therefore,ifagaugeforceisconcentratedoverasmallcycleitwillnotbedilutedoverthe
1 entiremanifold. Gravity,however,remainsdilutedoverthelarge volume. Thuslarge-volume, large
0 mass-gap extra dimensions with small cycles can remain hidden and result in a hierarchy between
2 gravity and the other forces. However, problematically, the cycles are required to be smaller than
n thehigher-dimensionalPlancklengthandthisraises concernoverquantumgravitycorrections. We
a speculate on possible cures.
J

1
A low-budget yet concrete observation about our uni- wouldbeweakenedrelativetothetruefundamentalscale.
1
verse is that it is four-dimensional, and although human For this reason, Standard Model gauge fields were local-

] beings by the billions confirm this observation daily, it ized on a brane while the gravitational field inhabited
h
might not be true. Our universe may have multiple the bulk. This split between habitats enforced a hierar-
t
- extra spatial dimensions and only appear to be four- chy between gravity and particle physics [1–3].
p
dimensional. Extra dimensions could hide if they are If we are to do away with the brane altogether and
e
h curled up so small that no observations to date could allow all fields to fill the bulk, then we need to save the
[ excite modes energetic enough to probe these directions. gaugecouplingsfromdilutionoverthelargeinternalvol-

Originally, interest in large-volume extra dimensions in- ume. Weshowherethatthisispossibleifthegaugefields
1
spired braneworlds as a new means to hide the extra di- are generated by the dimensional reduction of a purely
v
2 mensions – float our universe on a 3-brane and confine gravitational field – as in the original Kaluza-Klein re-
0 all Standard Model fields to that braneworld [1–3]. In duction [9, 10] – and there are some small Killing cycles
2 this article, we describe a braneless alternative that al- around the internal volume. In this picture, a photon is
2 lows us to hide large-volume extra dimensions. The in- reallyametricoscillationaroundanS1,andthestrength
1. ternalmanifoldshaveincommonthreeessentialfeatures: ofitscouplingisinverselyproportionaltothesizeofthat
0 (1) largevolume,(2) lowestmodes thatare energetically cycle. So while the volume is large, electromagnetic in-
1 expensive despite the large volume, and as we will see teractionsinvolveonlyonesmallcircleandnottheentire
1 (3) some small Killing cycles. However, the small cycles manifold. The dilution over the internal volume is com-
:
v are very small and therefore penetrate quantum gravity pensated by localizing all gauge interactions over small
i scales as we discuss shortly. cycles, instead of confining them to a brane.
X
We can lend intuition for why these three features are We will review the dimensional reduction of gravita-
ar essential. Partofthepicture waspresentedinaprevious tional fields from a higher-dimensional universe down to
article [4]. It is commonly assumed that the larger the a4-duniverseandshowthatthe conditions we wantour
volume, the lower the natural harmonics on the space. manifold to satisfy are the following:
The lower the notes, the less the effort that is required Large Volume: Underdimensionalreduction,thein-
to play them. If we were not confined to a brane, we tegratedvolume (inunitsofMN)isrelatedtotheratio
V
should expect to have observed a large internal volume oftheobserved4-dPlanckmass,mp,tothefundamental
already. However, this expectation is contradicted by higher-dimensional Planck scale, M TeV, through
∼
aninfinite numberofknownmanifolds thatdespitetheir
m 2
largevolumehaveno lownotes [4–8]. Suchspaceswould = p . (1)
V M
remainhidden despite their largesize because itremains
(cid:16) (cid:17)
too energetically expensive to probe them. ThisonlygeneratesahierarchyiftheHiggsmassissmall
Still, an expensive spectrum of modes is not sufficient comparedtom . Sincethe4-dmetricisnotwarped,any
p
to free the worldfrom incarcerationon a brane. If Stan- bulk scalar field added by hand will automatically have
dard Model fields were allowed to live in the bulk, it the mass it did in the bulk [1, 2]. Although vacuum
might seem that the strength of all the gauge interac- expectation values and couplings will be affected, com-
tions would be diluted over the large volume – just as binations of them lead to invariant masses [4]. So if the
the strength of gravity would be diluted – and all forces Higgs is a bulk scalarfield of mass M in the bulk, it will
2
reduce to a 4-d scalar field of mass M. excitations of the metric can be ignored at the energy
Large Mass Gap: The mass gap, set by the mini- scalesweareconsidering. Thenweusetheansatzforthe
mumnon-zeroeigenvalueoftheLaplacianontheinternal metric [11, 12]
manifold,mustbe largetosuppressKaluza-Kleinexcita-
tions, a condition we express as, g +AiAjξmξngˆ Aiξ
G = µν µ ν i j mn µ in , (7)
AB Ajξ gˆ
m &M. (2) (cid:18) ν jm mn (cid:19)
KK
where µ = 0,...,3 runs over 4-d coordinates x and m =
Small Cycles: Some of the 4-d gauge couplings, i.e.
