In this section, we give the proof of Lemma 3.6 for the case \(S=2\). The proof is similar to that of [11, Theorem 3.4]. We need to prove (for all n, m and uniformly in \(X_n,Y_m\)) Eq. (3.12) if Eqs. (3.10) and (3.11) are satisfied. To simplify notation we define

$$\begin & \gamma _\infty := \sup _ \sum _}^3} \left| }}_\sigma (k)\right| , \qquad I_g:= \sup _ \int _\Lambda \left| f_^2 - 1\right| = \sup _e \int _\Lambda |g_e|, \\ & \qquad I_\gamma := \sup _ \int _\Lambda \left| \gamma _\sigma \right| , \end$$

where as above \(}}_\sigma (k) = L^ \int _\Lambda \gamma _\sigma (x) e^ \,\text x\). Eq. (3.11) then reads that \(\gamma _\infty I_g (1+I_\gamma )\) is sufficiently small.

We give the proof in two steps. First we consider the case \(n=m=0\).

4.1 Absolute Convergence of the \(\Gamma \)-sum

In this section we show that

$$\begin \sum _ p,q\ge 0 \\ p+q \ge 2 \end} \frac \left| \sum _}_} \Gamma _D\right| < \infty \end$$

under the relevant conditions. Defining clusters as connected components of G we split the sum into clusters as in [11, Section 3.1]. Denoting the sizes of the clusters by \((n_\ell , m_\ell )\), \(\ell =1,\ldots ,k\) (meaning that the cluster \(\ell \) has \(n_\ell \) black vertices and \(m_\ell \) white vertices) we get

$$\begin \begin&\frac \sum _}_} \Gamma _D\\&\quad \!=\! \sum _^\infty \frac \sum _ n_1,\ldots ,n_k \!\ge \! 0 \\ m_1,\ldots ,m_k\ge 0 \\ \text \ \ell : n_\ell \!+\! m_\ell \!\ge \! 2 \end} \chi _ n_\ell \!=\! p)} \chi _ m_\ell = q)}\\&\qquad \times \frac^k n_\ell ! m_\ell !} \sum _}_} \idotsint \,\text X_ \,\text Y_ \left[ \prod _^k \prod _ g_e \right] \\&\qquad \times \left[ \sum _ \pi \in }_ \\ \tau \in }_ \end} (-1)^\pi (-1)^\tau \chi _)} \prod _^ \gamma _(x_i \!-\! x_) \prod _^ \gamma _(y_j - y_) \right] , \end\nonumber \\ \end$$

(4.1)

where \(}_\subset }_\) denotes the subset of connected graphs. The factorial factors arise from counting the possible labellings exactly as in Sect. 3.

The last line of Eq. (4.1) is what we will call the truncated correlation. We give a slightly more general definition for later use.

Definition 4.1

Let \(B_1,\ldots , B_k\) and \(W_1,\ldots , W_k\) be sets of distinct black and white vertices, respectively, such that for each \(\ell =1,\ldots ,k\) we have \(B_\ell \ne \varnothing \) and/or \(W_\ell \ne \varnothing \). Then the truncated correlation.Footnote 1 is defined as follows.

$$\begin \rho _t^= & \sum _ \pi \in }_ \\ \tau \in }_ \end} (-1)^\pi (-1)^\tau \chi _)}\nonumber \\ & \times \prod _ \gamma _(x_i - x_) \prod _ \gamma _(y_j - y_) \end$$

(4.2)

for any connected graphs \(G_\ell \in }_\). The definition does not depend on the choice of the graphs \(G_\ell \).

If the underlying sets \(B_1,\ldots , B_k, W_1, \ldots , W_k\) are clear we will also use the notation

$$\begin \rho _t^ = \rho _t^. \end$$

The truncated correlations are studied in [7, Appendix D]. To better compare to the definition in [7], we note the following.

