The Randall-Sundrum model

The Randall-Sundrum model Aleksandr Chatrchyan & Björn Jüliger After a brief non-technical introduction to the Hierarchy problem, we first discuss the old five-dimensional Kaluza-Klein model, its tower of states and the emergence of the electromagnetic gauge symmetry in it as an introduction to extra-dimensional theories. The main part consists of the description of the Randall-Sundrum model, which solves the Hierarchy problem by naturally generating the Planck and the weak scale from a single Planck-like scale by adding an orbifolded fifth dimension. We explain the generation of the scales as well as the stabilization of the size of the orbifolded dimension by a scalar field acquiring avev. 1Motivation:TheHierarchyproblem The Hierarchy problem is mainly the question why the strengths of the weak and the gravitational force are so vastly (by a factor / 10 32 ) di erent. More technically, the square of the (unbroken, tachyonic) Higgs mass in a theory with energy cuto gets quadratic quantum corrections to it from fermion loops. Therefore, one should expect it to be of order of the cuto, which is for the Standard Model believed to be somewhere around the Planck scale, unless new physics, e.g. supersymmetry, kick in before that. Therefore, the low value of the actually observed Higgs mass compared to the Planck scale means that the bare parameter for it is finely tuned to cancel the large quantum corrections and only leave the small actual value. While this is not, strictly speaking, a technical problem for the theory, it is aesthetically unsatisfying and seems to indicate that some as-of-yet unknown mechanism may actually produce the low Higgs mass. The aim of the Randall-Sundrum model is to have the di erences in coupling constants and masses emerge from a higher-dimensional theory in which such hierarchies are initially absent. 2Aprimeronextradimensions:Kaluza-Kleintheory A five-dimensional theory similar to the Randall-Sundrum model is the older Kaluza- Klein theory which obtains electromagnetism and four-dimensional gravity from fivedimensional gravity, where the fifth dimension is a circle of radius r, i.e. spacetime is taken to be M := R 1,3 S 1. A general five-dimensional metric g is written as g = g 44 (dx 4 + A µ dx µ ) 2 + g µ dx µ dx (1) where the Greek indices run over {0, 1, 2, 3} and we have set A µ = gµ4 2g 44. We now want to perform the so-called Kaluza-Klein reduction and look at how a five-dimensional theory of one free massless scalar field looks from the perspective of the four-dimensional 1

Minkowski space R 1,3 M. We suppose that the metric does not depend on the fifth dimension, so we may carry out the integral over the fifth dimension later on more easily. For simplicity, we will only consider a single real scalar field in five dimensions. To that end, we use that functions on an S 1 with radius R possess an expansion in a Fourier series as 1X (x) = n(x (4) )e i n R x4, n = n (2) n= 1 where with x (4) we denote the four-dimensional part of x 2Mand plug this into R the action for a massless free scalar field, i.e. 1 2 @M (x)@ M (x) p gd 5 x. 1 By orthogonality of the Fourier basis, we therefore get a four-dimensional action for the Fourier modes of by integrating over the fifth dimension: S = (2 R p Z 1 g 44 ) 2 @ µ 0 @ µ 0 + 1X n=1 @! µ n @ µ n + n2 R 2 n n p g(4)d 4 x (3) where we see that we now have a four-dimensional action for a countably infinite number of scalar fields, one of which is massless, and the others are massive complex scalar field of mass m n = n R. Therefore, the single massless scalar in five dimensions is reflected in an infinite number of ever more massive scalars in four dimensions. This infinity of massive scalar fields is known as the Kaluza-Klein tower. The smaller R is, the more massive the lowest-lying massive state of this tower is, meaning the Kaluza-Klein tower becomes harder and harder to detect (and equivalently more and more irrelevant) in the fourdimensional theory the smaller the extra dimension is. But the Kaluza-Klein tower is not the only thing we incur from considering our five-dimensional theory: We also get electromagnetism for free! Returning to fivedimensional gravity, di eomorphism invariance of our theory means that, in particular, x 4 7! x 4 + (x (4) ) for any scalar function of actual spacetime should be a symmetry of our theory. Looking at the metric (1), we see that this corresponds to a symmetry A 7! A +d (4) which is just an infinitesimal U(1) gauge symmetry. Therefore, the four-vector field A is a gauge field from the viewpoint of the four-dimensional space. This is not surprising, since M has the geometric form of a (trivial) U(1)-principal bundle over the four-dimensional spacetime, gauge fields locally descend from a 1-form on such principal bundles, and A µ dx µ is just such a form. There is a third element we have no yet assigned a four-dimensional meaning to: The metric component g 44. From the four-dimensional viewpoint, g 44 is a scalar since it does not have Greek indices. We can get the volume (or, more directly geometrically, the circumference) of the S 1 by Z Vol(S 1 p )= g44 dx 4 = p g 44 2 R (5) S 1 1 The capital Latin indices run over {0, 1, 2, 3, 4}. 2

