Institut für Theoretische Physik,

Freie Universität Berlin, Arnimallee 14,
14195 Berlin, Germany

Using strong-coupling quantum field theory
we calculate highly accurate critical exponents
following from new seven-loop expansions
in three dimensions. Our theoretical
value for the critical exponent
of the specific heat
near the -point
of superfluid helium is
,
in excellent agreement
with the space shuttle experimental value
.

**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 -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
dimensions [#!klep!#],
and is capable of interpolating
between the expansions
in
with those in
dimensions of the nonlinear -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
and
[#!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
slightly (by less than
).
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 ,
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
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
.

**2.**
We briefly recall the
available
expansions [#!anton!#] of the renormalized coupling
in terms of the bare coupling
for all O(*n*):

and of the critical exponents [#!not!#]

where . To save space we have omitted a factor 1/(

where . The growth parameter

(9) |

The quantity

(10) |

The prefactors in (6)-(8) require the calculation of the full fluctuation determinants. This yields

(11) |

The constants

(12) |

where

where

where the expression in brackets has to be optimized in the variational parameter . 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

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

They lead to the -values , the entries in this vector referring to

For the critical exponent we cannot use the same extrapolation procedure since the expansion (3) starts out with , so that there exists only an odd number of approximants . We therefore use two alternative extrapolation procedures. In the first we connect the even approximants and by a straight line and the odd ones by a slightly curved parabola, and vary

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

Alternatively, we connect the last odd approximants and also by a straight line and choose

as shown in Figs. 4, the dependences being somewhat weaker than in (17).

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

whose -dependence is the average of that in (17) and (18). For our extrapolation procedure, the power series for the critical exponent are actually better suited than those for , since they possess three even and three odd approximants, just as . Advantages of this expansion have been observed before [#!remark!#].

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

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

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

The new values are

lying reasonably close to the previous seven-loop results (17), (18) for the smaller -values (22). The first set yields the second . For we find the results

It is interesting to observe how the resummed values obtained from the extrapolated expansion coefficients in Table V continue to higher orders in

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 for

Since is of the order 2/3, this measurement is extremely sensitive to . It is therefore useful do the resummations and extrapolations for

in very good agreement with experiment. The extrapolated expansion coefficients for orders larger than 11 do not carry significant information on the critical exponent . 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 -plane has practically no effect upon the strong-coupling results at infinite . 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 in superfluid helium. Indeed, inserting and into the scaling relation we obtain , which differs by about 25% from the experimental .

[tbhp]

1999-05-31