In Eq. (4.2), we may view \((\pi ,\tau )\) together as a permutation of all the vertices (both black and white). Moreover, if we instead sum over all permutations \(\pi '\in }_\) we have that any \(\pi '\) not coming from two permutations \(\pi , \tau \) on the black (respectively, white) vertices contributes 0, since any \(\gamma \)-factor between vertices of different spins is 0. That is,

$$\begin \rho _t^ = \sum _}_} (-1)^\pi \chi _)} \prod _ \gamma _. \end$$

In [7, Equation (D.53)] is shown the formula for the truncated correlation

$$\begin \rho _t^ = \sum _}^} \prod _ \gamma _ \int \,\text \mu _A(r) \det }(r),\nonumber \\ \end$$

(4.3)

where \(}\) denotes the set of anchored trees, \(\mu _A\) is a probability measure and \(}(r)\) is an explicit matrix. The set \(}^\) of anchored trees is the set of all directed graphs on the vertices \(\cup _\ell B_\ell \cup \cup _\ell W_\ell \) such that each vertex has at most one incoming and at most one outgoing edge, and such that upon identifying all vertices inside each cluster the resulting graph is a (directed) tree. The matrix \(}(r)\) satisfies the bound

$$\begin \left| \det }(r)\right| \le \gamma _\infty ^. \end$$

(4.4)

This follows from [7, Equation (D.57)]. We give a sketch of the argument here, see also [7, Lemma D.2] and [11, Lemma 3.10].

Proof (sketch) of Eq. (4.4) Write \(\gamma _\sigma (z_\mu - z_\nu ) = \left\langle \alpha _\mu \Big \vert \beta _\nu \right\rangle _}^3)}\), where for \(k\in \frac}^3\)

$$\begin \alpha _\mu (k) = e^ \left| }}_\sigma (k)\right| ^ \frac}}_\sigma (k)}}}_\sigma (k)\right| }, \qquad \beta _\nu (k) = e^ \left| }}_\sigma (k)\right| ^, \end$$

with \(}}_\sigma (k) = L^ \int _\Lambda \gamma _\sigma (x) e^\,\text x\) the Fourier coefficients. Then by the Gram-Hadamard inequality [7, Lemma D.1]

$$\begin \left| \det [\gamma _\sigma (z_\mu - z_\nu )]_\right| \le \prod _ \left\| \alpha _\mu \right\| _}^3)} \left\| \beta _\mu \right\| _}^3)} \le \left[ \sum _}^3} \left| }}(k)\right| \right] ^p. \end$$

It is then explained in the proof of [7, Lemma D.6] how to adapt this argument to bound \(\det }(r)\).

Combining Eqs. (4.4) and (4.3) we conclude the bound

$$\begin \left| \rho _t^\right| \!\le \! \gamma _\infty ^ \sum _}^} \prod _ \left| \gamma _\right| .\nonumber \\ \end$$

(4.5)

With the truncated correlation we may write the last line of Eq. (4.1) as \(\rho _t^}, })}\), where

$$\begin }\!=\! (n_1,\ldots ,n_k), \ }\!=\! (m_1,\ldots ,m_k), \ (}, }) \!=\! ( (n_1,m_1),\ldots ,(n_k,m_k)). \end$$

That is,

$$\begin \begin \frac \sum _}_} \Gamma _D&= \sum _^\infty \frac \sum _ n_1,\ldots ,n_k\ge 0 \\ m_1,\ldots ,m_k\ge 0 \\ \text \ \ell : n_\ell + m_\ell \ge 2 \end} \chi _ n_\ell = p)} \chi _ m_\ell = q)} \frac^k n_\ell ! m_\ell !}\\&\quad \times \idotsint \,\text X_ \,\text Y_ \left[ \prod _^k \sum _}_} \prod _ g_e \right] \rho _t^}, })}. \end \end$$

To bound this we use the tree-graph bound [19], see also [16, Proposition 6.1]. By assumption Eq. (3.10) is satisfied and thus [19]

$$\begin \left| \sum _}_} \prod _ g_e \right| \le C_}^ \sum _}_} \prod _ |g_e|, \end$$

(4.6)

where \(}_ \subset }_\) denotes the subset of trees. (To see this note that \(}_\) (respectively, \(}_\)) is the set of connected graphs (respectively, trees) on \(p+q\) vertices with the colours of the vertices just serving as a handy reminder of the edge-weights \(g_e\).) By moreover using the bound on the truncated correlation in Eq. (4.5) we conclude that

(4.7)

To do the integrations, we note that the graph \(}\) with edges the union of (g-)edges in \(T_1,\ldots ,T_k\) and (\(\gamma \)-)edges in A is a tree on all the \(\sum _\ell n_\ell + \sum _\ell m_\ell \) many vertices. One then integrates the coordinates one leaf at a time (meaning that the index of the corresponding coordinate is a leaf of the graph \(}\)) and removes a vertex from the graph after integrating over its corresponding coordinate.

