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Critical Exponents from Seven-Loop Strong-Coupling $\phi^4$-Theory in Three Dimensions

Hagen Kleinert [*]


Institut für Theoretische Physik,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

Abstract:

Using strong-coupling quantum field theory we calculate highly accurate critical exponents $ \nu , \eta $ following from new seven-loop expansions in three dimensions. Our theoretical value for the critical exponent $ \alpha $ of the specific heat near the $ \lambda $-point of superfluid helium is $ \alpha =-0.01294\pm0.00060$, in excellent agreement with the space shuttle experimental value $ \alpha =-0.01285\pm0.00038$.

1. The accurate calculation of critical exponents from field theory presents a theoretical challenge since the relevant information is available only from divergent power series expansions. The results are also of practical relevance since they predict the the outcome many possible future experiments on many second-order phase transitions. In recent work [#!kl!#] we have developed a novel method for extracting these exponents from such expansions via a strong-coupling theory of scalar fields with a $\phi^4$-interaction. The fields are assumed to have n components with an action which is O(n)-symmetric. As an application, we have used available six-loop perturbation expansions of the renormalization constants in three dimensions [#!1!#,#!3!#,#!anton!#] to calculate the critical exponents for all O(n) universality classes with high precision. Strong-coupling theory works also in $4- \epsilon $ dimensions [#!klep!#], and is capable of interpolating between the expansions in $4- \epsilon $ with those in $2+\epsilon$ dimensions of the nonlinear $ \sigma $-model [#!klepp!#]. The purpose of this note is to improve significantly the accuracy of our earlier results in three dimensions [#!kl!#] by making use of new seven-loop expansion coefficients for the critical exponents $ \nu $ and $ \eta $ [#!MN!#] and, most importantly, by applying a more powerful extrapolation method to infinite order than before. The latter makes our results as accurate as those obtained by Guida and Zinn-Justin [#!GZ!#] via a more sophisticated resummation technique based on analytic mapping and Borel transformations, which in addition takes into account information on the large-order growth of the expansion coefficients. We reach this accuracy without using that information which, as we shall demonstrate at the end in Section 5, has practically no influence on the results, except for lowering $ \omega $ slightly (by less than $\sim 0.2 \%$). The reason for the little importance of the large-order information in our approach is that the critical exponents are obtained from evaluations of expansions at infinite bare couplings. The information on the large-order behavior, on the other hand, specifies the discontinuity at the tip of the left-hand cut which starts at the origin of the complex-coupling constant plane [#!PI!#]. This is too far from the infinite-coupling limit to be of relevance. In our resummation scheme for expansions in powers of the bare coupling constant , an important role is played by the critical exponent of approach to scaling $ \omega $, whose precise calculation by the same scheme is crucial for obtaining high accuracies in all other critical exponents . It is determined by the condition that the renormalized coupling strength g goes against a constant g* in the strong-coupling limit. The knowledge of $ \omega $ is more yielding than the large-order information in previous resummation schemes in which the critical exponents are determined as a function of the renormalized coupling constant g near g* which is of order unity, thus lying a finite distance away from the left-hand cut in the complex g-plane. Although these determinations are sensitive to the discontinuity at the tip of the cut, it must be realized that the influence of the cut is very small due to the smallness of the fugacity of the leading instanton, which carries a Boltzmann factor $e^{-{\rm const}/g}$.
2. We briefly recall the available expansions [#!anton!#] of the renormalized coupling ${\bar g}\equiv g/m$ in terms of the bare coupling $\bar g_0\equiv g_0/m$ for all O(n):

