Spontaneous Symmetry Breaking

Having developed in section 1.2 the ideas about the perturbative approximation for the observables of the $\lambda \varphi ^{4}$ model, we will now discuss the calculation of some of the observables of the model to first order in $\varepsilon$ which, in the cases to be examined here, is also known as the ``one-loop'' order^1.2. The first thing that we will try to calculate will be the position of the critical curve near the Gaussian point. In order to to this we will examine the expectation value of the field,

$$v_{R}=\langle\varphi\rangle,$$

which functions as an order parameter for the phase transition that exists in the model. Of course, if we have in the model a non-vanishing external source $j$, then we should expect that $v_{R}$ is also non-vanishing. The situation of spontaneous symmetry breaking is that in which we have $v_{R}\neq 0$ even when $j=0$. Therefore, we will consider here the case $j=0$ and try to verify whether or not it is possible to obtain solutions of the model with $v_{R}\neq 0$ in the limit in which $N\rightarrow\infty$. Observe that only in this limit of large lattices one can expect to obtain a situation of phase transition, with the existence of two distinct phases in the parameter plane of the model, separated by a phase-transition curve.

In the symmetrical phase we necessarily have that $v_{R}=0$, while in the broken-symmetrical phase we may have $v_{R}\neq 0$. If the phase transition is of second order with respect to this parameter, as it is to be expected, then the critical curve is the geometrical locus in the parameter plane $(\alpha,\lambda)$ of the model where the solution $v_{R}=0$ becomes the only possibility, when we move from the broken-symmetrical phase to the symmetrical phase in the parameter plane. What we will do is to determine the values of $(\alpha,\lambda)$ for which $v_{R}\neq 0$ is a possibility and then impose that $v_{R}=0$ be the only solution, so as to determine the critical curve. Along the process, a trivial $v_{R}=0$ solution that exists in all the parameter plane will be factored out and eliminated. In the broken-symmetrical phase this solution corresponds, to make an analogy with the classical case, to the unstable solution in which the system is at the local maximum of the potential at $\varphi=0$.

In order to perform this calculation we must use the separation of the action in the free and interaction parts given in equations (1.2.4) and (1.2.5), which are those that should be used in the broken-symmetrical phase. First of all we write the definition of $v_{R}$, that is, that it is the expectation value of the original field $\varphi=\varphi'+v_{R}$. Next we use the perturbative expansion given in equation (1.2.3) in order to write the expectation values involved, limiting ourselves to the terms of order zero and one. We choose arbitrarily the site of the lattice with integer coordinates $\vec{n}=\vec{0}$ in order to do the calculation, a choice which is possible due to the discrete translation invariance of the lattice. Doing all this we obtain

$$v_{R}=\langle\varphi'+v_{R}\rangle_{0} -\left[\langle(\varph... ...gle(\varphi'+v_{R})\rangle_{0}\langle S_{I}\rangle_{0}\right].$$

Since $\langle\varphi'\rangle_{0}=0$ by construction, several terms vanish and we obtain, up to this order, a very simple equation,

$$\left\langle\varphi'S_{I}\right\rangle_{0}=0.$$

Is we write this in detail, substituting the expression for $S_{I}$ and then using all the available symmetries in order to simplify the expression (problem 1.3.1), in particular the fact that the expectation values of odd powers of the field are zero due to the fact that $S_{0}$ is symmetrical by reflection of the fields, we obtain

$$\sum_{s}\left[v_{R}\left(\alpha+\lambda v_{R}^{2}\right) \la... ...ft\langle\varphi'^{3}(s)\varphi'(0)\right\rangle_{0}\right]=0.$$ (1.3.1)

This equation is simply the lattice version of the equation known as the ``tadpole'' equation in one-loop order. Since all the terms contain at least one factor of $v_{R}$, we may now cancel out one factor of $v_{R}$, which is the trivial $v_{R}=0$ solution which we mentioned before, obtaining

$$\left(\alpha+\lambda v_{R}^{2}\right) \left\langle\varphi'(0... ...t\langle\varphi'(0)\sum_{s}\varphi'^{3}(s)\right\rangle_{0}=0.$$

The calculation of the remaining expectation values involves only Gaussian integrals and we obtain for the first term (problem 1.3.2)

$$\left\langle\varphi'(0)\sum_{s}\varphi'(s)\right\rangle_{0}= \frac{1}{\alpha_{0}}.$$ (1.3.2)

For the other expectation value we obtain (problem 1.3.3)

$$\left\langle\varphi'(0)\sum_{s}\varphi'^{3}(s)\right\rangle_... ...pha_{0}}\sum_{\vec{k}} \frac{1}{\rho^{2}(\vec{k})+\alpha_{0}}.$$ (1.3.3)