To be more precise suppose that \(\nu _0\) is a leaf of \(}\). Then the variable \(z_\) appears exactly once in the integrand. Either in a factor \(g_\) (in which case the \(z_\)-integral gives \(\int |g| \le I_g\) by the translation invariance) or in a factor \(\gamma _\) (in which case the \(z_\)-integral gives \(\int |\gamma | \le I_\gamma \) by the translation invariance). The final integral gives \(L^3\) by the translation invariance. There are \(k-1\) factors of \(\gamma \) and \(\sum _\ell (n_\ell + m_\ell - 1) = p+q-k\) factors of g. Thus, we get

In [7, Appendix D.5], it is shown that

$$\begin \# }^ \le k! C^. \end$$

Moreover, \(}_ = (n+m)^ \le C^ (n+m)!\) by Cayley’s formula. Finally, we may bound the binomial coefficients \(\frac\le 2^\). Thus

$$\begin \begin&\sum _ p,q\ge 0 \\ p+q\ge 2 \end} \frac \left| \sum _}_} \Gamma _D\right| \\&\qquad \quad \le C L^3\gamma _\infty \sum _^\infty \left[ \sum _ n,m\ge 0 \\ n+m\ge 2 \end} \frac (C I_g \gamma _\infty )^\right] ^k (C I_\gamma )^ \\&\qquad \quad \le C L^3 \gamma _\infty \sum _^\infty \left[ \sum _^\infty \ell (CI_g \gamma _\infty )^\right] ^k (C I_\gamma )^ \\&\qquad \quad \le C L^3 \gamma _\infty ^2 I_g < \infty \end \end$$

for \(\gamma _\infty I_g\) and \(\gamma _\infty I_g I_\gamma \) small enough. This shows that \(\sum _ \frac\sum _}_}\Gamma _D\) is absolutely convergent under this assumption. Next, we bound the \(\Gamma ^\)-sum for \(n+m \ge 1\).

4.2 Absolute Convergence of the \(\Gamma ^\)-sum

In this section, we prove that (for \(n+m\ge 1\) and uniformly in \(X_n, Y_m\))

$$\begin \sum _ \frac \left| \sum _}_^} \Gamma _D^\right| \le C_ \gamma _\infty ^ < \infty \end$$

if Eq. (3.10) is satisfied and \(\gamma _\infty I_g (1+I_\gamma )\) is sufficiently small.

We do the same splitting into clusters (connected components of G) as in Sect. 4.1. There is, however, a slight complication: One needs to keep track of in which clusters the external vertices lie. This is exactly parametrized by the set \(\Pi ^_\kappa \) (defined in Eq. (3.4)). Denoting the sizes (number of internal vertices) of the clusters containing external vertices by \((n^*_\lambda , m^*_\lambda )\) and the sizes of clusters only containing internal vertices by \((n_\ell , m_\ell )\) and introducing \(}_^ \subset }_^\) as the subset of connected graphs (and similarly \(}_^ \subset }_^\), recall Definition 3.1) we get

(4.8)

For \(k=0\) the \(n_1,m_1,\ldots ,n_k,m_k\)-sum should be interpreted as an empty product, i.e. as a factor 1. Similarly for \(p=0\) and/or \(q=0\) the empty product of integrals should be interpreted as a factor 1.

The last line in Eq. (4.8) is the truncated correlation

$$\begin \rho _t^}^* + }^*, }^* + }^*) \oplus (}, })}, \end$$

where

$$\begin & }^* = (n^*_1, \ldots , n^*_\kappa ), \qquad }= (n_1, \ldots , n_k), \qquad }^* = (m^*_1, \ldots , m^*_\kappa ),\\ & \qquad }= (m_1,\ldots ,m_k) \end$$

and \(\oplus \) means concatenation of vectors, i.e.

$$\begin & (}^* + }^*, }^* + }^*) \oplus (},})\\ & \quad = ((B_1^* + n_1^*, W_1^* + m_1^*), \ldots , (B_\kappa ^* + n_\kappa ^*, W_\kappa ^* + m_\kappa ^*), (n_1, m_1), \ldots , (n_k, m_k) ), \end$$

where we abused notation slightly and wrote \(B_1^* + n_1^*\) for the union of the vertices \(B_1^*\) and the \(n_1^*\) internal black vertices of the graph \(G_1^*\) (similarly for the other terms.)

