Critique of Perturbative Renormalization

Using the techniques and ideas developed in section 1.2 we may try to calculate perturbatively the renormalized coupling constant $\lambda_{R}$, which we wrote in terms of observables of the model in section 3.1. Using equation (3.1.7) and calculating the observables involved to first order, with the same choice $\alpha_{0}=\alpha_{R}$ that we used for the calculation of the propagator, we obtain (problem 3.2.1) the classical result $\lambda_{R}=\lambda$. Naturally, this implies that $\Lambda_{R}=\Lambda$ to first order in $\varepsilon$, which means to first order in $\lambda$, which seems to indicate, in a superficial way, that we may have a non-vanishing renormalized coupling constant in the quantum theory. We will see, however, that this is an excessively superficial analysis and that the real situation is much more complex than what it indicates.

Let us recall that, in the calculation of the propagator, the calculation of order zero in $\varepsilon$ resulted in the classical propagator, that is, in the propagator of the Gaussian ensemble that we are using to approximate the ensemble of the complete theory, while the calculation to first order in $\varepsilon$ gave us quantum corrections to the classical result. Also, this first-order result was not a small correction of the zero-order result, but instead was qualitatively different from it. In the case of the coupling constant it is clear that the zero-order calculation results in $\lambda_{R}=0$, which is the value of this quantity in the Gaussian ensemble, while the first-order calculation gives us the classical result. This situation is to be expected, since we are now calculating a quantity which is, by definition, at least proportional to the expansion parameter $\lambda$, and that vanishes when the expansion parameter vanishes, unlike what was the case for $\alpha _{R}$. Unlike what happened in the case of $\alpha _{R}$, we are discussing here a quantity that does not exist at all in the Gaussian ensemble of the free theory. Just like what happened in the case of the propagator, it is possible that passing to the next order, which takes us away from the classical results, will make a qualitative difference.

These results that we refer to as ``classical'' correspond, in the traditional language, to Feynman diagrams with zero loops, that is, to the ``tree'' approximation. These results do not include the effects of the quantum fluctuations of the fields on the observables. Hence, the zero-loop approximations do not include the quantum effects contained in the theory, but only the effects of the classical dynamics of the fields. In order to include the effect of the quantum fluctuations it is necessary to do the calculations up to the lowest order of $\varepsilon$ which includes diagrams with one loop. In the case of $m_{R}$ this meant doing calculations up to the first order in $\varepsilon$, but in order to explore the effects of the quantum fluctuations of the fields on the renormalized coupling constant it is necessary to calculate $\lambda_{R}$ to second order in $\varepsilon$, thus including diagrams with up to one loop. These calculations to order $\varepsilon^{2}$ are considerably longer and more complex than those to the first order, and involve quantities with strong divergences that behave as either $N^{d}$ or as $N^{2d}$, all of which cancel out completely from the final results. Doing the calculation in the symmetrical phase, with vanishing momenta on the four external legs, we obtain (problem 3.2.2)

$$\lambda_{R}=\frac{\displaystyle \lambda\left[ 1-4\frac{\alph... ... \left[1-\frac{\alpha_{R}-\alpha_{0}}{\alpha_{0}}\right]^{4}},$$

where the dimensionless sum $s_{2}(\alpha_{0})$ that appears here is given, in terms of the dimensionless free propagator $g_{0}(\vec{n},\vec{n}')$, by

$$s_{2}(\alpha_{0})=\sum_{\vec{n}}g_{0}^{2}(\vec{0},\vec{n}) =... ...{1}{N^d}\sum_{\vec{k}}\frac{1}{(\rho_{k}^{2}+\alpha_{0})^{2}}.$$ (3.2.1)

In principle it is also possible to do the calculation in the broken-symmetrical phase (problem 3.2.3), but currently we do not yet know the answer for that case. Making the choice $\alpha_{0}=\alpha_{R}$ as before, we obtain

$$\lambda_{R}=\lambda[1-9\lambda s_{2}(\alpha_{R})].$$ (3.2.2)

