Effective potential of the three-dimensional Ising model: the pseudo-$\epsilon$ expansion study

TL;DR

This study uses pseudo-$\epsilon$ expansion techniques to accurately calculate universal ratios of coupling constants in the 3D Ising model, achieving results that agree well with lattice and RG analyses.

Contribution

It introduces pseudo-$\epsilon$ expansions for higher-order coupling ratios in the 3D Ising model and demonstrates their effectiveness with advanced resummation methods.

Findings

Accurate values for $R_6^*$ agree with lattice results.

Conformal Borel resummation yields reliable estimates for $R_8^*$.

Higher-order couplings like $R_{10}^*$ are challenging due to series growth.

Abstract

The ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar $\lambda\phi^4$ field theory (3D Ising model) within the pseudo-$\epsilon$ expansion approach. Pseudo-$\epsilon$ expansions for the critical values of $g_6$, $g_8$, $g_{10}$, $R_6 = g_6/g_4^2$, $R_8 = g_8/g_4^3$ and $R_{10} = g_{10}/g_4^4$ originating from the five-loop renormalization group (RG) series are derived. Pseudo-$\epsilon$ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Pad\'e approximants yields proper numerical results. Use of Pad\'e--Borel--Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values $R_6^* = 1.6488$ and $R_6^* = 1.6490$ which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-$\epsilon$ expansions is less favorable. Nevertheless, the conform-Borel resummation gives $R_8^* = 0.868$, the number being close to the lattice estimate $R_8^* = 0.871$ and compatible with the result of 3D RG analysis $R_8^* = 0.857$. Pseudo-$\epsilon$ expansions for $R_{10}^*$ and $g_{10}^*$ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.

Tables

Table 1. Table 1: Padé table for pseudo- ϵ italic-ϵ \epsilon expansion of the ratio R 6 ∗ superscript subscript 𝑅 6 R_{6}^{*} . Padé approximants [L/M] are derived for R 6 ∗ / τ superscript subscript 𝑅 6 𝜏 R_{6}^{*}/\tau , i. e. with factor τ 𝜏 \tau omitted. The lowest line (RoC) shows the rate of convergence of Padé estimates to the asymptotic value. Here the Padé estimate of k 𝑘 k -th order is that given by diagonal approximant or by the average over two near-diagonal ones when corresponding diagonal approximant does not exist.

$M\setminus L$	0	1	2	3	4
0	2	1.5117	1.7518	1.5368	1.7947
1	1.6075	1.6726	1.6383	1.6540
2	1.6896	1.6465	1.6502
3	1.6036	1.6504
4	1.7135
RoC	2	1.5596	1.6726	1.6424	1.6502

Table 2. Table 2: Padé-Borel-Leroy estimates of R 6 ∗ superscript subscript 𝑅 6 R_{6}^{*} resulting from pseudo- ϵ italic-ϵ \epsilon expansion (15) as functions of the shift parameter b 𝑏 b . Padé approximants [4/1], [3/2], [2/3] are used for analytical continuation of the Borel-Leroy transform while approximant [1/4] is spoilt by positive axis pole for all relevant values of b 𝑏 b . It is amazing that under b = 4 𝑏 4 b=4 the estimates provided by approximants [3/2] and [2/3] coincide, i. e. the curves R 6 ∗ ​ ( b ) [ 3 / 2 ] superscript subscript 𝑅 6 subscript 𝑏 delimited-[] 3 2 R_{6}^{*}(b)_{[3/2]} and R 6 ∗ ​ ( b ) [ 2 / 3 ] superscript subscript 𝑅 6 subscript 𝑏 delimited-[] 2 3 R_{6}^{*}(b)_{[2/3]} touch (do not cross) each other at this point.

$b$	[4/1]	[3/2]	[2/3]
0	1.6485	1.64786	1.64819
1	1.6494	1.64822	1.64832
2	1.6500	1.648475	1.648506
3	1.6505	1.648672	1.648678
4	1.6509	1.6488294	1.6488294
5	1.6512	1.648958	1.648961
6	1.6514	1.649065	1.649074
7	1.6516	1.649155	1.649173
10	1.6521	1.64936	1.64940
15	1.6526	1.64957	1.64964

Table 3. Table 3: Padé triangle for pseudo- ϵ italic-ϵ \epsilon expansion of the ratio R 8 ∗ superscript subscript 𝑅 8 R_{8}^{*} . Padé approximants [L/M] are constructed for R 8 ∗ / τ superscript subscript 𝑅 8 𝜏 R_{8}^{*}/\tau , i. e. with factor τ 𝜏 \tau omitted. The lowest line (RoC) demonstrates the rate of convergence of Padé estimates where Padé estimate of k 𝑘 k -th order is that given by diagonal approximant or by the average over two near-diagonal ones if diagonal approximant does not exist.