We use as in Sect. 4.1 the tree-graph bound and the bound on the truncated correlation in Eq. (4.5). For the clusters with external vertices, we add 0-weights to the disallowed edges as in [11, Section 3.1.3], i.e. for \(G\in }_^\) define

$$\begin }}_e = 0 & e = (i,j) \text i,j \text \\ g_e & \text . \end\right. } \end$$

Then, we may readily apply the tree-graph bound [19] with edge-weights \(}}_e\):

$$\begin \left| \sum _}_^} \prod _ g_e\right|= & \left| \sum _}_} \prod _ }}_e\right| \le C_}^ \sum _}_} \prod _ \left| }}_e\right| \\= & C_}^ \sum _}_^} \prod _ \left| g_e\right| , \end$$

where \(}_ \subset }_\) and \(}_^ \subset }_^\) denotes the subsets of trees. Thus

$$\begin \begin&\sum _ \frac \left| \sum _}_^} \Gamma _D^\right| \\&\quad \le \sum _^\infty \frac \sum _^ \frac \sum _}^*,}^*)\in \Pi _\kappa ^} \sum _ n_^*,\ldots ,n_^*\ge 0 \\ m_^*,\ldots ,m_^*\ge 0 \end} \sum _ n_1,\ldots ,n_k\ge 0 \\ m_1,\ldots ,m_k\ge 0 \\ \text \ \ell : n_\ell + m_\ell \ge 2 \end}\\&\qquad \times \frac^\ell n_^*! m_^*!} \frac^k n_\ell ! m_\ell !} \\&\qquad \times \sum _}^}^* + }^*, }^* + }^*) \oplus (},})}} \sum _ T^*_1,\ldots ,T^*_\kappa \\ T^*_\lambda \in }_^ \end} \sum _ T_1,\ldots ,T_k \\ T_\ell \in }_ \end} \\&\qquad \times \idotsint \,\text X_ \,\text Y_\\&\qquad \times \Bigg [ \prod _^\kappa \prod _ |g_e| \prod _^k \prod _ |g_e| \prod _ \left| \gamma _\right| \Bigg ] \\&\qquad \times (C_}\gamma _\infty )^ C_}^. \end\nonumber \\ \end$$

(4.9)

To do the integrations, we bound some g- and \(\gamma \)-factors pointwise. Recall first that there are \(\kappa \) clusters with external vertices. We split the anchored tree into pieces according to these clusters as follows.

We may view the anchored tree A as a tree on the set of clusters. If \(\kappa = 1\) set \(A_1 = A\). Otherwise iteratively pick a \(\gamma \)-edge on the path in A between any two clusters with external vertices and bound it by

$$\begin \left| \gamma _\sigma (z)\right| = \left| \sum _}^3} \hat_\sigma (k) e^\right| \le \gamma _\infty \end$$

and remove it from A. This cuts the anchored tree A into pieces. Doing this \(\kappa -1\) many times we get \(\kappa \) anchored trees \(A_1,\ldots ,A_\kappa \) with each exactly one cluster with external vertices. That is,

$$\begin \prod _ |\gamma _| \le \gamma _\infty ^ \prod _^\kappa \prod _ |\gamma _|. \end$$

Next, in each cluster with external vertices, say with label \(\lambda _0\), we do a similar procedure of splitting the cluster into pieces according to the external vertices.

In the cluster \(\lambda _0\) there are \(\# B_^* + \# W_^* \ge 1\) external vertices. If \(\# B_^* + \# W_^* = 1\) set \(T_^* = T_^*\). Otherwise iteratively pick a g-edge on the path in \(T_^*\) between any two external vertices and bound it by

$$\begin \left| g_e\right| = \left| f_e^2 - 1\right| \le \max \ \le C_}^2 \end$$

using Eq. (3.10) for \(q=2\). Remove the edge e from \(T_^*\). This cuts the tree \(T_^*\) into pieces. Doing this \(\# B_^* + \# W_^* - 1\) many times we get \(\# B_^* + \# W_^*\) trees \(T_^*, \ldots , T_^* + \# W_^*}^*\) with each exactly one external vertex. That is,

$$\begin \prod _^*} |g_e| \le C_}^^* + \# W_^* - 1)} \prod _^^* + \# W_^*} \prod _^*} |g_e|. \end$$

