Four-Loop Perturbative Expansion for the Lattice $N$-Vector Model

TL;DR

This paper calculates four-loop contributions to the beta function and anomalous dimension in the $O(N)$-vector model, providing refined analytic corrections to key physical quantities like correlation length and susceptibility.

Contribution

It presents the first four-loop perturbative expansion for the lattice $N$-vector model's beta function and anomalous dimension, advancing precision in theoretical predictions.

Findings

Computed four-loop beta function and anomalous dimension.

Derived second analytic corrections to correlation length.

Provided corrections for general spin-$n$ susceptibility.

Abstract

We compute the four-loop contributions to the $\beta$-function and the anomalous dimension of the field for the $O(N)$-invariant $N$-vector model. These results are used to compute the second analytic corrections to the correlation length and the general spin-$n$ susceptibility.

Tables

Table 1. Table 1: Numerical value of the lattice integrals appearing in the β 𝛽 \beta and γ 𝛾 \gamma functions. Their explicit definitions are reported in Appendix A.1.

Constants
$G_1$	$-$ 0.0461636
$R$	$-$ 0.0148430
$J$	$-$ 0.1366198
$K$	$-$ 0.0958870
$L_1$	$-$ 0.0029334
$V_1$	$-$ 0.0169610
$V_2$	$-$ 0.0011400
$V_3$	$-$ 0.0724300
$V_4$	$-$ 0.0013125
$V_5$	$-$ 0.0100630
$V_6$	$-$ 0.0175070
$W_1$	$-$ 0.0296860
$W_2$	$-$ 0.0022100

Table 2. Table 2: Ratio of the Monte Carlo results for ξ 𝜉 \xi and the theoretical predictions at two loops, three loops and four loops. N = 3 𝑁 3 N=3 data are taken from [ 18 ] , N = 4 , 8 𝑁 4 8 N=4,8 from [ 19 ] .

$N$	$\beta$	$\xi$	${ℛ}^{\left(2-loop\right)}$	${ℛ}^{\left(3-loop\right)}$	${ℛ}^{\left(4-loop\right)}$
3	1.85	89.16(21)	0.743(2)	0.781(2)	0.830(2)
3	2.25	1049(7)	0.861(6)	0.897(6)	0.934(6)
3	2.60	8569(92)	0.901(10)	0.934(10)	0.962(10)
3	3.00	94.6(1.6) $\cdot {10}^3$	0.930(16)	0.959(16)	0.981(17)
4	2.50	34.85(9)	0.911(2)	0.930(2)	0.953(2)
4	2.80	86.07(37)	0.927(4)	0.945(4)	0.964(4)
8	5.80	33.41(8)	0.985(2)	0.990(2)	0.995(2)

Table 3. Table 3: Values of the perturbative coefficients in the large- N 𝑁 N limit.

$n$	${\overline{a}}_n$	${\overline{b}}_n$	${\overline{e}}_n$	${\overline{f}}_n$
1	$-$ 0.01413	$-$ 0.06259	$-$ 0.005993	0.13650
2	$-$ 0.01289	$-$ 0.01079	$-$ 0.001568	0.02651
3	$-$ 0.00404	$-$ 0.00401	$-$ 0.000530	0.00929
4	$-$ 0.00145	$-$ 0.00146	$-$ 0.000221	0.00411
5	$-$ 0.00063	$-$ 0.00063	$-$ 0.000109	0.00212
6	$-$ 0.00031	$-$ 0.00030	$-$ 0.000063	0.00127

Equations

$${\cal H}^{latt}\,=\,-\sum_{x\mu}\mbox{\boldmath$\sigma$}_{x}\cdot\mbox{\boldmath$\sigma$}_{x+\mu}$$

$$Z\,=\,\int\exp(-\beta{\cal H}^{latt})\,\prod_{x}d^{N}\mbox{\boldmath$\sigma$}_{x}\,\delta(\mbox{\boldmath$\sigma$}_{x}^{2}-1)$$

$${\cal H}^{latt}\,=\,-\sum_{x\mu}\mbox{\boldmath$\sigma$}_{x}\cdot\mbox{\boldmath$\sigma$}_{x+\mu}-\,h\,\sum_{x}\mbox{\boldmath$\sigma$}_{x}^{1}$$

