# Feynman rules for nonlinear sigma models

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1. May 19, 2015

### hyperkahler

Nonlinear sigma models are particular field theories in which the fields take values in some nontrivial manifold. In the simplest cases this is equivalent to saying that the fields appearing in the lagrangian are subject to a number of constraints. Since the lagrangian fields are not independent from one another they cannot be used directly to construct Feynman diagrams in perturbation theory. The most general way to deal with the quantization of constrained systems is by use of the Faddeev-Popov procedure. The functional integral over the constrained set of fields is converted to a functional integral over unconstrained fields, the unconstrained fields including now, in addition to the original fields, also new auxiliary fields. The basic idea is to exponentiate the constraint, converting it to new lagrangian terms through the introduction of new fields. In YM theories this quantization procedure is well-known and leads to the appearance of so called ghost and anti-ghost fields: by imposing a gauge fixing condition, for example a Lorentz-gauge condition, one ends up with new scalar but anticommuting fields in the adjoint representation of the gauge group. These fields cannot appear as external states, since they violate the spin-statistics theorem, but must be included as internal propagating particles to preserve unitarity.

I was thinking about how to implement a similar idea for nonlinear sigma models so that standard perturbation theory methods via Feynman diagrams are applicable. In particular consider the $CP(N-1)$ model:

$$L= \frac{2}{g^2}[(\partial_\mu n^{\dagger})(\partial^\mu n)+(n^\dagger \partial_\mu n)^2]$$

where $n$ is a N-component vector of complex scalars with the constrained $n^\dagger n=1$. My problem is that this constraint leads to a new lagrangian term $\lambda(n^\dagger n-1)$ (introducing an auxiliary field $\lambda$ as lagrangian multiplier). The new field however is non propagating. Is there any way to correct this or do you know any other way how to quantize this type of system?

2. May 19, 2015

### fzero

The solution of the $CP^{N-1}$ model in 2d is discussed in the 2nd to last chapter of Coleman's Aspects of Symmetry. The idea is to introduce 2 auxiliary fields. The first is a vector $A_\mu$ that you use to rewrite the quartic term $(n^\dagger\partial_\mu n)^2$ term. The second is $\sigma$ which is the same as your $\lambda$, imposing the constraint. The resulting action is quadratic in $n$, so you can integrate $n$ out to give an effective action which is a functional of the auxiliary fields. The graphic expansion is in powers of $1/N$.

The physical interpretation of the result is that the interactions between the $n$ dynamically generate a gauge field $A_\mu$. The resulting long-range interaction is linear and so the original $n$ degrees of freedom are confined.