Phenomenologie des Cordes

Emilian Dudas
CERN-TH et CPhT-Ecole Polytechnique
Phenomenologie des Cordes
23 mars, 2006, GDR Xtra dims, Annecy
Outline
• Pourquoi la gravitation est differente
• Theories d’unification. Unification des couplages.
• Dimensions supplementaires et unification a
la Kaluza-Klein
• Theorie des cordes et quantification de la
gravitation
• Compactification de dix a quatre dimensions
et chiralite
• Anomalies et leur compensations
• Stabilisation des moduli et brisure de supersymetrie
• D-branes et modeles d’univers branaire
• Unification des interactions a basse energie
• Cordes : Succes et problemes
1. Pourquoi la gravitation est
differente
Il y a quatre interactions fondamentales dans
la nature :
Interaction
Description
Gravitation
Rel. gen.
Electromagn.
F orte
F aible
M axwell
Y ang − M ills (QCD)
W einberg − Salam
Portee
Inf inie
Inf inie
10−15m
10−17m
A l’exception de la gravitation, toutes les autres
interactions sont decrites par des theories quantiques des champs qui sont renormalisables.
theorie quantique des champs = relativite restreinte + mecanique quantique
Le calcul des observables physiques est essentiellement base sur la theorie des perturbations
Les interactions ponctuelles dans les diagrammes
de Feynman engendrent des divergences ultraviolettes (UV)
theorie renormalisable → les divergences UV
peuvent etre reabsorbees dans un nombre fini
des parametres.
La renormalisation predit la variation avec l’energie
de la constante de structure fine, qui a ete confirme experimentalement par les mesures de
precision au LEP, CERN, a Geneve.
La relativite generale d’Einstein est une theorie
classique.
Masse (energie) → geometrie d’espace temps
gμν et courbure de l’espace
Sa quantification
gμν = ημν + hμν
conduit a des divergences UV qui ne peuvent
pas etre reabsorbees dans un nombre fini des
parametres
→ theorie non-renormalisable
2. Theories d’unification. Unification des couplages
L’electromagnetisme, les interactions electrofaibles et fortes sont toutes incorpores par le
principe de jauge : le couplage des champs
μ
de matiere aux potentiels vecteurs Ai apparait
par le couplage minimal
μ
μ
μ
pμ → pμ − g1QA1 − g2T2A2 − g3T3A3 ,
(1)
ou g1, g2, g3 sont les couplages des trois interactions, Q est la charge electrique et T2, T3 sont
des generateurs des groupes de symetries internes SU (2) et SU (3). Le groupe de symetrie
interne du Modele Standard est
SU (3) × SU (2) × U (1)
(2)
L’histoire de la physique a montre a plusieures
reprises que des interactions superficiellement
differentes sont souvent des manifestations de
la meme lois fondamentale.
Ex :
• la chute des corps vers la terre + l’attraction entre
les planetes → l’attraction gravitationelle universelle
• l’electricite + le magnetisme → l’electromagnetisme
Question :
est-ce que les trois interactions
: electromagnetique, forte et faible, sont des
manifestations differentes d’une seule interaction unifiee ?
mais les couplages d’interactions mesures experimentalement sont tres differents ...
Georgi-Glashow,1974
SU (3) × SU (2) × U (1) devient SU (5) a haute
energie, avec un seul couplage d’interaction.
Les trois couplages evoluent avec l’energie et
s’unifient a une tres grande echelle d’energie
3. Dimensions supplementaires et
unification a la Kaluza-Klein
L’interaction electromagnetique et la gravitation peuvent etre decrites d’une maniere unifiee a partir d’une gravitation d’Einstein a 4 +
1 dimensions
cinq dimensions
gM N
quatre dimensions
gμν ,
(graviton)
(graviton)
Aμ = gμ5 ,
g55
photon
scalar
Mais si la nouvelle dimension d’espace est infinie, la force d’attraction gravitationelle est
F ∼ m1r3m2
au lieu de
F ∼ m1r2m2
→ la nouvelle dimension y doit etre compacte
(ex. circle de rayon R) et petite
Les particules observees sont les modes de vibration φ(m)
φ(xμ , y) =
∞
imy
e R
φ(m) (xμ)
m=−∞
Meme nombres quantiques (charge,spin, etc)
que les particules ordinaires φ(0). Masse
m2
2
Mm = 2
R
On n’a pas vu des photons ou electrons de
KK → typiquement seulement des dimensions
R < 10−17cm (M1 ≥ T eV ) sont permises.
