The Model

Let us start by giving the definition of the model, in the classical and quantum domains, and then quickly reviewing the Gaussian-Perturbative approximation. Consider then the Euclidean quantum field theories of an $SO(\mathfrak{N})$-symmetric set of scalar fields $\vec{\phi}(x_{\mu})$ defined within a periodical cubic box of side $L$ in $d$ dimensions by the classical action

$$\begin{eqnarray*} S\!\left[\raisebox{-0.3ex}{$\vec{\phi}$}\right] & = & \oint... ...mu})\right]^{2} - J_{0}\phi_{\mathfrak{N}}(x_{\mu}) \right\}, \end{eqnarray*}$$

where $d\geq 3$. This is the usual form of the $SO(\mathfrak{N})$-symmetric $\lambda\phi^{4}$ model in the classical continuum, with an external source $J_{0}$, which by assumption is a constant. The vector notation $\vec{\phi}(x_{\mu})$ is shorthand for

$$\vec{\phi}(x_{\mu}) = \left( \phi_{1}(x_{\mu}), \phi_{2}(x_{\mu}), \ldots, \phi_{\mathfrak{N}}(x_{\mu}) \right),$$

and the dot-product notation represents the scalar product of vectors in the internal $SO(\mathfrak{N})$ space, that is a sum over $i=1,2,\ldots,\mathfrak{N}$,

$$\vec{\phi}(x_{\mu})\cdot\vec{\phi}(x_{\mu}) = \sum_{i}^{\mathfrak{N}} \left[\phi_{i}(x_{\mu})\right]^{2}.$$

In this action the quantity $J_{0}$ is a homogeneous external source associated with the $\phi_{\mathfrak{N}}(x_{\mu})$ field component. Its introduction breaks the $SO(\mathfrak{N})$ symmetry, of course, and causes the generation of a non-zero expectation value for the $\phi_{\mathfrak{N}}(x_{\mu})$ field component.

In order to use the definition of the quantum theory on a cubical lattice of size $L$ with $N$ sites along each direction, with lattice spacing $a=L/N$, we consider the corresponding lattice action

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

where all quantities are now dimensionless, defined by the appropriate scalings,

$$\varphi_{i}(n_{\mu}) = a^{(d-2)/2}\phi_{i}(x_{\mu}),$$

$$n_{\mu} = a^{-1}x_{\mu},$$

$$\alpha = a^{2}m^{2},$$ (B.1)

$$\lambda = a^{4-d}\Lambda,$$

$$j_{0} = a^{(d+2)/2}J_{0}.$$

In order for the model to be stable we must have $\lambda\geq 0$ and, in addition to this, if $\lambda=0$ then we must also have $\alpha\geq 0$. Up to this point there are no further constraints on the real parameters $\alpha$ and $\lambda$.

When possible, the summations are notated, in the subscript, by the variable which is being summed over, and, in the superscript, by the number of terms in the sum. The integer coordinates $n_{\mu}$ are taken to vary as symmetrically as possible around the origin $n_{\mu}=0_{\mu}$, that is we have $n_{\mu}=n_{\rm min},\ldots,0,\ldots,n_{\rm max}$ with certain values of $n_{\rm min}$ and $n_{\rm max}$ that depend on the parity of $N$,

$$n_{\mu} = -\,\frac{N-1}{2},\ldots,0,\ldots,\frac{N-1}{2},$$

for odd $N$, and

$$n_{\mu} = -\,\frac{N}{2}+1,\ldots,0,\ldots,\frac{N}{2},$$

for even $N$, in either case for all values of $\mu=1,\ldots,d$.

In this paper we will perform the calculations of the critical line and of the renormalized masses in a situation in which we have, in terms of the dimensionfull field $\vec{\phi}(x_{\mu})$, for $i=1,\ldots,\mathfrak{N}-1$,

$$\left\langle\phi_{i}(x_{\mu})\right\rangle = 0,$$

and, for $i=\mathfrak{N}$,

$$\left\langle\phi_{\mathfrak{N}}(x_{\mu})\right\rangle = V_{0},$$

where $V_{0}$ is a constant with the physical dimensions of the field $\phi_{\mathfrak{N}}(x_{\mu})$. In terms of the dimensionless field $\vec{\varphi}(n_{\mu})$ we have for the only non-trivial condition

$$\left\langle\varphi_{\mathfrak{N}}(n_{\mu})\right\rangle = v_{0},$$

where the dimensionless constant is given by $v_{0}=a^{(d-2)/2}V_{0}$.