 
$\displaystyle {\bar g}/{\bar g}_0$ = $\displaystyle 1 \!-\! {\bar g_0}\,\left( 8\!+\!n \right) \!+\!
{{\bar g_0}^{2}}\,\left(2108/27 \!+\! 514n/27 \!+\! {n^{2}} \right)$  
    $\displaystyle \!+\!
{{\bar g_0}^{3}}\,\left( \!-\!878.7937193 \!-\! 312.63444671n \!-\! 32.54841303{n^{2}}
\!-\!
{ n^{3}} \right)$  
    $\displaystyle \!+\! {\bar g_0^{4}}\,
\left( 11068.06183 \!+\! 5100.403285n \!+\! 786.3665699{n^{2}} \!+\!
48.21386744{n^{3}} \!+\! {n^{4}} \right)$  
    $\displaystyle \!+\!
{{\bar g_0}^{5}}\,\left( \!-\!153102.85023 \!-\! 85611.9199...
...n^{2}} \!-\!
1585.1141894{n^{3}} \!-\! 65.82036203{n^{4}} \!-\! {n^{5}} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^6}\,\left( 2297647.148\, \!+\! 1495703.313\,n \!+\!
371103.0896{n^2} \!+\! 44914.04818{n^3}
\right.$  
    $\displaystyle \left.~~~~~~~\!+\! 2797.291579{n^4} \!+\!
85.21310501{n^5} \!+\! {n^6} \right),$ (1)

and of the critical exponents [#!not!#]
    
$\displaystyle \!\!\omega ({\bar g}_0)$ = $\displaystyle \!-\!1 \!+\!2 {\bar g_0}\,\left( 8 \!+\! n \right) \!-\!
{{{\bar g_0}}^{2}}\,\left( 1912/9 \!+\!452n/9\!+\!2{n^{2}} \right)$  
    $\displaystyle \!+\! {{{\bar g_0}}^{3}}\,\left( 3398.857964 \!+\! 1140.946693n \!+\!
95.9142896{n^{2}} \!+\! 2{n^{3}} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^{4}}\,\left( \!-\!60977.50127 \!-\! 26020.14956n \!-\!
3352.610678{n^{2}} \!-\! 151.1725764{n^{3}} \!-\! 2{n^{4}} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^{5}}\,\left( 1189133.101 \!+\! 607809.998n \!...
...n^{2}} \!+\! 7450.143951{n^{3}} \!+\! 214.8857494{n^{4}} \!+\!
2{n^{5}} \right)$  
    $\displaystyle \!+\! {{{\bar g_0}}^{6}}\,
\left( \!-\!24790569.76 \!-\! 14625241.87n \!-\!
3119527.967{n^{2}}\right.$  
    $\displaystyle \left.~~~~~~~~~\, \!-\! 304229.0255{n^{3}} \!-\!
14062.53135{n^{4}} \!-\! 286.3003674{n^{5}} \!-\! 2{n^{6}} \right),$ (2)
$\displaystyle \!\!\eta({\bar g})$ = $\displaystyle {{{\bar g_0}}^2}\,\left( 16/27 \!+\! 8n/27 \right) \!+\!
{{{\bar ...
...^3}\,\left( \!-\!9.086537459 \!-\! 5.679085912n \!-\!
0.5679085912{n^2} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^4}\,\left( 127.4916153 \!+\! 94.77320534n \!+\! 17.1347755{n^2} \!+\!
0.8105383221{n^3} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^5}\,\left( \!-\!1843.49199 \!-\! 1576.46676n \!-\! 395.2678358{n^2} \!-\!
36.00660242{n^3} \!-\! 1.026437849{n^4} \right) ,$  
    $\displaystyle \!+\!
{{{\bar g_0}}^6}\,\left( 28108.60398 \!+\! 26995.87962n \!+...
...2} \!+\!
1116.246863{n^3} \!+\! 62.8879068{n^4} \!+\! 1.218861532{n^5} \right),$ (3)
$\displaystyle \!\!\eta_m({\bar g})$ = $\displaystyle {\bar g_0}\,\left( 2 \!+\! n \right) \!+\!
{{{\bar g_0}}^2}\,\left( \!-\!523/27 \!-\! 316n/27 \!-\! {n^2} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^3}\,\left( 229.3744544 \!+\! 162.8474234n \!+\! 26.08009809{n^2}
\!+\!
{n^3} \right)$  
    $\displaystyle \!+\! {{{\bar g_0}}^4}\,
\left( \!-\!3090.996037 \!-\! 2520.848751n \!-\! 572.3282893{n^2} \!-\!
44.32646141{n^3} \!-\!{n^4} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^5}\,\left( 45970.71839 \!+\! 42170.32707n \!+...
...52.70675{n^2} \!+\!
1408.064008{n^3} \!+\! 65.97630108{n^4} \!+\! {n^5} \right)$  
    $\displaystyle \!+\!
{{{\bar g_0}}^6}\,\left( \!-\!740843.1985 \!-\! 751333.064n...
...575.57037{n^3} \!-\! 2842.8966{n^4} \!-\! 90.7145582{n^5} \!-\! {n^6} \right)
,$ (4)

where $\eta_m\equiv2- \nu ^{-1} $. To save space we have omitted a factor 1/(n+8)n accompanying each power $\bar g_0^n$ on the right-hand sides. The additional seventh-order coefficients have been calculated for $n=0,\,1,\,2,\,3$ and are [#!MN!#] (these without a factor 1/(n+8)7 on the right-hand side)
 