4. Theorie des cordes et quantification de la gravitation
2 ∼ nM 2
corde classique → modes vibration ωn
s
corde quantique → particules Mn2 ∼ nMs2
Conditions consistence → 10 dims d’espacetemps
→ six dimensions d’espace supplementaires
- Nombre egal de bosons et fermions, supersymetrie → supercordes : energie du vide zero
- Il y a 5 supercordes 10 dims., caracterises par
ls (Ms) = longueur (masse) de la corde
gs = eφ couplage de la corde, φ = dilaton
Il y a deux types des cordes :
cordes fermees
excitations : gravitons
cordes ouvertes
excitations : electrons,etc
Les cordes n’ont pas des interactions ponctuelles
→ pas des divergences UV !
Quelques proprietes remarquables des supercordes :
ls2
- symetrie R → R → ls = distance minimale ?
- symetries nonperturbatives gs → g1 , qui echangent
s
des etats perturbatifs aves des etats nonperturbatifs
2
M
s
>
|P, masse2 = gsMs2 > → |N P, masse2 =
gs
- En regime de couplage fort, une nouvelle
dimension peut apparaitre , avec
3/2
R ∼ gs
10 dims
etats :
Mn = gn
s
11 dims
n
Mn = R
- une theorie des cordes sur un espace courbe
(anti-de Sitter × une sphere)
↔
theorie des champs conforme a quatre dimensions. Theorie de corde faiblement couple ↔
theorie de champ fortement couple
→ nouvelles methodes de calculs nonperturbatifs en theorie des champs
→ holographie , cruciale dans la comprehension des trous noirs.
L’espace compact de la theorie des cordes est
dynamique
les composantes de la metrique gIJ sont des
champs scalaires appelles des moduli a quatre
dimensions.
Certains d’entre eux ont une masse zero et correspondent a des directions plates du potentiel
Des masses zero sont inacceptables pour la
gravitation et la cosmologie.
Un probleme central → est de donner des masses
(ou stabiliser les) champs de moduli.
5. Compactification de dix a quatre
dimensions et chiralite
The light fields in toroidal compactifications
are zero modes of the 10d fields, that depend
on the topology of the compact space.
m, n = six dim. internal indices
→
gAB : gμν (graviton) gmn (scalars) gμm (vect.) ,
BAC : Bμν (axion) Bmn (axions) Bμm (vect.)
where gmn are scalars describing the size and
the shape of the compact space.
Toroidal compactification of string theories to
4d gives rise to spectra with N = 4 SUSY.
Proof : decompose massless 10d states, representations of
SO(8)
→
SO(2) ⊗ SO(6)
Under this decomposition, a Majorana-Weyl
10d spinor (e.g. a supercharge) decomposes
as
8s = (2, 4)
and corresponds to four supercharges in 4d.
The number of 4d SUSY is given by the number of covariantly constant spinors a satisfying
∇μ a = 0
(3)
Their number N is governed by the holonomy
group of the compact space.
Imposing the
SU (3) subgroup of SO(6) to be the holonomy
group reduces to ONE the number of covariantly constant spinors, according to
(2, 4) = (2, 1) + (2, 3) ,
since only (2, 1) is SU (3) invariant.
A well-known example : Calabi-Yau spaces .
5.1 Orbifolds = A simple way of reducing
the number of SUSY and of producing fermion
chirality
• A d-dimensional orbifold Od is a d-dim. euclidean space Rd or the d-dimensional torus T d
with identified points as
Od = Rd/S = T d/P
,
where
• the space group S contains rotations θ and
translations v
• the point group P is the discrete group of rotations obtained from the space group ignoring
the translations.
A typical element of S acts on compact coordinates as
X → θX+v
and is denoted (θ, v).
• The subgroup of S formed by pure translations (1, v) = the lattice Γ of S.
T d = Rd/Γ
Points of T d can then be further identified under P to form the orbifold Od.
This is consistent only if P consists of rotations
which are automorphisms of the lattice Γ.
We are interested in N = 1 models, obtained
by orbifolding the six real (three complex) internal coordinates (x0, x1, x2, x3 = spacetime
coord.)
