Critical Exponents of the O(N)-symmetric $\phi^4$ Model from the $\varepsilon^7$ Hypergeometric-Meijer Resummation

We extract the $\varepsilon$-expansion from the recently obtained seven-loop $g$-expansion for the renormalization group functions of the $O(N)$-symmetric model. The different series obtained for the critical exponents $\nu,\ \omega$ and $\eta$ have been resummed using our recently introduced hypergeometric-Meijer resummation algorithm. In three dimensions, very precise results have been obtained for all the critical exponents for $N=0,1,2,3$ and $4$. To shed light on the obvious improvement of the predictions at this order, we obtained the divergence of the specific heat critical exponent $\alpha$ for the $XY$ model. We found the result $-0.01136$ compared to the famous experimental result of -0.0127(3) from the specific heat of zero gravity liquid helium while the six-loop Borel with conformal mapping resummation result in literature gives the value -0.007(3). For the challenging case of resummation of the $\varepsilon$-expansion series in two dimensions, we showed that our resummation results reflect an improvement to the previous six-loop resummation predictions.


I. INTRODUCTION
Quantum field theory (QFT) offers a successful way to study critical phenomena in many physical systems [1][2][3][4][5][6][7][8][9][10][11]. It is universality that is behind the scene where different systems sharing the same symmetry properties follow the conjecture that they ought to behave in a similar manner at phase transition. So it is not strange to have a fluid having the same critical exponents like a magnetic one when both lie in the class of universality. The  [12][13][14][15][16][17][18][19][20].
Besides, bootstrapping the model in three dimensions has been accomplished recently and researchers succeeded to obtain precise results [21][22][23][24][25][26]. The nonperturbative renormalization group has been applied to the same model and gives accurate results too [27]. Apart from these non-perturbative methods, the oldest way to tackle the critical phenomena in QFT is resummation techniques applied to resum the divergent perturbation series associated with that model. The most traditional algorithm is Borel and its extensions which have been widely used in literature [3,4,[7][8][9][28][29][30]. In fact, the recent progress in obtaining higher orders of the perturbation series stimulates the need for the application of the resummation techniques to investigate the theory. Regarding that, the six-loop of the renormalization group functions has been recently obtained [29] and then the seventh order has been obtained too [31]. These orders are representing the renormalization group functions within the minimal subtraction regularization scheme in D = 4 − ε dimensions.
The study of critical phenomena by finding an approximant to the perturbation series follows different routes. For instance, perturbative calculations at fixed D dimensions [4,28] are always giving better results specially in three dimensions. However, while exact results are known in two dimensions, the resummation of perturbation series did not give reliable results for some exponents [32,34]. This point has been studied in Refs. [29,33,34] and it has been argued there that the reason behind this is thought to be the non-analiticity of the β−function at the fixed point. For the ε-expansion on the other hand, perturbation series though possesses slower convergence [4,29], might not suffer from non-analiticity issues like the g-series [35]. In view of the recent seven-loops (g-expansion) calculations [31], one thus can aim to get improved results from resumming the corresponding seventh order of ε expansion in three dimensions (ε = 1) as well as get improved predictions for two dimensions ε = 2.
In previous articles [36,37], we introduced and applied the hypergeometric-Meijer resummation algorithm. What make our algorithm preferable is its simplicity and of having no free parameters like Borel and its extensions. Besides, it gives very competitive predictions when compared to the more sophisticated Borel with conformal mapping algorithm for instance. The algorithm has been applied successfully for the six-loop ε−series and for the seven-loop coupling series in Ref. [36]. For expansions in 4 − ε dimensions, however, it is always believed that the ε−series has better convergence than the coupling-series [4]. In fact one can speculate about this by considering the large order behavior for both series. For the coupling series, the large order behavior includes the term (−g c ) n (g c ≡ critical coupling) while the ε−series has the term (−σε) n with σ = 3 N +8 . For N = 1 and in three dimensions (for instance), at the fixed point the g-series behaves as (−0.47947) n (from seven-loops calculations) while the ε−series behaves as (−0.33333) n . So it is expected that the resummation of the ε−series has better convergence.
The recent resummation results of the six-loop ε-series [29,36] gave accurate predictions for the critical exponents ν, η and ω for the O(N)-symmetric φ 4 theory. However, the predictions of the relatively small exponents like divergence of specific heat exponent α are still far away from expected results. For the XY model for instance, our hypergeometric-Meijer algorithm gives the result α = −0.00885 [36] while Borel with conformal mapping result in Ref. [29] is −0.007(3) and the resummation of seven-loop g-series in Ref. [36] predicts the value −0.00860. All of these predictions are all not close enough to the result of the famous experiment in Ref. [38]. In that reference, the measurement of the specific heat of liquid helium in zero gravity yields the result −0.0127 (3). Accordingly, resumming the sevenloop ε-series represents an important point to monitor the improvement of the predictions of the critical exponents. With that in mind, our aim in this work is to first obtain the ε−series corresponding to the recent seven-loop coupling series for the β,γ m 2 and γ φ renormalization group functions and then apply our resummation algorithm to the series representing the critical exponents ν,η and ω for the O(N)−symmetric quantum field model. The organization of this paper is as follows. In Sec.II, a brief description of the hypergeometric-Meijer resummation algorithm is introduced. We present in Sec.III the extracted seven-loop ε-expansion of the renormalization group functions. The resummation of the different ε-series representing the critical exponents ν, η and ω is presented in Sec.IV.
