The Longitudinal Propagator

We will now complete our calculations with the expectation value of the observable

$${\mathbf O}[\vec{\varphi}'] = \varphi_{\mathfrak{N}}'(n_{\mu}') \varphi_{\mathfrak{N}}'(n_{\mu}'').$$

We call this the longitudinal propagator because it belongs to the field component which is in the direction of the external source in the internal $SO(\mathfrak{N})$ space. Once more the observable will be taken at two arbitrary points $n_{\mu}'$ and $n_{\mu}''$. The first-order Gaussian-Perturbative approximation for this observable gives

$$\begin{eqnarray*} \left\langle \varphi_{\mathfrak{N}}'(n_{\mu}') \varphi_{\ma... ...left\langle S_{V}[\vec{\varphi}'] \right\rangle_{0} \right\}, \end{eqnarray*}$$

where $g_{\mathfrak{N}}(n_{\mu}'-n_{\mu}'')$ is the two-point function with mass parameter $\alpha_{\mathfrak{N}}$. We must now calculate the two expectation values which appear in this formula. The first expectation value is the same we had before for the transversal propagator, and from Equation (A.5) we have,

$$\begin{eqnarray*} \left\langle S_{V}[\vec{\varphi}'] \right\rangle_{0} & = &... ...2} + \frac{3\lambda}{4}\, \sigma_{\mathfrak{N}}^{4} \right]. \end{eqnarray*}$$

The second expectation value is calculated in Appendix A, given in Equation (A.7),and the result is

$$\begin{eqnarray*} \lefteqn { \left\langle \varphi_{\mathfrak{N}}'(n_{\mu}') ... ...mu}'')} } { [\rho^{2}(k_{\mu})+\alpha_{\mathfrak{N}}]^{2} }. \end{eqnarray*}$$

The factor in front of $g_{\mathfrak{N}}(n_{\mu}'-n_{\mu}'')$ can now be verified to be exactly equal to $\left\langle S_{V}[\vec{\varphi}']\right\rangle_{0}$, and therefore once again this whole part cancels off from our observable. We may now write for the difference of expectation values that appears in it,

$$\begin{eqnarray*} \lefteqn { \left\langle \varphi_{\mathfrak{N}}'(n_{\mu}') ... ...mu}'')} } { [\rho^{2}(k_{\mu})+\alpha_{\mathfrak{N}}]^{2} }. \end{eqnarray*}$$

Finally, we can write the complete result,

$$\begin{eqnarray*} \lefteqn { \left\langle \varphi_{\mathfrak{N}}'(n_{\mu}') ... ...^{2} } { \rho^{2}(k_{\mu})+\alpha_{\mathfrak{N}} } \right], \end{eqnarray*}$$

where we once more wrote $g_{\mathfrak{N}}(n_{\mu}'-n_{\mu}'')$ in terms of its Fourier transform. Exactly as in the previous case, we see from the structure of this propagator that we can make $\alpha_{\mathfrak{N}}$ equal to the longitudinal renormalized mass parameter by choosing it so that the numerator of the second fraction vanishes. This gives the result

$$\alpha_{\mathfrak{N}} = 3\lambda v_{0}^{2} + \alpha +... ...-1) \sigma_{0}^{2} + 3 \sigma_{\mathfrak{N}}^{2} \right].$$ (C.6)

This result for $\alpha_{\mathfrak{N}}=a^{2}m_{\mathfrak{N}}^{2}$ is valid for a constant but possibly non-zero external source, in both phases of the model, where $m_{\mathfrak{N}}$ is the mass associated to the field component $\varphi_{\mathfrak{N}}'(n_{\mu})$.