The Coupling Constant

Having discussed in sections 1.2 and 1.3 the behavior of the $\lambda \varphi ^{4}$ model with respect to the one-point and two-point correlation functions, which are related respectively to the phenomena of spontaneous symmetry breaking and of the propagation of waves and particles, we will now consider the three-point and four-point functions, which are related to the phenomena of interaction between waves or between particles within the model. Our first task will be to discuss the nature of the renormalized (or physical) coupling constant $\lambda_{R}$, and of its dimensionfull version $\Lambda_{R}$, relating them to expectation values of observables of the model. In this way we will define these quantities and determine, at least in principle, the way to calculate them.

As we will see, the renormalized coupling constant is a quantity that vanishes in the Gaussian model and whose value measures how non-Gaussian the renormalized ensemble of the model under study is, thus determining its true degree of non-linearity and the existence or not, within the structure of the model, of phenomena of interaction between waves or between particles. Note that we are not talking here about the ensemble of the fundamental field, but rather about the ensemble of the physical variables associated to blocks, as was discussed in the section in reference [6], since these are the variables that are directly associated to the actual physical observables of the theory. Hence, we should expect the quantity of greater interest in this discussion to be the dimensionfull renormalized coupling constant $\Lambda_{R}$, since it is the dimensionfull quantities that scale in the correct way and thus are related to the block-variable observables, as we saw explicitly in the case of the propagator in the section in reference [6].

In order to be able to write the renormalized coupling constant in terms of observables of the model, we return to the discussion of the formalism of the generating functionals and of the effective action, which were introduced in the sections in references [39] and [45]. We saw in the section in reference [39] that the complete Green functions $g_{1,\ldots,n}=\left\langle\varphi_{1}\ldots\varphi_{n}\right\rangle$ of the theory in the absence of external sources can be obtained by means of multiple functional differentiations with respect to $j$ of the functional $Z[j]$ defined in the equation in reference [41], after which one makes $j=0$. In addition to this, the connected Green functions $g_{(c,j)1,\ldots,n}$ in the presence of external sources can be obtained by means of multiple functional differentiations with respect to $j$ of the functional $W[j]=\ln(Z[j])$. It is in these connected functions that the true correlations of the theory are encoded, including in the case in which we have non-vanishing external sources. The connected two-point function was calculated explicitly in the equation in reference [42] and the calculation of the corresponding three-point function was proposed in the problem in reference [43]. In that problem, starting from the definition of the three-point function in terms of $W[j]$,

$$\frac{\mbox{\boldmath$\mathfrak{d}$}^{3}W[j]}{\mbox{\boldmat... ...d}$}j_{2}\mbox{\boldmath$\mathfrak{d}$}j_{3}} =g_{(c,j)1,2,3},$$

one shows that this functions can be written in terms of the complete three-point function as

$$g_{(c,j)1,2,3}=g_{(j)1,2,3}-g_{(c,j)1,2}\;\varphi_{(c)3} -g_... ...varphi_{(c)2} -\varphi_{(c)1} \varphi_{(c)2} \varphi_{(c)3},$$

which corresponds to the subtraction from the complete function of all possible factorizations in terms of connected functions with a smaller number of points. Note that we returned here to the notation of the section in reference [39], denoting the expected value of the field by $\varphi _{(c)}$ instead of $v_{R}$ and the dependencies on the positions $\vec{n}_{i}$ of the sites by indices, $\varphi_{(c)}(\vec{n}_{1})=\varphi_{(c)1}$. Observe that in any circumstances in which $\varphi_{(c)}=0$, which naturally implies that $j=0$, we have for this connected function $g_{(c)1,2,3}=g_{1,2,3}$, which makes the definition of the renormalized coupling constant considerably simpler in models based on three-point interactions, as is the case, for example, for electrodynamics.