so g 44 controls the size of the compact dimension. Such fields controlling certain geometrical factors are often called moduli (fields). One may substitute the form of the metric (1) into a five-dimensional Einstein- Hilbert action to obtain a four-dimensional Einstein-Hilbert action, a free U(1) Yang- Mills action and a kinetic term for a field related to g 55, which is called a radion. 3 The Randall-Sundrum setup: An orbifold and a warped metric The Randall-Sundrum model (proposed in [RS99a; RS99b]) lives on a five-dimensional spacetime, where one spatial dimension is compact. In contrast to the old Kaluza-Klein model, which inferred gravity and electromagnetism from five-dimensional gravity, the compact dimension is not a manifold as is the Kaluza-Klein circle, but it is a circle whose upper and lower halves have been identified. Formally, such an object (which we might describe as a manifold with corners ) is called an orbifold, and in this case, it is described as O = S 1 /Z 2, where the action of the non-trivial element of the two-element group Z 2 on S 1 is given by S 1! S 1, e i 7! e i (6) We may also think of O as an interval [0,L R] where functions f on it are given by functions on the circle, i.e. f(y) =f(y +2 R), which are also even, i.e. f(y) =f( y), or just even functions defined on [ L, L]. We write down our five-dimensional theory of gravity as Z S = R 1,3 Z L L p g(m 3 Ri )dyd 4 x (7) where we account for the possibility of a cosmological constant and M is a fivedimensional fundamental mass scale and L is the length of the interval O. Ri is the Ricci scalar. We now ask what an actual (vacuum) solution to the Einstein equations here looks like. We want the theory to give the correct metric in the four non-compact dimensions, which we take for simplicity to be the Minkowski metric. We make the (not fully general) ansatz ds 2 =e 2A(y) µ dx µ dx +dy 2 (8) where we write y x 4. Plugging this into the Einstein equations 2 leads to A(y) =k y (9) where k 2 = 12M. Note that k is real for a negative cosmological constant, meaning 3 the five-dimensional space is anti-de Sitter, and imaginary for a positive cosmological 2 We will not calculate the Einstein tensor here. See [Gab06] for an explicit calculation. 3

constant. The Randall-Sundrum model is the case where k is real, and the so-called warp factor A(y) hence decaying, not oscillating. Denoting the derivative with respect to y with a prime, the four-dimensional part of the Einstein equations yields (6A 02 A 002 )g µ =6k 2 g µ (10) where the r.h.s. is the stress-energy tensor resulting from the cosmological constant. Now, A 0 (y) =k sgn(y) and A 00 (y) =2k( (y) (y L)). This is manifestly bad since the A 02 term perfectly cancels with the r.h.s., leaving us with the term incurred from the second derivative. The answer is to consider a slight modification to our setup: From a stringy point of view, it is clear that the boundaries of the five-dimensional space should be considered as dynamical objects of the theory in their own right - they are branes, 3-branes in this case, whole worldvolume constitutes the boundary. There are two of them - one at y = 0 and the other at y = L. 3 As dynamical objects, the branes contribute terms to the action (that is what we really mean by them being dynamical objects). In this case, we might consider simple analogs of the cosmological constant living on them, and write S i = Z p gi id 4 x (11) for i =0,Land g i (x (4) )=g(x, i). Intuitively, the i constitute energy densities on the branes (just as usual cosmological constants represent (vacuum) energy densities). If we want to write these as five-dimensional integrals so they become proper terms in the action, we just use 1 = R (y)dy. Now we can revisit (10) and see that 0 = L = 12M 3 k if we are to fulfill the Einstein equation with this, since instead of, we now have + 0 (y)+ L (y L) contributing to the stress-energy tensor. Note that we have this exact cancellation because our ansatz requires the four-dimensional cosmological constant in the bulk - the space between the branes - to vanish. 4Mattercontent:Generatingscales We now imagine a four-dimensional theory with matter living on one of the branes, which we think of as our spacetime we usually speak of. This will be the brane at y = L. The argument how to generate a weak scale from the Planck scale M is now straightforward: Consider a Higgs-like theory on the brane with action Z p S Higgs = gl g L ((Dh), (Dh)) apple(h h v 2 ) 2 d 4 x (12) 3 Since we have just said that they are dynamical objects, the question arises why they should have constant y. The answer is that the orbifold symmetry forbids them to move, since moving away from the fixed points of the symmetry would imply a doubling of the brane. 4

