The $\lambda$-point anomaly in view of the seven-loop Hypergeometric resummation for the critical exponent $\nu$ of the $O(2)$ $\phi^4$ model

In this work, we use a specific parameterization of the hypergeometric approximants ( the one by Mera et.al in Phys. Rev. Let. 115, 143001 (2015)) to approximate the seven-loop critical exponent $\nu$ for the $O(2)$-symmetric $\phi^4$ model. Our prediction gives the result $\nu=0.6711(7)$ which is compatible with the value $\nu=0.6709(1)$ from the famous experiment carried on the space shuttle Columbia. On the other hand, our result is also compatible with recent precise theoretical predictions that are excluding the experimental result. These theoretical results include non-perturbative renormalization group calculations ( $\nu=0.6716(6)$), the most precise result from Monte Carlo simulations ($\nu=0.67169(7)$) as well as the recent conformal bootstrap calculations ($\nu=0.67175(10)$). Although our result is compatible with experiment, the plot of renormalization group result versus the number of loops suggests that higher orders are expected to add significantly to accuracy and precision of the $\nu$ exponent in a way that may favor the theoretical predictions.


Slava Rychkov wrote a nice commentary [1] on the recent conformal bootstrap (CB)
prediction for the critical exponent ν of the O(2) model [2] which can describe the 4 He superfluid phase transition. The CB work in Ref. [2] confirmed the most precise result of Monte Carlo (MC) simulations in Ref. [3] but excluded the experimental result in Ref. [4].
These facts have been summarized by Slava Rychkov where he outlined the current status of the predictions of both theoretical and experimental results. The CB and MC results, in conjunction with the recent result from non-perturbative renormalization group (NPRG) [5], have asserted what can be called the λ-point dispute between theory and experiment which lasts for a decade. Resummation of the perturbation series for the associated renormalization group (RG) function, on the other hand, so far is not precise enough to favor either experiment or the mentioned non-perturbative theoretical calculations.
The λ-shape behavior describing the change in specific heat vs. temperature for helium superfluid transition is characterized by the critical exponent α. In fact, the specific heat critical exponent α is related to the exponent ν by the hyper scaling relation α = 2 − 3ν. Using this relation, the critical exponent ν for the 4 He super-fluid transition can be extracted from the α value in Ref. [4] as ν = 0.6709(1). As mentioned above, this result is in contradiction with both recent Monte Carlo (MC) simulation result of ν = 0.67169(7) [3] and bootstrap calculations of ν = 0.67175 (10) [1,2]. Note that in Refs. [5,23], it has been stated that the recent NPRG result (ν = 0.6716(6)) also excludes the experimental result.
In other words, while MC, CB and NPRG predictions are compatible with each other, they exclude the result of the famous experiment by Lipa et. al. in Ref. [4] that leads to the above result for ν (all these results and others are listed in Fig.1).
The best known result from resummation of renormalization group functions at fixed (15) [6]. It is clear that this result is not precise enough and has not been improved since its appearance in 1998 as explained by Slava Rychkov in Ref. [1]. Also, the resummation of the ε-expansion of RG functions is known to have slower convergence than the resummation of RG functions at fixed dimensions [7]. This may explain the significant difference (see Fig.1) between theoretical (MC,CB,NPRG) as well as experimental results and the recent six-loop (ε-expansion) resummation result ( ν = 0.6690(10) ) [8]. In fact, although the fixed dimension and the ε-expansion RG results have uncertainties of the same order of magnitude, one can realize that the experimental as well as MC and CB results exclude the prediction of the six-loop ε-expansion in Ref. [8]. In In this graph, we show the predictions form seven-loop resummation (this work), the recent Monte Carlo prediction (MC) from Ref. [3], conformal bootstrap result [2], non-perturbative renormalization group (NPRG) [5], experimental result from Ref. [4], six-loop resummation from Ref. [8] and the five-loop result from Ref. [6].
this work, we show that (unlike the six-loop) the seven-loop ε-expansion for the exponent ν of the O(2)-model gives precise result that enables it to play a role in the λ-point dispute. To do that, we use a modified parametrization of the hypergeometric resummation algorithm [9].