$M\setminus L$	0	1	2	3	4
0	$-9$	8.864	$-6.986$	10.713	$-13.961$
1	$-3.015$	0.466	1.376	0.407
2	$-1.743$	1.910	0.879
3	$-1.131$	0.095
4	$-0.831$
RoC	$-9$	2.925	0.466	1.643	0.879

Table 4. Table 4: Padé-Borel estimates extracted from the pseudo- ϵ italic-ϵ \epsilon expansion for R 8 ∗ superscript subscript 𝑅 8 R_{8}^{*} . Padé approximants [L/M] are used for analytical continuation of the Borel transform. The lowest line (RoC) shows the rate of convergence of Padé-Borel estimates to the asymptotic value. Padé-Borel estimate of k 𝑘 k -th order is that originating from diagonal approximant for R 8 ∗ / τ superscript subscript 𝑅 8 𝜏 R_{8}^{*}/\tau or by average over two near-diagonal ones when corresponding diagonal approximant does not exist.

$M\setminus L$	1	2	3	4	5
0	$-9$	8.864	$-6.986$	10.713	$-13.961$
1	$-3.6473$	1.1000	0.88541	0.89064
2	$-2.4658$	0.90410	0.89048
3	$-2.0048$	0.89164
4	$-1.7788$
RoC	$-9$	2.608	1.1000	0.8948	0.89048

Table 5. Table 5: The values of the universal ratio R 10 subscript 𝑅 10 R_{10} obtained in this work, found by resummation of 3D RG series and ϵ italic-ϵ \epsilon -expansions, extracted from the lattice calculations (LC) and given by the exact-renormalization-group analysis (ERG).

This work	$1.45$,	$-1.76$
3D RG	$-2.06\pm 1.3$ [16],	$-2.3\pm 1.6$ [19]
$ϵ$-expansion	$-1.06\pm 0.1$ [16],	$-1.1\pm 0.1$ [19],	$-1.8\pm 1.4$ [23]
LC	$-0.75\pm 0.38$ [18],	$-1.2\pm 0.4$ [26],	$-0.97\pm 0.17$ [30],	$-1.39\pm 0.04$ [34]
ERG	$-1.65\pm 0.4$ [17],	$-1.152$ [33]

Equations

$$F(z,m)-F(0,m)={\frac{m^{3}}{g_{4}}}\Biggl{(}{\frac{z^{2}}{2}}+z^{4}+R_{6}z^{6}+R_{8}z^{8}+R_{10}z^{10}...\Biggr{)},$$

$${\chi }_4$$

$${\chi }_6$$

$${\chi }_8$$

$${\chi }_{10}$$

$$H={1\over 2}\int d^{3}x\Biggl{(}m_{0}^{2}\varphi^{2}+\nabla\varphi^{2}+{\lambda\over 12}\varphi^{4}\Biggr{)},$$

$$\displaystyle G_{R}^{-1}(0,m,g_{4})=m^{2},\qquad\quad{{\partial G_{R}^{-1}(p,m,g_{4})}\over{\partial p^{2}}}\bigg{\arrowvert}_{p^{2}=0}=1,$$

$${\Gamma }_R(0,0,0,m,g)=m^2{g}_4,{\Gamma }_R^{1,2}(0,0,m,g_4)=1.$$

$$\displaystyle g_{6}={\frac{9}{\pi}}g_{4}^{3}\Bigl{(}1-{\frac{3}{\pi}}g_{4}+1.38996295g_{4}^{2}-2.50173246g_{4}^{3}+5.275903\ g_{4}^{4}\Bigr{)},$$

$$\displaystyle g_{8}=-{\frac{81}{{2\pi}}}g_{4}^{4}\Bigl{(}1-{\frac{65}{6\pi}}g_{4}+7.77500131g_{4}^{2}-18.5837685g_{4}^{3}+48.16781g_{4}^{4}\Bigr{)},$$