Observe that, since the sum is a positive quantity, the correction is always negative, tending to decrease the magnitude of the positive classical result. We may try to evaluate the behavior of the sum $s_{2}$ for large values of $N$ by means of approximations by integrals, as we did before in other cases. Doing this (problem 3.2.4) we obtain the following results, for the usual values of the dimension $d$:

$$\begin{array}{ll} \rule{0em}{3ex} d=1: & s_{2}\approx \frac{... ...aystyle \Omega_{d}}{\displaystyle (2\pi)^{4}(d-4)}, \end{array}$$ (3.2.3)

where $\Omega_{d}$ is the total solid angle of $d$-dimensional space. We see therefore that the second-order result is divergent for $d=3$ and $d=4$, and finite for $d\geq 5$. Observe that, in order for the results in $d=3$ and $d=4$ to make any sense, it is necessary that we make $\lambda\rightarrow 0$ in the continuum limit, thus forcing us to return to the Gaussian point. In $d=4$ this takes us back to the free theory but, curiously, despite this limitation the result is still of some interest in the case $d=3$, because in this case the dimensionfull coupling constant is given by $\Lambda_{R}=\lambda_{R}/a=N\lambda_{R}/L$ and we therefore have, in terms of the dimensionfull quantities, a finite expression,

$$\Lambda_{R}=\Lambda[1-9\Lambda S_{2}],$$ (3.2.4)

where the dimensionfull quantity $S_{2}=s_{2}L/N$ has a finite limit, so long as we have a finite and non-vanishing renormalized mass $m_{R}$,

Figure 3.2.1: Behavior of the sum $s_{2}$ with $N$ in the case $d=1$.

$$S_{2}=\frac{s_{2}L}{N}\approx\frac{\displaystyle 1}{\displaystyle 2\pi^{2}m_{R}}.$$

This seems to indicate that in $d=3$ there are non-trivial limits, with $\Lambda_{R}\neq 0$, that approach the Gaussian point, where both $\lambda$ and $\lambda_{R}$ are zero. We can use this $d=3$ result to exhibit explicitly renormalization flows $[\alpha(N),\lambda(N)]$, in the parameter space of the corresponding model, that approach the Gaussian point in such a way that both $m_{R}$ and $\Lambda_{R}$ are different from zero in the limit (problem 3.2.5).

Figure 3.2.2: Behavior of the sum $s_{2}$ with $N$ in the case $d=2$.

Of course this type of limit constitutes a small subset of all possible alternatives, in which we approach, in the limit, some other arbitrary point of the critical curve, rather than the Gaussian point. About these other possibilities our perturbative approximation has nothing to say, but note that, is we assume that there is at least one such limit for each one of these points, in which $\Lambda_{R}$ is finite, it follows immediately that it is necessary that $\lambda_{R}\rightarrow 0$ over the whole critical curve, when we make $N\rightarrow\infty$. This means that it is highly likely that the ensemble of the renormalized theory becomes Gaussian over the critical curve in the continuum limit. Since $\lambda_{R}$ is a dimensionless quantity that measures, just like $\alpha _{R}$, a moment of the distribution of the renormalized model, it is very reasonable to think that both should have the same particular type of behavior in the locus of the parameter plane of the model where the critical transition takes place. In other words, it is reasonable to think that $\lambda_{R}$ should always go to zero over the critical curve in the continuum limit, as part of the critical behavior of the model.

Figure 3.2.3: Behavior of the sum $s_{2}$ with $N$ in the case $d=3$.

We may also try to extract some information of the result in equation (3.2.2) in the case $d\geq 5$. In this case our result seems to indicate that we will have a finite and non-vanishing $\lambda_{R}$ as a function of $\alpha$ and $\lambda$, since the sum $s_{2}$ is finite and non-vanishing. It is very reasonable to think that the dimensionless quantities like $\lambda_{R}$ will always be finite in the limit, when we make $N\rightarrow\infty$, since they do not scale with $N$. Since we have that $\Lambda_{R}=a^{d-4}\lambda_{R}$, for dimensions $d\geq 5$ we have that a finite $\lambda_{R}$ implies a vanishing $\Lambda_{R}$ in the limit. We establish in this way the expectation that the model is completely trivial in $d\geq 5$, with $\lambda_{R}$ finite and $\Lambda_{R}$ vanishing over the critical curve. Of course this conclusion depends on the higher-order terms of the perturbative series of $\lambda_{R}$ being all finite, besides the series being convergent. Obviously, none of these two things is guaranteed. One might even imagine that the series could end up converging to zero, a hypothesis which would not change the physical meaning of the theory, since that meaning is defined in terms of $\Lambda_{R}$, which would still vanish in the limit. However, that hypothesis would make sense in terms of the critical behavior of the dimensionless quantity $\lambda_{R}$, as we discussed above in the case $d=3$.