Observe how we avoided infrared problems in both cases, by the introduction of the non-zero parameter $\alpha_{0}$. In these calculations all the strong divergences due to the behavior of $S_{I}$, which consist of terms proportional to $N^{d}$, cancel out. This fact corresponds, in the usual language of the traditional approach to the theory directly in the continuum, to the cancellation of the so-called ``vacuum bubbles'', and is a direct consequence of the fact that we are expanding a ratio of two functional integrals. We have therefore for our tadpole equation

$$\frac{\left(\alpha+\lambda v_{R}^{2}\right)}{\alpha_{0}}+ \f... ...a_{0}}\sum_{\vec{k}} \frac{1}{\rho^{2}(\vec{k})+\alpha_{0}}=0.$$

We recognize now that the sum over the momenta is our already well-known quantity $\sigma_{0}^{2}$, the square of the width of the local distribution of the fields in the measure of $S_{0}$. We obtain therefore, substituting in terms of $\sigma_{0}^{2}$ and cancelling the factor of $1/\alpha_{0}$,

$$\lambda v_{R}^{2}+\alpha+3\lambda\sigma_{0}^{2}=0.$$ (1.3.4)

This equation gives us $v_{R}$ for small values of $-\alpha$ and $\lambda$ in the broken-symmetrical phase.

Let us consider here the issue of the dependence of this result on $\alpha_{0}$. Observe that the result does not depend explicitly on $\alpha_{0}$, but it may depend on this parameter through the squared width $\sigma_{0}^{2}$. For finite $N$ the width does indeed depend on $\alpha_{0}$ but, as was shows in the section in [11], in the continuum limit it does not depend on this parameter, so long as we make it go to zero sufficiently fast. More precisely, it suffices that we make $\alpha_{0}=m_{0}^{2}L^{2}/N^{2}$, for some finite $m_{0}$, for the limit to be completely independent of the value of $m_{0}$. The mass parameter $m_{0}$ could even be chosen to have the same value as the renormalized mass $m_{R}$ of the model, but there is no need for this coming from this calculation, all we know up to now is that $m_{0}$ must be finite. Note that the need to choose $\alpha_{0}$ dependent on $N$ in a certain way in order to make the results independent of $m_{0}$ is already a first indication that the perturbative expansion is not completely well-behaved, since there should be no dependence at all on $\alpha_{0}$.

Going back to the analysis of the critical behavior of the model, if we impose now that the only possible value for $v_{R}$ be zero, we obtain from equation (1.3.4), by setting $v_{R}=0$ in it, the equation of the critical curve, to wit

$$-\alpha_{c}=3\lambda_{c}\sigma_{0}^{2},$$ (1.3.5)

where $\alpha>-3\lambda\sigma_{0}^{2}$ corresponds to the symmetrical phase and $\alpha<-3\lambda\sigma_{0}^{2}$ to the broken-symmetrical phase. We see that this equation has the same form of the heuristic estimate that we proposed in section 1.1, differing from it only by the numerical factor $\sqrt{3}\sim 1.73$ involved in the evaluation of the relation between the width $\sigma_{0}$ of the local distribution and the parameter $\sqrt{-\alpha/\lambda}$ of the potential well. In other words, the result coincides with our heuristic estimate if we choose for the numerical constant $C_{0}$ introduced in that section the value $C_{0}=\sqrt{3}$. We may now write our perturbative result for $v_{R}$ in terms of the expression in the equation of the critical line as

$$v_{R}=\sqrt{\frac{-(\alpha+3\lambda\sigma_{0}^{2})}{\lambda}},$$ (1.3.6)

which is only real in the broken-symmetrical phase, as expected, and which shows explicitly how $v_{R}$ goes to zero when one approaches the critical line from the broken-symmetrical phase.

At this point it is important to point out, quite emphatically, that we have just found one more worrisome property of the perturbative approximation technique. We have found here a definite result for the position of the critical curve for the model in a box with periodical boundary conditions, for any value of $N$, either finite or not. In addition to this, this position of the critical curve has the curious property of depending weakly on the irrelevant parameter $\alpha_{0}$ if $N$ is finite, and of becoming independent of the same parameter if $N\rightarrow\infty$. Taken in this superficial way, our result seems to indicate that, given a value of $\alpha_{0}$, the system displays a completely well-defined phase transition on finite lattices with periodical boundary conditions.

However, it is a well-known fact that there is no possibility of existence of a phase transition on finite lattices with periodical boundary conditions in systems of the type that we are examining here. In this kind of system, with couplings only between next-neighbors and without external borders, the phase transition can be realized only in the $N\rightarrow\infty$ limit. We can only presume that the curious dependence on $\alpha_{0}$ for finite $N$ is somehow related to this fact, effectively indicating, at best, that there can be a kind of ``approximate critical behavior'' for finite $N$. This is one more circumstance in which we verify that this method of approximation has rather singular properties and that it should only be used with the greatest care.