$$\displaystyle g_{10}={\frac{243}{{\pi}}}g_{4}^{5}\Bigl{(}1-{\frac{20}{\pi}}g_{4}+23.1841758g_{4}^{2}-74.2747105g_{4}^{3}+238.6138g_{4}^{4}\Bigr{)},$$

$$g_4^{\ast }$$

$$\displaystyle g_{6}^{*}={8\pi^{2}\over 81}\tau^{3}\bigl{(}1+0.600823045\tau+0.104114939\tau^{2}-0.023565414\tau^{3}-0.01838783\tau^{4}\bigr{)}$$

$$\displaystyle g_{8}^{*}=-{8\pi^{3}\over 81}\tau^{4}\bigl{(}1-0.717421125\tau-0.201396988\tau^{2}-0.70623903\tau^{3}+0.8824349\tau^{4}\bigr{)}$$

$$\displaystyle g_{10}^{*}={32\pi^{4}\over 243}\tau^{5}\bigl{(}1-2.33196159\tau+1.84782991\tau^{2}-3.0485209\tau^{3}+6.816764\tau^{4}\bigr{)}.$$

$$\displaystyle R_{6}^{*}=2\tau\bigl{(}1-0.244170096\tau+0.120059430\tau^{2}-0.1075143\tau^{3}+0.1289821\tau^{4}\bigr{)}.$$

$$\displaystyle R_{8}^{*}=-9\tau\bigl{(}1-1.98491084\tau+1.76113570\tau^{2}-1.9665851\tau^{3}+2.741546\tau^{4}\bigr{)}.$$

$$\displaystyle R_{10}^{*}=54\tau\bigl{(}1-4.02194787\tau+7.55009811\tau^{2}-11.784685\tau^{3}+20.05363\tau^{4}\bigr{)}.$$

$$f(x)=\sum _{i=0}^{\infty }{c}_i{x}^i=\underset{0}{\overset{\infty }{\int }}{e}^{-t}{t}^{b}F(xt)dt,F(y)=\sum _{i=0}^{\infty }\frac{c_i}{\Gamma (i+b+1)}{y}^i=\sum _{i=0}^{\infty }{F}_i{y}^i,$$

$$R_6^{\ast }=1.6488\pm \mathrm{0.0014.}$$

$$w(y)=\frac{\sqrt{1+\alpha y}-1}{\sqrt{1+\alpha y}+1}\ ,\qquad\quad y(w)=\frac{4}{\alpha}\frac{w}{(1-w)^{2}}\$$

$$F(y(w))=\sum _{i=0}^{\infty }{F}_i{\left(\frac{4}{\alpha }\right)}^i\frac{w^i}{(1-w{)}^{2i}}=\sum _{i=0}^{\infty }{W}_i{w}^i(y),$$

$$W_0=B_0,W_n=\sum _{i=1}^n{F}_i{\left(\frac{4}{\alpha }\right)}^i\frac{(i+n-1)!}{(n-i)!(2i-1)!}$$

$$F^{(N)}(y)=\sum _{i=0}^N{F}_i{y}^i=\sum _{i=0}^N{W}_i{w}^i(y).$$

$$f^{(N)}(y)=\sum\limits_{i=0}^{N}W_{i}\int\limits_{0}^{\infty}e^{-t}t^{b}\Biggl{(}\frac{\sqrt{1+\alpha yt}-1}{\sqrt{1+\alpha yt}+1}\Biggr{)}^{i}dt.$$

$$\alpha =\mathrm{0.147774232.}$$

$$R_6^{\ast }=1.6490$$

$$R_6^{\ast }=1.570\pm 0.007(n=2),R_6^{\ast }=1.500\pm 0.005(n=3).$$

$$R_8^{\ast }=\mathrm{0.868.}$$

Full text

††journal: Nuclear Physics B

1 Introduction

The critical behavior of the systems undergoing continuous phase transitions is characterized by a set of universal parameters including, apart from critical exponents, renormalized effective coupling constants $g_{2k}$ and the ratios $R_{2k}=g_{2k}/g_{4}^{k-1}$. These ratios enter the small-magnetization expansion of free energy (the effective potential) and determine, along with renormalized quartic coupling constant $g_{4}$, the nonlinear susceptibilities of various orders:

[TABLE]

[TABLE]

where $z=M\sqrt{g_{4}/m^{1+\eta}}$ is a dimensionless magnetization, renormalized mass $m\sim(T-T_{c})^{\nu}$ being the inverse correlation length, $\chi$ is a linear susceptibility while $\chi_{4}$, $\chi_{6}$, $\chi_{8}$ and $\chi_{10}$ are nonlinear susceptibilities of fourth, sixth, eighth and tenth orders.

