Introduction to (Large) Extra Dimensions A. Lionetto Department of Physics & INFN Roma Tor Vergata SLAC “Dark Matter & Exotic Physics WG” – p. 1/39 Outline • Introduction • Small vs Large Extra Dimensions (LED) • Braneworld • Kaluza-Klein (KK) Particles • Orbifolds • Low Energy Theory and Phenomenology • Cosmology & Astrophysics • Extra Dimensions and GUT SLAC “Dark Matter & Exotic Physics WG” – p. 2/39 Introduction General framework: class of models with n extra space-time (usually space-like) dimensions D = 4 + n First attempt ( 1920) Kaluza and Klein ∼ ) Unification of the electromagnetism with the Einstein gravity with a compact 5th dimension. The photon comes from the extra components of the metric. SLAC “Dark Matter & Exotic Physics WG” – p. 3/39 Small Extra Dimensions Early assumption: extra dimensions are “small” and are compactified to manifolds (typical S1 S1) of small “radii” of the order of the × · · · × Planck length 1=2 8πG ~ − l = N 10 33 cm P c3 ∼ related to the Planck mass (the D=4 Physics relevant quantum gravity/string scale) D=4+n Physics ~c 1=2 M = 2:4 1018 GeV P 8πG ∼ × N SLAC “Dark Matter & Exotic Physics WG” – p. 4/39 Large Extra Dimensions (ADD Model) Gravity coupling in higher dimensional model G∗. In D = 4 G∗ = GN , the Newton constant. Let us assume a toroidal compactification of radius R (M 4 T n). The higher dimensional Einstein-Hilbert action× is 1 4+n Sgrav = d x g(4+n) R(4+n) 16πG∗ Z q where g(4+n) is the D = 4 + n metr ic 2 M N ds = gMN dx dx with M; N = 0; 1; ; 3 + n · · · SLAC “Dark Matter & Exotic Physics WG” – p. 5/39 The dimension of a scalar field φ is obtained from the 4 n A kinetic term d xd y@ φ∂Aφ that implies • [φ] = [MassR]1+n=2 Dimensional Analysis Assuming g(4+n) dimensionless. − • 2 2 R(4+n) = [Length] = [Mass] − • 2 [GN ] = [Mass] −(2+n) • [G∗] = [Mass] SLAC “Dark Matter & Exotic Physics WG” – p. 6/39 Dimensional Analysis Assuming g(4+n) dimensionless. − • 2 2 R(4+n) = [Length] = [Mass] − • 2 [GN ] = [Mass] −(2+n) • [G∗] = [Mass] The dimension of a scalar field φ is obtained from the 4 n A kinetic term d xd y@ φ∂Aφ that implies • [φ] = [MassR]1+n=2 SLAC “Dark Matter & Exotic Physics WG” – p. 6/39 • gµν is the D = 4 metric that depends only by the D = 4 coordinates xµ, µ = 0; 1; 2; 3 • a b a δabdy dy gives the line element of the bulk, y a = 1; ; n · · · + g(4+n) = g(4) R(4+n) = R(4) q q Dimensional Reduction In order to obtain the D = 4 effective action let us assume a flat ansatz (vs warped ansatz) ds2 = g (x)dxµdxν δ dyadyb µν − ab SLAC “Dark Matter & Exotic Physics WG” – p. 7/39 • a b a δabdy dy gives the line element of the bulk, y a = 1; ; n · · · + g(4+n) = g(4) R(4+n) = R(4) q q Dimensional Reduction In order to obtain the D = 4 effective action let us assume a flat ansatz (vs warped ansatz) ds2 = g (x)dxµdxν δ dyadyb µν − ab • gµν is the D = 4 metric that depends only by the D = 4 coordinates xµ, µ = 0; 1; 2; 3 SLAC “Dark Matter & Exotic Physics WG” – p. 7/39 + g(4+n) = g(4) R(4+n) = R(4) q q Dimensional Reduction In order to obtain the D = 4 effective action let us assume a flat ansatz (vs warped ansatz) ds2 = g (x)dxµdxν δ dyadyb µν − ab • gµν is the D = 4 metric that depends only by the D = 4 coordinates xµ, µ = 0; 1; 2; 3 • a b a δabdy dy gives the line element of the bulk, y a = 1; ; n · · · SLAC “Dark Matter & Exotic Physics WG” – p. 7/39 Dimensional Reduction In order to obtain the D = 4 effective action let us assume a flat ansatz (vs warped ansatz) ds2 = g (x)dxµdxν δ dyadyb µν − ab • gµν is the D = 4 metric that depends only by the D = 4 coordinates xµ, µ = 0; 1; 2; 3 • a b a δabdy dy gives the line element of the bulk, y a = 1; ; n · · · + g(4+n) = g(4) R(4+n) = R(4) q q SLAC “Dark Matter & Exotic Physics WG” – p. 7/39 For the torus V = Rn ) n The action Sgrav is precisely the standard gravity action upon the identification G∗ GN = Vn Effective Action The effective D = 4 action is Vn 4 Sgrav = d x g(4) R(4) 16πG∗ Z q where Vn is the volume of the extra space . SLAC “Dark Matter & Exotic Physics WG” – p. 8/39 The action Sgrav is precisely the standard gravity action upon the identification G∗ GN = Vn