$\displaystyle \eta^{(7)}
=\left\{
\begin{array}{r}
- 0.2164239372\\
- 0.239546...
...~~~~
\left\{ \begin{array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}~%
\right\} .$     (5)

3. It is instructive to see how close the new coefficients are to their large-order limiting values derived from instanton calculations, according to which the expansion coefficients with respect to the renormalized coupling $\bar g$ should grow for large order k as follows [#!parisi!#]:
    
$\displaystyle \omega ^{(k)}$ = $\displaystyle \gamma_ \omega (-a)^k k! k \Gamma(k+b_ \omega
)
\left(1+\frac{c_ \beta ^{(1)}}{k} +\frac{c_ \beta ^{(2)}}{k^2}+\dots\right),$ (6)
$\displaystyle \eta ^{(k)}$ = $\displaystyle \gamma _\eta (-a)^k k! k \Gamma(k+b_\eta)
\left(1+\frac{c_ \eta ^{(1)}}{k} +\frac{c_ \eta ^{(2)}}{k^2}+\dots\right),$ (7)
$\displaystyle \bar{\eta}^{(k)}$ = $\displaystyle \gamma _{\bar{\eta}} (-a)^k
k! k \Gamma (k+b_{ \bar{\eta}})
\left...
...c{c_ {\bar{\eta}} ^{(1)}}{k} +\frac{c_ { \bar{\eta}} ^{(2)}}{k^2}+\dots\right).$ (8)

where $ \bar{\eta}\equiv \eta + \nu ^{-1}-2$. The growth parameter a proportional to the inverse euclidean action of the classical instanton solution $\varphi_c ({\bf x})$ to the field equations
$\displaystyle %
a =
(D-1)\frac{16\pi}{I_4}\frac{1}{N+8}=0.14777423 \frac{9}{N+8}.$     (9)

The quantity I4 denotes the integral $I_4= \int d^D x [\varphi_c ({\bf x})]^4$. Its numerical values in two and three dimensions D are listed in Table I. The growth parameters $b_\beta,b_ \eta , b_{ \bar{\eta}} $, are directly related to the number of zero-modes in the fluctuation determinant around the instanton, which is D+N (associated with D translations, n-1 rotations, and one dilation). Their values are
$\displaystyle %
b_ \omega =b_\beta+1 = \frac{1}{2} (D+5+n) , ~~~
b _\eta = \frac{1}{2} (D+1+n),~~~
b _{\bar{\eta}} = \frac{1}{2} (D+3+n)$     (10)

The prefactors $\gamma_ \beta ,\gamma_ \eta , \gamma_{ \bar{\eta}} $ in (6)-(8) require the calculation of the full fluctuation determinants. This yields
$\displaystyle %
\gamma_ \beta$ $\textstyle \equiv$ $\displaystyle \frac{(n+8)2^{(n+D-5)/2} 3^{-3(D-2)/2}}
{\pi ^{3+D/2}\Gamma \left...
...) ^2
\left( \frac{I_6}{I_4}-1\right) ^{D/2}
D_L^{-1/2} D_T^{-(n-1)/2} e^{-1/a}.$ (11)