For the three-dimensional (3D) Ising model, the effective potential and nonlinear susceptibilities are intensively studied during several decades. In particular, renormalized coupling constants $g_{2k}$ and the ratios $R_{2k}$ were evaluated by a number of analytical and numerical methods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Estimating universal critical values of $g_{4}$, $g_{6}$ and $R_{6}$ by means of field-theoretical renormalization group (RG) approach in fixed dimensions has shown that RG technique enables one to get accurate numerical estimates for these quantities. For example, four- and five-loop RG expansions resummed by means of Borel-transformation-based procedures lead to the values for $g_{6}^{*}$ differing from each other by less than $0.5\%$ [15, 16] while the three-loop RG approximation turns out to be sufficient to provide an apparent accuracy no worse than $1.6\%$ [15, 24]. In principle, this is not surprising since the field-theoretical RG approach proved to be highly efficient when used to estimate critical exponents, critical amplitude ratios, marginal dimensionality of the order parameter, etc. for numerous phase transition models [3, 4, 11, 19, 20, 28, 31, 32, 36, 37, 38].

To obtain proper numerical estimates from diverging RG expansions the resummation procedures have to be applied. Most of those being used today employ Borel transformation which avoids factorial growth of higher-order coefficients and enables one to construct converging iteration schemes. This transformation has paved the way to a great number of high precision numerical estimates. There exists, however, alternative technique turning divergent perturbative series into more suitable ones, i. e. into expansions that have smaller lower-order coefficients and much slower growing higher-order ones than those of original series. We mean the method of pseudo-$\epsilon$ expansion invented by B. Nickel (see Ref. 19 in the paper of Le Guillou and Zinn-Justin [4]). The pseudo-$\epsilon$ expansion approach has been shown to be very efficient when used to estimate critical exponents and other universal quantities for various 3D and 2D systems [4, 35, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].

In this paper, we study renormalized effective coupling constants and universal ratios $R_{2k}$ of the three-dimensional Ising model with the help of pseudo-$\epsilon$ expansion technique. The pseudo-$\epsilon$ expansions ($\tau$-series) for renormalized coupling constants $g_{6}$, $g_{8}$ and $g_{10}$ will be calculated on the base of five-loop RG expansions obtained by R. Guida and J. Zinn-Justin [16] for scalar field theory of $\lambda\varphi^{4}$ type. Along with the higher-order couplings, universal critical values of ratios $R_{6}=g_{6}/g_{4}^{2}$, $R_{8}=g_{8}/g_{4}^{3}$ and $R_{10}=g_{10}/g_{4}^{4}$ will be found as series in $\tau$ up to $\tau^{5}$ terms. The pseudo-$\epsilon$ expansions obtained will be processed by means of Padé, Padé–Borel–Leroy and conformal mapping resummation techniques as well as by direct summation when it looks reasonable. The numerical estimates for the universal ratios will be compared with numerous results deduced from the higher-order $\epsilon$-expansions, perturbative RG expansions in physical dimensions and extracted from the lattice calculations and some conclusions concerning the numerical power of the pseudo-$\epsilon$ expansion approach will be formulated.

The paper is organized as follows. In the next section the pseudo-$\epsilon$ expansions for $g_{6}^{*}$, $g_{8}^{*}$, $g_{10}^{*}$, $R_{6}^{*}$, $R_{8}^{*}$ and $R_{10}^{*}$ are derived from 3D RG series and known $\tau$-series for the Wilson fixed point location. Section III contains numerical estimates for the sextic coupling resulting from the pseudo-$\epsilon$ expansion for $R_{6}^{*}$. Sections IV deals with the renormalized octic coupling and numerical estimates for $R_{8}^{*}$ obtained within various resummation techniques are presented here. In Section V the tenth-order coupling constant and the ratio $R_{10}^{*}$ are evaluated and relevant numerical results are discussed. The last section contains a summary of the results obtained.