A particularly interesting aspect of the structure of the model that we can obtain from equation (1.3.5) is the slope $\partial\lambda_{c}/\partial\alpha_{c}$ of the critical curve near the Gaussian point, which is given by

$$\frac{\partial\lambda_{c}}{\partial\alpha_{c}} =-\frac{1}{3\sigma_{0}^{2}}.$$ (1.3.7)

We may ask here how close to the truth this result can be. Note that it depends neither on $\alpha$ nor on $\lambda$, and let us recall that the dependence on $\alpha_{0}$ vanishes in the continuum limit. Hence, if the perturbative technique establishes at least a first-order approximation for the result of the complete model, then this result should be exact at the Gaussian point (problem 1.3.4). We will see later on that it is consistent with the results obtained by means of mean-field techniques and of stochastic simulations. In the case of the stochastic simulations realized so far, it has been verified that it is particularly difficult to execute them close to the Gaussian point, due to the fact that in that region the potential wells of the model become very shallow, which makes it more difficult to control the statistical errors. Therefore, up to now it has not been possible to do more than to confirm qualitatively this result with the stochastic simulations.

Using the asymptotic values of $\sigma_{0}^{2}(N)$ in the dimensions $d=3$ to $d=5$, presented in the section in [11], we obtain the results shown in table 1.3.1 for the slope, where $\theta$ is the smaller angle that the tangent line to the critical curve at the Gaussian point makes with the negative $\alpha$ semi-axis. It is interesting to observe that in the cases $d=1$ and $d=2$, since $\sigma_{0}(N)$ diverges, the slopes go to zero in the limit and the critical curve collapse onto the negative $\alpha$ semi-axis, where the model does not exist because this semi-axis is part of the unstable region. One might consider the interpretation that this is the perturbative way of verifying that the $\lambda \varphi ^{4}$ model does not really exist as a quantum field theory for $d<3$.

Table 1.3.1: Table of the slopes of the critical curves at the Gaussian point.

$d$	$\tan(\theta)$	$\theta$ (degrees)
$3$	$\simeq 1.3189$	$\simeq 52.83$
$4$	$\simeq 2.1515$	$\simeq 65.07$
$5$	$\simeq 2.8828$	$\simeq 70.87$

Our next objective is to calculate the propagator of the model, which we will do first in the symmetrical phase. We will denote the dimensionless two-point function of the complete model by

$$g(\vec{n}_{1},\vec{n}_{2})= \langle\varphi(\vec{n}_{1})\varphi(\vec{n}_{2})\rangle.$$

To order zero we simply have that $g(\vec{n}_{1},\vec{n}_{2})=g_{0}(\vec{n}_{1},\vec{n}_{2})$, where

$$g_{0}(\vec{n}_{1},\vec{n}_{2})= \langle\varphi(\vec{n}_{1})\varphi(\vec{n}_{2})\rangle_{0},$$

so that in this order we have the results of the free theory, $\alpha_{R}=\alpha$ and a simple pole with its residue equal to one (apart from the normalization factor of $1/N^{d}$) in the region of imaginary momenta $\rho^{2}(\vec{k})=-\alpha_{R}$. Note that this first-order result is not sufficient to allow us to take the continuum limit, because we know that $\alpha$ must become negative in the limit, while $\alpha _{R}$ cannot be negative. In the next-order approximation, using equation (1.2.3) up to first order, we will have a result that we shall denote by $g_{1}(\vec{n}_{1},\vec{n}_{2})$, with

$$\langle\varphi(\vec{n}_{1})\varphi(\vec{n}_{2})\rangle\approx g_{1}(\vec{n}_{1},\vec{n}_{2}),$$

and where

$$g_{1}(\vec{n}_{1},\vec{n}_{2})=g_{0}(\vec{n}_{1},\vec{n}_{2}... ...g_{0}(\vec{n}_{1},\vec{n}_{2})\langle S_{I}\rangle_{0}\right].$$

The expectation values that appear here are the zero-order propagator, as we calculated it before in the theory of the free field,

$$g_{0}(\vec{n}_{1},\vec{n}_{2})=\frac{1}{N^d}\sum_{\vec{k}} e... ...c{n}_{1}-\vec{n}_{2})} \frac{1}{\rho^{2}(\vec{k})+\alpha_{0}},$$

and the expectation values containing $S_{I}$. The first one of these can be easily calculated (problem 1.3.5) in terms of expectation values that we have discussed and calculated before in the sections in [12] and [11], yielding

$$\langle S_{I}\rangle_{0}=\frac{1}{2} \left(\alpha-\alpha_{0}+\frac{3}{2}\lambda\sigma_{0}^{2}\right) \sigma_{0}^{2}N^{d}.$$ (1.3.8)