$$\mbox{\boldmath$\sigma$}_{x}\,=\,\left(\sqrt{1-\mbox{\boldmath$\pi$}^{2}_{x}},\mbox{\boldmath$\pi$}_{x}\right)$$

$${\Gamma }_{latt}^{(n)}(p_1,\dots ,p_n;\beta ,h;1/a)=Z_2^{n/2}{\Gamma }_{\overline{MS}}^{(n)}(p_1,\dots ,p_n;Z_1^{-1}\beta ,Z_1{Z}_2^{-1/2}h;\mu )$$

$$W^{\overline{MS}}(Z_1^{-1}\beta )$$

$${\gamma }^{\overline{MS}}(Z_1^{-1}\beta )$$

$$W^{scheme}(\beta )=-\frac{w_0}{{\beta }^2}-\frac{w_1}{{\beta }^3}-\frac{w_2^{scheme}}{{\beta }^4}-\frac{w_3^{scheme}}{{\beta }^5}+O({\beta }^{-6})$$

$${\gamma }^{scheme}(\beta )=\frac{{\gamma }_0}{\beta }+\frac{{\gamma }_1^{scheme}}{{\beta }^2}+\frac{{\gamma }_2^{scheme}}{{\beta }^3}+\frac{{\gamma }_3^{scheme}}{{\beta }^4}+O({\beta }^{-5})$$

$$w_0$$

$$w_1$$

$${\gamma }_0$$

$$w_2^{\overline{MS}}$$

$$w_3^{\overline{MS}}$$

$${\gamma }_1^{\overline{MS}}$$

$${\gamma }_2^{\overline{MS}}$$

$${\gamma }_3^{\overline{MS}}$$

$$w_2^{latt}=\frac{N-2}{(2\pi {)}^3}\left[\left(\frac{1}{2}+\frac{{\pi }^2}{8}-4{\pi }^2{G}_1\right)(N-2)+1+\frac{\pi }{2}-\frac{5{\pi }^2}{24}\right]$$

$${\gamma }_1^{latt}$$

$${\gamma }_2^{latt}$$

$$(N-1)h\{\frac{1}{3072{\pi }^3}(3N-5)(5N-9){\log }^3\frac{h{a}^2}{32}$$

$$+\frac{1}{512{\pi }^3}\left[4N^2-23N+29-2\pi (3N-5)\right]{\log }^2\frac{h{a}^2}{32}$$

$$-\frac{1}{4\pi }[\frac{1}{192}(3N-17)+\frac{1}{64\pi }(5N-11)-\frac{1}{128{\pi }^2}(N-3{)}^2-$$

$$\frac{1}{2}{G}_1(N-2)]\log \frac{h{a}^2}{32}+A_3\}+$$

$$p^2\{\frac{1}{384{\pi }^3}(2N-3)(N-1){\log }^3\frac{h{a}^2}{32}+$$

$$+\frac{1}{256{\pi }^3}\left[11N^2-41N+36-2\pi (N-1)\right]{\log }^2\frac{h{a}^2}{32}$$

$$-\frac{1}{4\pi }[-\frac{1}{64{\pi }^2}(5N^2-42N+65)+\frac{1}{32\pi }(7N-13)-$$

$$R(2N-3)(N-2)]\log \frac{h{a}^2}{32}+B_3\}$$

$$A_3$$

$$B_3$$

$$h(N-1)$$

$$+p^2$$

$$Z_1$$

$$Z_2$$

$$Z_{13}$$

$$Z_{23}$$

$$w_3^{latt}$$

$${\gamma }_3^{latt}$$

$$\frac{w_3^{latt}}{N^3}$$

$$\frac{{\gamma }_3^{latt}}{N^3}$$

$$\xi$$

Full text

1 Introduction

Non-linear $\sigma$-models have been and are being investigated in theoretical physics for a variety of reasons: in condensed matter physics the non-linear $N$-vector model describes the critical behaviour of systems with an $N$-component order parameter [1, 2]; in elementary particle physics two-dimensional $\sigma$-models serve as a playground for testing ideas which are relevant to four dimensional gauge theories: indeed they are asymptotically free and can be studied with a weak-coupling perturbative expansion [3, 4, 5, 6].