2 Pseudo-$\epsilon$ expansions for higher-order coupling constants

The critical behavior of 3D Ising model is described by Euclidean field theory with the Hamiltonian:

[TABLE]

where $\varphi$ is a real scalar field, bare mass squared $m_{0}^{2}$ being proportional to $T-T_{c}^{(0)}$, $T_{c}^{(0)}$ – mean field transition temperature. The $\beta$-function for the model (6) has been calculated within the massive theory [3] with the propagator, quartic vertex and $\varphi^{2}$ insertion normalized in a conventional way:

[TABLE]

Later, the five-loop RG series for renormalized coupling constants $g_{6}$, $g_{8}$ and $g_{10}$ of this model were obtained [16] and the six-loop pseudo-$\epsilon$ expansion for the Wilson fixed point location was reported [35]. The series mentioned are:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Combining these expansions one can easily arrive to the $\tau$-series for the higher-order coupling constants at criticality. They are as follows:

[TABLE]

[TABLE]

[TABLE]

Corresponding pseudo-$\epsilon$ expansions for the universal ratios $R_{2k}$ read:

[TABLE]

[TABLE]

[TABLE]

These $\tau$-series will be used for evaluation of higher-order effective couplings near the critical point.

3 Sextic effective interaction at criticality

In this Section we find numerical estimates for the critical asymptote of the ratio $R_{6}$ from the pseudo-$\epsilon$ expansion obtained. Since the series (15) has small higher coefficients direct summation of this series looks more or less reasonable. Within third, fourth and fifth orders in $\tau$ it gives 1.752, 1.537 and 1.795 respectively, i. e. the numbers grouping around the estimates 1.644 and 1.649 extracted from advanced field-theoretical and lattice calculations [16, 34]. It is interesting that the value 1.537 obtained by truncation of the series (15) by the smallest term (optimal truncation [52]) differs from the estimates just mentioned by 6% only. Moreover, direct summation of $\tau$-series for $g_{6}^{*}$ (12) having very small higher-order coefficients gives the value $g_{6}^{*}=1.621$ which under $g_{4}^{*}=0.9886$ [3] results in the estimate $R_{6}^{*}=1.659$ looking rather optimistic. These fact confirms the conclusion that the pseudo-$\epsilon$ expansion itself may be considered as a resummation method [35, 52, 53, 54, 55].

Much more accurate numerical value of $R_{6}^{*}$ can be obtained from the pseudo-$\epsilon$ expansion (15) using Padé approximants [L/M]. Padé triangle for $R_{6}^{*}/\tau$, i. e. with the insignificant factor $\tau$ neglected is presented in Table I.

Along with the numerical values given by various Padé approximants the rate of convergence of Padé estimates to the asymptotic value is shown in this Table (the lowest line). Since the diagonal and near-diagonal Padé approximants are known to possess the best approximating properties the Padé estimate of $k$-th order is accepted to be given by the diagonal approximant or by the average over two near-diagonal ones when corresponding diagonal approximant does not exist. As seen from Table I, the convergence of Padé estimates is well pronounced and the asymptotic value $R_{6}^{*}=1.6502$ is close to the 3D RG estimate $R_{6}^{*}=1.644\pm 0.006$ [16] and, in particular, to the value $R_{6}^{*}=1.649\pm 0.002$ given by the advanced lattice calculations [34].

Let us apply further the more powerful, Padé–Borel–Leroy resummation technique. It employes the Borel–Leroy transformation of the original diverging series

[TABLE]

with subsequent analytical continuation of the Borel-Leroy transform $F(y)$ with a help of Padé approximants. The shift parameter $b$ is commonly used for optimization of the resummation procedure. To resum the series (15) we address all the non-trivial Padé approximants apart from the approximant [1/4] which turned to be spoilt by a positive axis pole for the relevant values of $b$. The Padé–Borel–Leroy estimates of $R_{6}^{*}$ given by approximants [4/1], [3/2], and [2/3] under various $b$ are listed in Table II. Since near-diagonal approximants [3/2] and [2/3] are expected to lead to the most reliable results we concentrate on the last two columns of this Table. It is seen that the estimates these approximants result in are very close at any $b$. Moreover, under $b=4$ these estimates coincide – the curves $R_{6}^{*}(b)_{[3/2]}$ and $R_{6}^{*}(b)_{[2/3]}$ touch (do not cross) each other at this point. So, the value $R_{6}^{*}=1.6488$ corresponding to this touch point may be considered as a final Padé–Borel–Leroy estimate for the effective sextic coupling at criticality. This estimate is remarkably close to the value $1.649\pm 0.002$ obtained recently in the course of comprehensive lattice calculations [34].