Figure 3.2.4: Behavior of the sum $s_{2}$ with $N$ in the case $d=4$.

Figure 3.2.5: Behavior of the sum $s_{2}$ with $N$ in the case $d=5$.

In the case $d=4$ we cannot conclude anything about the behavior of the theory from our results, since in this case $s_{2}$ diverges logarithmically while $\Lambda_{R}=\lambda_{R}$. We can confirm our analytical estimates of the dependencies of $s_{2}$ on $N$ and calculate approximately the relevant coefficients, performing numerically the sum on finite lattices (problem 3.2.6). The results obtained in this way, for $m_{R}L=1$, are shown in the graphs found in figures from 3.2.1 to 3.2.5. Simple but good-quality curve fittings, with only a single parameter in the cases $d=1$, $d=2$ and $d=3$, and with three parameters in the cases $d=4$ and $d=5$, give the approximate results

$$\begin{eqnarray*} s_{2}(d=1) & \approx & 1.0N^{3}, s_{2}(d=2) & \approx & 1.0... ...s_{2}(d=5) & \approx & 0.19+1.0\frac{1}{N}-0.014\frac{1}{N^{2}}. \end{eqnarray*}$$

One may also consider calculating the expectation value of the field and of the propagator up to order $\varepsilon^{2}$, which corresponds to the inclusion of diagrams with up to two loops. In some cases one may still be able to extract from these calculations some useful information such as, for example, for the determination of the critical curve (problem 3.2.7) and, in some cases, for the determination of the renormalized mass (problem 3.2.8). However, in general the results of these calculations include divergent sums and it is not possible to use them in a systematic way to establish a system of successive approximations, which hopefully would be increasingly precise, for all the observables of the theory.

Therefore, we see that we do not really have a complete expansion for the $\lambda \varphi ^{4}$ model, but only a set of approximations that work reasonably well in some cases. What we have here is not a consistent and systematic series development, but only a set of isolated approximations whose validity can only be verified, ultimately, by direct comparison with numerical results or other non-perturbative approximations. Observe that this behavior of the perturbative expansion is due to the exchange of the order of the two limits involved, the continuum limit and the series summation limit. For finite $N$ all the terms of the expansion are finite and the series may be convergent, or at least asymptotic, but in the $N\rightarrow\infty$ limit the individual terms become infinite and nothing can be done to salvage the situation in general, except making the parameters of the model converge sufficiently fast to the Gaussian point. In order to make use of the series it is imperative to first sum it on finite lattices and only then take the continuum limit, one cannot invert the order of the limits.

In conclusion, we verify that in the case of the coupling constant we do not have a perturbative approximation as successful as in the cases of the renormalized mass and of the expectation value of the field. It allows us to go so far as to formulate the conjecture that, in any dimension $d\geq 3$, in any limit where $N\rightarrow\infty$, $\alpha(N)\rightarrow\alpha_{(c)}$ and $\lambda(N)\rightarrow\lambda_{(c)}$, where $(\alpha_{(c)},\lambda_{(c)})$ is a point over the critical curve, we have $\lambda_{R}\rightarrow 0$. However, it does not allow us to make concrete predictions about the behavior of the model away from the Gaussian point, as we were able to do for the renormalized mass and the expectation value of the field. It seems, therefore, that the discussion of the perturbative approximation should be separated in two parts, the first one being relative to the calculation of the quantities involving, at most, the second moment of the distribution of the model, as is the case for $v_{R}$ and $\alpha _{R}$, while the second one relates to the quantities involving the higher-order moments, such as $\lambda_{R}$.