The simplest example is the $\sigma$-model where the fields take values in the sphere $S^{N-1}$ and where the action is invariant under global $O(N)$ transformations. Besides perturbation theory, it can be studied using different techniques. It can be solved in the $N=\infty$ limit [7, 8] and $1/N$ corrections can be systematically computed [9, 10, 11]. Moreover an exact $S$-matrix can be computed [12, 13] and, using the thermodynamic Bethe ansatz , the exact mass-gap of the theory in the limit $\beta\to\infty$ can been obtained [14, 15, 16]. The model has also been studied numerically: extensive simulations have been performed for $N=3$ [17, 18], $N=4$ [19, 20] and $N=8$ [19]. The results for the correlation length agree with the conventional predictions — including the nonperturbative coefficient — to within about 4% for $N=3$ (at $\xi\approx 10^{5}$), 6% for $N=4$ (for $\xi\approx 100$) and 1% for $N=8$ (for $\xi\approx 30$). The remaining deviations are not much larger than the three-loop correction: for $N=3$ (resp. 4, 8), at the largest $\beta$ where Monte Carlo data are available, the three-loop correction is about 3% (resp. 2%, 0.5%). For this reason we expect the inclusion of the four-loop term to improve sensibly the agreement with the conventional predictions.

In this paper we compute the $\beta$-function and the anomalous dimension of the field up to four-loops, thus extending previous work by Falcioni and Treves [21]. From this computation we obtain the second analytic coefficients in the perturbative expansion of the correlation length $\xi$ and of the vector susceptibility $\chi$ and the third analytic correction to the ratio $\chi/\xi^{2}$. Using results obtained in [22] we can also compute the second correction to the general spin-$n$ susceptibility. Some technical details concerning the computation are reported in Appendix A. A check of the results is provided by the $1/N$ results of [11]: in Appendix B we have checked the correctness of the large-$N$ limit of our results. We have finally compared our four-loop prediction to the available data for the correlation length (a much more detailed comparison for $N=3$, which includes also the susceptibilities, will appear in [23]): we find that the discrepancy between theory and experiment at the largest $\beta$ today available is now reduced to 2%, 4%, 0.5% for $N=3,4,8$, the four-loop correction being of order 2%, 2%, 0.5% in the three cases. The remaining difference should be ascribed to higher-loop corrections: for $N=8$, using the large-$N$ results, we have indeed verified that, if all analytic corrections were included up to eight loops, the discrepancy should be \mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 316\relax}}0.1%.

2 Four-loop RG Functions

In this paper we consider the nearest-neighbor lattice $N$-vector model in two dimensions. The fields are unit-length spins \mbox{\boldmath\sigma}_{x}\in R^{N} and the hamiltonian is given by

[TABLE]

The partition function is given by

[TABLE]

As it is well known, the perturbative expansion of this model in two dimensions is plagued by infrared divergences. We will not discuss this problem here and we will adopt the common technique of adding a magnetic field $h$ to the hamiltonian as an infrared regulator. Thus, if the magnetic field points in the first direction we have the hamiltonian

[TABLE]

The perturbative expansion is then obtained by considering small fluctuations around the direction of the magnetic field. Thus one sets

[TABLE]

and expands the Hamiltonian in powers of $\pi$.

We will now compute the four-loop $\beta$-function and anomalous dimension of the field $\sigma$. In principle this can be done through a direct lattice computation. However it is much simpler to take advantage of the fact that the four-loop calculation has already been done for the continuum theory in dimensional regularization [24, 25, 26]. This allows us to compute the four-loop contribution by performing a lattice computation at three loops. The idea is to compute the finite renormalization constants $Z_{1}(\beta,\mu a)$ and $Z_{2}(\beta,\mu a)$ which relate the Green’s functions in the $\overline{MS}$-scheme and on the lattice.