What may be referred to as a measure of accuracy of the numerical result just obtained? The choice for such a measure which looks natural is a variation of the most stable Padé–Borel–Leroy estimate – $R_{6}^{*}(b)_{[2/3]}$ – under $b$ varying within the whole range $[0,\infty)$. For $b$ running from 0 to infinity $R_{6}^{*}(b)_{[2/3]}$ grows from 1.6482 to 1.6502. Hence, we adopt

[TABLE]

The inaccuracy bar accepted, being really small, is, in fact, rather conservative since it covers as well a range of variation of the less stable Padé–Borel–Leroy estimate $R_{6}^{*}(b)_{[3/2]}$.

Although the estimate just obtained looks quite satisfactory, to confirm its reliability and accuracy it is reasonable to resum the series (15) employing some alternative procedure for the analytical continuation of the Borel–Leroy transform. We address here the conformal mapping technique which, being widely used in the theory of critical phenomena, is known to demonstrate high numerical efficiency. This technique is based on the knowledge of the closest singularity of the Borel–Leroy transform (18) from the large-order behavior of the original series coefficients. For the Borel–Leroy image $F(y)$ it assumes an analyticity in the complex $y$-plane with a cut from $-1/\alpha$ to $-\infty$. Since the series for $F(y)$ converges within the circle $|y|<1/\alpha$ the integration of (18) requires its analytical continuation. It is performed by means of a conformal mapping $y=f(w)$ that should transform the plane with a cut into the unit circle $|w|<1$ and shift the singularities lying on the negative real axis on the circumference of the circle [2]. The conformal mapping

[TABLE]

is easily seen to satisfy the aforementioned requirements. Thus, we arrive to a new series:

[TABLE]

[TABLE]

that converges for any $y$. In our case only first $N$ coefficients of the expansion are known, i. e. we have to work with the truncated series for Borel–Leroy transform:

[TABLE]

Correspondingly, the conform-Borel-resummed series for $f(x)$ is given by the following expression:

[TABLE]

For 3D Ising model the constant $\alpha$ was evaluated in a course of the large-order perturbative analysis [31]:

[TABLE]

In principle, parameter $b$ for various universal quantities may be taken from the same source [58]. On the other hand, the five-loop approximation is obviously can not be thought of as lengthy enough to match the Lipatov’s asymptotics. That is why it looks natural to consider $b$ as a fitting parameter which may be used for optimization of the resummation procedure. We mean here the acceleration of the series convergence and, especially, the stability of the numerical estimate under the variation of the fitting parameter.

In Fig. 1 the numerical value of $R_{6}^{*}$ given by conform-Borel resummed series (15) is shown as a function of $b$. As is seen, the region where the universal ratio demonstrates a minimal sensitivity (maximal stability) with respect to $b$ is centered near $b\approx 3.5$. More precisely, the curve $R_{6}^{*}(b)$ has an extremum at $b=3.4434$ which corresponds to $R_{6}^{*}=1.64898$. Thus we adopt the value

[TABLE]

as a final estimate the conformal mapping technique yields. This number is seen to be in a complete agreement with results obtained by 3D RG analysis and advanced lattice calculations.

It is worthy to note that an account for the five-loop terms in the RG expansion and $\tau$-series for $R_{6}^{*}$ shifts the numerical value of this universal ratio only slightly. Indeed, the Padé–Borel–Leroy resummation of the four-loop RG series for $g_{6}$ leads to $R_{6}^{*}=1.648$ [25] while Padé resummed four-loop $\tau$-series for $R_{6}^{*}$ and $g_{6}$ result in $R_{6}^{*}=1.642$ and $R_{6}^{*}=1.654$, respectively [35]. This may be considered as an extra manifestation of the fact that the RG expansion and $\tau$-series for $R_{6}^{*}$ have a structure rather favorable from the numerical point of view. It is especially true for the pseudo-$\epsilon$ expansion (15) which being alternating and having small higher-order coefficients turns out to be very convenient for getting numerical estimates.