However, the polynomial models of scalar fields, such as the $\lambda \varphi ^{4}$ model that we are studying here, are based on interactions involving four or more points, so that we must go at least up to the four-point function in order to be able to define the renormalized coupling constant. It is necessary, therefore, to take a fourth and last derivative of $W[j]$ (problem 3.1.1), which results, after a long but straightforward algebraic calculation, in a relation between the connected four-point function and the corresponding complete function,

$$g_{(c,j)1,2,3,4} = g_{(j)1,2,3,4}$$

$$- [g_{(c,j)2,3,4}\;\varphi_{(c)1} +g_{(c,j)1,3,4}\;\varphi_{(c)2} +g_{(c,j)1,2,4}\;\varphi_{(c)3} +g_{(c,j)1,2,3}\;\varphi_{(c)4}]$$

$$- [g_{(c,j)1,2} g_{(c,j)3,4} +g_{(c,j)1,3} g_{(c,j)2,4} +g_{(c,j)1,4} g_{(c,j)2,3}]$$

$$- [g_{(c,j)1,2}\;\varphi_{(c)3}\;\varphi_{(c)4} +g_{(c,j)1,3}\;\varphi_{(c)2}\;\varphi_{(c)4} +g_{(c,j)1,4}\;\varphi_{(c)2}\;\varphi_{(c)3}$$

$$+g_{(c,j)2,3}\;\varphi_{(c)1}\;\varphi_{(c)4} +g_{(c,j)2,4}\;\varphi_{(c)1}\;\varphi_{(c)3} +g_{(c,j)3,4}\;\varphi_{(c)1}\;\varphi_{(c)2}]$$

$$- \varphi_{(c)1} \varphi_{(c)2} \varphi_{(c)3} \varphi_{(c)4}.$$ (3.1.1)

As was the case for the three-point function, this time we also obtain, as one can see, the subtraction from the complete function of all the possible factorizations in terms of connected functions with a smaller number of points. The expression consists of a relatively small number of terms with different structures, each one accompanied of all the possible permutations of the position indices. Observe that in any circumstances in which $\varphi_{(c)}=0$, corresponding necessarily to $j=0$, we obtain a much simpler relation, that can be written as

$$g_{(c)1,2,3,4} =g_{1,2,3,4}-[g_{1,2} g_{3,4}+g_{1,3} g_{2,4}+g_{1,4} g_{2,3}],$$ (3.1.2)

since for $j=0$ we have that $g_{(c)i,j}=g_{i,j}$. Just as $g_{(c,j)i,j}$ gives us the true two-point correlations, $g_{(c,j)i,j,k,l}$ gives us the true four-point correlations of the model, that is, those which are not just superpositions of two-point correlations. While the two-point function is related to the propagation of waves and particles, the four-point function is related to the interaction between these waves and between these particles. It gives us the part of the complete four-point function which is not just a product of two-point functions. The part of the complete four-point function that can be decomposed into such a product of two-point functions corresponds to two waves or particles propagating together in a region of space-time, but that superpose linearly, passing transparently through each other, without interacting with one another. This is in fact all that happens in the theory of the free scaler field. One can show (problem 3.1.2) that in that theory, where everything is linear, the connected four-point function vanishes identically, a fact which corresponds to the lack of interactions between waves or between particles in that theory.

In order to relate this function with the renormalized coupling constant we must go back to the discussion of the concept of the effective action, which was introduced in the section in reference [39] and discussed in detail in the section in reference [45]. The renormalized coupling constant is one of the parameters that appears in the expression of the effective action, it is the parameter that relates most directly to the connected four-point function and that encodes in the most concise way the structure of interactions of the theory. As we saw, the effective action $\Gamma[\varphi_{(c)}]$ is a functional of $\varphi _{(c)}$ defined from $W[j]$ by a Legendre transformation. As we saw in the equation in reference [47], the double functional derivative of $\Gamma[\varphi_{(c)}]$ with respect to $\varphi _{(c)}$ is related to the inverse of the propagator. Starting from that equation we may write, with some changes of indices and an additional sum over the lattice, the equation

$$\sum_{3,4}g_{(c,j)1,3}\;g_{(c,j)2,4} \frac{\mbox{\boldmath$\... ...)3}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)4}}=g_{(c,j)1,2}.$$

Figure 3.1.1: Diagrammatic representation of the equation for the three-point function.