The constants I1, I2, I6 are generalizations of the above integral I4: $I_p= \int d^D x [\varphi_c ({\bf x})]^p$, and DL and DT are found from the longitudinal and transverse parts of the fluctuation determinants. Their numerical values are given in Table I. The constant $ \gamma _ \beta $ is the prefactor of growth in the expansion coefficients of the $ \beta $-function (the integral over $ \omega (\bar g))$: $ \beta ^{(k)}\approx
\gamma_ \beta (-a)^k k! \Gamma(k+b_ \beta)
$. The prefactors in $\gamma_ \omega $, $\gamma_ \eta$, and $ \gamma_{ \bar{\eta}} $ in (6)-(8) are related to $ \gamma _ \beta $ by
$\displaystyle \gamma _ \omega =-a \gamma _{ \beta },~~~~ ~
\gamma _\eta = \gamm...
...ma _{\bar{\eta}} = \gamma _ \beta \frac{n+2}{n+8} (D-1)
4\pi \frac{I_2}{I_1^2}.$     (12)

where I2 = (1-D/4)I4 and H3 are listed in Table I. The numerical values of all growth parameters for are listed in Table II. In Figs. 1 we show a comparison between the exact coefficients and their asymptotic forms (8).
  
Figure: Precocity of large-order behavior of coefficients of the expansions of the critical exponents $ \omega $, $ \bar \eta\equiv \nu^{-1}+ \eta -2 $, and $ \eta $ in powers of the renormalized coupling constant. The dots show the relative deviations exact/asymptotic-1. The curves are plots of the asymptotic expressions in Eqs. (6)-(8) listed in Table III. The curve for $ \omega $ is the smoothest, promising the best extrapolation to the next orders, with consequences to be discussed in Section 5.
\begin{figure}
~\\
~~~~~~%
\special{! /image { 3 -1 roll pop false 3 1 roll i...
...$\eta ~~~n=3$ }
\put(430,21){\scriptsize$k$ }
\end{picture}
\\
\end{figure}

 
4. The critical exponents are derived from the divergent expansions (1)-(5) by going to the limit $\bar g_0\rightarrow \infty$. In a theory with scaling behavior, the renormalized coupling constant $\bar g$ tends to a limiting value $\bar g^*$ as follows:

 \begin{displaymath}\bar g(\bar g_0)=\bar g^*-\frac{{\rm const}}{\bar g_0^{ \omega/ \epsilon}}+\dots~,
\end{displaymath} (13)

where g* is commonly referred to as the infrared-stable fixed point, and $ \omega $ is called the critical exponent of the approach to scaling. The same exponent governs the approach to scaling of every function $G(\bar g)$ which behaves like $G(\bar g)=G(\bar g^*)+G'(\bar g^*)\times {{\rm const}}/{\bar g_0^{ \omega}}+\dots~.$ How do we recover the $\bar g_0\rightarrow \infty$ -limits of a function $f(\bar g_0)$ if we know the first N terms of its asymptotic expansion $ f_N(\bar g_0)=\sum_{n=0}^N a_n \bar g_0 ^n$? Extending systematically the behavior (13) we shall assume that $f(\bar g_0)$ approaches its constant limiting value f* in the form of an inverse power series [#!confl!#] $f_M(\bar g_0)= \sum_{m=0}^M b_m (\bar g_0 ^{- \omega }) ^m$. This strong-coupling expansion has usually a finite convergence radius gs (see [#!kl!#,#!PI!#,#!int!#]). The Nth approximation to the value f* is obtained from the formula
 
    $\displaystyle f_N^* =\mathop{\rm opt}_{\hat{g}_0}\left[
\sum_{j=0}^N a_{j}^{\rm...
...\left( \begin{array}{c}
- j/ \omega \\  k
\end{array}\right)
(-1)^{k} \right] ,$ (14)

where the expression in brackets has to be optimized in the variational parameter $\hat g_0$. The optimum is the smoothest among all real extrema. If there are no such extrema, which happens for the even approximants, the turning points serve the same purpose. From the theory [#!kl!#], we expect the exact values to be approached exponentially fast with the order N of the available expansions, with the error decreasing like $e^{-cN^{1- \omega }}$. In order to extrapolate our results to $N=\infty$, we plot the data against the variables $x_N=e^{-cN^{1- \omega }}$ (see also Addendum to Ref. [#!kl!#]). This is done separately for even and odd approximants, since the former stem from extrema, the latter from turning points. The unknown constants c c are determined by fitting to each set of points a slightly curved parabola and making them intersect the vertical axis at the same point, which yields the extrapolated critical exponent listed on top of each figure (together with the seventh-order value in parentheses, and the optimal parameter c).
  
Figure: Strong-coupling values for the critical exponent $ \nu ^{-1}$ obtained from expansion (4) via formula (14), for increasing orders $N=2,3,\dots,7$ of the approximation. The exponents are plotted against the variable $x_N=e^{-cN^{1- \omega }}$ and should for large n lie on a straight line. Here at finite N, even and odd approximants may be connected by slightly curved parabolas whose common intersection with the vartical exis determines the critical exponents for $N=\infty$. More details on the determination of the constant c are given in the text. The numbers on top give the extrapolated critical exponents and, in parentheses, the highest approximants, to illustrate the extrapolation distance.
\begin{figure}
~\\
\special{! /image { 3 -1 roll pop false 3 1 roll imagemask...
...=3$ }
\end{picture}
~\\ ~\\ ~\\ ~\\ ~\\ ~\\ ~\\ [-3mm]
~\\ ~\\
\end{figure}