We will consider continuum limits in which we have both $N\to\infty$ and $L\to\infty$. In order to do this we will choose to make $L$ increase as $\sqrt{N}$ and $a$ decrease as $\sqrt{N}$, so that we still have $a=L/N$. The calculations on finite lattices will be performed with periodical boundary conditions, with the understanding that at the end of the day such a limit is to be taken.

Observe that we are specifying the value of the expectation value $v_{0}$ of $\varphi_{\mathfrak{N}}(n_{\mu})$ rather than the value of the corresponding external source $j_{0}$. What we are doing here is to assume that there is some external source present such that we have the expectation value specified. It follows that one of the expected results of our calculations is the determination, at least implicitly, of the form of the external source in terms of $v_{0}$.

Our first calculational task in preparation for the Gaussian-Perturbative calculations is to rewrite the action in terms of a shifted field, which has a null expectation value. We thus define a new field variable $\vec{\varphi}'(n_{\mu})$ such that

$$\vec{\varphi}(n_{\mu}) = \vec{\varphi}'(n_{\mu}) + (0,0,\ldots,v_{0}),$$

so that we have $\left\langle\vec{\varphi}'(n_{\mu})\right\rangle=0$ for all $n_{\mu}$, with $\mu=1,\ldots,d$. We must now determine the form of the action in terms of $\vec{\varphi}'(n_{\mu})$. If we write each term of the action in terms of the shifted field we get

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

We will now eliminate all field-independent terms, since they correspond to constant factors that cancel off in the ratios of functional integrals which give the expectation values of the observables. Doing this we get the equivalent action

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

where except for the kinetic part the terms have been ordered by increasing powers of the field.

The last task we have to perform, in preparation for the Gaussian-Perturbative calculations, is the separation of the action in two parts. Since the symmetry is broken by the introduction of the external sources, besides the fact that depending on the values of the parameters $\alpha$ and $\lambda$ it might be spontaneously broken as well, this separation involves two new mass parameters, $\alpha_{0}$ for $\varphi_{1}'(n_{\mu}),\ldots,\varphi_{\mathfrak{N}-1}'(n_{\mu})$, and $\alpha_{\mathfrak{N}}$ for $\varphi_{\mathfrak{N}}'(n_{\mu})$. Note that an $SO(\mathfrak{N}-1)$ symmetry subgroup is left over after the $SO(\mathfrak{N})$ symmetry breakdown. We therefore adopt for the Gaussian part of the action

$$S_{0}[\vec{\varphi}'] = \sum_{n_{\mu}}^{N^{d}} \left\{ \rule{0em}{4ex} \frac{1}{2} \sum_{... ...{\varphi}'(n_{\mu}) \cdot \Delta_{\nu}\vec{\varphi}'(n_{\mu}) \right] + \right.$$

$$\hspace{2.0em} \left. \rule{0em}{4ex} + \frac{\alpha_{0}}{2} \lef... ..._{\mathfrak{N}}-\alpha_{0}}{2}\, \varphi_{\mathfrak{N}}'^{2}(n_{\mu}) \right\},$$ (B.2)

where there are no constraints on the parameters introduced other than $\alpha_{0}\geq 0$ and $\alpha_{\mathfrak{N}}\geq 0$. Note that, despite the way in which this is written, we do in fact have here just an $\alpha_{0}$ mass term for each field component $\varphi_{i}'(n_{\mu})$, for $i=1,\ldots,\mathfrak{N}-1$, and an $\alpha_{\mathfrak{N}}$ mass term for the field component $\varphi_{\mathfrak{N}}'(n_{\mu})$. It follows that the non-Gaussian part of the action is

$$S_{V}[\vec{\varphi}'] = \sum_{n_{\mu}}^{N^{d}} \left\{ \rule{0em}{3ex} v_{0} \left[\alpha... ...rphi_{\mathfrak{N}}'(n_{\mu}) - j_{0}\varphi_{\mathfrak{N}}'(n_{\mu}) + \right.$$

$$\hspace{2.23em} \left. + \frac{\alpha-\alpha_{0}+\lambda v_{0}^{2... ...rak{N}}+2\lambda v_{0}^{2}}{2}\, \varphi_{\mathfrak{N}}'^{2}(n_{\mu}) + \right.$$

$$\hspace{2.0em} \left. \rule{0em}{3ex} + \lambda v_{0} \left[\vec{... ... \left[\vec{\varphi}'(n_{\mu})\cdot\vec{\varphi}'(n_{\mu})\right]^{2} \right\},$$ (B.3)

which has its terms now written strictly in the order of increasing powers of the field.