If we differentiate this once more with respect to $j$, using then the chain rule in order to rewrite the derivatives as derivatives with respect to $\varphi _{(c)}$, as we did in the derivation of the equation in reference [47], and recalling also that the double functional derivative of $\Gamma[\varphi_{(c)}]$ is the inverse of the propagator, we obtain (problem 3.1.3)

$$\sum_{4,5,6}g_{(c,j)1,4}\;g_{(c,j)2,5}\;g_{(c,j)3,6} \frac{\... ...mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}} =-g_{(c,j)1,2,3}.$$

Figure 3.1.2: Diagrammatic representation of the equation for the four-point function.

This shows that the triple functional derivative of $\Gamma[\varphi_{(c)}]$ is related to the three-point connected function. We can see that the connected three-point function is obtained from the triple functional derivative of $\Gamma[\varphi_{(c)}]$ by means of a type of triple transformation in which the transformation function is the propagator. In a diagrammatic language, we can say that the triple functional derivative is a vertex to which are connected three external legs representing the three propagators that act as transformation functions, this whole set of elements being equivalent to the connected three-point function. This is therefore a new type of decomposition, a way of decomposing the connected three-point function into simpler, more fundamental parts. These simpler parts are denominated one-particle irreducible or ``1pi'' functions. The corresponding diagram is illustrated in figure 3.1.1. If we differentiate this expression a fourth and last time, using the same techniques and ideas, we obtain, after long algebraic passages (problem 3.1.4), the relation

$${\sum_{5,6,7,8}g_{(c,j)1,5} \;g_{(c,j)2,6}\;g_{(c,j)3,7}\;g_{(c,j... ...ath$\mathfrak{d}$}\varphi_{(c)7}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)8}}}$$

$$=-g_{(c,j)1,2,3,4} +\sum_{5,6}g_{(c,j)1,2,5} \frac{\mbox{\boldmath$\mathfrak{d}$}^{2... ...k{d}$}\varphi_{(c)5}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}}g_{(c,j)3,4,6}$$

$$+\sum_{5,6}g_{(c,j)1,3,5} \frac{\mbox{\boldmath$\mathfrak{d}$}^{2... ...k{d}$}\varphi_{(c)5}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}}g_{(c,j)2,4,6}$$

$$+\sum_{5,6}g_{(c,j)1,4,5} \frac{\mbox{\boldmath$\mathfrak{d}$}^{2... ...{d}$}\varphi_{(c)5}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}}g_{(c,j)2,3,6}.$$ (3.1.3)

This relation has an interesting diagrammatic representation, which we illustrate in figure 3.1.2, where the symbol $\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)}$ represents the inverse of the propagator, as defined in the equation in reference [48]. The last three parts of this diagram correspond to all the possible ways to build a four-point process with the connected three-point functions and at most a single internal $\raisebox{0.6ex} {\mbox{\fbox{\rule{0ex}{0.2ex}\rule{0.2ex}{0ex}}}}_{(c)}$. We see therefore that the left side of the equation corresponds to the difference between the connected four-point function and these constructions. It is due to this that the functions generated directly by $\Gamma[\varphi_{(c)}]$ are called ``one-particle irreducible'' or ``1pi'', that is, they are irreducible functions in the sense that they cannot be separated into other functions by the elimination of an internal line of the corresponding the diagram. In somewhat more physical terms, the 1pi four-point function is the part of the four-point interaction that cannot be built out of two three-point interactions. The dimensionless versions of these 1pi functions will be denoted by $\gamma$, with position variables as arguments,

$$\frac{\mbox{\boldmath$\mathfrak{d}$}^{n}\Gamma[\varphi_{(c)}... ...} =\gamma(\vec{n}_{1},\ldots,\vec{n}_{n})=\gamma_{1,\ldots,n}.$$

These functions are also referred to as truncated, meaning that the propagators corresponding to the external legs are absent from the functions $\gamma_{1,\ldots,n}$. It is for this reason that they are also called ``vertices'', meaning that they represent only the central vertices that connect together the external legs of the diagrams, without the inclusion of the external legs themselves. All this diagrammatic nomenclature is mentioned here only to make contact with what one sees in the more traditional ways to approach the theory, since it will not have much importance for our approach in these notes.