Following this procedure, we find from the expansion (4) for $ \nu ^{-1}$ the approximants $ \nu_N ^{-1}$ via formula (14). Extrapolating separately even and odd approximants $ \nu _N$, we determine the limiting value $ \nu $, as shown in Fig. 2. The $ \omega $-values used for this extrapolation are those of Ref. [#!klep!#], listed in the last colum of Table IV:

 \begin{displaymath}\omega_6
=\left\{
\begin{array}{r}
0.810\\ 0.805\\ 0.797\...
...ray}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\}.~~ %
\end{displaymath} (15)

They lead to the $ \nu $-values $\nu_7=\left\{
0.5883,
0.6305,
0.6710,
0.7075\right\} $, the entries in this vector referring to n=0,1,2,3. Since these results depend on the critical exponents $ \omega $, it is useful to study the dependence of the extrapolation on $ \omega $, with the result

 \begin{displaymath}\nu_7
=\left\{
\begin{array}{r}
0.5883+0.0417\times ( \om...
...array}{l}
N=0\\
N=1\\
N=2\\
N=3
\end{array}\right\} .%
\end{displaymath} (16)

For the critical exponent $ \eta $ we cannot use the same extrapolation procedure since the expansion (3) starts out with $\bar g_0^2$, so that there exists only an odd number of approximants $ \eta _N$. We therefore use two alternative extrapolation procedures. In the first we connect the even approximants $ \eta _2$ and $ \eta _4$ by a straight line and the odd ones $ \eta _3, \eta _5, \eta _7$ by a slightly curved parabola, and vary c until there is an intersection at x=0. This yields the critical exponents $ \eta $ shown in Figs. 3.
  
Figure: Strong-coupling values for the critical exponent $ \eta $ obtained from the expansion (3) via formula (14). for increasing orders $N=3,\dots,7$ of the approximation. The exponents are plotted against $x_N=e^{-cN^{1- \omega }}$. Even approximants are connected by straight line and odd approximats by slightly curved parabolas, whose common intersection determines the critical exponents expected for $N=\infty$.
\begin{figure}
~\\ [-4mm] %
\special{! /image { 3 -1 roll pop false 3 1 roll im...
...=3$ }
\end{picture}
~\\ ~\\ ~\\ ~\\ ~\\ ~\\ ~\\ [-3mm]
~\\ ~\\
\end{figure}

Allowing for the inaccurate knowledge of $ \omega $, the results may be stated as

 \begin{displaymath}\eta_7
=\left\{
\begin{array}{l}
0.03215+0.1327\times ( \...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} ,%
\end{displaymath} (17)

Alternatively, we connect the last odd approximants $ \eta _5$ and $\eta _7$ also by a straight line and choose c to make the lines intersect at x=0. This yields the exponents

 \begin{displaymath}\eta _7
=\left\{
\begin{array}{l}
0.03010+0.08760\times (...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} ,%
\end{displaymath} (18)

as shown in Figs. 4, the $ \omega $ dependences being somewhat weaker than in (17).
  
Figure: Plot analogous to Fig. 3 but the extrapolation is found from the intersection of the straight lines connecting the last two even and odd approximants. The resulting critical exponents differ only little from those obtained in Fig. 3, the differences given an estimate for the systematic error of our results.
\begin{figure}
~\\ %
\special{! /image { 3 -1 roll pop false 3 1 roll imagemask...
...=3$ }
\end{picture}
~\\ ~\\ ~\\ ~\\ ~\\ ~\\ ~\\ [-3mm]
~\\ ~\\
\end{figure}

Combining the two results and using the difference to estimate the systematic error of the extrapolation procedure, we obtain for $ \eta $ the values

 \begin{displaymath}\eta_7
=\left\{
\begin{array}{l}
0.0311\pm0.001\\
0.034...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} ,%
\end{displaymath} (19)

whose $ \omega $-dependence is the average of that in (17) and (18). For our extrapolation procedure, the power series for the critical exponent $ \gamma = \nu (2- \eta )$ are actually better suited than those for $ \eta $, since they possess three even and three odd approximants, just as $ \nu ^{-1}$. Advantages of this expansion have been observed before [#!remark!#].
  
Figure: Strong-coupling values for the critical exponent $ \gamma = \nu (2-\eta)=(2- \eta )/(2- \eta _m)$ obtained from a combination of the expansions (3) and (4) via formula (14). for increasing orders $N=2,3,\dots,7$ of the approximation. The exponents are plotted against the variable $x_N=e^{-cN^{1- \omega }}$ and should lie on a straight line in the limit of large N. Even and odd approximants are connected by slightly curved parabolas whose common intersection determines the critical exponents expected for $N=\infty$. The determination of the constant c is described in the text.
\begin{figure}
~\\ %
\special{! /image { 3 -1 roll pop false 3 1 roll imagemask...
...=3$ }
\end{picture}
~\\ ~\\ ~\\ ~\\ ~\\ ~\\ ~\\ [-3mm]
~\\ ~\\
\end{figure}

The associated plots are shown in Fig. 5. The extrapolated exponents are, including the $ \omega $-dependence,

 \begin{displaymath}\gamma_7
=\left\{
\begin{array}{r}
1.161-0.049\times ( \o...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} .%
\end{displaymath} (20)

Unfortunately, the exponent $ \gamma = \nu (2- \eta )$ is not very insensitive to $ \eta $ since this is small compared to 2, so that the extrapolation results (17) are more reliable than those obtained from $ \gamma $ via the scaling relation $\eta=2- \gamma / \nu $. By combining (16) and (17), we find from $ \gamma = \nu (2- \eta )$:

 \begin{displaymath}\gamma_7
=\left\{
\begin{array}{r}
1.1589\\
1.2403\\
...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} ,%
\end{displaymath} (21)

the difference with respect to (20) showing the typical small errors of our approximation, which are of the same order as those of the exponents obtained in Ref. [#!GZ!#]. As mentioned in the beginning, the knowledge of the large-order behavior does not help to improve significantly the accuracy of the approximation. In our theory, the most important exploited information is the knowledge of the exponentially fast convergence which leads to a linear behavior of the resummation results of order N in a plot against $x_N=e^{-cN^{1- \omega }}$. This knowledge, which allows us to extrapolate our approximations for N=2,3,4,5,6,7 quite well to infinite order N, seems to be more powerful than the knowledge of the large-order behavior exploited by other authors (quoted in Table IV). The complete updated list of exponents is shown in Table IV, which also contains values for the other critical exponents $ \alpha =2- D \nu $ and $ \beta = \nu (D-2+ \eta )/2$.  
5. Let us now show that the large-order information is indeed rather irrelevant to the critical exponents within strong-coupling theory. For this purpose we choose the coefficients c(i) in the asymptotic formulas (6)-(8) to fit exactly the six known expansion coefficients of $ \omega (\bar g)$ and the seven of $ \bar{\eta} (\bar g)$ and $ \eta (\bar g)$. The coefficients are listed in Table III, and the associated fits are shown in Figs. 1. Since even and odd coefficients $ \eta ^{(k)} $ lie on twoseparate smooth curves, we fit the two sets separately. These fits permit us to extend the presently available coefficients and predict the results of future higher-loop calculations, listed in Table V up tp order 25. The errors in these predictions are expected to be smallest for $ \omega ^{(k)}$, as illustrated in Figs. 6. At this place we observe an interesting phenomenon: According to Table V, the expansion coefficients $ \omega ^{(k)}$ of $ \omega (\bar g)$ have alternating signs and grow rapidly, reaching precociously their asymptotic form (6), as we have seen in Figs. 1. Now, from $ \omega (\bar g)$ we can derive the so called $ \beta $-function $ \beta (\bar g)\equiv \int d\bar g\,\omega (\bar g) $, and from this the expansion for the bare coupling constant $\bar g_0(\bar g)=- \int d\bar g/ \beta (\bar g)$, with coefficients $\bar g_{0}^{(k)}$ listed in Table VI. From the standard instanton analysis [#!PI!#], we know that the function $\bar g_{0}(\bar g)$ has the same left-hand cut in the complex $\bar g$-plane as the functions $ \omega (\bar g),\bar \eta(\bar g), \eta (\bar g)$, with the same discontinuity proportional to $e^{-{\rm const}/g}$ at the tip of the cut. Hence the coefficients $\bar g_{0}^{(k)}$ must have asymptotically a similar alternating signs and a factorial growth. Surprisingly, this expectation is not borne out by the explicit seven-loop coefficients $\bar g_{0}^{(k)}$ following from (6) in Table VI. If we, however, look at the higher-order coefficients derived from the extrapolated $ \omega ^{(k)}$ sequence which are also listed in that table, we see that sign change and factorial growth do eventually set in at the rather high order 11. Before this order, the coefficients $\bar g_{0}^{(k)}$ look like those of a convergent series. Thus, if we would make a plot analogous to those in Fig. 1 for $\bar g_{0}^{(k)}$, we would observe huge deviations from the asymptotic form up to an order much larger than 10. In contrast, the inverse series $\bar g(\bar g_{0})$ has expansion coefficients $\bar g_{k}$ which do approach rapidly their asymptotic form, as seen in Table VII. This is the reason why our resummation of the critical exponents $ \omega ,\bar \eta , \eta$ as power series in $\bar g_0$ yields good results already at the available rather low order seven. Given the extrapolated list of expansion coefficients in Table V, we may wonder how much these change the seven-loop results. In Figs. 7 we show the results. The known six-loops coefficients of $ \omega(\bar g_0) $ and $ \eta(\bar g_0) $ were extended by one extrapolated coefficient, since this produces an even number of approximants which can be most easily extrapolated to infinite order. For $\bar \eta (\bar g_0)$ we use two more coefficients for the same reason. The extrapolations are shown in Figs. (7). The resulting $ \omega _8$-values are lowered somewhat with respect to $ \omega _6$ from (15) to

 \begin{displaymath}\omega_8
=\left\{
\begin{array}{r}
0.7935\\ 0.7916\\ 0.79...
...ray}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\}.~~ %
\end{displaymath} (22)


  
Figure: Relative errors in predicting the kth expansion coefficient by fitting the strong-coupling expansions (6)-(8) for $ \omega $, $ \bar \eta\equiv \nu^{-1}+ \eta -2 $, and $ \eta $ to the first k-1 expansion coefficients.
\begin{figure}
~\\
~~~~~~%
\special{! /image { 3 -1 roll pop false 3 1 roll i...
...$\eta ~~~n=3$ }
\put(430,60){\scriptsize$k$ }
\end{picture}
\\
\end{figure}