In this section a comparison with predictions from other methods for N = 0, 1, 2, 3 and 4 is listed in different tables for each N individually. The study of the challenging twodimensional case will follow in Sec.V. The last section in this paper (Sec.VI) is dedicated for the summary and conclusions.

II. THE HYPERGEOMETRIC-MEIJER RESUMMATION ALGORITHM
To make the work self consistent, we summarize in this section the hypergeometric-Meijer resummation algorithm that was firstly introduced in Ref. [37] and then applied to the sixloop (ε-expansion) and seven-loop g-expansion in Ref. [36]. Now, consider a perturbation series of a physical quantity Q for which the first M + 1 terms are known: Assume that the asymptotic large-order behavior for the series is also known to be of the form: As shown in Ref. [37], the hypergeometric series p F p−2 (a 1 , a 2 , ...., a p ; b 1 , b 2 , ....b p−2 ; −σx) can reproduce the same large-order behavior with constraint on its numerator and denominator parameters as: So the hypergeometric series p Once parametrized by matching with the given series, the divergent hypergeometric series is now known up to any order and can be resummed by using its representation in terms of the Meijer-G function of the form [39]: 1−a 1 ,...,1−ap Note that for M even, M equations are generated by matching with the available orders from the given perturbation series to solve for M = (2p − 2) unknown parameters in the hypergeometric function. In the odd M case, we employ the constraint in Eq.(3) to get M + 1 equations to solve for the M + 1 unknown parameters. In any case, we always need an even number of equations to determine the 2p − 2 unknown parameters.
To give an example, consider the lowest order approximant (two-loops) 2 F 0 (a 1 , a 2 ; ; −σx) when matched we get the results: These equations are solved for the unknown parameters a 1 and a 2 provided that the parameter σ is known from the large-order behavior. Then we use the Meijer G-function representation given by: to obtain an approximant for the quantity Q(x) in Eq.(1) for M = 2.
For the M = 3 approximant 3 F 1 (a 1 , a 2 , a 3 ; b 1 ; −σx) we have the equations: to be solved for the four unknowns a 1 , a 2 , a 3 and b 1 . Thus we get the approximation of Q(x) as: Our aim in this work is to resum the ε 7 series for the critical exponents of the O(N)symmetric model. Up to the best of our knowledge, the ε-expansion for these exponents (for all N cases studied here and for the same exponents) is not available so far in literature. So in the following section, we shall extract them first from the recent seven-loop calculations in Ref. [31].
In 4 − ε dimensions within the minimal subtraction technique, Oliver Schnetz has obtained the seven-loops order (g-expansion) for the renormalization group functions β, γ m 2 and γ φ [31]. Here β is the famous β-function that determines the flow of the coupling in terms of mass scale, γ m 2 is the mass anomalous dimensions and γ φ represents the field anomalous dimensions. In the following subsections, we list the corresponding seven-loop ε-expansion for each individual exponent for the cases N = 0, 1, 2, 3, 4, respectively.
III.1. The seven-loop ε-expansion for self-avoiding walks (N = 0) For N = 0, we have the results [31]: The recipe to extract the corresponding ε-expansion is direct where we solve the equation β (g) = 0 (fixed point) for the critical coupling g c as a function of ε and then substitute in the equations for γ φ (g c ) and γ m 2 (g C ). Note that the critical exponents ν and η are obtained from the relations ν = [2 + γ m 2 (g c (ε))] −1 and η (ε) = 2γ φ (g c (ε)) while the correction to scaling exponent ω is given as ω = β ′ (g c ). In Ref. [40], the method of Lagrange inversion has been used to get the exact seven-loop ε-expansion coefficients and has been applied to the N = 1 case but there they obtained the series for ν while here we list the series for ν −1 .
However, here we will obtain the ε−series by solving the equation β (g) = 0 implicitly and then expand the implicit solution as a power series in ε keeping only orders up ε 7 . As we will see, our results coincide with those obtained in Ref. [40] for η and ω for N = 1. For ν −1 , η and ω for N = 0, 1, 2, 3 and 4, we found that our results are compatible with the five-loop results available in Ref. [4] and six-loop series (after proper scaling) in Ref. [29].