We see that the reason why the approximation does not work so well for the coupling constant is the fact that it is not possible to impose, in this case, the equivalent of the condition $\alpha_{0}=\alpha_{R}$ used in the case of the renormalized mass, which transformed the first-order perturbative approximation into a self-consistent Gaussian approximation. This is due, of course, to the fact that the Gaussian model, the only one we know how to solve exactly, has no moments of order greater than two, whose coefficients may be adjusted so as to reproduce faithfully the characteristics of the ensemble of the complete model. When we choose $\alpha_{0}=\alpha_{R}$ we are making the moment of order two of the Gaussian ensemble reproduce in a perfect way the corresponding moment of the ensemble of the complete model, so that the difference between the two distributions can in fact be considered a small perturbation of the Gaussian distribution, in so far as that observable is concerned. However, independently of any choices of parameters, the fourth-order moment of the Gaussian ensemble is always zero and cannot be adjusted to reproduce the corresponding moment of the complete ensemble. Therefore, the fourth-order moment can never be understood as a small perturbation when we are dealing with observables that only exist when this moment is not zero.

It is important that we discuss here the role of the traditional scheme of perturbative renormalization, in the context of the calculations on the lattice^3.1. First of all, there can be no doubt that the definition of the model on the lattice implies the existence of well-defined relations $\alpha_{R}(\alpha,\lambda)$ and $\lambda_{R}(\alpha,\lambda)$ between these renormalized quantities and the parameters of the model, for any values of these parameters within the stable region of the parameter plane of the models and in any dimension $d$, for finite lattices, and over the critical curve in any dimension $d\geq 3$, in the continuum limit. Analogously, for any observable ${\cal O}$ that is physically relevant we have a well-defined relation ${\cal O}(\alpha,\lambda)$, under the same conditions. What we discover when we work out the development of perturbation theory is that the perturbative approximations for the relations between the observables ${\cal O}$, $\alpha _{R}$ and $\lambda_{R}$ and the parameters $\alpha$ and $\lambda$ are singular, in the sense that they contain quantities that diverge in the continuum limit.

The traditional perturbative renormalization scheme consists of giving up, at this point, any effort of extracting from the theory the relations $\alpha_{R}(\alpha,\lambda)$, $\lambda_{R}(\alpha,\lambda)$ and ${\cal O}(\alpha,\lambda)$ that it contains. In addition to this, on each finite lattice, where all quantities are finite, we may consider rewriting ${\cal O}$ directly in terms of $\alpha _{R}$ and $\lambda_{R}$, eliminating the parameters $\alpha$ and $\lambda$ from the picture in favor of their renormalized counterparts, and thus obtaining a relation ${\cal O}(\alpha_{R},\lambda_{R})$ that, possibly, will not contain any quantities that diverge in the continuum limit. If it is possible to do this, then the relation obtained is a well-behaved perturbative approximation of ${\cal O}$ in terms of $\alpha _{R}$ and $\lambda_{R}$, and we may then take the continuum limit without stumbling on any singularities. If it is possible to do this for the perturbative calculations to all orders, then we say that the theory is perturbatively renormalizable, and the scheme produces a complete perturbative series in the continuum limit, with finite individual terms, which may or may not be convergent. Since $\alpha$ and $\lambda$ are not directly observable, while $\alpha _{R}$ and $\lambda_{R}$ presumably are, the resulting function ${\cal O}(\alpha_{R},\lambda_{R})$ is a direct relation between observables of the theory, so that not much seems to be lost when one does this.

Of course, rewriting ${\cal O}$ in terms of $\alpha _{R}$ and $\lambda_{R}$ on finite lattices may not be easy, in fact, it may not be possible at all in closed form, so one may be compelled to re-expand the expressions that appear when one tries to do this, possibly neglecting higher-order terms in order to keep consistent powers of the expansion parameters. All this considerably complicates the whole argument and makes it more difficult to understand what is really going on when one does all this. Let us try to exemplify this in the context of the calculations that we made here, at first in a very simple and possibly incomplete way. We have calculated one-loop approximations for $\alpha _{R}$ and $\lambda_{R}$, obtaining expressions of the form

$$\alpha_{R}=f_{\alpha}(\alpha,\lambda)\mbox{   and   } \lambda_{R}=f_{\lambda}(\alpha,\lambda).$$

We saw that, while the one-loop propagator is entirely finite, the renormalized mas parameter being given by