The hypergeometric approximants suggested by Mera et.al in Ref. [9] uses the hypergeometric function 2 F 1 (a 1 , a 2 ; b 1 ; −σx) to approximate a divergent series with n! growth factor (zero radius of convergence). Although this approximant can give accurate results [11][12][13][14][15][16], it has been realized (by the same authors) that it suffers from a genuine problem as the expansion of 2 F 1 (a 1 , a 2 ; b 1 ; −σx) has a finite radius of convergence [10,11,13]. To overcome this issue, the tip of the branch cut is forced to lie at the origin [10,13]. In Ref. [17], we have shown that the numerator parameters (a 1 , a 2 ) are representing the strong-coupling asymptotic behavior and thus knowing them can accelerate the convergence. On the other hand, employing parameters from the large-order asymptotic behavior of the given perturbation series are well known to accelerate the convergence of resummation algorithms too [7,18]. For the problem under consideration in this work, the large-order parameters are more important than the strong-coupling parameters as the first are well known in quantum field problems while the second are not. However, the approximant 2 F 1 (a 1 , a 2 ; b 1 ; −σx) can not accommodate the large-order parameters because its expansion does not have the same from of large-order behavior that the given perturbation series has. In fact, as stated by Mera et.al, the parameter σ ought to take large values to account for the missing n! factorial growth factor [10,11]. Apart from this, one needs to find a way to change the parametrization of the approximant to make it able to give explicitly the n! growth factor of the given series.
In Ref. [19], we suggested the parametrization of the form 2 F 1 (a 1 , a 2 ; b 1 ; −b 1 σx). If the parameter b 1 takes large values, then we have the following limit: In fact the expansion of 2 F 0 (a 1 , a 2 ; −σx) has an n! growth factor [25] and thus the parameter σ can be taken from large-order behavior of the given series.
For more explanation of how to apply the new parametrization, consider a quantity Q with the first four perturbative terms are known: This quantity can be approximated by the hypergeometric approximant The three unknown parameters (a 1 , a 2 , b 1 ) are then obtained by solving the above set of equations.
In case one knows M +1 orders, one can approximate the series for Q (x) by the generalized hypergeometric function: with M = 2k + 1. This strategy has been followed in Ref. [19] and led to accurate results for different examples.
In this work, however, instead of solving for the parameter b, we shall make it as an input and set it as large as possible and then solve for the other parameters. This setting is very suitable to approach the limit in Eq.(1).
Before we use the algorithm to tackle the problem of the critical exponent ν of the model under consideration, let us first apply it to an example for which exact value is known. To do that, consider the seven-loop of the reciprocal of critical exponent ν for the Ising case of the O(N )-vector model (N = 1) [25]: with σ = 1 3 [7]. The exact value for ε = 2 (two dimensions ) is ν = 1. Taking b = 170 (the maximum value one can use due to Γ functions defining the coefficients of the hypergeometric series), one can approximate this series by: We find the following values for the parameters: For ε = 2, these parameters lead to the result ν (2) = 0.977203. To see how our idea improved the prediction of the algorithm, we obtained ν −1 by using the original hypergeometric approximant (with parametrization as presented in Ref. [9]) which has the parametrization: This parametrization gives the result ν (2) ≈ 0.964952. Note that, the more sophisticated Borel with conformal mapping resummation for the six-loop of the same series in Ref. [8] gives the result ν (2) ≈ 0.952 (14). Although the algorithm we follow here in this work might be the simplest one, it gives accurate result as shown above. One can even refine the result by taking larger values for the parameter b but in this case to overcome the machine limit one can use the limiting case: In such a case one has to resort to the representation of the hypergeometric function a 2 , a 3 , a 4 ; b 2 , b 3 ; − 1 3 ε) in terms of Meijer-G function [20]. Using this and the value of b = 10 5 , we get the value ν (2) = 0.98499. This example shows clearly that the simple algorithm we use in this work can give challenging results and thus is trusted to tackle the λ-point anomaly for 4 He superfluid phase transition as our main problem. Now consider the seven-loop critical exponent for the O(2) case given by [25]: which again can be approximated by (σ = 3/10):  [21]. However, as we mentioned above, a more precise bootstrap result of ν = 0.67175 (10) has been recently appeared in Refs. [1,2].
To know what is the situation of the precision of our resummation result among all other predictions, one needs to offer an estimate for the size of the error in the calculated result.