For models with $j=0$ and in which there is symmetry by the reflection of the fields, not only we have $\varphi_{(c)}=0$ but the symmetry also implies that all the functions with an odd number of points vanish. In particular, $g_{(c)1,2,3}=0$ and there are, therefore, no three-point interactions. In this case equation (3.1.3) can be simplified to

$$\sum_{5,6,7,8}g_{(c)1,5}\;g_{(c)2,6}\;g_{(c)3,7}\;g_{(c)4,8}... ...\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)8}}=-g_{(c)1,2,3,4}.$$ (3.1.4)

It is clear that the four-fold functional derivative has the effect of extracting from the quartic term of $\Gamma[\varphi_{(c)}]$ its coefficient, which will be proportional to the renormalized coupling constant. Therefore, the last step we must take in this sequence of calculations it to isolate $\gamma_{5,6,7,8}$ in the relation above between this 1pi functions and the connected correlation function $g_{(c)1,2,3,4}$, which we already know how to write directly in terms of expectation values of products of the fields. In order to do this we will rewrite this equation in momentum space, performing Fourier transforms on the four variables $\vec{n}_{1},\ldots,\vec{n}_{4}$ which are not added over. Taking the four-fold Fourier transform of equation (3.1.4) and using the discrete translation invariance of the lattice (problem 3.1.5), we obtain

$$\widetilde g_{(c)1}\;\widetilde g_{(c)2}\;\widetilde g_{(c)3... ...^{4d}\;\widetilde\gamma _{1,2,3,4}=-\widetilde g_{(c)1,2,3,4},$$

Figure 3.1.3: The effective potential as a function of the classical field $\varphi _{(c)}$.

where we are denoting the momentum coordinates by indices and where $\widetilde g_{(c)1}=\widetilde g_{(c)}(\vec{k}_{1})$ is the momentum-space propagator, which depends on only a single momentum coordinate, due to the discrete translation invariance. Since the propagators in momentum space are never zero, we may now isolate the 1pi function, writing it as

$$\widetilde\gamma _{1,2,3,4}=\frac{-1}{N^{4d}}\;\frac{\wideti... ...widetilde g_{(c)2}\;\widetilde g_{(c)3}\;\widetilde g_{(c)4}}.$$ (3.1.5)

Figure 3.1.4: The effective potential as a function of the shifted classical field $\varphi '_{(c)}$.

We have here the 1pi function written in terms of expectation values of the model in momentum space.

In order to relate this to the renormalized coupling constant, we will have to make some assumptions about the form that the effective action $\Gamma[\varphi_{(c)}]$ may have, which will be based on the symmetries of the model. Let us recall then that our polynomial model is defined by the action given in equation (1.1.1), which we reproduce here,

$$S[\varphi]=\frac{1}{2}\sum_{\ell}(\Delta_{\ell}\varphi)^{2} ... ...um_{s}\varphi^{2}(s) +\frac{\lambda}{4}\sum_{s}\varphi^{4}(s).$$

We will be interested primarily in analyzing the low-momentum regime of the model, because this is enough for obtaining the value of the renormalized coupling constant, since it appears as part of a local potential, which exists even for fields which are constant over the lattice, having therefore infinite wavelength and vanishing momenta. In addition to this, we will assume that the effective action has the same symmetries that the fundamental action which defines the model has. Naturally, at this point it is necessary to consider for a while the issue of the possibility of spontaneous symmetry breaking that we know to exist in this model.