5,...,4+ N runs over the internal coordinates y. The
those of the Standard Model, must be of order one. The
ξm(y) are the Killing vectors of that under the Lie
4-d gauge couplings can be expressed as i N
bracket obey the algebra of the isometry group of the
−1 internal manifold,
g s2 1/2M 1/2 , (3)
4d
∼ h i V
(cid:16) (cid:17) [ξi,ξj]µ =Cijkξkµ , (8)
where s2 1/2 is the root mean square of the circumfer-
h i
ence of the corresponding Killing cycle over the internal with C k the canonical ( 1) structure constants of the
ij ∼
space (as we review below [11]). Setting this 1 gives algebra. In words, the internal spacetime symmetries
≃
appear to us in 4-d to be proper gauge symmetries, and
s2 1/2 (M 1/2)−1 . (4) the off-diagonal excitations of the metric camouflage as
h i ∼ V
gauge fields.
Whenallthreeconditionsaremet,wehavelarge-volume
After dimensional reduction, we get 4-d gravity with
extra dimensions that can be hidden without invoking a metric g , 4-d Yang-Mills gauge fields Ai with gauge
µν µ
brane while still affecting a hierachy between the weak-
group isomorphic to , and scalar field moduli from the
ness of gravity relative to particle interactions. G
internal metric gˆ ,
mn
However, problematically, by Eq. (1) in Eq. (4),
M2+N 1
hs2i1/2 ∼m−p1 . (5) d4+Nx√−G 2 [R(g)−gˆmnξimξjn4FµiνFjµν +...
Z (cid:21)
(9)
The cycles corresponding to gauge interactions are
smaller by a factor of the hierarchy than the higher-
where Fi = ∂ Ai ∂ Ai +C iAjAk is the standard
dimensional Planck length, leading to curvature invari- µν µ ν − ν µ jk µ ν
non-abelianYang-Millscurvature. The...indicatesaddi-
ants (or analogous topological invariants) that are large
tionalmoduli terms, including curvature terms like R(gˆ)
andsusceptibletouncontrolledquantumcorrections. We
that serve as a potential for the moduli of the internal
willconsiderinternalmanifoldsthatareadirectproduct
dimension and/or contribute to the cosmological con-
of submanifolds as well as internal manifolds that are
stant. (While both moduli stabilization and the cosmo-
warped products of submanifolds. Although the inter-
logical constant are important problems for any higher-
nally warped spaces seem promising in that the small
cycles are of order M−1, the warping shrinks the cycles dimensionalmodel, we defer to the richliterature on the
subjects.)
overthespanofthemanifoldsotheyaremetricallysmall
in places. Consequently, we run into trouble with large Integrating the Einstein-Hilbert term over y, we see
curvature invariants and cannot claim controlled quan- that the (4+N)-d Planck constant is related to to 4-d
tum gravity corrections. Planck constant by
m 2
Gravity Reduction MN dNy gˆ= p . (10)
V ≡ M
The gravityreduction ofKaluza andKlein[9, 10] pro-
Z p (cid:16) (cid:17)
videdaremarkable,explicitdemonstrationofunification:
This is the first condition, Eq. (1). The internal volume
a metric flux around a circle in 5-d appeared to the 4-d
underdimensionalreductionmustbelargerelativetothe
world as the photon. Since then, all manner of gauge
fundamental scale M.