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

the one-loop coupling constant contains the divergent sum $s_{2}$, being given by

$$\lambda_{R}=\lambda[1-8s_{2}\lambda].$$

We proceed then to a change of variables, introducing a new parameter $\lambda_{0}$ in place of $\lambda$, defined by the relation

$$\lambda=\lambda_{0}+9s_{2}\lambda_{0}^{2}.$$

Note that this mixes powers of $\lambda$ and corresponds, therefore, to a reorganization of the perturbative expansion. Substituting this expression for $\lambda$ in the result for $\lambda_{R}$ we verify that the divergent terms of order $\lambda_{0}^{2}$ cancel out, so that we obtain

$$\lambda_{R} =\lambda_{0}-162s_{2}^{2}\lambda_{0}^{3}-729s_{2}^{3}\lambda_{0}^{4}.$$

We now argue that we can neglect in this equation the terms of orders $\lambda_{0}^{3}$ and $\lambda_{0}^{4}$, not because they are small, since they are clearly divergent in the limit, but under the allegation that they will cancel out with the remaining higher-order terms that have not yet been explicitly included in this analysis. This is the first condition involved in the criterion of perturbative renormalizability in a weak sense, term by term in the perturbative expansion, without preoccupation with its convergence. What we are requiring here is that the divergent terms cancel out, not in the original series, but after its reorganization by the change of variables from $\lambda$ to $\lambda_{0}$. Under these conditions we have

$$\lambda_{R}=\lambda_{0},$$

which shows that our change of variables is in fact a change from an expansion in terms of the basic parameter $\lambda$ to another expansion in terms of the renormalized parameter $\lambda_{R}$. If we now write $\lambda$ in terms of $\lambda_{R}$,

$$\lambda=\lambda_{R}+9s_{2}\lambda_{R}^{2},$$

we see that $\lambda$ should diverge in order for this relation to be valid. This is the opposite of what we saw in the original perturbative expansion, where we verified that $\lambda$ must go to zero in order for the perturbative approximation to be valid. However, since by now we have given up obtaining from the theory the relations between the basic quantities and the renormalized quantities, we might as well just not worry about this any more, and simply disregard the equation above. Of course, one cannot avoid the strong impression that this whole procedure is mired with guesswork and arbitrariness. It certainly looks like it would be very difficult to show that all the facts assumed do indeed hold to all orders and hence to establish the results of this procedure on a firm logical basis.

Anyway, up to this point the change of variables has not really been of any use, since it simply introduced another parameter $\lambda_{0}$ that ended up being another name for $\lambda_{R}$. No additional information about the relation between $\lambda_{R}$ and the basic parameters of the model was obtained. It is important to emphasize that this fact is no more than a limitation of the perturbative method and that this relation undoubtedly exists in the model defined by means of the lattice. In order to show the possible usefulness of the perturbative renormalization scheme, we may now consider the calculation of a third observable ${\cal O}$, at first in terms of $\alpha$ and $\lambda$, resulting in a relation of the type

$${\cal O}=f_{{\cal O}}(\alpha,\lambda),$$

which presumably contains some terms with divergent factors. We may now substitute $\lambda$ for $\lambda_{0}$, re-expanding the resulting expression and neglecting once more the higher-order terms that appear. With some more manipulation we may also substitute $\alpha$ for $\alpha _{R}$, thus obtaining a new relation

$${\cal O}=\bar{f}_{{\cal O}}(\alpha_{R},\lambda_{R}),$$

which, so long as the model is perturbatively renormalizable, should not contain any divergences. In this way we extract from the model a well-behaved relation between $\alpha _{R}$, $\lambda_{R}$ and ${\cal O}$, although the fundamental perturbative expansion in terms of $\alpha$ and $\lambda$ is not well behaved. As an example of such an observable, we may consider the coupling constant for a non-vanishing momentum $\vec{k}$, a quantity which is related in a direct way to the scattering cross-sections. Calculating the coupling constant for the same momentum $\vec{k}$ on all the four external legs, up to order $\lambda^{2}$, we obtain (problem 3.2.9)

$$\lambda_{R}(\vec{k})=\lambda\left\{1-3\lambda \left[2s_{2}(\vec{0},\alpha_{0})+s_{2}(\vec{k},\alpha_{0})\right] \right\},$$

where the sum $s_{2}(\vec{k},\alpha_{0})$ is given by

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

Proceeding with the substitution of $\lambda$ by $\lambda_{0}=\lambda_{R}=\lambda_{R}(\vec{k}=\vec{0})$ we obtain

$$\lambda_{R}(\vec{k})=\lambda_{0}\left\{1+3\lambda_{0} \left[... ...(\vec{0},\alpha_{0})-s_{2}(\vec{k},\alpha_{0})\right]\right\}.$$ (3.2.5)