The new $ \eta $ values are

 \begin{displaymath}\eta _8
=\left\{
\begin{array}{r}
0.02829-0.01675\times (...
...array}{l}
n=0\\
n=1\\
n=2\\
n=3
\end{array}\right\} ,
\end{displaymath} (23)

lying reasonably close to the previous seven-loop results (17), (18) for the smaller $ \omega $-values (22). The first set yields $ \eta_8
=\left\{
0.0300, 0.0356, 0.0360,0.0354\right\} ,
$ the second $ \eta_8
=\left\{
0.0315 0.0342, 0.0349, .0345\right\} $. For $\bar \eta $ we find the results

 \begin{displaymath}\bar \eta _9
=\left\{
\begin{array}{r}
-0.2711+0.0400\tim...
...array}{l}
N=0\\
N=1\\
N=2\\
N=3
\end{array}\right\} .%
\end{displaymath} (24)

It is interesting to observe how the resummed values $ \omega _N,\bar \eta _N, \eta _N$ obtained from the extrapolated expansion coefficients in Table V continue to higher orders in N This is shown in Figs. 8. The dots converge against some specific values which, however, are different from the extrapolation results in Figs. 7 based on the theoretical convergence behavior error $\approx e^{-cN^{1- \omega }}$. We shall argue below that these results are worse than the properly extrapolated values.
  