Observe that all the terms diverge strongly in the continuum limit, containing factors of $N^{d}$. The calculation of the last expectation value (problem 1.3.6) is longer and, after some work, we may write it in the form

$$\langle\varphi(\vec{n}_{1})\varphi(\vec{n}_{2})S_{I}\rangle_{0} = \frac{1}{2} \left(\alpha-\alpha_{0}+\frac{3}{2}\lambda\sigma_{0}^{2}\right) \sigma_{0}^{2}N^{d}\;g_{0}(\vec{n}_{1},\vec{n}_{2})$$

$$+(\alpha-\alpha_{0}+3\lambda\sigma_{0}^{2})\sum_{\vec{n}} \;g_{0}(\vec{n}_{1},\vec{n})\;g_{0}(\vec{n},\vec{n}_{2}).$$ (1.3.9)

At this point we have everything written in terms of the propagator of the free theory. Observe that here also we have terms with strong divergences, involving factors of $N^{d}$. The sums over position space may be rewritten in momentum space and manipulated in such a way that, when all the terms are brought together, one verifies that all the terms with strong divergences cancel out, resulting in the final expression for the first-order propagator in position space,

$$\begin{eqnarray*} g_{1}(\vec{n}_{1},\vec{n}_{2})=\frac{1}{N^d}\sum_{\vec{k}} e^{... ...bda\sigma_{0}^{2}} {[\rho^{2}(\vec{k})+\alpha_{0}]^{2}}\right\}. \end{eqnarray*}$$

The expression within braces is the form of the propagator in momentum space. Observe that this time the result depends significantly on $\alpha_{0}$. On the other hand, we may use our freedom in principle, of choosing $\alpha_{0}$ in any way we wish within the stability bounds, in order to simplify this expression, by eliminating the second term, which contains a double pole. In order to do this is suffices to choose

$$\alpha_{0}=\alpha+3\lambda\sigma_{0}^{2}.$$

We may do this only so long as the resulting $\alpha_{0}$ remains positive and so long as it goes to zero in the continuum limit. Examining the expression in the right-hand side of this equation we recognize it as the expression in the equation of the critical curve that we calculated before, which shows that it does in fact go to zero in the $N\rightarrow\infty$ limit, so long as we take the limit in such a way that the parameters of the theory approach the critical curve. We therefore have here some perturbative evidence that the phase transition of the model is indeed of second order and we see once more why it is necessary to take the system to the critical situation in the continuum limit. In addition to this, as we already discussed before in this section, the expression in the right-hand side of the equation is positive in the symmetrical phase, showing that $\alpha_{0}$ will be approaching zero by positive values and thus establishing the consistency of this choice for $\alpha_{0}$. Observe that, with this choice for $\alpha_{0}$, we may write the result for the propagator as

$$g_{1}(\vec{n}_{1},\vec{n}_{2})=\frac{1}{N^d}\sum_{\vec{k}} e... ...c{n}_{1}-\vec{n}_{2})} \frac{1}{\rho^{2}(\vec{k})+\alpha_{R}},$$ (1.3.10)

that is, we get a propagator with form identical to that of the propagator of the free theory, with a renormalized mass $m_{R}$, where we see that $\alpha_{R}=\alpha_{0}$ and the renormalized mass is given by

$$m_{R}^{2}=\lim_{N\rightarrow\infty}N^{2}\alpha_{R}/L^{2}.$$

Formally, we may try to understand the expression $\alpha_{R}=\alpha+3\lambda\sigma_{0}^{2}$ for the renormalized mass parameter $\alpha _{R}$ as the sum of a zero-order term $\alpha$ and a first-order term proportional to $\lambda$. However, in truth this is misleading, because we must recall that the parameter $\alpha$ is in fact negative in any continuum limit and hence that the first-order term cannot be considered as a small correction to the situation in the theory of the free field, in which $\alpha$ must be positive. We see that, in spite of the fact that we have developed this approximation technique in the lines of an expansion in a perturbative series, the resulting object has a character rather different from the expected.

We will see later on that the results of this process of approximation for the renormalized mass agree surprisingly well with the results of the stochastic simulations. In particular, note that the result indicates a unit residue for the pole of the propagator, exactly as in the free theory, $\alpha _{R}$ being the only non-trivial parameter that appears. This unit residue is also found in all the stochastic simulations, within the statistical errors. Judging by the form of the propagator, one would say that the spectrum of the theory seems to be that of free particles with mass $m_{R}$. One might consider interpreting this as an indirect perturbative indication related to the underlying triviality of the model. At least, the result for the residue is compatible with it.

