Calculation of Expectation Values

In this section we calculate in detail the expectation values which are needed for the evaluation of the observables discussed in this paper. These are all expectation values in the measure of the Gaussian action given in Equation (2),

$$\begin{eqnarray*} S_{0}[\vec{\varphi}'] & = & \sum_{n_{\mu}}^{N^{d}} \left\{... ...alpha_{0}}{2}\, \varphi_{\mathfrak{N}}'^{2}(n_{\mu}) \right\}, \end{eqnarray*}$$

which is even on the fields. They will all involve the non-Gaussian or ``interacting'' part of the action, which is given in Equation (3),

$$\begin{eqnarray*} S_{V}[\vec{\varphi}'] & = & \sum_{n_{\mu}}^{N^{d}} \left\{... ...phi}'(n_{\mu})\cdot\vec{\varphi}'(n_{\mu})\right]^{2} \right\}. \end{eqnarray*}$$

There are field-odd and field-even terms in this action. Since the expectation values will single out one of these parities, it is convenient to write explicitly the field-odd and field-even parts of the non-Gaussian part of the action,

$$\begin{eqnarray*} S_{V,{\rm odd}}[\vec{\varphi}'] & = & \sum_{n_{\mu}}^{N^{d}... ...phi}'(n_{\mu})\cdot\vec{\varphi}'(n_{\mu})\right]^{2} \right\}. \end{eqnarray*}$$

It is also convenient, for use in the calculations, to write versions of these expressions in which the terms containing the $\varphi_{\mathfrak{N}}'(n_{\mu})$ field component are written explicitly, and one version in which the terms containing the $\varphi_{1}'(n_{\mu})$ field component are written explicitly as well, rather than as part of the scalar products,

$$S_{V,{\rm odd}}[\vec{\varphi}'] = \sum_{n_{\mu}}^{N^{d}} \left\{ \rule{0em}{4.0ex} v_{0} \left[\alp... ...rphi_{\mathfrak{N}}'(n_{\mu}) - j_{0}\varphi_{\mathfrak{N}}'(n_{\mu}) + \right.$$

$$\hspace{2.0em} \left. \rule{0em}{4.0ex} + \lambda v_{0} \left[ \s... ...ak{N}}'(n_{\mu}) + \lambda v_{0} \varphi_{\mathfrak{N}}'^{3}(n_{\mu}) \right\},$$ (A.1)

$$S_{V,{\rm even}}[\vec{\varphi}'] = \sum_{n_{\mu}}^{N^{d}} \left\{ \rule{0em}{5ex} \frac{\alpha-\alph... ...a v_{0}^{2}}{2} \sum_{i=1}^{\mathfrak{N}-1} \varphi_{i}'^{2}(n_{\mu}) + \right.$$

$$\hspace{2.2em} \left. + \frac{\alpha-\alpha_{\mathfrak{N}}+3\lambda v_{0}^{2}}{2}\, \varphi_{\mathfrak{N}}'^{2}(n_{\mu}) + \right.$$ (A.2)

$$\hspace{2.0em} \left. \rule{0em}{5ex} + \frac{\lambda}{4} \left[ ... ...}(n_{\mu}) + \frac{\lambda}{4}\, \varphi_{\mathfrak{N}}'^{4}(n_{\mu}) \right\},$$

$$S_{V,{\rm even}}[\vec{\varphi}'] = \sum_{n_{\mu}}^{N^{d}} \left\{ \rule{0em}{5ex} \frac{\alpha-\alpha_{0}+\lambda v_{0}^{2}}{2}\, \varphi_{1}'^{2}(n_{\mu}) + \right.$$

$$\hspace{2.2em} \left. + \frac{\alpha-\alpha_{0}+\lambda v_{0}^{2}}{2} \sum_{i=2}^{\mathfrak{N}-1} \varphi_{i}'^{2}(n_{\mu}) + \right.$$

$$\hspace{2.2em} \left. + \frac{\alpha-\alpha_{\mathfrak{N}}+3\lambda v_{0}^{2}}{2}\, \varphi_{\mathfrak{N}}'^{2}(n_{\mu}) + \right.$$ (A.3)

$$\hspace{2.0em} \left. \rule{0em}{5ex} + \frac{\lambda}{4}\, \varp... ...ft[ \sum_{i=2}^{\mathfrak{N}-1} \varphi_{i}'^{2}(n_{\mu}) \right]^{2} + \right.$$

$$\hspace{2.0em} \left. \rule{0em}{5ex} + \frac{\lambda}{2}\, \varp... ...}(n_{\mu}) + \frac{\lambda}{4}\, \varphi_{\mathfrak{N}}'^{4}(n_{\mu}) \right\}.$$