where v will be the vacuum expectation value of the Higgs, D is some gauge covariant derivative and apple a coupling constant determining the strength of the Higgs self-interaction. It is v that determines all fermion and boson masses in the Standard Model through gauge and Yukawa coupling, so v is, for our purposes, the weak scale whose size we aim to explain. Since we know the Randall-Sundrum metric, we can directly calculate p g L = e 4kL and g L ((Dh), (Dh)) = e 2kL ((Dh), (Dh)). Plugging this in leads to an action with a non-canonical kinetic term, so we redefine the physical Higgs field by ĥ := e kl h to obtain Z S Higgs = ((Dĥ), (Dĥ)) apple(ĥ ĥ e 2kL v 2 ) 2 d 4 x (13) where we see that the e ective VEV that is seen by the QFT on the brane is v e = e kl v. If M is the Planck scale (as is often imagined), and v is initially of the order of M in the five-dimensional theory (i.e. the hierarchy problem is absent), then the right size of kl is able to warp down the large v to a v e of order of the weak scale. Through the exponential, the large discrepancy of order 10 16 between the Planck and the weak scale is transformed into a factor kl ln(10 16 ) 37 (14) whose origin we have to explain (without simply tuning it, since that would defeat the whole purpose of the model). The stabilization of the radius L at a certain value is achieved through the Goldberger-Wise mechanism. We should also check what happens to the four-dimensional Planck scale: We can perturb the metric (8) with a four-dimensional perturbation h µ,plugthisintothe action (7) and obtain a term e 2k y p g (4) Ri (4) (15) and we can carry out the integral in the action over the compact dimension: Z L L e 2k y dy =2 Z L 0 e 2ky dy = 1 k (e 2kL 1) (16) Plugging in and identifying the prefactor of Ri (4) as the square of the four-dimensional Planck scale, we get MPl 2 = M 3 k (1 e 2kL ) (17) meaning that while the weak scale depends strongly on L, for su ently large L the Planck scale depends only very weakly on the size of the extra dimension. 5Thesize:Stabilizingtheradion The idea of the Goldberger-Wise mechanism is to fix the size of the extra dimension through a field acquiring an expectation value. The field providing this modulus 5

stabilization will be a real massive scalar field with action Z Z L S = L p g(@ A @ A m 2 2 ) p g0 0 ( 2 v 2 0) 2 (y) p gl L ( 2 v 2 L) (y L) dyd 4 x (18) or, more generally, we may replace the specific terms on the branes by unspecified potentials V i ( ) and add a potential V ( )inthebulk. Since we do not want to be visible in the four-dimensional theory, it should only vary in the extra dimension: (x, y) (y). The equation of motion for this scalar field is 1 p p @ M g@ M = @ ( V V 0 V L ) (19) g Since we constrained the scalar field to only depend on the extra dimension, this simplifies to @ 2 y 4@ y A@ y = @ V + @ V 0 (y)+@ V L (y L) (20) Besides the e.o.m. for the scalar field, we have to consider the influence of the bulk potential on the five-dimensional Einstein equations, i.e. we have to modify the procedure that led us to the solution for the warp factor A. The Einstein equations now look like (we denote the derivative with respect to y with a prime again) 2A 02 A 00 = 1 2M 3 A 02 = 1 2M 3 1 6 1 12 02 1 6 V ( ) 02 1 3 (V + V 0 (y)+v L (y L)) (21a) (21b) where we can solve (21a) for 0 and substitute it in (21b) to obtain A 00 = 1 6M 3 (V 0 (y)+v L (y L)) (22) Note that integration of these equations on infintesimal neighbourhoods of the branes will yield lim lim 0 = @ V i (23a) y!i + y!i lim lim A 0 = 1 y!i + y!i 6M 3 V i (23b) which then are boundary/consistency equations for our solutions with the orbifold geometry. To solve this system in general is quite hard. However, to demonstrate the possibility of the modulus stabilisation, we only need to find a special case where these e.o.m. determine the size of the dimension reasonably well without fine-tuning. Goldberger and Wise in their original paper [GW99] consider quartic potentials on the branes and 6

none in the bulk initially, then derive an e ective four-dimensional potential and do various approximations to get their result. We, however, follow [Gab06] s approach and simply write down a convenient form of the bulk potential: V = 1 8 (@ W 12M 3 2 )2 (24) W for some other potential W ( ). Comparing this to (21a), we can immediately write down 0 = 1 2 @ W (25a) A 0 = 1 12M 3 W as well as a five- and choosing a potential W that will lead to a mass term for dimensional cosmological constant, we can pick (25b) W = 12M 3 k µ 2 2 (26) 0 and see that = µ 2, which leads us to (y) = 0 e µ2 y and (L) = 0 e µ2l, which gives us L = 1 µ 2 ln 0 (27) (L) Since 0 is determined by the boundary conditions (23a), it is now the parameter µ that controls the size of the extra dimension. From (14), we see that µ must only be tuned on an order of O(100) to provide a good size for the extra dimension to solve the Hierarchy problem, which is an improvement over the initial large discrepancy between the strong and the weak scale. References [Gab06] [GW99] [RS99a] [RS99b] Maxim Gabella. The Randall-Sundrum model. http://www-thphys.physics. ox.ac.uk/people/maximegabella/rs.pdf. 2006. Walter D. Goldberger and Mark B. Wise. Modulus Stabilization with Bulk Fields. In: Phys. Rev. Lett. 83 (1999). arxiv: hep-ph/9907447v2. Lisa Randall and Raman Sundrum. A Large Mass Hierarchy from a Small Extra Dimension. In: Phys. Rev. Lett. 83 (1999). arxiv: hep-ph/9905221v1. Lisa Randall and Raman Sundrum. An Alternative to Compactification. In: Phys. Rev. Lett. 83 (1999). arxiv: hep-th/9906064v1. 7