It is interesting to try to understand in clearer physical terms the nature of the approximation technique that we have developed. The crucial point for the success of the technique is the choice of $\alpha_{0}$, which ends up being equivalent to a preliminary implicit choice $\alpha_{0}=\alpha_{R}$, to be resolved after the end of the calculation, a possibility that was suggested in section 1.2. From the very beginning we are trying to approximate the expectation values of the complete model by expectation values of a Gaussian model, which is characterized by only two independent quantities, the expectation value of the field $v_{R}$, which is related to the first-order moment (observables with a single power of the field) of the statistical distribution of the model, and the renormalized mass $m_{0}$, which is related to the second-order moment (observables with two powers of the field). In the case in which there is a non-vanishing $v_{R}$ in the complete model, the shift from the field $\varphi$ to the field $\varphi'$ can be understood as a way to make identical the first-order moments of the two distributions, that of the complete model and that of the Gaussian model used for the approximation. In a similar way, the choice $\alpha_{0}=\alpha_{R}$ can be understood as a way to make identical the second-order moments of the two distributions. Both are implicit conditions which are resolved in a self-consistent way at the end of the calculations.

We see therefore that what we are dealing with here is, much more than part of a perturbative expansion, a Gaussian approximation technique, which is not at all an expansion, but rather a single-step self-consistent type of approximation. Since the Gaussian does not have any moments with order greater than two, we cannot expect that this technique can be successfully used to approximate observables that are related to the higher moments of the distribution of the complete model. In particular, we should not expect that it will be useful to examine the issue of the renormalized coupling constant and the phenomenon of the interaction between particles within the structure of the quantum theory, which are related to the moments of order four and larger. In addition to this, we should not expect that it will be possible to improve on the results obtained here by the inclusion in the calculations of the terms of higher order of the expansion given in equation (1.2.3) since, when we adjust the only two independent moments existing in the Gaussian distribution so as to make them identical to the corresponding moments of the distribution of the complete model, we are already doing the best that can be done in terms of approximate a non-Gaussian distribution by a Gaussian distribution.

As our last objective in this section, we calculate the propagator of the model in the broken-symmetrical phase. The calculations are all very similar to the corresponding calculations in the symmetrical phase, except for the need of the use in this case of the shifted field $\varphi'$. In particular, in this case the same type of cancellation of all the terms with strong divergences takes place. After some work (problem 1.3.7) we obtain in this phase for the first-order propagator, which we denote by $g'_{1}(\vec{n}_{1},\vec{n}_{2})$, with

$$\langle\varphi'(\vec{n}_{1})\varphi'(\vec{n}_{2})\rangle\approx g'_{1}(\vec{n}_{1},\vec{n}_{2}),$$

the result

$$g'_{1}(\vec{n}_{1},\vec{n}_{2})=\frac{1}{N^d}\sum_{\vec{k}} ... ...c{n}_{1}-\vec{n}_{2})} \frac{1}{\rho^{2}(\vec{k})+\alpha_{R}},$$

where the renormalized mass is now defined in terms of the dimensionless parameter

$$\alpha_{R}=-2\left(\alpha+3\lambda\sigma_{0}^{2}\right),$$ (1.3.11)

which is a positive quantity in this phase. Once again the expression in the equation of the critical curve appears, showing once more that $\alpha _{R}$ will go to zero when we approach this curve in the continuum limit, this time by the other side, from the broken-symmetrical phase. The factor of $2$ that appears in this result confirms once more our heuristic expectations and, as we will see later on, it also matches with surprising precision the numerical results in this phase.

Observe that, since $\sigma_{0}^{2}$ is a function of $\alpha _{R}$, both this result and the result for the symmetrical phase are not explicit solutions for $\alpha _{R}$ but rather equations that determine $\alpha _{R}$ in an implicit way,

$$\alpha_{R}-\alpha-3\lambda\sigma_{0}^{2}(\alpha_{R})=0$$

in the symmetrical phase and

$$\alpha_{R}+2\alpha+6\lambda\sigma_{0}^{2}(\alpha_{R})=0$$

in the broken-symmetrical phase, where

Figure 1.3.1: Perturbative results for the renormalized mas parameter $\alpha _{R}$. The curves on the left part are in the broken-symmetrical phase. In this graph $\alpha$ and $\lambda$ are represented by the equivalent parameters $r$ and $\theta$, which are defined in the text. The values of the angle $\theta$ are given in degrees.

$$\sigma_{0}^{2}(\alpha_{R}) =\frac{1}{N^{d}}\sum_{\vec{k}}\frac{1}{\rho^{2}(\vec{k})+\alpha_{R}}.$$

It is not difficult to determine the existence, the number and the character of the solutions of these equations, if one separates from the sum in $\sigma_{0}^{2}$ the term containing the zero mode, and to find these solutions on finite lattices by numerical means (problems 1.3.8, 1.3.9, 1.3.10 and 1.3.11). We show in figure 1.3.1 a graph with some of the numerical solutions, illustrating their behavior for lattices of increasing size. In this graph, instead of the usual Cartesian coordinates $\alpha$ and $\lambda$ in the parameter plane of the model, we use polar coordinates centered at the Gaussian point, with the radius $r$ given by $\sqrt{\alpha^{2}+\lambda^{2}}$, and the angle $\theta$ defined as the angle between the radius vector $(\alpha,\lambda)$ and the negative $\alpha$ semi-axis.