More precisely, define $\Gamma^{(n)}_{latt}(p_{1},\ldots,p_{n};\beta,h;1/a)$ as the lattice $n$-point one-particle-irreducible correlation function for the $\pi$-field and $\Gamma^{(n)}_{\overline{MS}}(p_{1},\ldots,p_{n};\beta,h;\mu)$ its counterpart in the $\overline{MS}$ scheme. Then the general results of [28] imply

[TABLE]

It follows that the $\beta$-functions $W(\beta)$ and the anomalous dimensions $\gamma(\beta)$ in the two schemes are related by

[TABLE]

In general we expand, on the lattice as well as in the ${\overline{\hbox{\scriptsize MS}}}$ scheme,

[TABLE]

and

[TABLE]

The coefficients $w_{0}$, $w_{1}$ and $\gamma_{0}$ are universal in the sense that they do not depend on the renormalization procedure and for this reason we have not added the superscript $scheme$. They are explicitly given by

[TABLE]

All other terms instead are scheme-dependent. In ${\overline{\hbox{\scriptsize MS}}}$ they are explicitly given by [24, 25, 26, 27]

[TABLE]

where $\zeta(3)\approx 1.2020569$. On the lattice we have [21, 22]

[TABLE]

and

[TABLE]

where $G_{1}\approx 0.0461636$.

We will now compute $w^{latt}_{3}$ and $\gamma_{3}^{latt}$. We must first of all compute the three-loop self-energy on the lattice for the $\pi$-field. The Feynman graphs are reported in Fig. (1). We get

[TABLE]

where

[TABLE]

Here $V_{1}$, $\ldots,V_{6}$, $G_{1}$, $R$, $J$ and $W_{2}$ are finite lattice integrals. Their definition is reported in Appendix A.1 and their numerical value in table 1.

Most of the graphs can be computed with a limited effort: using the table of integrals appearing in Appendix A.2, all graphs but the last one in Fig. (1) have been reduced automatically using the symbolic language mathematica. The computation of the last graph was much more difficult and involved. The whole computation has been done independently by the two authors, and many intermediate results have been checked numerically. Some technical details can be found in Appendix A.

We must also compute the same correlation function in the continuum theory. In the ${\overline{\hbox{\scriptsize MS}}}$-scheme we get

[TABLE]

From these expressions it is easy to obtain the renormalization constants $Z_{1}$ and $Z_{2}$. We expand

[TABLE]

The terms proportional to $\beta^{-1}$ and $\beta^{-2}$ have already been computed in [22]. For the three-loop terms we get

[TABLE]

Then, using (2.6) and (2.7), we finally obtain

[TABLE]

A check of these results is provided by the $1/N$-results of [11]. In the large-$N$ limit we get from the previous expressions

[TABLE]

In Appendix B we have checked that (2.31)/(2.32) agree with the predictions of the $1/N$ expansion.

3 Long-Distance Quantities

We will now use $w_{3}^{latt}$ and $\gamma_{3}^{latt}$ to compute the second analytic correction to the correlation length $\xi$ and spin susceptibility $\chi$.

Let us begin with $\xi$. In general we have

[TABLE]

The constant $C_{\xi}$ is non-perturbative and its value depends on the explicit definition of the correlation length. For the isovector exponential correlation length $\xi_{V}^{(exp)}$, which controls the large-distance behaviour of the two-point function \langle\mbox{\boldmath\sigma}_{0}\cdot\mbox{\boldmath\sigma}_{x}\rangle, an explicit expression has been obtained using the thermodynamic Bethe ansatz [14, 15, 16]. Explicitly

[TABLE]

Other possibilities are the second-moment correlation length

[TABLE]

in the isovector channel, or the analogous quantities in higher-isospin channels. For instance, in studies of mixed $O(N)$/$RP^{N-1}$ [23], the isospin-2 correlation lengths associated with the correlation \langle(\mbox{\boldmath\sigma}_{0}\cdot\mbox{\boldmath\sigma}_{x})^{2}-1/N\rangle were introduced: the exponential isotensor correlation length $\xi_{T}^{(exp)}$ and the corresponding second-moment correlation length $\xi_{T}^{(2)}$. Using the fact that in the $O(N)$ $\sigma$-model no bound states exist [12], we have immediately

[TABLE]

For the second-moment correlation lengths no exact expressions exist. However, in the large-$N$ limit we get [29]

[TABLE]

Let us now consider the perturbative corrections to the universal behaviour. The first coefficient $a_{1}$ was computed in [21]:

[TABLE]

We can now compute $a_{2}$ to get

[TABLE]

Analogous expressions can be derived for the isovector susceptibility \chi_{V}=\sum_{x}\langle\mbox{\boldmath\sigma}_{0}\cdot\mbox{\boldmath\sigma}_{x}\rangle. We get

[TABLE]

The (non-universal) constant $C_{\chi}$ cannot be computed in perturbation theory, and no exact expression is available at present. We can evaluate $C_{\chi}$ in the large-$N$ limit. Using the $1/N$ results of [11] we obtain the following expression:

[TABLE]

where $\gamma_{E}\approx 0.5772157$ is Euler’s constant and

[TABLE]

We will also consider the ratio $\chi_{V}/\xi^{2}$ as in this case we can compute an additional analytic correction. We write

[TABLE]

where $R_{V}=C_{\chi}/C^{2}_{\xi}$ is a non-perturbative universal quantity.