groups have been shown to result from the dimensional
Foragivensimple partofthe gaugegroup,the kinetic
reduction of pure gravity over higher-dimensional mani-
coefficientofthegaugefieldscanbechosendiagonal,and
folds. We consider pure (4+N)-d gravity
its coefficient will be related to the 4-d gauge coupling
M2+N g4d,
S = d4+Nx√ GR(G), (6)
2 −
Z M2 1
V gˆ ξmξn = δ , (11)
on a product space , where is 4-d and the 2 h mn i ji g2 ij
M × N M 4d
internal N-d manifold has isometry group . Let us
consider only zero-modNes under dimensional rGeduction, where ϕ(y) =MN −1 dNy√gˆϕ(y) indicates anaver-
h i V
which is equivalent to assuming that the Kaluza-Klein age of a function ϕ(y) over the internal volume.
R
3
The action is then the canonicalactionfor 4-dgravity The argument also generalizes to the case of a U(1)
with 4-d gauge fields: groupfactor,thoughone hasto couplein matter to read
offthecouplingstrength[11]. Weakorstronggaugecou-
S = d4x√ g m2pR(g) 1 Fi 2 +... (12) plings correspond to large or small cycles respectively.
Z − " 2 − 4g42d µν #
(cid:0) (cid:1) Direct Product Spaces: No Warping
The massive gravitons corresponding to Kaluza-Klein Consider as a first example a direct product of 4-d
modes for the metric that would appear in the action with an N-dimensional manifold built from a string
have masses m2 corresponding to eigenvalues of the MSn ... Sn, of D hyperspheres of radius R [4] taken
∼ k × × 1
scalar Laplacian on the higher-dimensional manifold, inproduct with one muchsmaller n-sphereofradius R ,
2
so N = n(D+1). The gauge fields will come from the
∇2(N)ψk =−m2kψk. (13) smallersphere, andwe have hs2i1/2 ∼R2. By our 3 con-
ditions, the internal space is subject to the constraints:
To suppress Kaluza-Klein modes, we require non-zero
eigenvalues of the Laplacian to be large, = D (R M)Dn(R M)n (m /M)2,
V VS1VS2 ∼ 1 2 ∼ p
mkmin &M . (14) mKK ∼R1−1 ∼M,
g (R M 1/2)−1 (R m )−1 1 . (20)
This is the second condition, Eq. (2). 2 ∼ 2 V ∼ 2 p ∼
To obtain the third condition, Eq. (4), we give a min-
Choosing R m−1, R & M−1 and D 1 easily
imal review of the argument in [11], to show that gauge 2 ∼ p 1 ≫
satisfiesall3conditions. ThegaugefieldsfromR couple
couplingconstantsinthelowerdimensionaltheory,given 2
with strength g 1 while the string of large Sn’s will
by Eq. (11), can be interpreted as averaged circumfer- 2 ∼
couplewithastrengthsuppressedbyafactorofM/m
ences over the compact internal manifold. Given a com- 10−16. p ∼
pact, simple Lie group acting on a compact manifold,
Ingeneralwecanbuildtheinternalmanifoldasaprod-
there is a Killing vector ξm corresponding to each Lie
i uctofanylarge-volume,largemassgapspacewithhighly
algebra generator T . A generic Killing vector ξm gen-
i
dilutedgaugecouplingstimesasmallmanifoldwithundi-
erates closed orbits Ym(λ), parametrized by some λ, in
lutedgaugecouplings. Anotherinterestinginternalman-
the corresponding compact manifold,
ifoldisprovidedbyasquashedT2inproductwithasmall
d space. Unlike the previous example, this manifold does
Ym(λ)=ξm(Y(λ)). (15)
notrequirelargedimensionality. Thelargevolume,large
dλ
mass gap comes from the squashed T2 as was shown in
Given a starting point y for the orbit, the solution is [13, 14] while the undiluted gauge coupling could come
from a small internal manifold such as CP2 S2 S1,
Ym(λ,y)=eλξn(y)∂nYm(0,y), (16) which has the isometries of the Standard Mo×del g×auge
group [15].