The equation for the symmetrical phase has two solutions, but only one of them is positive and hence physically meaningful. While the positive solution remains finite and non-vanishing in the $N\rightarrow\infty$ limit, the negative solution becomes identically zero in the limit. Note that this equation has solutions over the whole parameter plane and not only in the symmetrical phase. The curves corresponding to this solution are the ones with their maximums at the right in figure 1.3.1. Of course this solution only has meaning in the symmetrical phase, but since this is not a well-defined concept on finite lattices, in order to determine the range of validity of the solution on finite lattices, from the point of view of perturbation theory, we must first discuss the solutions in the broken-symmetrical phase. The equation for the broken-symmetrical phase has two positive solutions, a small one and a large one, but only for certain values of parameters, thus defining a perturbative broken-symmetrical phase even on finite lattices. In the complementary region of the parameter plane, which we might call the perturbative symmetrical phase, the equation has no real solutions. On each finite lattice the position of the curve separating these two regions can be determined numerically, and we might call this curve the perturbative critical curve. The small solution becomes identically zero in the $N\rightarrow\infty$ limit, showing that we should not attribute to it any physical meaning. Once more this seems to be just a perturbative ghost associated to an unstable solution sitting at the maximum that the potential has at the origin. The curves corresponding to the large solution are the ones with their maximums at the left in figure 1.3.1.

One can see that it is the large positive solutions in either phase that carry the expected physical meaning by noting that for $\theta$ equal to $180^{\rm o}$ we are over the positive $\alpha$ semi-axis and therefore have the result for the free theory, $\alpha_{R}=\alpha$, since in this case $\alpha=r=0.1$. For $\theta$ close to $0^{\rm o}$ we approach the negative $\alpha$ semi-axis where, as we discussed before, the two potential wells acquire a large separation from each other and the local distribution of the fields sits at the minimum of one of them. As we saw before, the minimum of the potential can then be approximated by a parabola with a positive quadratic coefficient $-2\alpha$, and the model once more approaches the free theory on finite lattices, with this new value for the dimensionless mass parameter. As one can see in figure 1.3.1 in this case the large solution does indeed approach $\alpha_{R}=-2\alpha$ as expected, where $-2\alpha=2r=0.2$. Just as in the case $\theta=180^{\rm o}$ the solution seems to be exact in this case, because for $\theta$ approaching $0^{\rm o}$ we have a vanishing $\lambda$ and the distribution tends to become purely Gaussian, so that the Gaussian approximation tends to become a perfect one. It is therefore to the two large solutions that we should attribute physical meaning, using the broken-symmetrical solution where it exists and using the symmetrical solution where the broken-symmetrical solution does not exist, and therefore truncating in this way the symmetrical solution. Note that there is a discontinuity between the two solutions at the transition point, but this discontinuity vanishes in the $N\rightarrow\infty$ limit. In this limit the edges of the two curves approach the value $\alpha_{R}=0$ at the critical line.

In the $N\rightarrow\infty$ limit $\sigma_{0}^{2}$ becomes independent of $\alpha _{R}$, the critical transition becomes completely well-defined and the relations between $\alpha _{R}$ and the parameters $(\alpha,\lambda)$ become linear, in either phase. If we define $\sigma_{\infty}^{2}=\sigma_{0}^{2}(N\rightarrow\infty)$, then we have in the symmetrical phase

$$\alpha_{R}=\alpha+3\sigma_{\infty}^{2}\lambda,$$

and in the broken-symmetrical phase

$$\alpha_{R}=-2\alpha-6\sigma_{\infty}^{2}\lambda,$$

which are equations of planes over the $(\alpha,\lambda)$ parameter plane. The three-dimensional graph of $\alpha_{R}(\alpha,\lambda)$ over the parameter plane is composed of pieces of two planes that intersect within the $(\alpha,\lambda)$ plane at the critical line. The first one is a piece of the plane defined by the critical line and the line $\alpha_{R}=\alpha$ within the vertical $(\alpha,\alpha_{R})$ plane, the second one is a piece of the plane defined by the critical line and the line $\alpha_{R}=-2\alpha$ in that same vertical $(\alpha,\alpha_{R})$ plane. The relevant part of the first plane is the part that stands over the symmetrical phase, the relevant part of the second plane is the part that stands over the broken-symmetrical phase.

Note that both on finite lattices and in the $N\rightarrow\infty$ limit the perturbative solution for the expectation value $v_{R}$ of the field is proportional to the value of the renormalized mass in the broken-symmetrical phase, that is, we have

$$v_{R}^{2}=\frac{\alpha_{R}}{2\lambda}.$$

Due to the extra dependence on $\lambda$, we can have independent values of the two dimensionless renormalized quantities, $v_{R}$ and $\alpha _{R}$. Whether or not the same is true for the corresponding dimensionfull quantities $V_{R}$ and $m_{R}$ depends on the dimension $d$ of space-time, because the dimensions of the field and therefore of $V_{R}$ depend on it (problem 1.3.12).