As was recently shown [35], the pseudo-$\epsilon$ expansions of $R_{6}^{*}$ for the systems with $n$-vector order parameter (easy-plane and Heisenberg ferromagnets, etc.) have smaller higher-order coefficients than those for the Ising ($n=1$) model. This implies that for $n>1$ the iteration procedure based on the Padé–Borel–Leroy resummation technique should converge faster and give better numerical results than for $n=1$. So, we believe that for $n>1$ the four-loop pseudo-$\epsilon$ expansions [35] will give the numerical estimates of $R_{6}^{*}$ practically as precise as that given by (unknown) five-loop $\tau$-series provided the Padé–Borel–Leroy resummation is made. Here we present such four-loop Padé–Borel–Leroy estimates for $XY$ ($n=2$) and Heisenberg ($n=3$) models, i. e. for the systems most interesting from the physical point of view:

[TABLE]

These numbers are in a good agreement with other field-theoretical and lattice estimates [25, 27, 35, 59, 60].

4 Octic coupling: resummation and numerical estimates

As seen from Eqs. (13) and (16), for the renormalized octic coupling we have pseudo-$\epsilon$ expansions with less favorable structure. The series for $R_{8}^{*}$ being alternating have big higher-order coefficients. To estimate this ratio we’ll apply various resummation procedures – the Padé, Padé–Borel–Leroy and conformal mapping techniques. Higher-order coefficients of the $\tau$-expansion for $g_{8}^{*}$ are much smaller but have irregular signes. This series will be also processed within the techniques mentioned aiming to find the universal value of $R_{8}$ via the relation $R_{8}=g_{8}/g_{4}^{3}$.

Let us start estimating the universal value of the octic coupling from the Padé approximant approach. Padé triangle for pseudo-$\epsilon$ expansion of the ratio $R_{8}^{*}/\tau$, i. e. with the insignificant factor $\tau$ omitted is presented in Table 3. The rate of convergence of Padé estimates to the asymptotic value is also shown at the bottom of this table. As one can easily see the convergence of the Padé estimates for $R_{8}^{*}$ is much less pronounced than in the case of $R_{6}^{*}$. At the same time, the simple method employed gives the asymptotic value $R_{8}^{*}=0.879$ that is in a good agreement with the result of lattice calculations $R_{8}^{*}=0.871\pm 0.014$ [34] and with the number $R_{8}^{*}=0.857\pm 0.086$ given by the 3D RG analysis [16].

The next method of the resummation which we will address is the Padé–Borel–Leroy technique. This technique reduces to the Padé–Borel resummation procedure when one put the fitting parameter $b$ equal to zero. Let us first present the estimates of $R_{8}^{*}$ that this simpler machinery yields. They are collected in Table 4. As is seen, use of the Borel transformation significantly accelerates the convergence of numerical estimates to the asymptotic value, but this value itself – $R_{8}^{*}=0.890$ – turns out to be slightly further from the results of 3D RG analysis and advanced lattice calculations than its Padé counterpart.

Can the situation be improved by making the fitting parameter $b$ active? To clear up this point, we sum up the series (16) by means of the Padé–Borel–Leroy technique using as working three Padé approximants – [3/2], [2/3] and [4/1]. The results are presented at Fig. 2 where corresponding estimates as functions of $b$ are shown.

One can see from this figure that the pseudo-$\epsilon$ expansion (16) has much worse approximating properties than that of the $\tau$-series for $R_{6}^{*}$. Indeed, only one approximant – [3/2] – gives the estimates that are stable with respect to the variation of $b$. They are grouped around the number 0.9 that therefore may be thought of as close to the true value. On the other hand, working approximants [2/3] and [4/1] result in the different estimate: curves $R_{8}^{*}(b)_{[2/3]}$ and $R_{8}^{*}(b)_{[4/1]}$ intersect under $b\approx 1.5$ thus yielding $R_{8}^{*}=0.8$, the value which also looks plausible. Since the numbers just found disagree with each other by more than 10$\%$ none of them can be considered as reliable.

In such situation an alternative resummation procedure should be applied. As before, we use the conformal mapping technique. The results of the resummation of series (16) by means of conform-Borel approach under varying $b$ are presented in Fig. 3.

The curve $R_{8}^{*}(b)_{cB}$ is seen to have a smooth extremum and, correspondingly, a wide enough region where the numerical estimate for $R_{8}^{*}$ is stable with respect to $b$. The value of octic coupling at the extremum $b=6.638$, i. e. at the point of maximal stability is

[TABLE]

As was already noted, the universal ratio $R^{*}_{8}$ may be also found via evaluation of the octic coupling constant $g_{8}$ at criticality and use of the relation $R_{8}=g_{8}/g_{4}^{3}$. Since the coefficients of the pseudo-$\epsilon$ expansion (13) are considerably smaller than those of the $\tau$-series for $R^{*}_{8}$ this way seems to be promising. However, the expansion (13) has rather irregular structure and all attempts to sum up this series with a help of above methods have failed to yield satisfactory results.