1
z1 = √ (x4 + ix5)
2
1
z3 = √ (x8 + ix9)
2
by the twist
,
1
z2 = √ (x6 + ix7)
2
,
θ (z1, z2, z3) = (e2iπv1 z1, e2iπv2 z2, e2iπv3 z3)
• v ≡ (v1, v2, v3) is called the twist vector
• for a ZN orbifold θ N = 1.
The action of the orbifold on a 10d MajoranaWeyl spinor denoted as |s1s2s3s4 >, where si =
,
±1/2 are the helicities in the spacetime and the
three compact torii, is
θ |s1s2s3s4 > = e2πi (v1s2+v2s3+v3s4) ) |s1s2s3s4 >
= eπi (±v1±v2±v3)) |s1s2s3s4 >
•If v1 ± v2 ± v3 = 0 with some fixed sign choice,
with all vi = 0, one 4d spinor is invariant and
the model has N = 1 SUSY
5.2 Branes at angles : Intersecting
brane worlds
Rotate the branes in the compact space.
There are three angles θ1, θ2, θ3 that D6 brane(s)
can make with the horizontal axis x4, x6, x8 of
the three torii of the compact space.
The relevant quantities are the relative angles
(12)
θi
(1)
= θi
(2)
− θi
. The number of unbroken
SUSY (supercharges) is
(12)
= 0 , θ1
(12)
± θ2
(12)
± θ2
θ3
θ1
θ1
(12)
(12)
± θ3
(12)
± θ3
(12)
± θ2
= 0
(12)
= 0
(12)
= 0
→ N = 2 SUSY
→ N = 1 SUSY
→ N = 0 SUSY
In the compact space, there are two important
additional ingredients :
• the rotation of branes in the compact space
is quantized, according to
(a)
(a)
tan θi
(a)
(mi
=
mi
Ri2
(a)
ni Ri1
(4)
(a)
, ni ) = integers, the wrapping numbers
of the brane(s) D(a) along the compact torus
Ti2.
- The total internal volume of the brane D (a)
is then
V (a) = (2π)3
3 (a),2 2
(a),2 2
Ri2 + ni
Ri1
mi
i=1
(5)
For two stacks of branes D(a) and D(b), it can
easily be shown geometrically that the number
of times they intersect in the compact torus
Ti2 is given by the intersection number
(ab)
Ii
(a) (b)
(a) (b)
ni
− ni mi
= mi
(6)
The remarkably simple property of the intersecting brane constructions is that they easily
generate chirality.
• simplest example with two sets of Ma coincident D(a) and Mb coincident D(b) D6 intersecting branes :
- the gauge group is U (Ma) ⊗ U (Mb)
- the D(a) − D(a) and D(b) − D(b) open spectra
are non-chiral
- the strings stretched between the two sets of
D-branes have a chiral fermionic spectrum in
the representation
(Ma, M̄b)
(7)
of the gauge group
- multiplicity = total number of times the branes
D(a) and D(b) intersect in the compact space
I (ab) =
3
i=1
(ab)
Ii
=
3
(a) (b)
(a) (b)
ni − ni mi ) .
(mi
i=1
(8)
• Various quasi-realistic models were constructed
in the last couple of years.
- The generic construction contains four (or
more) stacks, containing D-branes with a minimal gauge group
U (3) × U (2) × U (1)2 = SU (3) × SU (2) × U (1)4.
- Out of the four abelian gauge factors, three
are anomalous and get masses of the order the
string scale.
- One linear combination is massless and is to
be identified with the hypercharge.
- The quarks and leptons come typically from
the byfundamental states of the open strings
stretched between the various D-brane stacks.
- Right-handed neutrinos are usually part of
the massless spectrum
- The number of Higgs scalars is typically large,
but it can be reduced.
Anomalies et leur compensations
- A consistent quantum field theory should have
no gauge anomalies.
T r(Qi Qj Qk )
=
0
Qi is the generator of a local (abelian or nonabelian) gauge symmetry ; the trace is over the
whole chiral fermionic spectrum of the theory.
- A consistent string theory is also anomalyfree, but anomaly cancelation can be achieved
in a non-trivial way. This is mainly due to axionic type fields with nonlinear gauge transformations. Their couplings to gauge fields produce local gauge variations compensating the
triangle anomalies.
•The simplest example is the anomalous U (1)X
factor present in heterotic string compactifications.