The difference of the two sums can be evaluated for small values of the momentum $\vec{k}$ with the help of approximations by integrals and, doing this in $d=4$ and for large values of $N$ (problem 3.2.10), we obtain a finite result,

$$\lambda_{R}(\vec{k})=\lambda_{R}\left[1+ 24\pi^{2}\lambda_{R}\frac{k^{2}}{L^{6}m_{R}^{6}}\right].$$

We say that the theory is perturbatively renormalizable if it is possible to do this in each order of perturbation theory, and hence to obtain predictions with arbitrarily high precision for ${\cal O}$, given values of $\alpha _{R}$ and $\lambda_{R}$. Note that, if we imagine that the theory is in fact trivial, then we see that this result is not wrong, but that is is simply rather irrelevant, because in this case the only possible value for $\lambda_{R}$ is zero and the relation simply shows that $\lambda_{R}(\vec{k})=0$ for any $\vec{k}$. We can see now that there is in fact a rather subtle problem behind all this. When we do this kind of manipulation we are giving up obtaining from the theory the relations between $(\alpha_{R},\lambda_{R})$ and $(\alpha,\lambda)$ and, instead of that, we implicitly assume that certain values of $\alpha _{R}$ and $\lambda_{R}$ are possible in the context of the model defined in a non-perturbative way by means of the lattice. This seems to be a very reasonable thing to do in a model which is defined with two free parameters, and we certainly know which values are or are not possible for $\alpha$ and $\lambda$. However, we do not have now any information about which values are in fact possible for the renormalized parameter $\lambda_{R}$, according to the non-perturbative definition of the model. Therefore, we do now know which values we may in fact use for $\lambda_{R}$ in this perturbative renormalization scheme.

It is implicitly assumed, in the traditional perturbative renormalization scheme, that the possible values for $(\alpha_{R},\lambda_{R})$ are the same which are possible for $(\alpha,\lambda)$. However, in general it is possible that this is not true, and that there are restrictions for the images of the relations $\alpha_{R}(\alpha,\lambda)$ and $\lambda_{R}(\alpha,\lambda)$ determined by the non-perturbative definition of the models. One restriction that we already know to exist in this model is that $\alpha_{R}\geq 0$, while the parameter $\alpha$ can be either positive or negative on finite lattices, and must become negative in the continuum limit, as we saw in section 1.3. Another fact, which is even more important than this one, is that there certainly are important restrictions for $\lambda_{R}$ in a model that ends up being trivial, in which the only possible value for $\lambda_{R}$ in the $N\rightarrow\infty$ limit is zero. We can always determine beforehand which values are possible for $(\alpha,\lambda)$, but we cannot do the same for $(\alpha_{R},\lambda_{R})$. Triviality implies that the usual implicit hypothesis, that the possible values for $(\alpha_{R},\lambda_{R})$ are the same which are possible for $(\alpha,\lambda)$, is false. To continue with the usual perturbative renormalization scheme under these conditions can only produce fictitious results, without any physical or mathematical relevance.

The conclusion is that a model satisfying the criterion of perturbative renormalizability is not sufficient to guarantee the usefulness of its perturbative expansion, renormalized in the usual way. It is also necessary to determine the values which are actually possible for the renormalized parameters, in terms of which one chooses to write the renormalized perturbative expansion. In regard to this aspect of the structure of the theory it is important to emphasize the profound difference between a truly physical theory, such as quantum electrodynamics, and models that have only the role of mathematical laboratories, such as the polynomial models. In quantum electrodynamics we can go to the laboratory and determine experimentally the values of the renormalized mass and of the renormalized coupling constant, that is, of the mass and charge of the electron, thus establishing that certain values are possible for these quantities. On the other hand, in the laboratory models we are limited to what we can calculate analytically or numerically and we must extract this type of information from the relations that the models establish between the renormalized quantities and the parameters involved in their definitions. Since perturbation theory is not able to give us these relations in a complete form, it only remains for us to try non-perturbative methods, such as computational stochastic simulations, as tools to establish the possible values for the renormalized parameters. Another way to characterize this profound difference is to say that, in the case of a truly physical theory, we have access to the use of the ultimate computer: the fundamental laws of physics at play in nature.