We do this procedure for all the \(\kappa \) many clusters with external vertices. Then, the graph \(}\) with edges the union of all (g- or \(\gamma \)-)edges in \(T_^*, T_\ell , A_\lambda \) (for \(\lambda \in \\), \(\ell \in \\) and \(\nu \in \\)) is a forest (disjoint union of trees) on the set of vertices \(V_\) with each connected component (tree) having exactly one external vertex. Moreover, we have the bound

$$\begin \begin&\idotsint \,\text X_ \,\text Y_\\&\qquad \times \Bigg [ \prod _^\kappa \prod _ |g_e| \prod _^k \prod _ |g_e| \prod _ \left| \gamma _\right| \Bigg ] \\&\quad \le C_}^ \gamma _\infty ^ \left[ \prod _^\kappa \prod _^ \idotsint \prod _^*} |g_e|\right. \\&\qquad \times \left. \prod _ |\gamma _| \prod _ \prod _|g_e| \right] , \end\nonumber \\ \end$$

(4.10)

where \(T_\ell \sim A_\lambda \) means that \(T_\ell \) and \(A_\lambda \) share a vertex. (Equivalently they are part of the same connected component of \(}\).)

Since each connected component of \(}\) is a tree we may do the integrations one leaf at a time exactly as for the \(\Gamma \)-sum in Sect. 4.1. To bound the value, we count the number of \(\gamma \)- and g-factors that are left.

The number of \(\gamma \)-integrations is exactly the number of \(\gamma \)-factors. There are \(k+\kappa \) many clusters, so A has \(k+\kappa - 1\) many edges. In constructing \(A_1,\ldots ,A_\kappa \) we cut \(\kappa -1\) many edges, thus there is k many \(\gamma \)-factors left and so there are k many \(\gamma \)-integrations in Eq. (4.10). The remaining \(\sum _\lambda (n_\lambda ^* + m_\lambda ^*) + \sum _\ell (n_\ell + m_\ell ) - k\) integrations are of g-factors. The integrals may be bounded by \(\int \left| \gamma \right| \le I_\gamma \) and \(\int \left| g\right| \le I_g\) as in Sect. 4.1. Moreover, since each connected component of \(}\) has one external vertex, which is not integrated over, there are no volume factors from the last integrations in any of the connected components of \(}\). That is,

$$\begin \begin&\idotsint \,\text X_ \,\text Y_\\&\qquad \quad \times \Bigg [ \prod _^\kappa \prod _ |g_e| \prod _^k \prod _ |g_e| \prod _ \left| \gamma _\right| \Bigg ] \\ &\qquad \quad \le C_}^ \gamma _\infty ^ I_g^ I_\gamma ^. \end \end$$

We use this to bound the integrations in Eq. (4.9). Additionally, we need to bound the number of (anchored) trees. In [7, Appendix D.5], it is shown that

$$\begin \# }^}^* + }^*, }^* + }^*) \oplus (},})} \le (k+\kappa )! C^, \end$$

since we have \(k+\kappa \) many clusters and \(n + m + \sum _\lambda (n^*_\lambda + m^*_\lambda ) + \sum _\ell (n_\ell + m_\ell )\) many vertices in total. Moreover, \(\# }_^ \le \# }_ = (p + q+n+m)^ \le (p+q+n+m)! C^\) by Cayley’s formula as in Sect. 4.1. These bounds together with Eq. (4.9) then gives

$$\begin \begin&\sum _ \frac \left| \sum _}_^} \Gamma _D^\right| \\&\quad \le (C\gamma _\infty )^ \sum _^\infty \sum _^ \frac \sum _}^*,}^*)\in \Pi _\kappa ^} \sum _ n_^*,\ldots ,n_^*\ge 0 \\ m_^*,\ldots ,m_^*\ge 0 \end} \sum _ n_1,\ldots ,n_k\ge 0 \\ m_1,\ldots ,m_k\ge 0 \\ \text \ \ell : n_\ell + m_\ell \ge 2 \end} \\&\qquad \times \left[ \prod _^\kappa \frac \right] \left[ \prod _^k \frac\right] \\&\qquad \times (C I_g \gamma _\infty )^(n_\ell + m_\ell - 1)} (CI_\gamma )^k. \end\nonumber \\ \end$$