Let us recall that on finite lattices the symmetry is always broken, with $v_{R}\neq 0$. If we introduce into the model an infinitesimal constant external source $\delta j$, the field will spontaneously orient itself in the direction of the external source, be it positive or negative, without the system presenting any resistance to this change. For a sufficiently small $\delta j$, this happens without any significant change in the ``energy'' (in fact, the action) of the system, so that the effective potential of the theory, the part of the effective action that does not involve derivatives of $\varphi _{(c)}$, must be completely flat in a region around the point $\varphi_{(c)}=0$, as figure 3.1.3 illustrates. We use in this figure the quantity $v_{R}$ without arguments as the value of $v_{R}(\alpha,\lambda,j)$ for $j\rightarrow 0$ by positive values. Both $\alpha _{R}$ and $\lambda_{R}$ are renormalized parameters that are related to the form of the graph of the effective potential in the regions $\varphi_{(c)}\geq v_{R}$ and $\varphi_{(c)}\leq-v_{R}$.

If we rewrite the effective potential in terms of the shifted classical field $\varphi'_{(c)}=\varphi_{(c)}-v_{R}$, recalling that $v_{R}$ changes sign when $\delta j$ changes sign, then we can represent the effective potential as shown in figure 3.1.4. Observe that in the continuum limit we have $v_{R}\rightarrow 0$, since it is necessary that we approach the critical curve in the limit, where $v_{R}=0$ since the phase transition is second-order. Hence, if we will end up by taking the limit, we can do the analysis either in terms of $\varphi '_{(c)}$ or in terms of $\varphi _{(c)}$. For simplicity, we will limit ourselves here to the derivation of the relation between $\lambda_{R}$ and the observables of the model in the case in which $j=0$, but it is not difficult to generalize the result (problem 3.1.6).

Since we will assume that the effective action, when written in terms of $\varphi '_{(c)}$, has the same symmetries that the fundamental action which defines the model has, it follows that $\Gamma[\varphi'_{(c)}]$ must be composed of terms that have the same symmetries of the terms existing in $S[\varphi]$, that is, that it must have the general form

$$\Gamma[\varphi'_{(c)}]=\frac{1}{\zeta}\left\{ \frac{1}{2}\su... ...}}{4}\sum_{s}\varphi'^{4}_{(c)}(s) +\mbox{ (others) }\right\},$$

where we wrote explicitly the terms which are relevant for the analysis of the low-momentum regime of the model, $\zeta$ is the residue of the pole of the propagator and ``others'' indicates terms with more than four powers of the field and terms with more than two derivatives. Terms with many derivatives do not contribute significantly to the low-momentum regime and terms with more than four powers of the field do not contribute to the four-point function. Based on the numerical experience with this model, we may assume that $\zeta=1$, which seems to be true with significant precision in all cases examined so far.

When taking the functional derivatives of $\Gamma[\varphi'_{(c)}]$, and considering that we are interested in the case $j=0$, we should realize that it is implicit that we should put $\varphi'_{(c)}=0$ at the end of the calculations, because this is the value of $\varphi '_{(c)}$ that corresponds to the condition $j=0$ in this model. It is clear that the quadratic terms will vanish anyway when we take the derivatives, while the terms with powers larger than four will vanish due to the condition $\varphi'_{(c)}=0$. Therefore, we may consider only the terms of the effective action that contain exactly four powers of the field and no derivatives, and so we are reduced to considering only the term

$$V_{(4)}[\varphi'_{(c)}]=\frac{\lambda_{R}}{4}\sum_{0}\varphi'^{4}_{(c)0},$$

which is the term of the effective potential which is relevant for zero momentum. Taking the first derivative we get

$$\frac{\mbox{\boldmath$\mathfrak{d}$}V_{(4)}}{\mbox{\boldmath... ...arphi'^{3}_{(c)0}\delta_{0,1} =\lambda_{R}\varphi'^{3}_{(c)1}.$$