Problems

Note: Some of the calculations contained in some of these problems are really very long, and a considerable amount of organization and care is needed to get to the end without errors. The problems containing such long calculations are marked with three stars.

1. Calculate $\lambda_{R}$ using equation (3.1.7) and calculating the observables involved to first order, with the choice $\alpha_{0}=\alpha_{R}$, thus obtaining the classical result $\lambda_{R}=\lambda$. Do the calculations both in the symmetrical phase and in the broken-symmetrical phase. Note that, since $\lambda_{R}$ itself is already a first-order quantity in $\lambda$, in order to keep consistent orders of the expansion parameter the numerator of equation (3.1.7) should be calculated to first order, while it is enough to calculate the denominator to order zero.

2. ($\star$) Calculate $\lambda_{R}$ to second order, using equation (3.1.7), with the choice $\alpha_{0}=\alpha_{R}$, in the symmetrical phase, obtaining the result quoted in the text,

   
   

   $$\lambda_{R}=\frac{\displaystyle \lambda\left[ 1-4\frac{\alph... ... \left[1-\frac{\alpha_{R}-\alpha_{0}}{\alpha_{0}}\right]^{4}}.$$
   
   

   Note that, since $\lambda_{R}$ itself is already a first-order quantity in $\lambda$, in order to keep consistent orders of the expansion parameter the numerator of equation (3.1.7) should be calculated to second order, while it is enough to calculate the denominator to first order.

3. ( $\star\star\star$) Repeat the calculation of $\lambda_{R}$ to second order proposed in problem 3.2.2, this time in the broken-symmetrical phase^3.2.

4. Evaluate the asymptotic behavior of the sum $s_{2}$ given in equation (3.2.1), for large values of $N$, approximating it by integrals over the momenta, as we did before in the case of the quantity $\sigma_{0}^{2}$ related to the propagator, for each relevant value of $d$. Whenever it becomes necessary to use a minimum but non-vanishing value of the modulus of the momentum as a lower integration limit, use $m_{R}$ as that value.

5. Use the result in equation (3.2.4) for the dimensionfull renormalized coupling constant $\Lambda_{R}$ in $d=3$, as well as the one-loop result for the renormalized mass $m_{R}$ obtained in section 1.3, in order to exhibit explicitly flows $[\alpha(N),\lambda(N)]$ that approach the Gaussian point in the continuum limit and for which both $m_{R}$ and $\Lambda_{R}$ have finite and non-vanishing limits. Assume, if necessary, that $\Lambda_{R}$ is small compared to $m_{R}$. The solutions should tend asymptotically to the line tangent to the critical curve at the Gaussian point, and both $\alpha(N)$ and $\lambda(N)$ should go to zero as $1/N$. Hint: try

   
   

   $$\alpha=\frac{A}{N}+\frac{B}{N^{2}}\mbox{   and   }\lambda=\frac{C}{N}.$$
   
   

6. Write programs to calculate numerically the sum $s_{2}$ given in equation (3.2.1) in dimensions from $d=1$ to $d=5$ and confirm the asymptotic results obtained in problem 3.2.4. These sums should be calculated with the same numerical techniques that were used for the calculation of the sums that appear in the quantity $\sigma_{0}^{2}$, which were calculated in the section in reference [49]. In fact, it suffices to make small changes in the programs written for that case in order to produce the programs needed in this case.

