1 Week 7: Non linear sigma models and pion lagrangians Reading material from the books Burgess-Moore, Chapter 9.3 Donoghue, Golowich, Holstein Chapter 4, 6 Weinberg, Chap. 19 1 Goldstone boson lagrangians As we have discussed already, the spontaneous breaking of global continuous symmetries leads to a spectrum of massless particles: the Goldston bosons. The number of such massless particles is the dimension of the coset G/H, where H is the unbroken group of symmetries. The spaces of the form G/H are many times homogeneous spaces: spaces of constant curvature and with simple geometry. The idea of pion lagrangians is to exploit the geometry of these spaces and the symmetry of the theory to make general predictions. Let us consider an example first. Assume that you have a theory with N + 1 real scalars φ i, i = 0, 1,...N, and with a symmetry O(N + 1) of rotations of these scalars. The most general Lagrangian with canonical kinetic terms and up to two derivatives in the potential will be given by L = 1 2 δ ij µ φ i µ φ j V (φ 2 ) (1) where φ 2 = φ i φ j δ ij. If we have spontaneous symmetry breaking, without loss of generality we can give a vev to φ 0 and no other field (we can always rotate the fields to have this property). If we label the vev φ 0 = v, we can ask what is the physics of the system at center of mass energies that are much lower than v. For these energies we can only excite the Goldstone bosons, but not other fields. Thus it makes sense that if we are thinking about this problem experimentally, we would discover the Goldstone bosons, but not the massive 1 c David Berenstein 2009 1
fields, and we would still like to know what are the predictions of the theory for these energies. The idea is to write the theory in spherical coordinates, so that a general field configuration will explore the bottom of the potential, but it will not have energy to excite the rest. Remember that v depends on other couplings like masses and the general couplings appearing in V. However, v is physical in that the current acting on the vacuum has an amplitude proportional to v to excite a one particle state. Since the current is part of a non-abelian theory of symmetries, we normalize it to the structure constants of the group. This is, we normalize it so that [Q a, j b µ(x)] if abc j c µ(x) (2) where Q a d 3 x(j 0 ) a. The vev v is also what we have been calling f π. We now write φ i (v + h)y i (θ), where the h field is a fluctuation of the radial direction, and the Y are a basis of spherical harmonics of spin one. The action for the angular pieces will be given by the following general form 1 2 g αβ(θ) µ θ α µ θ β (3) with some geometry that depends on the coordinates that are chosen. In this case the geometry of the valley of the field configurations is a sphere, and the size of the sphere is controlled by v The vev v breaks the symmetry from O(N + 1) to O(N), so we should have dim(o(n + 1)/O(N)) Goldstone bosons. Indeed, dim(o(n + 1)) = N(N + 1)/2, dim(o(n)) = N(N 1)/2 (4) and we have N Goldstone bosons. Indeed, the space O(N + 1)/O(N) S N is exactly the N-dimensional sphere. What we are finding is that the metric of the space G/H is exactly the metric appearing in the lagrangian description of the Goldstone bosons alone. This is a general feature. The action 3 is called the non-linear sigma model action. Let us specialize to a simple situation, where N = 3, so that G = SO(4) SO(3) SO(3) and H = SO(3). The space G/H SO(3) SO(3)/SO(3) SO(3) 2
SU(2), so that the sphere is also a group. This is not strictly necessary, but it simplifies the form of the Lagrangian and it is the type of model that appears in the theory of the strong interactions, where we will get SU(N) SU(N)/SU(N) SU(N), where N can take different values. In the special case where the Goldstone bosons are valued in a group G, rather than a coset, we can think of the position on the manifold associated to G as a general element U of G, where the vacuum is chosen at U = 1 (this is the identity element of the group). Also, the configuration space for the path integral is maps µ : R 3,1 G. If we are very near the identity, it is convenient to expand in the Lie algebra of G, by saying that U 1 + iτ a δθ a (5) where the τ are hermitian. The lagrangian with two derivatives should be of the form 1 2 κ ab µ δθ a µ δθ b (6) Notice that the group G has an action of G G on it (by left and right multiplication). The identity is fixed by a diagonal copy of G embedded into G G: left multiply by g and right multiply by g 1. Thus, the δθ a are acted on linearly (in a canonical way) by one copy of G. This action is the adjoint action. The metric κ ab should be invariant under this action of G, and for simple G it is unique: this is the Killing metric on the Lie algebra of G. The group G is homogeneous. This means that any point on G is equivalent to the origin. Thus, we can find a unique metric on G which is translation invariant. This is the metric that leads to the Haar measure on G. The metric κ ab δθ a δθ b tr(δθδθ) where tr is a trace in any irreducible representation of G, where the only difference is the normalization. This normalization is representation dependent and is called C 2 (R) (the second Cassimir of the representation). The general lagrangian becomes L = 1 2 f2 πtr((u 1 µ U)(U 1 µ U) (7) where U 1 U is a vector field at U translated to the origin. This is equivalent to L = 1 2 f2 πtr( µ U 1 µ U) (8) 3