The final conclusion of this effort is that the perturbative technique of Gaussian approximation allows us to calculate in a useful way the observables related to the aspects of propagation of particles and to the aspects of spontaneous symmetry breaking in the quantum theory of the $\lambda \varphi ^{4}$ model. As we shall see later on, these results are surprisingly precise in some cases and, by and large, give us a qualitatively correct picture of the critical behavior of the model. On the other hand, it is doubtful that the technique can be extended in an effective way to other observables and aspects of the model. As far as one can verify up to this point, the model seems to contain particles of mass $m_{R}$, which we may adjust freely, in addition to being able to generate a non-vanishing expectation value $v_{R}=\langle\varphi\rangle$ for the dimensionless field. In the continuum limit $v_{R}$ vanishes, since we must approach the critical curve where the phase transition is of second order, with $v_{R}=0$ over the curve, but it is possible to adjust things so that the dimensionfull field has a non-vanishing expectation value $V_{R}=\langle\varphi\rangle$ in the limit (problem 1.3.12). Hence, up to this point the model seems to contain only the phenomena of propagation and of spontaneous symmetry breaking. Whether or not it contains anything beyond this is an issue for further exploration and discussion (problems 1.3.13 and 1.3.14).

Problems

1. Write the expectation value $\left\langle\varphi'S_{I}\right\rangle_{0}$ in detail and derive equation (1.3.1).

2. Calculate in detail the expectation value shown in equation (1.3.2).

3. Calculate in detail the expectation value shown in equation (1.3.3).

4. Show that the result expressed by equation (1.3.5) implies that the result in equation (1.3.7) for the slope of the critical curve at the origin is in fact exact. In order to show this, take the limits involved with due care: take first the limit $N\rightarrow\infty$ under the condition $\alpha_{0}=m_{0}^{2}/N^{2}$ for finite $m_{0}$, and then take the limit in which $\alpha_{c}\rightarrow 0$ and $\lambda_{c}\rightarrow 0$ along the critical curve, and in which the ratio $\lambda_{c}/\alpha_{c}$ is kept finite and non-zero. Obtain the final result in the form

   
   

   $$\tan(\theta)= \lim_{N\rightarrow\infty}\frac{1}{3\sigma_{0}^{2}(\alpha_{0})},$$
   
   

   where $\theta$ is the angle between the negative $\alpha$ semi-axis and the tangent to the critical curve at the Gaussian point.

5. Calculate in detail the expectation value shown in equation (1.3.8).

6. ($\star$) Calculate in detail the expectation value shown in equation (1.3.9).

7. ($\star$) Calculate in detail the propagator in the broken-symmetrical phase, arriving at the result shown in equation (1.3.11).

8. Show that the perturbative equation which determines the renormalized mass parameter $\alpha _{R}$ in the symmetrical phase, as a function of $\alpha$ and $\lambda$,

   
   

   $$\alpha_{R}-\alpha-3\lambda\sigma_{0}^{2}(\alpha_{R})=0,$$
   
   

   were we recall that

   
   

   $$\sigma_{0}^{2}(\alpha_{R}) =\frac{1}{N^{d}}\sum_{\vec{k}}\frac{1}{\rho^{2}(\vec{k})+\alpha_{R}},$$
   
   

   has a single positive solution for each pair of values $(\alpha,\lambda)$ in the stable region of the parameter plane of the model. In order to do this, remember that the parameter $\alpha _{R}$ has to be positive or zero and consider the behavior of the left side of the equation when $\alpha_{R}\rightarrow 0$ and when $\alpha_{R}\rightarrow\infty$. Remember that the sum that defines the quantity $\sigma_{0}^{2}$ includes the zero mode and write it as

   
   

   $$\sigma_{0}^{2}=\sigma'^{2}_{0}+\frac{1}{N^{d}\alpha_{R}},$$
   
   

   where $\sigma'^{2}_{0}$ has a finite limit for $\alpha_{R}\rightarrow 0$. Show also that there is a second solution, which is negative (and hence destitute of any physical meaning) and which becomes identically zero in the $N\rightarrow\infty$ limit.

9. ($\star$) For given $\alpha$, $\lambda$ and $N$, write a program to solve numerically the equation

   
   

   $$\alpha_{R}=\alpha+3\lambda\sigma_{0}^{2}(\alpha_{R})$$
   
   

   for $\alpha_{R}(\alpha,\lambda)$.