Figure: Extrapolation of resummed $ \omega ,\bar \eta , \eta$-values if one $( \omega , \eta )$ or two $(\bar \eta )$ more expansion coefficients of Table V are taken into account. The fat dots show the resummed values used for extrapolation, the small dots indicate higher resummed values not used for the extrapolation. The numbers on top specify the extrapolated values and the values of the last approximation, corresponding to the leftmost fat dot.
\begin{figure}
~\\
~~~~~~%
\special{! /image { 3 -1 roll pop false 3 1 roll i...
...e$\eta ~~~n=3$ }
\put(430,61){\scriptsize$k$ }
\end{picture}
\\
\end{figure}


  
Figure: Direct plots of the resummed $ \omega ,\bar \eta , \eta$-values for all resummed values from all extrapolated expansion coefficients of Table V. The line is fitted to the maximum of all dots at the place specified by the number on top. Fat and small dots distinguish the resummed exponents used in the previous extrapolations from the unused ones.
\begin{figure}
~\\
~~~~~~%
\special{! /image { 3 -1 roll pop false 3 1 roll i...
...ze$\eta ~~~n=3$ }
\put(430,2){\scriptsize$k$ }
\end{picture}
\\
\end{figure}

All the above numbers agree reasonably well with each other and with other estimates in the literature listed in Table IV. The only comparison with experiment which is sensitive enough to judge the accuracy of the results and the reliability of the resummation procedure is provided by the measurement of $ \nu $ for n=2, where the critical exponent $ \alpha =2-3 \nu$ has been extracted from the singularity $C\propto \vert 1-T/T_c\vert^{- \alpha }$ in the specific heat at the $ \lambda $-point of superfluid helium with high accuracy [#!ahl!#]:

 \begin{displaymath}\alpha =-0.01285\pm0.00038.
\end{displaymath} (25)

Since $ \nu $ is of the order 2/3, this measurement is extremely sensitive to $ \nu $. It is therefore useful do the resummations and extrapolations for N=2 directly for the approximate $ \alpha $-values $ \alpha_N =2-3 \nu_N$, once for the six-loop $ \omega $-value $ \omega =0.8$, and once for a neighboring value $ \omega= 0.790$, to see the $ \omega $-dependence. The results are shown in Figs. 9. The extrapolated values for our $ \omega =0.8$ in Table (IV) yield

 \begin{displaymath}\alpha =-0.01294\pm0.00060,
\end{displaymath} (26)

in very good agreement with experiment. The extrapolated expansion coefficients for orders larger than 11 do not carry significant information on the critical exponent $ \nu $. The fact that the extrapolated expansion coefficients should lie rather close to the true ones as expected from the decreasing errors in the plots in Fig. (6) does not imply the usefulness of the new coefficients in Table V for obtaining better critical exponents. The errors are only relatively small with respect to the huge expansion coefficients. The resummation procedure removes the factorial growth and becomes extremely sensitive to very small deviations from thes huge coefficients. This is the numerical consequence of the fact discussed earlier that the information residing in the exponentially small imaginary part of all critical exponents near the tip of the left-hand cut in the complex $\bar g_0$-plane has practically no effect upon the strong-coupling results at infinite $\bar g_0$. Note also that the critical exponents which one would obtain from a resummation of the extrapolated expansion coefficients of high order in Table V and their naive extrapolation performed in Figs. 8 yield a slightly worse result for $ \alpha $ in superfluid helium. Indeed, inserting $\bar \eta =-0.47366$ and $ \eta =0.0331$ into the scaling relation $ \alpha =2-3/(2+\bar \eta - \eta )$ we obtain $ \alpha =-0.0091$, which differs by about 25% from the experimental $ \alpha $.
\begin{figure}
~\\
~~~~~~%
\special{! /image { 3 -1 roll pop false 3 1 roll i...
....7900$ }
\put(430,65){\scriptsize$k$ }
\end{picture}
\\
~~~~~~%
\end{figure}
[tbhp]  
      
\begin{picture}(48.5,92.75)
\put(-15,-25){\renewcommand {\epsfsize}[2]{0.45...
...,50){\scriptsize$ \omega =0.7900$ }
\put(430,65){\scriptsize$k$ }
\end{picture}

      

 
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Hagen Kleinert
1999-05-31