where the partial derivative in the exponent is with re-
There are an infinite number of 2-surfaces that could
spect to y. This generates the exponential map on the
participate in this construction. In [4], compact hy-
group manifold, and since the generators are normalized
perbolic 2-surfaces were considered. Surfaces of arbi-
canonically (structure constants are 1), and the group
∼ trarily large genus, and therefore arbitrarily large area,
is compact, the curve comes back to its starting point
A = 4π(g 1)R2, were shown to have minimum eigen-
after some order one range of λ (usually λmax =2π). value of ro−ughly k 1/(2R) [16–21]. With R M−1
The circumference of this curve is thus min ∼ ∼
and g 1, these qualify as large mass-gap, large-
≫
λmax dYmdYn volume manifolds. Additionally, hyperbolic spaces have
s(y)= dλ gˆ (Y) . (17) no Killing vectorsandso wouldnotcontribute any addi-
mn
dλ dλ
Z0 r tional,unwantedgaugefields. Wecouldequallywelltake
the internalspace to be a product of these large-volume,
By differentiating the quantity in the square root with
large mass-gap manifolds with a small internal manifold
respect to λ, using Eq. (15), and using Killing’s equa-
whose isometries generate the Standard Model.
tion gˆ = 0, it is straightforward to check that the
ξ µν
integrLand is actually independent of λ, so we have Thesedirectproductspaceshaveaniceinterpretation:
There is a large internal volume but gauge fields corre-
s(y) gˆ ξmξn. (18) spond to excitations along small cycles and so do not
mn
∼ requireringingthe wholebigmanifold,justasmallpiece
Taking the average as py varies over the submanifold of it. Therefore the gauge coupling is not diluted, while
, we have from Eq. (11) gravity is.
N
Despite this nice interpretation, these examples are
1 flawed. Oneofthegeometricscales,m−1,ismanyorders
g . (19) p
∼ M 1/2 s2 ofmagnitude smaller thanthe fundamental length scale,
V h i
p
4
M−1, with the considerable disadvantage that small cy- in that inner product, an additional factor of f is intro-
cles might force us into quantum gravity arenas, un- duced so that the gauge couplings consistent with Eq.
dermining the consistency of the analysis. The con- (19) are set by V s2 = VNGR2M2L2 (f2/y2)dzdy =
cRe2r,nRiµsνRthµaνt...hbigehceorm-deimsiegnnsiifiocnaanlt.opBereaftoorreswoef stpheecufolartme oαf2tVhNeGu(nL-Mwa)r2pRe2d(c1(cid:10)y−clǫ(cid:11)e)ownhtehreesRubims tahneifocRhldaraGct.erIitstfoicllosiwzes
onpossibleresolutions,weturntowarpedinternalspaces that N
next, althoughwe will see that these also have the prob-
−1
lem of small cycles. g αM2LR 1/2 . (23)
∼ VNG
(cid:16) (cid:17)
Warped Internal Spaces
Consider now a product of with an internal space For VNG ∼ LM ∼ RM ∼ α ∼ 1, we have an O(1)
coupling.
M
= H2 that is a product of a swath of the hy-
G The hierarchy between fundamental scales has been
Nperbolic p×laNne, H2, with coordinates y,z, and a space
shifted to a hierarchy between geometric scales. As with
with coordinates xˆm and metric gˆ with isometry
G mn the direct product, the warped internal product dilutes
N
group the desired gauge group. Let carry a warp
G gravity over a large internal volume while gauge fields
N
factor f(y) dependent on only the y coordinate of the
correspond to excitations over small cyles. Unfortu-
hyperbolic plane (note that the warping is internal and
nately, also like the direct product, the cycles vary by
that the 4-d metric carries no warp factor),
f(y)Randaremetricallysmallinplaces. The hazards
ofquantumgravitythusreappearasthebulkRicciscalar
L2 p
ds2 =g dxµdxν+f(y)gˆ dxˆmdxˆn+ dy2+dz2 . on the internal space is largerby the warp factor than is
µν mn y2 tolerable, with a contribution of the form f−1R(gˆ).