Let us end this section by recalling the calculational techniques that will be involved. Given an arbitrary observable ${\mathbf O}[\vec{\varphi}']$ its expectation value is defined by

$$\left\langle{\mathbf O}[\vec{\varphi}']\right\rangle = \f... ...i]\, e^{-S_{0}[\vec{\varphi}']-\xi S_{V}[\vec{\varphi}']} },$$

which is a function of $\xi$, where $[{\rm d}\varphi]$ denotes the flat measure and hence integrals from $-\infty$ to $+\infty$ over all the field components at all sites. The expectation values of the model are obtained for $\xi=1$, and the corresponding expectation values in the Gaussian measure of $S_{0}[\vec{\varphi}']$ are those obtained for $\xi=0$. The Gaussian-Perturbative approximation consists of the expansion of the right-hand side in powers of $\xi$ to some finite order, around the point $\xi=0$, and the application of the resulting expression at $\xi=1$. The first-order Gaussian-Perturbative approximation of the expectation value of the observable ${\mathbf O}[\vec{\varphi}']$ is given by

$$\begin{eqnarray*} \left\langle{\mathbf O}[\vec{\varphi}']\right\rangle & = & ... ... \left\langle S_{V}[\vec{\varphi}']\right\rangle_{0} \right\}, \end{eqnarray*}$$

where the subscript $0$ indicates the expectation values in the measure of $S_{0}[\vec{\varphi}']$. These expectation values are most easily calculated in momentum space, where they involve only uncoupled Gaussian integrals. Therefore, let us also recall here the transformations to and from the momentum space representation of the model. We have for the field and its Fourier transform $\widetilde\varphi _{i}'(k_{\mu})$

$$\begin{eqnarray*} \widetilde\varphi _{i}'(k_{\mu}) & = & \frac{1}{N^{d}}\sum_... ...\sum_{\mu}^{d}k_{\mu}n_{\mu}}\,\widetilde\varphi _{i}'(k_{\mu}), \end{eqnarray*}$$

where the sums over $k_{\mu}$ are taken in as symmetric a way as possible around $k_{\mu}=0_{\mu}$, just as we did for $n_{\mu}$. In other words, we have $k_{\mu}=k_{\rm min},\ldots,0,\ldots,k_{\rm max}$ with the same values of $k_{\rm min}$ and $k_{\rm max}$, depending on the parity of $N$, that were used for $n_{\rm min}$ and $n_{\rm max}$,

$$k_{\mu} = -\,\frac{N-1}{2},\ldots,0,\ldots,\frac{N-1}{2},$$

for odd $N$, and

$$k_{\mu} = -\,\frac{N}{2}+1,\ldots,0,\ldots,\frac{N}{2},$$

for even $N$, in either case for all values of $\mu=1,\ldots,d$. The orthogonality and completeness relations of the Fourier base are given by

$$\begin{eqnarray*} \sum_{n_{\mu}}^{N^{d}} e^{\pm\mbox{\boldmath$\imath$}(2\pi/N... ...u}(n_{\mu}-n_{\mu}')} & = & N^{d}\delta^{d}(n_{\mu},n_{\mu}'). \end{eqnarray*}$$

A typical Gaussian expectation value in momentum space, and possibly the most fundamental one, is given for a generic field component by

$$\left\langle \widetilde\varphi _{i}'(k_{\mu}) \widetilde\... ... = \frac{1}{N^{d}}\, \frac{1}{\rho^{2}(k_{\mu})+\alpha_{i}},$$

where $\alpha_{i}$ is either $\alpha_{0}$ or $\alpha_{\mathfrak{N}}$, depending on the field component involved, and where $\rho^{2}(k_{\mu})$ are the eigenvalues of the discrete Laplacian on the lattice, which are given by

$$\rho^{2}(k_{\mu}) = 4 \left[ \sin^{2}\!\left(\frac{\pi ... ...ldots + \sin^{2}\!\left(\frac{\pi k_{d}}{N}\right) \right].$$

This and several other expectation values, Gaussian integration formulas and lattice sums can be found in Appendix B.