The explicit values of $b_{1}$, $c_{1}$ and $c_{2}$ are reported in [21, 22]. Explicitly

[TABLE]

Here we will compute $b_{2}$ and $c_{3}$. They are given by:

[TABLE]

A check of these expressions is provided by the $1/N$-expansion. In Appendix B we have verified that these expressions are in agreement with the $1/N$ results of [11].

Numerically we have

[TABLE]

Using the results of [22] we can also compute the second analytic correction to all non-derivative dimension-zero operators. A suitable basis is given by

[TABLE]

where “Traces” must be such that ${Y^{\alpha_{1},\ldots,\alpha_{n}}_{n}}$ is completely symmetric and traceless. These polynomials are irreducible $O(N)$-tensors of rank $n$ and thus they renormalize multiplicatively with no off-diagonal mixing. We define the spin-$n$ susceptibility as

[TABLE]

Standard renormalization group arguments give [30]

[TABLE]

The non-universal constant $C_{\chi}^{(n)}$ cannot be estimated in perturbation theory. A general expression is available only in the large-$N$ limit. We have

[TABLE]

Explicitly $C^{(2)}_{\chi}=\pi/32$, $C^{(3)}_{\chi}=\pi^{3}R/4$. For $n=2$ we also computed the first $1/N$ correction to get [29]

[TABLE]

where $\zeta^{\prime}(2)=-0.9375482$. Let us now consider the analytic corrections. In [22] we considered the ratio

[TABLE]

and computed $c_{1}^{(n)}$ and $c_{2}^{(n)}$. Their expression is

[TABLE]

Then we get immediately

[TABLE]

Numerically

[TABLE]

We want now to compare our four-loop prediction with the available Monte Carlo data for the correlation length. We define

[TABLE]

where $\xi_{MC}(\beta)$ is the Monte Carlo value of the correlation length 111We consider here the isovector exponential correlation length $\xi_{V}^{(exp)}$. Notice that the data in [18] for $N=3$ refer instead to $\xi^{(2)}_{V}$. The two quantities differ however by less than 0.1% [31] and thus we will ignore the difference. and $\xi_{th}^{n-loop}(\beta)$ is the theoretical $n$-loop prediction given by (3.33) and (3.34). In table (2) we report $\cal{R}$ for some selected values of $N$ and $\beta$. It is evident that the inclusion of the four-loop correction improves the agreement between theory and “experiment”. The remaining discrepancy at the largest $\beta$-values available today is now 2% for $O(3)$, 4% for $O(4)$ and 0.5% for $O(8)$ and it can presumably be ascribed to the neglected higher-loop corrections.

We want now to try to keep into account the higher-loop corrections using the large-$N$ results of [11]. Let us first define

[TABLE]

The coefficients $a_{n}^{(1/N)}$ are the leading contribution to $a_{n}$ in the limit $N\to\infty$ and $\bar{a}_{n}$ are numerical coefficients which can be computed using the $1/N$ expansion. Their explicit value for $n=1\ldots 6$ is reported in Table (3). For $n=1$ and $n=2$ we can compare $a_{n}^{(1/N)}$ with the exact value $a_{n}$. One finds that in both cases the ratio $a_{n}/a_{n}^{(1/N)}$ is a decreasing function of $N$ reaching the limiting value of one for $N\to\infty$. The convergence is however slow. For $N=4$ we have indeed

[TABLE]

while for $N=8$ we get

[TABLE]

The approximation is good at the 10% level only for N\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 318\relax}}50 (resp. N\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\mathchar 536\relax}\hss}\raise 2.0pt\hbox{\mathchar 318\relax}}35) for $a_{1}$ (resp. $a_{2}$). Nonetheless we can try to use the $1/N$ results to get a rough idea of the role of higher loop corrections.