Denoting the gauge group as
G = a Ga ⊗ U (1)X
an explicit computation by using the massless spectrum shows that there can be nonvanishing mixed gauge anomalies
U (1)X G2
a
:
Ca
1
2
T
r(Q
a X)
2
4π
=
1
3
T
r(X
)
2
4π
1
2
U (1)X SO(1, 3) : Cgrav =
T rX ,
192π 2
where the last anomaly is the mixed gaugeU (1)3
X
:
CX =
gravitational anomaly.
- All the other gauge anomalies have to vanish.
• the values of the mixed anomalies are not
independent, they are related through the relation
δGS
=
Ca
ka
=
CX
kX
=
1
T rX
2
192π
(9)
ka (rational numbers) = Kac-Moody levels, define the gauge couplings.
In superspace notation, the gauge kinetic function is
1
d θ ka S W α,a Wα,a + h.c.
4
2
where S is the universal dilaton-axion superfield and Wα,a denotes the Ga chiral gauge superfield.
The anomalies (9) define a consistent theory
since the Kahler potential of S is of the form
K(S, S̄) = − ln (S + S̄ − δGS VX )
and contains a Stueckelberg mixing
μ
δGS AX ∂μ ImS
between the axion ImS and the gauge field.
The supergauge transformations which leave
invariant the Kähler potential are
VX → VX + Λ + Λ̄
,
S → S + δGS Λ .
The gauge variation of the whole effective action is then
δS = −
1
4
d2θ Λ
(CA−δGS kA )W α,A Wα,A +h.c
A=a,x
which vanishes precisely when (9) holds.
Anomaly cancelation in orientifold models involves several axions. Abelian gauge fields →
Stueckelberg mix with the axions which render
the corresponding, “anomalous” gauge fields,
massive.
If their mass is in the TeV range, they can
behave like Z gauge bosons. However, these
massive gauge bosons can and do have anomalous couplings which naively break gauge invariance.
An important role is played by local and gauge
non-invariant terms called generalized ChernSimons terms
• Relevant terms in the effective action
1
1
μν
I
I i 2
F
F
−
(∂
a
+
M
Aμ ) ,
S = −
μ
i,μν
i
i
2
2 I
i 4gi
1
1
I
IF F +
+
C
a
Eij,k AiAj Fk ,
i j
ij
2
2
24π
48π
- Ai are abelian gauge fields, aI are axions with
Stueckelberg couplings which render massive
(some of ) the gauge fields
Anomaly cancellation conditions read
I = 0 ,
tijk + Eijk + MiI Cjk
where tijk = T r(Qi Qj Qk ) and Qi is the generator for Ai.
Axionic exchanges = nonlocal contributions,
whereas the GCS terms are local terms → the
sum : triangle diagrams, axionic exchange and
GCS terms is gauge invariant but non vanishing, and leads to anomalous three gauge boson
couplings at low energy.
Stabilisation des moduli et brisure
de supersymetrie
Moduli stabilization is one of the major problems in string phenomenology.
- In SUSY compactifications, moduli fields are
massless and correspond to flat directions in
the scalar potential → unacceptable.
- SUSY breaking generate typically runaway
moduli potentials which are unacceptable due
to the induced time dependence and rolling
of the moduli towards uninteresting configurations.
Stabilisation of the dilaton φ is a nonperturbative phenomenon, since
gs = eφ
is the string coupling constant, defining perturbation theory.
Two examples of moduli stabilisation
1) Nonperturbative effects (gaugino condensation) in a hidden sector
The hidden sector is an asymptotically free
gauge theory, ex. super Yang-Mills, dynamical scale Λ
−
Λ = MP e
1
2b0 g 2
0
where b0= beta function of the hidden sector
gauge theory.
In string (or brane) context, gauge couplings
are vev’s of moduli fields, say T = t + ia
1
= t
g02
then we generate nonperturbative moduli potentials
non − SUSY
SUSY
:
:
V (t) ∼ Λ
4
− b2t
= e
0
3T
− 2b
W (T ) ∼ e
0
The typical examples in string effective supergravity are
• Heterotic strings
K = −3 ln(T + T̄ ) − ln(S + S̄)
3S
− 2b
W = W0 + e
0
• type II strings
K = −3 ln(T + T̄ )
3T
− 2b
W = W0 + e
0
2) Fluxes in the compact space (type II strings)
Ex. :
2
2
+ e−φ Hijk
+ ···
L = eφFijk
where (i, j, k= compact indices )
Fijk = ∂iCjk + · · ·
,
Hijk = ∂iBjk + · · ·
It is consistent (but subject to quantization
conditions) to have
Fijk = c1ijk
1
2πα
,
1
Fijk = 2πn ,
2πα
Hijk = c2ijk
Hijk = 2πm
(10)
V (φ)
∼
φ
2 −φ
c2
1 e + c2 e
→
stabilisation at
eφ = cc2
1
In practice, several moduli fields → several stabilisation methods used simultaneously.