10. Show that the perturbative equation which determines the renormalized mass parameter $\alpha _{R}$ in the broken-symmetrical phase, as a function of $\alpha$ and $\lambda$,

    
    

    $$\alpha_{R}+2\alpha+6\lambda\sigma_{0}^{2}(\alpha_{R})=0,$$
    
    

    has two different real and positive solutions for some pairs of values $(\alpha,\lambda)$ in the stable region of the parameter plane of the model, and no real solutions for other pairs of values. Use the same ideas and techniques that were suggested in problem 1.3.8. Show that the condition on $\alpha$ and $\lambda$ for the existence of solutions can be written in an implicit way, which depends on $\alpha _{R}$ on finite lattices, as

    
    

    $$(\alpha+3\lambda\sigma'^{2}_{0})^{2}\geq\frac{6\lambda}{N^{d}},$$
    
    

    and interpret the meaning of this condition on the continuum limit. Show that when the equality holds in the condition above the renormalized mass parameter is given by $\alpha_{R}=\sqrt{6\lambda/N^{d}}$, and hence that it goes to zero at the critical curve in the continuum limit. Show also that the smaller of the two solutions becomes identically zero in the $N\rightarrow\infty$ limit.

11. ($\star$) For given $\alpha$, $\lambda$ and $N$, write a program to solve numerically the equation

    
    

    $$\alpha_{R}=-2[\alpha+3\lambda\sigma_{0}^{2}(\alpha_{R})]$$
    
    

    for $\alpha_{R}(\alpha,\lambda)$.

12. Verify, in dimensions from $d=3$ to $d=5$, whether or not there are any continuum limits in which $V_{R}=\langle\varphi\rangle$ is finite. If there are, identify them and verify what values the renormalized mass $m_{R}$ can have in such limits. In particular, consider limits in which it is required that both $V_{R}$ and $m_{R}$ remain finite. Show that in $d=3$ this requirement forces us to go to the Gaussian point in the limit, that in $d=4$ we can satisfy it at any point along the critical line, and that in $d=5$ it forces us to make $\lambda$ tend to infinity, a limit which is also known as the sigma-model limit.

13. ($\star$) Calculate, using the first-order perturbative approximation scheme presented in the text, and making the choice $\alpha_{0}=\alpha_{R}$, in each one of the two phases of the model, the quantity $\sigma_{4}$ given by $\sigma_{4}^{4}=\langle\varphi^{4}\rangle$ in the symmetrical phase and by $\sigma_{4}^{4}=\langle\varphi'^{4}\rangle$ in the broken-symmetrical phase. Show that, in either case, one obtains

    
    

    $$\sigma_{4}^{4}\simeq 3\sigma_{0}^{4}-6\lambda S_{4},$$
    
    

    where the sum $S_{4}$ is given in terms of the free propagator by

    
    

    $$S_{4}=\sum_{\vec{n}}g_{0}^{4}(\vec{0},\vec{n}).$$
    
    

14. ($\star$) Evaluate, in each one of the dimensions $d=3$ to $d=5$, the behavior of the sum $S_{4}$ that appears in problem 1.3.13, using approximations by integrals or numerical methods. Determine the conditions under which $S_{4}$ goes to zero in the continuum limit, which causes the factorization rule $\sigma_{4}^{4}=3(\sigma_{0}^{2})^{2}$ to hold, just as is the case of the free theory. Observe that this implies that, to first order, $\langle S_{I}\rangle=\langle S_{I}\rangle_{0}$ in the continuum limit, thus showing that the exchange of the complete distribution by the Gaussian distribution does not affect appreciably the singular character of the action. Observe also that this factorization shows that the local distribution of values of the field at a site tends to become Gaussian in the continuum limit, that is, the model becomes progressively more similar to the free theory.

15. Analyze the behavior in the continuum limit of the equations that determine $\alpha _{R}$ in the two phases of the model, discussed in problems 1.3.8 and 1.3.10, verifying that both lead to the same critical curve.

16. Show, using the first-order perturbative results obtained in the text, that in the complete model the observable $\sigma_{1}$ given by $\sigma_{1}^{2}=\langle\varphi^{2}\rangle$ in the symmetrical phase and by $\sigma_{1}^{2}=\langle\varphi'^{2}\rangle$ in the broken-symmetrical phase, is equal, in either case, to the observable $\sigma_{0}^{2}$ of the free theory. Observe that this shows that the field fluctuates in a similar way in either model, undergoing fluctuations with the same typical size.

17. Show that it is possible to take continuum limits to the Gaussian point over the positive $\lambda$ semi-axis, that is, keeping $\alpha=0$ constant during the limit. Determine how to take this type of limit so that the renormalized mass is finite and non-zero, that is, discover how $\lambda(N)$ must be so that $m_{R}$ has a finite and non-zero limit. This type of limit, which produces a non-zero renormalized mass without involving any parameters with dimensions of mass from the corresponding classical theory, is related to what has been conventionally called the phenomenon of ``dimensional transmutation''.