$N=8$ should be the case where the approximation works better. In this case we will assume that the coefficients $a_{n}(N=8)$ for $n\geq 3$ are given by

[TABLE]

where $\bar{a}_{n}$ are defined in Table (3) and $k_{n}$ is a number that we will vary between about 1 and 2. In this way we can get an estimate for ${\cal R}^{(8-loop)}$. For $\beta=5.80$ (see Table (2)) we obtain for $k_{n}\equiv 1$ ${\cal R}^{(8-loop)}=0.998$; while for $k_{n}\equiv 2$ we have ${\cal R}^{(8-loop)}=1.001$. The eight-loop correction is of order $3k_{6}\cdot 10^{-4}$. Thus at this order we would expect an agreement at the order of 0.1% and this is indeed what we get from this rough approximation.

We can try the same rough approximation for $N=4$ , writing, for $n\geq 3$, $a_{n}(N=4)=k_{n}4^{n-1}\bar{a}_{n}$. For $\beta=2.80$ we get ${\cal R}^{(8-loop)}=0.970$ (resp. 0.994) for $k_{n}\equiv 1$ (resp. 5). Although in this case it is very difficult to make any quantitative statement this calculation shows that the numbers are in the correct ball-park.

From this analysis we can thus conclude that the theoretical prediction of [14, 15] is in very good agreement with the Monte Carlo data. We do not discuss here other long-distance quantities like the vector and tensor susceptibility. A detailed comparison with Monte Carlo data will appear elsewhere [23].

Acknowledgments

We thank Massimo Campostrini, Paolo Rossi and Alan Sokal for many useful comments.

Appendix A Technical Details

A.1 Notations

In this Appendix we introduce the notations we have used in the explicit computation of $w_{3}$ and $\gamma_{3}$. The one-loop perturbative results will be written in terms of the integral

[TABLE]

where $\hat{p}_{\mu}=2\sin(p_{\mu}/2)$, $\hat{p}^{2}=\sum_{\mu}\hat{p}^{2}_{\mu}$ and

[TABLE]

The integral $I(h)$ is logarithmically divergent for $h\to 0$. Explicitly

[TABLE]

where $K$ is a complete elliptic integral of the first kind. We will also use

[TABLE]

We need also some basic two-loop and three-loop integrals. To simplify the notation let us first introduce

[TABLE]

We will use the following integrals which already appear in our previous work:

[TABLE]

We have moreover introduced a set of 8 new constants . The quantities $V_{1},\ldots V_{6}$ correspond to lattice infrared-finite integrals and are explicitly given by

[TABLE]

We introduce also $W_{1}$ and $W_{2}$ as the finite part of infrared-divergent integrals:

[TABLE]

The numerical value of all the constants is reported in Table 1.

The numerical computation of $V_{1},\ldots$, $V_{6}$ does not present any difficulty as the integrals are infrared-finite. More tricky is handling the integrals leading to $W_{1}$ and $W_{2}$. In this case we have used a method which was introduced in the context of the $1/N$ expansion in [33] . Let us consider the case of $W_{1}$. First of all let us determine the divergent terms which must be subtracted from the original integral. If we introduce

[TABLE]

we can rewrite

[TABLE]

Then we determine the behaviour of $d_{\mu}(q,h)$ for $q\to 0$ by writing an integral representation for $d_{\mu}(q,h)$. Using the technique presented in [33] we get

[TABLE]

where $E(\zeta)$ and $K(\zeta)$ are complete elliptic integrals [32] and

[TABLE]

Using (A.1) we can now compute the expansion of $d_{\mu}(q,h)$ for $q^{2}\to 0$, $h\to 0$, with $q^{2}/h$ arbitrary. We get

[TABLE]

where

[TABLE]

Then we rewrite (A.94) as

[TABLE]

The first integral is infrared finite. Thus we can take the limit $h\to 0$ obtaining

[TABLE]