Multiplying this equation by $f_{1}(1)$, where, in order to simplify the notation, we are denoting the mode functions of the Fourier basis as

$$f_{i}(j)=e^{\imath\frac{2\pi}{N}\vec{n}_{i}\cdot\vec{k}_{j}},$$

then adding over the variable $\vec{n}_{1}$ and differentiating a second time we get

$$\sum_{1}f_{1}(1) \frac{\mbox{\boldmath$\mathfrak{d}$}^{2}V_{... ...}_{(c)1}\delta_{1,2} =3\lambda_{R}f_{2}(1)\varphi'^{2}_{(c)2}.$$

Multiplying now by $f_{2}(2)$, adding over the variable $\vec{n}_{2}$ and differentiating a third time we get

$$\begin{eqnarray*} \sum_{1,2}f_{1}(1)f_{2}(2) \frac{\mbox{\boldmath$\mathfrak{d}$... ...elta_{2,3} & = & 6\lambda_{R}f_{3}(2)f_{3}(1)\varphi'_{(c)3}. \end{eqnarray*}$$

Repeating the procedure a fourth and last time we get

$$\begin{eqnarray*} \sum_{1,2,3}f_{1}(1)f_{2}(2)f_{3}(3) \frac{\mbox{\boldmath$\ma... ...3}(1)\delta_{3,4} & = & 6\lambda_{R}f_{4}(3)f_{4}(2)f_{4}(1). \end{eqnarray*}$$

Multiplying by $f_{4}(4)$ and adding over $\vec{n}_{4}$ we obtain for the fourth functional derivative of $V_{4}$

$$\sum_{1,2,3,4}f_{1}(1)f_{2}(2)f_{3}(3)f_{4}(4) \frac{\mbox{\... ...{(c)4}} =6\lambda_{R}\sum_{4}f_{4}(4)f_{4}(3)f_{4}(2)f_{4}(1).$$

Since this fourth functional derivative is equal to the 1pi four-point function, we obtain

$$\sum_{1,2,3,4}f_{1}(1)f_{2}(2)f_{3}(3)f_{4}(4)\gamma_{1,2,3,4} =6\lambda_{R}\sum_{4}f_{4}(4)f_{4}(3)f_{4}(2)f_{4}(1).$$

In the left-hand side of this equation we have $N^{4d}$ times the four-fold Fourier transform of $\gamma_{1,2,3,4}$, while in the right-hand side, recalling that the mode functions $f_{i}(j)$ are exponentials that satisfy orthogonality and completeness relations, we have $N^{d}$ times a Kronecker delta function that expresses the conservation of momentum, that is,

$$N^{4d}\;\widetilde\gamma _{1,2,3,4}=6\lambda_{R}N^{d}\delta^{d}_{1+2+3+4}.$$

Combining now this equation with equation (3.1.5) we obtain the final relation between the renormalized coupling constant and the connected correlation functions,

$$\lambda_{R}\delta^{d}_{1+2+3+4}= -\frac{1}{6N^{d}}\;\frac{\w... ...widetilde g_{(c)2}\;\widetilde g_{(c)3}\;\widetilde g_{(c)4}}.$$ (3.1.6)

For combinations of momenta that satisfy the conservation condition $\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3}+\vec{k}_{4}=\vec{0}$ the delta function is simply $1$ and, assuming implicitly the conservation of momentum, we may write

$$\lambda_{R}=-\frac{1}{6N^{d}}\;\frac{\widetilde g_{(c)1,2,3,... ...widetilde g_{(c)2}\;\widetilde g_{(c)3}\;\widetilde g_{(c)4}}.$$

Naturally, since we neglected the terms in $\Gamma[\varphi'_{(c)}]$ with larger powers of the momenta, this relation only makes sense for small or vanishing momenta. We take, therefore, the zero-momentum case $\vec{k}_{1}=\vec{k}_{2}=\vec{k}_{3}=\vec{k}_{4}=\vec{0}$ in order to obtain, substituting the connected functions in terms of the complete functions,

$$\lambda_{R}=\frac{1}{6N^{d}} \;\frac{3\langle\vert\widetilde... ...{\langle\vert\widetilde\varphi (\vec{0})\vert^{2}\rangle^{4}}.$$ (3.1.7)

If we recall the factorization relations of the free theory for the correlation functions in momentum space, that were introduced in the section of reference [44], we immediately see that this quantity vanishes identically in the free theory.