So, we accept the number (28) as a final result of our pseudo-$\epsilon$ expansion analysis. It turns out to be close to the value $R_{8}^{*}=0.871\pm 0.014$ extracted from the most recent lattice calculations [34] and compatible with the 3D RG estimate $R_{8}^{*}=0.857\pm 0.086$ [16].

5 Universal ratio $R_{10}$

Comparing original RG expansion (10) with the series (14) and (17) one can see that the application of the pseudo-$\epsilon$ expansion technique certainly improves the structure of the series for $g^{*}_{10}$ and $R^{*}_{10}$: it significantly diminishes the coefficients leaving the series alternating. At the same time, the coefficients of the pseudo-$\epsilon$ expansions remain big and fast growing what makes direct or Padé summation of the series (14) and (17) meaningless. Attempting to arrive to proper numerical estimates we sum up the series (17) with a help of the Padé–Borel–Leroy procedure. The results thus obtained are shown in Fig. 4. As is seen, the values of $R^{*}_{10}$ given by four relevant approximants strongly depend on the shift parameter $b$ and markedly differ from each other. There exist, however, two points on the $b$ axis that may be thought of as corresponding to some meaningful results.

The first one – $b=7.33$ – is the point of consensus of three working approximants, i. e. the point where use of approximants [4/1], [3/2] and [1/4] yields practically the same value. This value $R^{*}_{10}=1.45$, however, is in an obvious disagreement with the numbers obtained by other methods; they are collected in Table 5.

The second point – $b=0.647$ – is the point of maximal stability of the estimate obtained on the base of the near-diagonal approximant [3/2]. Addressing this point leads to $R^{*}_{10}=-1.76$, the value which is compatible with many of the results presented in Table 5. This value, nevertheless, can not be referred to as fair since it is in conflict with the former result $R^{*}_{10}=1.45$ obtained within the same technique. The attempts to get reasonable estimates for $R^{*}_{10}$ using conform-Borel technique or via resummation of the series (14) for the coupling constant $g^{*}_{10}$ itself also turned out to be unsuccessful.

So, we see that although the pseudo-$\epsilon$ expansion machinery is able to transform strongly divergent 3D RG expansions into series with smaller and slower growing coefficients, in the case of $R_{10}$ it is not powerful enough to provide acceptable numerical estimates for its universal value.

6 Conclusion

To summarize, we have calculated pseudo-$\epsilon$ expansions for the universal values of renormalized coupling constants $g_{6}$, $g_{8}$, $g_{10}$ and of the universal ratios $R_{6}$, $R_{8}$, $R_{10}$ for 3D Euclidean scalar $\lambda\phi^{4}$ field theory. Numerical estimates for $R_{6}^{*}$ have been found using Padé, Padé–Borel–Leroy and conform–Borel resummation techniques. The pseudo-$\epsilon$ expansion machinery has been shown to lead to high-precision value of $R_{6}^{*}$ which is in very good agreement with the numbers obtained by means of other methods including advanced lattice calculations. For the octic coupling this technique was shown to be less efficient: numerical estimates extracted from $\tau$-series for $R_{8}^{*}$ by means of the Padé–Borel–Leroy and conform-Borel resummations slightly differ from each other and from their lattice and 3D RG counterparts. Corresponding differences, however, are small, especially between conform-Borel (0.868) and advanced lattice (0.871) estimates, indicating that pseudo-$\epsilon$ expansion technique provides precise enough numerical results for this coupling. Pseudo-$\epsilon$ expansion for the ratio $R_{10}^{*}$ has been also analyzed. It has been shown that the pseudo-$\epsilon$ expansion approach improves the structure of series for $g_{10}^{*}$ and $R_{10}^{*}$ but such an improvement turns out to be insufficient to make the series suitable for getting numerical estimates.

Acknowledgment

We gratefully acknowledge the support of Saint Petersburg State University via Grant 11.38.636.2013, of the Russian Foundation for Basic Research under Project No. 15-02-04687, and of the Dynasty foundation.