Although this integral is finite its numerical evaluation is complex as the integrand is a difference of two divergent quantities. To get stable results we have split the integration domain in two parts: a disk $D_{\epsilon}$ of radius $\epsilon=0.1$ around the origin and the remaining region $R_{\epsilon}=[-\pi,\pi]^{2}-D_{\epsilon}$. The integration over $R_{\epsilon}$ is done numerically; to compute the integral over $D_{\epsilon}$ we have first expanded the integrand up to $O(q^{12})$ and then we have performed the integration analytically. In the implementation a useful check is provided by the expression of $d_{\mu}(q,0)$ along the diagonal, i.e. for $q=(l,l)$ which can be computed exactly

[TABLE]

Let us now compute the second integral in (A.100) which is still infrared divergent. We first change the integration domain: if $\Lambda>\pi\sqrt{2}$ we rewrite it as

[TABLE]

The second integral is easily computed numerically, while the first gives

[TABLE]

In the calculation the choice of $\Lambda$ is completely arbitrary. We have chosen $\Lambda=\sqrt{32}$ as in this case it is simple to replace $\log h$ with $I(h)$.

The calculation of $W_{2}$ is completely analogous. Introducing

[TABLE]

we must compute

[TABLE]

First of all we compute an integral representation for $d_{\mu\nu}(q,h)$. We get

[TABLE]

where $d_{1}(q,h)$ is given in (A.1) and

[TABLE]

and

[TABLE]

From these integral representations we thus get in the limit $q^{2}\to 0$, $h\to 0$ with $q^{2}/h$ arbitrary

[TABLE]

The computation is analogous to the previous case. The relevant integral is now

[TABLE]

where $D_{\Lambda}$ is a disk of radius $\Lambda$ around the origin.

In order to check the manipulations of the extremely cumbersome expression for $\bar{d}_{1}(q,h)$ we found very useful the expression of $d_{1}(q,h)$ for $q=(l,l)$ and $h=0$ given by

[TABLE]

A.2 Some Lattice Integrals

In this section we report a list of integrals we have used in our computation.

Two-loop integrals:

[TABLE]

Three-loop integrals:

[TABLE]

A.3 Analytic

evaluation of one-loop integrals

In this appendix we discuss the evaluation of the most general one-loop integral. Let us introduce the notation

[TABLE]

for $n_{i}\geq 0$, $r>0$, and $x,y$ all different. In the following when one of the arguments $n_{i}$ is zero it will be omitted as argument of ${\cal B}$.

Using a technique we developed for four-dimensional integrals [34] all these integrals can be reduced to a sum of $I(h)$ and $I_{2}(h)$.

We will firstly generalize (A.123) by considering the following more general integrals

[TABLE]

where $r$ is a positive or negative integer and $\delta$ a real number which is introduced in order to avoid singular cases at intermediate stages of the computation and which will be set to zero at the end.

The first thing we want to show is that each integral $\tilde{\cal B}_{\delta}(r;n_{x},n_{y})$ can be reduced through purely algebraic manipulations to a sum of integrals of the same type with $n_{x}=n_{y}=0$.

Indeed the integrals $\tilde{\cal B}_{\delta}$ satisfy the following recursion relations:

[TABLE]

which can be obtained by the insertion of the trivial identity

[TABLE]

and by keeping into account the cases in which the index $q$ equals one of the other indices. Furthermore, when $m>1$ we can write

[TABLE]

Then, integrating by parts, we obtain the recursion relation:

[TABLE]

These relations allow to reduce every integral $\tilde{\cal B}_{\delta}(r;n_{x},n_{y})$ to a sum of the form

[TABLE]

For $p\geq 1$, $\lim_{\delta\to 0}$ $a_{p}(\delta)$ is finite while for $p<1$ $a_{p}(\delta)$ may behave as $1/\delta$ when $\delta$ goes to zero, meaning that we need to compute $\tilde{\cal B}_{\delta}(p)$ including terms of order $\delta$ when $p\leq 0$.

Now let us show that all $\tilde{\cal B}_{\delta}(p)$ can be expressed in terms of $\tilde{\cal B}_{\delta}(1)$ and $\tilde{\cal B}_{\delta}(2)$. Indeed let us consider the trivial identity

[TABLE]

If we apply the reduction procedure to the integrals appearing in this relation we get the following two relations

[TABLE]

If we now apply the first relation for $q<0$ and the second for $q>2$ we express every integral as

[TABLE]

A careful analysis of the structure of the recursion (A.3) shows that $B(\delta)$ and $C(\delta)$ are finite for $\delta\to 0$. As the l.h.s. is also finite for $\delta\to 0$ we get that also $A(\delta)$ is finite in this limit. Thus we can set $\delta=0$ to get

[TABLE]

As a final remark, notice that the whole procedure is completely algebraic and can be easily implemented on a computer using a symbolic language.