In fact, the sources of errors differ from a theoretical method to another. For the algorithm we use, the error is due to the unknown higher terms in the perturbation series as well as the arbitrary parameters in the resummation algorithm. Also for MC calculations, errors are due to Monte Carlo statistical errors and systematic errors associated with the correction to scaling [1]. For other methods like NPRG and CB they also have their own sources of errors.
So it is very natural to have different precision from different methods. An improvement of some calculation can be decided by comparison with previous calculations within the same method. For instance, looking at Fig.1, one can realize that our seven-loop prediction shows a significant improvement for both accuracy and precession when compared to the five-loop and six-loop predictions.
The algorithm we follow seems to have no arbitrary parameters which one can optimize and thus find the uncertainty in the result. However, deep understanding of the simple algorithm we follow can find some implicit arbitrariness in its parametrization. As an example, one can find different hypergeometric functions that can approximate a given order of perturbation series. For instance, the seven-loop in Eq.(8) can be approximated by: All these approximants are legal and use the same content of information and having the n! growth factor at the limit b → ∞. Of course they give different approximations and the question is which one shall we select? To answer this question, one also notices that the six-loop can also be approximated by different hypergeometric functions. A natural choice is then to select a pair of approximants for six and seven loops that has the fastest convergence or equivalently we select the pair that minimizes the difference (∆) defined as: where k defines the hypergeometric approximant ( k F k−1 ) used while superscripts for the number of loops involved. We found that the approximant: a 1 , a 2 , a 3 , a 4 for the seven-loop and the approximant for the six-loop give the smallest difference of ∆ = 0.0004. This could be used as an uncertainty and our seven-loop resummation result can be taken as ν = 0.67094(4). This method of error calculation has been used in different references ( see for instance sec. 16.6.1 in Ref. [7]).
We have another source of arbitrariness which can be taken from the form of the largeorder behavior of the given perturbation series as: In fact, the parameter α for the given model is known and has the value α = 5.892 × 10 −4 [24] but none of our equations have been constrained to account for it. The explicit form of α (Eqs. (4.19,4.10) in Ref. [24]) depends on 1 N +8 = σ 3 which means that one can account for its variation by varying the parameter σ and then find the variance of ν 7 4 defined as [8]: The width w = y −x is chosen according to the stability region in the curve of the seven-loop exponent ν 7 k (σ) (see Fig.2). Based on the shape of the curve in Fig.2, we choose w = 0.2 where we vary σ around its exact value of 0.3 from 0.2 to 0.4. We use an adapted form of method detailed in Ref. [8] where our error can be taken as the minimum of : This form of error is closer in shape to the one used in Ref. [8] but adapted in view of the shape in Fig.2 as well as the behavior of six-loop result vs. σ (not shown in figures). The error value obtained is ∆ min ≈ 7 × 10 −4 while the predicted value of ν is 0.6711 (7). As shown in Fig.1, this result shows a significant improvement to the resummation results of the same series in literature. In fact, one can easily realize that moving from six to seven loop, the RG result is more precise and accurate as it has been shifted toward the MC and BC results. Our prediction is compatible with both experiment as well as MC, BC and NPRG results.
However, in view of Fig. 3, we see that the seven-loop result is not sufficiently leveling off the curve for ν versus the number of loops n. In fact, the tangent of the curve is going smaller as function of n but not small enough to claim a stable shape. This shape of the curve is thus suggesting a possibility for smaller error as well as higher value in the exponent ν to come from future higher orders.   Ironed by the success of the modified hypergeometric algorithm in the two-dimensional case, we tackled the controversial three dimensional case for ν of the O(2)-model. In fact, for ε = 1, one expects even better convergence than the two dimensional case. We calculate the exponent ν for the O(2)-symmetric case and get the value ν = 0.6711 (7) which is compatible with the experimental result (ν = 0.6709(1)) as well as the theoretical calculations from NPRG method (ν = 0.6716 (6)), the more precise CB result (ν = 0.67175 (10)) and MC result (ν = 0.67169 (7)). Note that that NPRG, CB and MC results are excluding the experimental result.
The plot of the exponent ν versus the number of loops suggesting that the seven-loop result in this work might not the most precise as well as accurate prediction that one can obtain from resummation of RG perturbations. The shape of the curve expecting a more accurate as well as precise result from the future eight-loop series. In other words, there is still a room for RG result to agree with both MC and CB predictions but excluding the experimental result.