(4.11)

Multinomial coefficients may be bounded as \(\frac \le k^\). Moreover, \(\# B_\lambda ^* \le n\) and \(\# W_\lambda ^*\le m\). Thus, we may bound

$$\begin (n^*_\lambda + \# B_\lambda + m^*_\lambda + \# W_\lambda )! \le (n^*_\lambda + m^*_\lambda + n + m)! \le 4^ n! m! n^*_\lambda ! m^*_\lambda !. \end$$

We conclude the bound

$$\begin&\sum _ \frac \left| \sum _}_^} \Gamma _D^\right| \nonumber \\&\quad \le (C\gamma _\infty )^ \sum _^\infty \sum _^ 2^ \left[ \sum _ C_ (C I_g \gamma _\infty )^\right] ^\kappa \nonumber \\&\quad \times \left[ C I_\gamma \sum _ n_0, m_0 \ge 0 \\ n_0 + m_0 \ge 2 \end} (C I_g \gamma _\infty )^ \right] ^k. \end$$

(4.12)

For some \(c_ > 0\) we have that if \(\gamma _\infty I_g (1+I_\gamma ) < c_\) the sums are convergent and we get

$$\begin \sum _ \frac \left| \sum _}_^} \Gamma _D^\right| \le C_\gamma _\infty ^ < \infty . \end$$

This shows the desired. We conclude the proof of Lemma 3.6 for the case \(S=2\).

Remark 4.2

(Higher spin). For the case of higher spin \(S\ge 3\), the computations are essentially the same.

For later use, we define for all diagrams some values characterizing their sizes.

Definition 4.3

Let \(D\in }^_\). Define the number \(k=k(D)\) as the number of clusters entirely within internal vertices (i.e. the same k as in the computations above) and \(\kappa = \kappa (D)\) as the number of clusters containing at least one external vertex (i.e. the same \(\kappa \) as in the computations above). Define then \(\nu ^* = \nu ^*(D)\) and \(\nu = \nu (D)\) as

$$\begin \nu ^* = \sum _^\kappa (n_\lambda ^* + m_\lambda ^*), \qquad \nu = \sum _^k (n_\ell + m_\ell ) - 2k, \end$$

where \(n^*_\lambda , m^*_\lambda , n_\ell , m_\ell \) are the sizes of the different clusters exactly as in the computations above. (Then \(\nu + \nu ^ + 2k= p+q\).)

For a diagram D the number \(\nu + \nu ^*\) is the “number of added vertices” in the following sense. A diagram with \(n+m\) external vertices and k clusters entirely within internal vertices has at least \(n+m+2k\) many vertices, since each cluster (with only internal vertices) has at least 2 vertices. Then, \(\nu + \nu ^*\) is the number of vertices a diagram has more than this minimal number.

Note that in the special case of consideration with the scattering functions \(f_s, f_p\) and the one-particle density matrices \(\gamma _^\) we have

$$\begin \gamma _\infty \le \rho , \qquad I_g \le Cab^2, \qquad I_\gamma \le C s (\log N)^3 \end$$

by Eqs. (3.13) and (2.9), see also the proof of Theorem 3.2. Then, by following the arguments above (see in particular Eqs. (4.12) and (4.11)), we have (for \(p + q =2k_0 + \nu _0\))

$$\begin \frac \left| \sum _ D \in }_^ \\ k(D)= k_0 \\ \nu (D) + \nu ^*(D) = \nu _0 \end} \Gamma _D^ \right| \le C_\rho ^ (Cab^2\rho )^ (Cs (\log N)^3)^\nonumber \\ \end$$

(4.13)

for any n, m with \(n+m\ge 1\). We think of s as \(s \sim (a^3\rho )^\) for some small \(\varepsilon > 0\). Thus increasing \(\nu _0\) by 1 we decrease the size of the diagram by \((a^3\rho )^\), and increasing \(k_0\) by 1 we decrease the size of the diagram by \((a^3\rho )^\). (Recall that \(b=\rho ^\).)