We may also write this result in terms of the dimensionfull quantities, using the appropriate scaling relations to transform $\varphi$ in $\phi$ and $\lambda_{R}$ in $\Lambda_{R}$, thus obtaining

$$\Lambda_{R}=\frac{1}{6L^{d}} \;\frac{3\langle\vert\widetilde... ...e} {\langle\vert\widetilde\phi (\vec{0})\vert^{2}\rangle^{4}}.$$

This is the quantity whose value determines whether or not there exists in this model the phenomenon of non-linear interaction between waves, or between particles. Naturally, this quantity is of great physical interest and we will dedicate some time to the examination of its properties.

Problems

1. Using the definition of the connected four-point correlation function

   
   

   $$g_{(c,j)1,2,3,4}= \frac{\mbox{\boldmath$\mathfrak{d}$}^{4}W[... ...math$\mathfrak{d}$} j_{3}\mbox{\boldmath$\mathfrak{d}$}j_{4}},$$
   
   

   in a theory with a non-vanishing external source $j$, show that it is related to the complete functions of four, three and two points by the formula

   $$\begin{eqnarray*} g_{(c,j)1,2,3,4} & = & g_{(j)1,2,3,4} & - & [g_{(j)2,3,4}\;... ...;\varphi_{(c)1} \varphi_{(c)2} \varphi_{(c)3} \varphi_{(c)4}. \end{eqnarray*}$$
   
   
   
   

   Substituting the complete functions of three and two points in terms of the corresponding connected functions, show that the connected four-point function is related to the complete four-point function by means of

   $$\begin{eqnarray*} g_{(c,j)1,2,3,4} & = & g_{(j)1,2,3,4} & - & [g_{(c,j)2,3,4}... ... \varphi_{(c)1} \varphi_{(c)2} \varphi_{(c)3} \varphi_{(c)4}, \end{eqnarray*}$$
   
   
   
   

   which corresponds to the subtraction from the complete function of all the possible factorizations in terms of connected functions with a smaller number of points.

   Observe that a significant part of the long algebraic passages involved in this problem has already been executed before in the problem in reference [40], relative to the three-point function. A simpler alternative way to obtain the results shown above is to start from the final result of that problem, doing an additional differentiation and using once more the same result to substitute the complete three-point function where necessary.

2. Show that in the theory of the free scalar field, that is, in the $\lambda \varphi ^{4}$ model for the case $\lambda=0$ and $\alpha\geq 0$, the connected four-point function given in equation (3.1.1) vanishes identically. Recall the results related to the factorization of the correlation functions of the free theory in momentum space, discussed in the section in reference [44], they will be very useful here.

3. Starting from the equation in reference [46] show that we can write that equation in the form

   
   

   $$\sum_{3,4}g_{(c,j)1,3}\;g_{(c,j)2,4} \frac{\mbox{\boldmath$\... ...)3}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)4}}=g_{(c,j)1,2}.$$
   
   

   Then differentiate this equation once more with respect to $j$, using the chain rule to rewrite the derivatives as derivatives with respect to $\varphi _{(c)}$, thus obtaining

   $$\begin{eqnarray*} \sum_{4,5,6}g_{(c,j)1,4}\;g_{(c,j)2,5}\;g_{(c,j)3,6} \frac{\mb... ...ox{\boldmath$\mathfrak{d}$}\varphi_{(c)5}} & = & g_{(c,j)1,2,3}. \end{eqnarray*}$$
   
   
   
   

   Next, use the fact that the second functional derivative of $\Gamma[\varphi_{(c)}]$ is the inverse of the propagator and rearrange the terms in order to obtain the final relation in the form