Effective Action The effective D = 4 action is Vn 4 Sgrav = d x g(4) R(4) 16πG∗ Z q where Vn is the volume of the extra space . For the torus V = Rn ) n SLAC “Dark Matter & Exotic Physics WG” – p. 8/39 Effective Action The effective D = 4 action is Vn 4 Sgrav = d x g(4) R(4) 16πG∗ Z q where Vn is the volume of the extra space . For the torus V = Rn ) n The action Sgrav is precisely the standard gravity action upon the identification G∗ GN = Vn SLAC “Dark Matter & Exotic Physics WG” – p. 8/39 • r R the torus effectively disappear for the D= 4 obser) ver m m U(r) = G 1 2 − N r • r R the D = 4 observer is able to feel the presence ) of the bulk m1m2 U∗(r) = G∗ − r1+n Newton Law in Higher Dimensions Let us assume a couple of particles m1 and m2 located on the hypersurface ya = 0 and separated by a distance r. There are two regimes SLAC “Dark Matter & Exotic Physics WG” – p. 9/39 • r R the D = 4 observer is able to feel the presence ) of the bulk m1m2 U∗(r) = G∗ − r1+n Newton Law in Higher Dimensions Let us assume a couple of particles m1 and m2 located on the hypersurface ya = 0 and separated by a distance r. There are two regimes • r R the torus effectively disappear for the D= 4 obser) ver m m U(r) = G 1 2 − N r SLAC “Dark Matter & Exotic Physics WG” – p. 9/39 Newton Law in Higher Dimensions Let us assume a couple of particles m1 and m2 located on the hypersurface ya = 0 and separated by a distance r. There are two regimes • r R the torus effectively disappear for the D= 4 obser) ver m m U(r) = G 1 2 − N r • r R the D = 4 observer is able to feel the presence ) of the bulk m1m2 U∗(r) = G∗ − r1+n SLAC “Dark Matter & Exotic Physics WG” – p. 9/39 Planck Scale In higher dimensional theory the fundamental (quantum) gravity scale is given in terms of G∗ 1=(2+n) ~1+nc5+n M∗c2 = 8πG∗ Turning to natural units we can see the relation between the D = 4 and the D = 4 + n Planck scale 2 2+n MP = M∗ Vn SLAC “Dark Matter & Exotic Physics WG” – p. 10/39 If the volume of the extra space is large enough the fundamental scale could be as low as mEW + No hierarchy in the fundamental scale of physics but now we have to explain why the extra dimensions are large! Fundamental Interactions From particle physics there is no evidence of quantum gravity, supersymmetry or string effects up to energies around few hundred GeV + M∗ 1TeV ≥ SLAC “Dark Matter & Exotic Physics WG” – p. 11/39 but now we have to explain why the extra dimensions are large! Fundamental Interactions From particle physics there is no evidence of quantum gravity, supersymmetry or string effects up to energies around few hundred GeV + M∗ 1TeV ≥ If the volume of the extra space is large enough the fundamental scale could be as low as mEW + No hierarchy in the fundamental scale of physics SLAC “Dark Matter & Exotic Physics WG” – p. 11/39 Fundamental Interactions From particle physics there is no evidence of quantum gravity, supersymmetry or string effects up to energies around few hundred GeV + M∗ 1TeV ≥ If the volume of the extra space is large enough the fundamental scale could be as low as mEW + No hierarchy in the fundamental scale of physics but now we have to explain why the extra dimensions are large! SLAC “Dark Matter & Exotic Physics WG” – p. 11/39 Size of the Compact Dimensions Assuming a toroidal compactification with all the S1 of the same size R and M∗ m in the relation (*) ∼ EW 1+2=n − 1 TeV R 1030=n 17 cm ∼ × m EW n R 1 1011 m 2 0:2 mm .. .. SLAC “Dark Matter & Exotic Physics WG” – p. 12/39 Size of the Compact Dimensions Assuming a toroidal compactification with all the S1 of the same size R and M∗ m in the relation (*) ∼ EW 1+2=n − 1 TeV R 1030=n 17 cm ∼ × m EW n R 1 1011 m 2 0:2 mm .. .. current limit of short distance gravity experiments SLAC “Dark Matter & Exotic Physics WG” – p. 12/39 + SM particles can not freely propagate into large extra dimensions + they must be constrained on a d = 4 submanifolds, a brane Usually at fixed points (*) of the compact space breaking of translational+ invariance Braneworld Electroweak physics tested up to small distances − − m 1 10 18cm EW ∼ SLAC “Dark Matter & Exotic Physics WG” – p.