Appendix B Comparison with the $1/N$ expansion

In this appendix we want to compare our four-loop result with the $1/N$ results of [11]. In the perturbative limit we have

[TABLE]

with

[TABLE]

and [11]

[TABLE]

where

[TABLE]

and

[TABLE]

A point must be noted in the solution (B.140), (B.141): both $\xi_{1}$ and $\chi_{1}$ are expressed in terms of differences of two integrals which have a non-integrable singularity for $k=0$. This notation is however symbolic and it must be interpreted in the following way: for two functions $f(k)$ and $g(k)$ we define

[TABLE]

where $B$ is the domain $[-\pi,\pi]^{2}$, $C=D_{\sqrt{32}}-B$ and $D_{r}$ the disk centered in the origin of radius $r$. It is easily checked that with this definition everything is well-defined for $k=0$ as in both cases the singularity cancels in the difference. From (B.140) and (B.141) we can easily derive the large-$N$ behaviour of $a_{2}$, $b_{2}$ and $c_{3}$. We get

[TABLE]

In order to compare these expressions with our results we must reexpress these integrals in terms of our basic constants. In the Appendix F of [33] 222Notice the change in notation: $A_{\#}^{(\alpha)}=2A_{\#}$, $B_{\#}^{(\alpha)}=2B_{\#}$ one can find the basic relations which are needed to compare the results of the $1/N$ expansion with the standard three-loop result. As we will use them in the following we report them here:

[TABLE]

We must now derive analogous expressions for the integrals appearing in $a_{2}$, $b_{2}$ and $c_{3}$.

Let us begin by considering the quantity

[TABLE]

In this case we start from the three-loop integral

[TABLE]

whose value is reported in (A.120). It can also be computed using the method we presented in Appendix A.1 when we considered $W_{1}$ and $W_{2}$. If we define

[TABLE]

we can rewrite $T$ as

[TABLE]

Now for $q\to 0$, $h\to 0$, $q^{2}/h$ fixed, $d_{0}(q,h)$ becomes its continuum counterpart

[TABLE]

where $\rho$ is defined in (A.99). Thus we rewrite

[TABLE]

The second integral is easily done and gives

[TABLE]

For the term in curly brackets let us notice that

[TABLE]

is integrable for $k\to 0$ for all values of $h$. Thus we can expand the integrand for $h\to 0$. Using the fact that for $h\to 0$ at $k$ fixed we have

[TABLE]

and the relations (B.149)/(B.150), we get at the end

[TABLE]

Comparing with (A.120) we get

[TABLE]

Simple algebraic manipulations give also

[TABLE]

The last integral which is needed is

[TABLE]

This quantity can be handled exactly in the same way starting from

[TABLE]

which can be computed by derivation with respect to $h$ from (A.121). The relevant integral is

[TABLE]

We finally get

[TABLE]

We have checked numerically (B.163), (B.164) and (B.168). This provides also a check of the integrals (A.120) and (A.121).

Using (B.146), (B.147) and (B.148) we finally get for the coefficients $a_{2}$, $b_{2}$ and $c_{3}$

[TABLE]

From these expressions, using the renormalization group relations (3.33)/(3.41), we can compute the large-$N$ contribution to $w_{3}^{latt}$ and $\gamma_{3}^{latt}$. These expressions agree with (2.31)/(2.32).

From the general expressions (B.140) and (B.141) it is also possible to compute the value of higher-loops coefficients. Defining

[TABLE]

we report their values in Table(3). We can also compute, in view of the possibility of using *improved * expansions, the perturbative expansion of the isovector energy E_{V}=\langle\mbox{\boldmath\sigma}_{0}\cdot\mbox{\boldmath\sigma}_{1}\rangle. From [11] we get

[TABLE]

We can thus rewrite for the correlation length

[TABLE]

where $\beta_{E}=(N-1)/(4(1-E_{V}))$ and

[TABLE]

The coefficients $\bar{e}_{n}$ and $\bar{f}_{n}$ are reported in Table (3).