   
   

   $$\sum_{4,5,6}g_{(c,j)1,4}\;g_{(c,j)2,5}\;g_{(c,j)3,6} \frac{\... ...mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}} =-g_{(c,j)1,2,3}.$$
   
   

4. Starting from the final result of equation (3.1.3), differentiate it once again with respect to $j$, using the chain rule to rewrite the derivatives as derivatives with respect to $\varphi _{(c)}$, thus obtaining
   $$\begin{eqnarray*} \sum_{5,6,7,8}g_{(c,j)1,5}\;g_{(c,j)2,6}\;g_{(c,j)3,7}\;g_{(c,... ...{\boldmath$\mathfrak{d}$}\varphi_{(c)7}} & = & g_{(c,j)1,2,3,4}. \end{eqnarray*}$$
   
   
   
   

   Using again the final result of problem 3.1.3 and, once more, the fact that the second functional derivative of $\Gamma[\varphi_{(c)}]$ is the inverse of the propagator, obtain the final relation

   $$\begin{eqnarray*} \lefteqn{\sum_{5,6,7,8}g_{(c,j)1,5} \;g_{(c,j)2,6}\;g_{(c,j)3,... ...c)5}\mbox{\boldmath$\mathfrak{d}$}\varphi_{(c)6}}g_{(c,j)2,3,6}. \end{eqnarray*}$$
   
   
   
   

5. Starting from equation (3.1.4), write it in the form

   
   

   $$\sum_{5,6,7,8}g_{(c)1,5}\;g_{(c)2,6}\;g_{(c)3,7}\;g_{(c)4,8} \;\Gamma_{5,6,7,8}=-g_{(c)1,2,3,4}$$
   
   

   and execute four Fourier transforms on the external variables $\vec{n}_{1},\ldots,\vec{n}_{4}$, using the corresponding variables $\vec{k}_{1},\ldots,\vec{k}_{4}$ in momentum space, and recalling that, for a function $F$ of $n$ position variables $\vec{n}_{i}$,

   
   

   $$\frac{1}{N^{nd}}\sum_{1,\ldots,n}\;f_{1}(1)\ldots\;f_{n}(n) \;F_{1,\ldots,n}={\widetilde F}_{1,\ldots,n},$$
   
   

   where, to simplify the notation, we are denoting the mode functions of the Fourier basis as

   
   

   $$f_{i}(j)=e^{\imath\frac{2\pi}{N}\vec{n}_{i}\cdot\vec{k}_{j}},$$
   
   

   besides indicating the momentum coordinates by indices on the remaining functions in momentum space, as we have done before for the position coordinates. Use also the fact that, due to the discrete translation invariance of the lattice, we have

   
   

   $$\frac{1}{N^{d}}\sum_{1}f_{1}(1)\;g_{(c)1,2}=f_{2}(1)\;\widetilde g_{(c)1},$$
   
   

   where $\widetilde g_{(c)1}$ is the propagator in momentum space, which depends only on a single momentum coordinate, due the the discrete translation invariance, in order to write the final result

   
   

   $$\widetilde g_{(c)1}\;\widetilde g_{(c)2}\;\widetilde g_{(c)3... ...^{4d}\;\widetilde\Gamma _{1,2,3,4}=-\widetilde g_{(c)1,2,3,4}.$$
   
   

6. Derive the expression for $\lambda_{R}$ in terms of the zero-momentum correlation functions of the model, for the general case in which $j\neq 0$ and $v_{R}\neq 0$.

7. Show that the expression of the coupling constant in terms of correlation functions with a given constant momentum $\vec{k}$ that enters in the direction of the vertex in two of the four external legs and goes out in the opposite direction in the other two legs is

   
   

   $$\lambda_{R}(\vec{k})=\frac{1}{6N^{d}} \;\frac{2\langle\vert\... ...{\langle\vert\widetilde\varphi (\vec{k})\vert^{2}\rangle^{4}},$$
   
   

   both in the case in which $j=0$ and $v_{R}=0$ and in the general case.