Phenomenologie des Cordes
Transcription
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 !