and remember also that U 1 = U for unitary groups. The action of the group G G on U is given by U guh 1. The objects U 1 U are invariant under the left multiplications by g. These are left-invariant vector fields, and can also be thought of as being intrinsic differential forms on the group manifold G (the left invariant differentiable forms E). The action is manifestly invariant under G G (these are global transformations), and because of our arguments it is the unique action with two derivatives that is compatible with the symmetries (The metric on a group is unique). This is true in general for homogeneous isotropic spaces: the metric is unique and characterized by G/H. What is important is that at this stage we have a lot of choices on how to parametrize the elements of G. We can use Euler angles, or an exponential map, or any of a myriad of different coordinate systems for G. However the physics should not depend on those choices, although the individual terms in the action etc do. This means that we can and should choose a simplest system to describe interactions. This is provided by not fixing any coordinate system until the end, and using the objects P µ = U 1 µ U instead. The symmetries of the system mean that the action can only depends on powers of P µ and derivatives of P µ which is Lie-algebra valued. To next order in powers of P, we can have tr(p µ P ν )tr(p µ P ν ), [tr(p µ P µ )] 2, tr(p µ P ν P µ P ν ), tr(p µ P µ P ν P ν ) (9) tr( µ P ν ν P µ ), tr( µ P µ ν P ν ) (10) modulo total derivatives and the equations of motion. If we include parity violating effects, we could have also elements with ǫ µνρσ included. We will discuss these later on. Another presentation of the action is to use products of traces of the form modulo the equations of motion. tr(d [α1 ]Ud [α2 ]U 1 d [α3 ]Ud [α4 ]U 1...) (11) 1.1 Power counting If we consider the action given so far, the field associated to U is dimensionless, but derivatives have powers of momenta. If we calculate a propagator 4
for pions, we get a power of fπ 2 for each pion propagator. All dimensionful constants should be normalized with respect to f π so that they are dimensionless. As such, a term with 2k derivatives has a power of fπ 4 2k appearing in front of it. Because the fields appearing in U are not canonically normalized, each external leg also carries a power of fπ 1. Also, each derivative carries one unit of momentum. The simplest 2 2 scattering results in various types of contributions at tree level: those with 2 derivatives, and those with 4, 6, 8, dots. There are no three particle vertices if we choose the metric correctly, so the Feynman diagram of the amplitude is just addition of two pieces. The one with two derivatives will go like p 2 fπ 2f π 4, while the next one will behave as p4 fπ 4 and so on. Each higher order will bring an extra power of p 2 /fπ. 2 At low energies (small p << f π only the first one contributes). At intermediate energies p < f π we get a power series in energy, so assuming some amount of convergence, we can truncate to finitely many pieces. If we take a 1PI one loop calculation for example, where the loop has s propagators, we get a power of fπ 2s for the loop, and a power of fπ 4 2k for each vertex (each such vertex has 2k legs and at least k = 1. There also have to be s vertices), so we get a term that scales as an inverse power of f π. This means that infinities that show up in this calculation would renormalize terms with higher numbers of derivatives up to some fixed number determined by the k. If the loops are done in dimensional regularization, only logarithmic divergences appear, and the powers of f π in front of it tell us that they renormalize the terms with higher derivatives in the action only. If we remember the quadratic divergences and higher, these infinities are absorbed in the bare coupling constants and do not affect the physics. The logarithmic divergences are important as they control the RG flow of the theory and result in non-polynomial behavior in the momenta p. The standard one loop integral that is logarithmically divergent is the following µ 4 d d 4 ǫp 1 (2π) 4 ǫ (p 2 + m 2 ) 1 2 4π 2 log(m2 /µ 2 ) (12) In the end, each loop will bring an extra factor of 1/(4π 2 fπ 2 ). This means that usually we expect the radius of convergence (or validity) of the perturbation expansion to be controlled by energies of order 2πf π. There are other 5
combinatoric factors that usually show up that suggest that 4πf π is more appropriate. For the pion system we have that f π 92MeV in these conventions, so that the pion lagrangian can be a god description of physics up to about 1200MeV 1.2GeV. By contrast, the mass of the pion is of order 130 MeV, which suggest that the theory has of order 10% corrections to tree level results. 2 The SU(2) SU(2)/SU(2) sigma model One of the problems one usually faces when writing the non-linear sigma modes is to choose a parametrization of the coordinates of the manifold expanded around some particular point. For the 3-sphere we can use various parametrizations. We can use U = exp(iσ a θ a ) as our set of coordinates on S 3 SU(2), where we have an exponential form of a rotation and the a are Pauli matrices. taking derivatives µ U in this expression is technically difficult, because we can not just take derivatives in the exponent and put them in front: matrices don t commute. Instead, for general unitary matrices we can write U = ŨS S = ŨS...ŨS (13) where ŨS = exp(iσ a θ a /S) where S is integer and then we can take the limit when S. When S, we have that U S (1 + iσ a θ a /S), so that we can take derivatives easily, and we get that in the end µ U 1 0 dy exp(iσ a τ a y)(i µ θ a σ a ) exp(iθ a σ a (1 y)) (14) So the calculations can be done, but they are somewhat cumbersome. We can also just expand the exponential and take derivatives. So long as we are doing perturbation theory, this approach also works. For SU(2) this works rather well. The reason for this is that the exponential is simple. If we take θ a τ a where τ a are the Pauli matrices, then (θ a τ a ) 2 = θ 2 /4, where θ is the angle of rotation. 6
As such we have that U = cos(θ/2) + i sin(θ/2) 2σa θ a θ (15) And since θ appears in an even function, differentiation is straightforward. There is another representation of the sphere in terms of different coordinates in R 3, so that the metric is given by ds 2 = dx 2 (1 + x 2 ) 2 (16) This has manifest SO(3) symmetry of rotations of the x. The north pole is at x =. The θ a also have this same symmetry, so it makes sense to have a map of the form θ a x a f( x 2 ) (17) This can be done by realizing that we can compute distances in both the metric of the sphere in the θ coordinates and the x coordinates, starting from the origin and going to some point where θ 2 = θ 3 = 0. This distance is is proportional to θ 1. By contrast, we have to match this with x 0 dy = Arctan(x) (18) 1 + y2 The north pole of the sphere is the rotation by 2π, so we find that we should normalize θ = 4Arctan(x). Obviously, the full map is θ a = xa Arctan(x) (19) x which explains why the metric in the θ coordinates can be messy. Given the metric, it is now possible to compute scattering matrix elements. 7