Theory and satellite experiment for critical exponent α of λ-transition in superfluid helium

On the basis of recent seven-loop perturbation expansion for ν−1 = 3/(2 − α) we perform a careful reinvestigation of the critical exponent α governing the power behavior |Tc −T |−α of the specific heat of superfluid helium near the phase transition. With the help of variational strong-coupling theory, we find α = −0.01126 ± 0.0010, in very good agreement with the space shuttle experimental value α =−0.01056± 0.00038.  2000 Elsevier Science B.V. All rights reserved. 1. The critical exponent α characterizing the power behavior |Tc − T |−α of the specific heat of superfluid helium near the transition temperature Tc is presently the best-measured critical exponent of all. A microgravity experiment in the Space Shuttle in October 1992 rendered a value with amazing precision [1]: (1) αss =−0.01056± 0.00038. This represents a considerable change and improvement of the experimental number found a long time ago on earth by Ahlers [2]: (2) α =−0.026± 0.004, in which the sharp peak of the specific heat was broadened to 10−6 K by the tiny pressure difference between top and bottom of the sample. In space, the temperature could be brought to within 10−8 K close to Tc without seeing this broadening. E-mail address: [email protected] (H. Kleinert). 1 Tel./fax: 0049-30-8383034, URL: http://www.physik.fu-berlin.de/∼kleinert. The exponent α is extremely sensitive to the precise value of the critical exponent ν which determines the growth of the coherence length when approaching the critical temperature, ξ ∝ |T − Tc|−ν . Since ν lies very close to 2/3, and α is related to ν by the scaling relation α = 2− 3ν, a tiny change of ν produces a large relative change of α. Ahlers’ value was for many years an embarrassment to quantum field theorists who never could find α quite as negative — the field theoretic ν-value came usually out smaller than νAhl = 0.6753± 0.0013. The space shuttle measurement was therefore extremely welcome, since it comes much closer to previous theoretical values. In fact, it turned out to agree extremely well with the most recent theoretical determination of α by strong-coupling perturbation theory [3] based on the recent seven-loop power series expansions of ν [4], which gave [5] (3) αsc =−0.0129± 0.0006. The purpose of this note is to present yet another resummation of the perturbation expansion for ν−1 and for α = 2− 3ν by variational perturbation theory applied in a different way than in [5]. Since it is 0375-9601/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S0375-9601(00) 00 68 85 206 H. Kleinert / Physics Letters A 277 (2000) 205–211 a priori unclear which of the two results should be more accurate, we combine them to the slightly less negative average value with a larger error (4) αsc =−0.01126± 0.0010. Before entering the more technical part of the paper, a few comments are necessary on the reliability of error estimates for any theoretical result of this kind. They can certainly be trusted no more than the experimental numbers. Great care went into the analysis of Ahlers’ data [2]. Still, his final result (2) does not accommodate the space shuttle value (1). The same surprise may happen to theoretical results and their error limits in papers on resummation of divergent perturbation expansions, since there exists so far no safe way of determining the errors. The expansions in powers of the coupling constant g are strongly divergent, and one knows accurately only the first seven coefficients, plus the leading growth behavior for large orders k like γ (−a)kk!k0(k + b). The parameter b is determined by the number of zero modes in a solution to a classical field equation, a is the inverse energy of this solution, and γ the entropy of its small oscillations. The shortness of the available expansions and their divergence make estimates of the error range of the result a rather subjective procedure. All publications resumming critical exponents such as α calculate some sequences of N th-order resummed approximations αN , and estimate an error range from the way these tend to their limiting value. While these estimates may be statistically significant, there are unknown systematic errors. Otherwise one should be able to take the expansion for any function f̃ (g) ≡ f (α(g)) and find a limiting number f (α) which lies in the corresponding range of values. This is unfortunately not true in general. Such reexpansions can approach their limiting values in many different ways, and it is not clear which yields the most reliable result. One must therefore seek as much additional information on the series as possible. One such additional information becomes available by resumming the expansions in powers of the bare coupling constant g0 rather than the renormalized one g. The reason is that any function of the bare coupling constant f (g0) which has a finite critical limit approaches this limit with a nonleading inverse power of g 0 , where ω is called the critical exponent of approach to scaling, whose size is known to be about 0.8 for superfluid helium. Any resummation method which naturally incorporates his power behavior should converge faster than those which ignore it. This incorporation is precisely the virtue of variational perturbation theory, which we have therefore chosen for the resummation of α. For a second additional information we take advantage of our theoretical knowledge on the general form of the large-order behavior of the expansion coefficients: (5) γ (−a)kk!k0(k + b) ( 1+ c (1) k + c (2) k2 + · · · )