(cid:0) (cid:1)(21)
We have used the upper half-plane representation of the
Speculation
hyperbolic plane, with L the curvature scale. We take a
It may be that there is a no-go theorem that ensures
square swath on the plane between the limits 0 < z < 1
thesmallcyclesweneedarealwayscatastrophicallysmall
and ǫ < y < 1 where ǫ is small but non-zero in order to
if there are no branes. On the other hand, although we
render lengths and areas finite.
did not present the calculation here, we find the dimen-
In order for all components of the higher-dimensional
sional reduction of pure gravity where the external 4-d
metric to transform properly under the gauge transfor-
space is warped as in Randall-Sundrum – as opposed to
mations and create the illusion of gauge bosons, we re-
theinternalwarpingdetailedabove–doesallowforgauge
quiretheKillingvectorsthatgeneratethemtobeKilling
fieldstoliveinthebulkwithorderunitygaugecouplings
vectors of the entire metric, not just of the submanifold
following this prescription. However, this construction
. Butbecausethewarpfactordependsonlyontheco-
oNrGdinates of H2, Killing vectors of are automatically is less novel, and furthermore, the hierarchy requires the
NG Higgstobe constrainedto abranesodoes notqualify as
Killingvectorsoftheentireinternalmanifold. Finitevol-
braneless.
umehyperbolicmanifoldsdonothaveKillingvectorsand
Finally, it is a celebrated result of string theory that
so do not introduce additional gauge fields.
small cycles are controlled in the UV theory as extra
Choosing f(y)=αy, with α some order one constant,
light degrees of freedom resurface and naturally resolve
and a submanifold of dimension 2 and un-warped
volume VNG ∼1 in uNnGits of M2, the 4-d Planck mass by astnryingmyeftorricmaulldaitvioenrg,eansciemsil[a2r2]r.esoWluetisopnemcualyataellothwaat fiunllya
Eq. (10) is
braneless model with small cycles and good gauge cou-
plings. If string theory is the UV completion, there is
m2 1 1
Mp2 =VNGM2L2Z0 dzZǫ dy y−2f(y) acalslolytnhoenw-torrivryialthsmatawllicnydcilnegcamnobdeecsoamroeulnigdhta,shionmceotthopeiir-
=αVNG(LM)2ln(1/ǫ). (22) mass ∼ R/α′. This is not a problem if the Killing cy-
cles are homotopically trivial, as in our example with
The 4-d Planck mass is then large if we take ǫ very near spheres Sn. We also mention that there are explicit ex-
zero. We thereby meet condition Eq. (1) to create a amplesofhyperbolic3-foldswithlargevolumeandsmall
weakenedstrengthofgravitycomparedtoafundamental geodesics (see Snappea [23]). The curvature invariants
scale. couldbestabilizedatM−1 whilesustaininglargevolume
To check the mass gap of Eq. (4), we note that the and small cycles. Additional operators based on curva-
KK modes on should be of order M−1. Also, the tureinvariantswouldbe controlled,skirtingthe problem
G
N
eigenmodes on the swath of the hyperbolic plane should of small cycles.
be subcurvaturemodes andthereforethe eigenvaluesare Thereareofcourseotherissuesthatmusttraditionally
bounded from below by 1/(2L) so the mass gap is com- be confronted in any Kaluza-Klein scenario, such as the
fortably large for L M−1. incorporationofchiralfermionsandstabilizationofmod-
∼
The gauge couplings are set by the normalization of uli. In the meantime, it is encouraging that a braneless
the Killing vectors, Eq. (11). Due to the factor of gˆ cosmosmighthidelarge-volumeextradimensions. While
mn
5
all fields are smeared out over the large-volume,interac- BrianGreene,DanielKabat,DavidKagan,AlbertoNico-
tions with gauge fields are concentrated over relatively lis, Massimo Porrati,and Dylan Thurston for invaluable
smallcyclesandtherebymanagetoremainundiluted. It conversations. KH acknowledges support from NASA
is intriguing to imagine that we live smeared out over a ATP grant NNX08AH27G, NSF grant PHY-0930521,
largedrumandour experienceofforces otherthan grav- DOE grant DE-FG05-95ER40893-A020and the Univer-
ityareanillusioncreatedbythecadenceofsmallhidden sity of Pennsylvania. JL acknowledges support from an
subspaces. NSF Theoretical Physics grant, PHY - 0758022. JL also
gratefully acknowledges a KITP Scholarship.
Acknowledgements: WethankGregoryGabadadze,
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