Gravity mediated SUSY breaking ↔ Moduli Ti
mediated SUSY breaking
Fi ∼ m3/2 MP
(11)
Soft masses in the observable (MSSM) sector
are
m̃M SSM ∼
Fi
∼ m3/2
MP
(12)
In this case, generically all masses (moduli masses,
soft masses for MSSM fields) are
mi ∼ m̃M SSM ∼ m3/2 ∼ TeV
(13)
• Moduli can be much lighter for other mechanisms to break SUSY
5. D-branes et modeles d’univers
branaire
La theorie des cordes a des hyper-surfaces a
p dims. d’espace appelles des D-branes, qui
contiennent des champs de jauge (couplage g)
et des champs de matiere
Univers Branaire = les trois interactions de
jauge (cordes ouvertes) sont localisees sur une
D3 brane
La gravitation (cordes fermees) vit partout (“dans
le bulk”).
Les (six) dimensions supplementaires perpendiculaires peuvent etre de taille macroscopique
Rperp ≤ 10−1 mm,
contrainte venant des mesures des eventuelles
deviations de la loi de Newton .
Dans ce cadre, les relations
MP2 =
1
Vperp Ms8
gs
g 2 = gs
avec 2 dims. de taille extreme Rperp ∼ 10−1 mm
donnent une echelle de masse de la corde
Ms ∼ 3 − 10
T eV
La gravitation devient une interaction forte a
des energies comparables a Ms.
→ des effets observables dans le collisionneur
LHC au CERN-Geneve en 2007 !
Contraintes experimentales :
- dimensions paralleles : Rpar ≤ 10−17 cm
-dimensions perpendiculaires : Rperp ≤ 10−1
mm
- Si brisure SUSY sur les branes, echelle MSU SY ∼
Ms
2
MSU
SY ∼ 10−3 eV
mbulkmoduli ∼
MP
La cosmologie est completement differente dans
l’univers primordial
→ signatures observables dans le rayonnement
cosmic de fond (CMB) ?
6. Unification des interactions a
basse energie
• L’unification des couplages semble etre difficilement reconciliable avec une basse echelle
fondamentale Ms.
Comment obtenir une unification plus rapide ?
- Les particules elementaires : electron, quarks,
etc se propagent dans les dimensions paralleles . Leur etats de Kaluza-Klein produisent une
evolution acceleree des couplages.
→ unification acceleree.
Unification of gauge couplings in the presence
of extra spacetime dimensions. We consider
two representative cases: R−1 = 105 GeV
(left), R−1 = 108 GeV (right).
Cordes : Succes et Problemes
• La theorie des cordes permet des calculs de
gravitation quantique
• Les theories de jauges et la gravitation
d’Einstein sont retrouvees a des energies
E << Ms
• Les dualites des cordes ont ameliore
considerablement l’etude des effets
nonperturbatifs en theorie des champs
• Les D-branes → univers branaires, qui ont
ramene les supercordes a la portee des
collisionneurs
• L’unification n’est pas necessairement un
reve a des energies inaccessibles
Espoirs, problemes
• La singularite de big-bang ne peut pas etre
adressee dans la relativite generale (RG). La
theorie des cordes peut traiter certaines
singularites.
Grande Courbure d’espace → au-dela de la
RG
• La technologie actuelle des cordes est tres
limite : premiere quantification, des fonds
tres particuliers
• Les supercordes possedent une
supersymetrie d’espace-temps. LHC va
chercher une supersymetrie spontanement
brisee a basse energie. Est-ce possible de
briser la supersymetrie ?
• La stabilisation des fluctuations de l’espace
compact (moduli) demande necessairement
des methodes nonperturbatives
• Le nombre de vide des supercordes est
enorme. Qui/comment a choisi notre univers
? Principe anthropique ?
• Est-ce que tout est calculable dans les
cordes ? La theorie a un seul parametre libre
!