7. ($\star\star$) Calculate $v_{R}$ to order $\varepsilon^{2}$ and thus obtain the two-loop evaluation of the equation of the critical curve. Start by calculating $v_{R}$ in the broken-symmetrical phase and obtain the result
   $$\begin{eqnarray*} 0 & = & \left(\lambda v^{2}+\alpha+3\lambda\sigma_{0}^{2}\righ... ...+6\lambda^{2}\alpha_{0}\sum_{\vec{n}}g_{0}^{3}(\vec{0},\vec{n}). \end{eqnarray*}$$
   
   
   
   

   Next evaluate the asymptotic behavior of the new sum that appears,

   
   

   $$s_{3}(\alpha_{0})=\sum_{\vec{v}}g_{0}^{3}(\vec{0},\vec{n}) =... ...)+\alpha_{0}] [\rho^{2}(\vec{k}_{1}+\vec{k}_{2})+\alpha_{0}]}.$$
   
   

   Finally, recalling that $\alpha_{0}$ must go to zero as $1/N^{2}$ in the limit, make $v_{R}=0$ and show that, in the continuum limit, one recovers the one-loop result for the equation of the critical curve,

   
   

   $$\left(\alpha+3\lambda\sigma_{0}^{2}\right)^{2}=0.$$
   
   

   Therefore, we conclude that the equation of the critical curve does not contain corrections of order $\lambda^{2}$ and that any correction to the order-$\lambda$ result must be at least of order $\lambda^{3}$.

8. ( $\star\star\star$) Try to calculate the renormalized mass $m_{R}$ in the symmetrical phase up to order $\varepsilon^{2}$. Start by calculating the propagator to this order, obtaining the result
   $$\begin{eqnarray*} g_{2}(\vec{n}_{1},\vec{n}_{2}) & = & g_{0}(\vec{n}_{1},\vec{n}... ..._{0}^{3}(\vec{n}_{3},\vec{n}_{4})g_{0}(\vec{n}_{4},\vec{n}_{2}), \end{eqnarray*}$$
   
   
   
   

   which, in momentum space, can be written as

   $$\begin{eqnarray*} N^{d}\widetilde g_{2}(\vec{k}) & = & \frac{1}{\rho^{2}(\vec{k}... ...}{[\rho^{2}(\vec{k})+\alpha_{0}]^{2}} s_{3}(\vec{k},\alpha_{0}), \end{eqnarray*}$$
   
   
   
   

   where the sum $s_{3}$ is given by

   
   

   $$s_{3}(\vec{k},\alpha_{0})=\frac{1}{N^{2d}}\sum_{\vec{k}_{1},... ..._{0}] [\rho^{2}(\vec{k}-\vec{k}_{1}-\vec{k}_{2})+\alpha_{0}]}.$$
   
   

   Observe that, since $\alpha _{R}$ must still vanish over the critical curve, and the equation of that curve did not change up to the order $\varepsilon^{2}$, as we saw in problem 3.2.7, we should expect that the renormalized mass also does not change up to this order. Discover whether or not it is possible to choose $\alpha_{0}$ in an appropriate way and thus show that the renormalized mass also does not change up to this order, establishing therefore the consistency of the two calculations^3.3.

9. ($\star$) Calculate $\lambda_{R}(\vec{k})$ for equal non-vanishing momenta $\vec{k}$ on all the external legs, entering in two of them and going out in the other two, using for this purpose the expression for this quantity that results from problem 3.1.7, to second order, with the choice $\alpha_{0}=\alpha_{R}$, in the symmetrical phase, obtaining the result quoted in the text,

   
   

   $$\lambda_{R}(\vec{k})=\lambda\left\{1-3\lambda \left[2s_{2}(\vec{0},\alpha_{0})+s_{2}(\vec{k},\alpha_{0})\right] \right\},$$
   
   

   where the sum $s_{2}$ is given by

   
   

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

   Note that, since $\lambda_{R}(\vec{k})$ itself is already a first-order quantity in $\lambda$, in order to keep consistent orders of the expansion parameter the numerator of the equation that defines $\lambda_{R}(\vec{k})$ should be calculated to second order, while it is enough to calculate the denominator to first order.

10. Evaluate the asymptotic behavior of the difference of sums given in equation (3.2.5), for large values of $N$, approximating the sums by integrals over the momenta, as we did before in problem 3.2.4. Whenever it becomes necessary to use a minimum but non-vanishing value of the modulus of the momentum as a lower integration limit, use $m_{R}$ as that value.