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Non Linear Sigma Models and Pion Lagrangians 1 Goldstone Boson

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Non Linear Sigma Models and Pion Lagrangians 1 Goldstone Boson Week 7: Non linear sigma models and pion lagrangians 1 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 φi, 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 1 L = − δ ∂ φi∂µφj − V (φ2) (1) 2 ij µ 2 i j where φ = φ φ δij. If we have spontaneous symmetry breaking, without loss of generality we can give a vev to φ0 and no other field (we can always rotate the fields to have this property). If we label the vev hφ0i = 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 ex- perimentally, 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 nor- malize it to the structure constants of the group. This is, we normalize it so that a b abc c [Q , jµ(x)] ∼ if jµ(x) (2) where Qa ∼ d3x(j0)a. The vev vR is also what we have been calling fπ. We now write φi ∼ (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 − g (θ)∂µθα∂ θβ (3) 2 αβ µ 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) ∼ SN 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 : R3,1 → 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 aδθa (5) where the τ are hermitian. The lagrangian with two derivatives should be of the form 1 − κ ∂ δθa∂µδθb (6) 2 ab µ 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 δθa are acted on linearly (in a canonical way) by one copy of G. This action is the adjoint action. The metric κab 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 equiva- lent 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. a b The metric κabδθ δθ ∼ 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 C2(R) (the second Cassimir of the representation). The general lagrangian becomes 1 L = − f 2tr((U −1∂ U)(U −1∂µU) (7) 2 π µ where U −1∂U is a vector field at U translated to the origin. This is equivalent to 1 L = − f 2tr(∂ U −1∂µU) (8) 2 π µ 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 differ- ential forms on the group manifold G (the left invariant differentiable forms E). The action is manifestly invariant under G × G (these are global trans- formations), 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 fix- −1 ing any coordinate system until the end, and using the objects Pµ = U ∂µU instead. The symmetries of the system mean that the action can only depends on powers of Pµ and derivatives of Pµ which is Lie-algebra valued. To next order in powers of P , we can have µ ν µ 2 µ ν µ ν tr(P P )tr(PµPν), [tr(P Pµ)] , tr(P P PµPν), tr(P PµP Pν) (9) ν µ µ ν tr(∂µP ∂ν P ), tr(∂µP ∂νP ) (10) modulo total derivatives and the equations of motion. If we include parity violating effects, we could have also elements with ǫµνρσ included. We will discuss these later on. Another presentation of the action is to use products of traces of the form −1 −1 tr(d[α1]Ud[α2]U d[α3]Ud[α4]U . ) (11) modulo the equations of motion. 1.1 Power counting If we consider the action given so far, the field associated to U is dimension- less, but derivatives have powers of momenta. If we calculate a propagator 4 −2 for pions, we get a power of fπ for each pion propagator. All dimensionful constants should be normalized with respect to fπ so that they are dimen- 4−2k sionless. As such, a term with 2k derivatives has a power of fπ appearing in front of it. Because the fields appearing in ∂U are not canonically normalized, each −1 external leg also carries a power of fπ . 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 2 2 −4 4 −4 derivatives will go like p fπ fπ , while the next one will behave as p fπ 2 2 and so on.
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