For N = 0 and after solving the equation β(g(ε))=0, we get the result: and The ε-expansion for Ising-like model(N = 1) In this case the seven-loop β−function is presented in Ref. [31] as: and Solving the equation β (g) = 0, we get the critical coupling as: Substituting this form in γ m 2 (g c ) and keep orders up to ε 7 only we get: Similarly, the forms for η and ω can be obtained as: III.3. The ε-expansion for N = 2 (XY universality class) For N = 2, the renormalization group functions are obtained in Ref. [31] as : We extracted from these equations the following forms for g c and the exponents ν, η and ω : and III.4. The ε-expansion for the exponents ν, ω and η in the Heisenberg universality The renormalization group functions can be generated using the maple package in Ref. [31] as: and From these functions one can obtain the following forms for the critical quantities g c , ν, η and ω as: and ω = 1.0000ε − 0.57025ε 2 + 1. For N = 4, we have the seven-loops g-series as : and Our prediction for the corresponding ε−expansion for the critical coupling and exponents are of the form: and It is well known that the ε−series is divergent and has an asymptotic large-order behavior of the type shown in Eq. (2) where [4]: Accordingly, the suitable hypergeometric approximant is of the form In this section we present the resummation results for the ε 7 -series but numerical values in this section are for ε = 1. For the series representing ν −1 in Eqs. (12,18,25,32,38), the suit- with the corresponding Meijer-G function representation : For the series representing the critical exponent ω in Eqs. (14,20,27,34,40), the suitable approximant is: while based on the type of series given for the exponent η in Eqs. (13,19,26,33,39), it can be easily shown that the suitable hypergeometric approximant takes the form: In case there is no solution found for the set of equations that determine the parameters, one resort to successive subtractions which can lead to another approximant [36]. So, in the following subsections, we will list the resummation results for the seven-loop perturbation series where for each case we list the suitable approximants.
For the critical exponent η, it has the seven-loop perturbation series in Eq. (12) and the suitable approximant is:   for ν from Ref. [14,29] and η from Ref. [15]. Also the predictions of the the resummation of six-loop series using Borel with conformal mapping (BCM) algorithm (ε 6 ) from Ref. [29] and five-loop (ε 5 ) from same reference is included. The perturbation series for critical exponent ν(ν −1 ) of the Ising-like model up to ε 7 is given by Eq. (18). The suitable approximant for this order is 2 5 F 3 (a 1 , ... which predicts the value ν ≈ 0.62973. To get an idea about how accurate this result is, we list her results from recent non-perturbative methods like the Monte Carlo simulations which gives the result ν = 0.63002(10) [12], the recent non-perturbative renormalization group (NPRG) method [27] which turns the result ν = 0.63012 (16) as well as the recent conformal bootstrap result ν = 0.62999(5) in Ref. [23]. In view of these non-perturbative calculations and in looking at table II, one can realize that the seven-loop resummation results add to the improvement of the six-loop results either in Ref. [36] or Ref. [29]. In fact, this is a general trend in all of the seven-loop calculations in this work.
For the exponent η, the seven-loop perturbation series is given by Eq. (19) while the suitable approximant is:  and ω of the Ising-like model (N = 1). Here we compare with our results from Ref. [36] for ε 6 (ε 6 : HM) and seven-loop gexpansion (HMg) from the same reference. The recent SC resummation results are listed for comparison. Also we list conformal bootstrap calculations from Ref. [23] and Monte Carlo simulation (MC) from Ref. [12]. In this table also, we list the six-loop (ε 6 ) resummation results from the Borel with conformal mapping (BCM) from Ref. [29] and five-loops (ε 5 ) from same reference.
The recent results from the non-perturbative renormalization group (NPRG) method [27] is listed last.  (13) 0.8303 (18) 0.832 (6) 0.820 (7) 0.818 (8) 0.832 (14) The microgravity experiment on the other hand gives the result α = −0.0127(3) [38]. What makes our prediction for α so impressive is that the six-loop resummation result using Borel with conformal mapping in Ref. [29] gives the value α = −0.007 (3). In using our algorithm to resum the same six-loop series in Ref. [36] we get the value α = −0.00885 while in the same reference the seven-loop g-expansion gives the result α = −0.00860. This shows that our resummation result that is very close to experiment reinforces the expectation that the ε-series has better convergence than the g-series. Besides, the resummation of seventh order in this work clearly improves the ε 6 resummation results.
In Similar to the above cases, the seven-loop perturbation series for the exponent ν has been obtained in the previous section in Eq. (38). As in the previous cases, we resummed it using the approximant 2 5 F 3 (a 1 , ...a 5 ; b 1 , b 2 , b 3 ; −σε) which gives the result ν ≈ 0.74391.
This result is compatible with the NPRG prediction of ν = 0.7478 (9) in Ref. [27]. Also the conformal bootstrap result is ν = 0.751(3) from Ref. [24] and Monte Carlo simulations gives the result ν = 0.750(2) [13].   [36]. Then we list the SC resummation results from Ref. [46]. The recent results from conformal bootstrap calculations are listed also where the values of ν and η are taken from Ref. [25] while ω from Refs [24,29]. For MC simulations ω is taken from from Ref. [17] while ν and η are taken from from Ref. [13]. The six-loop BCM resummation is taken from Ref. [29] and five-loops from same reference. As in all of above tables, we list in the last row the very recent calculations from NPRG method [27] (up to O(∂ 4 )).  (7) 0.797 (7) 0.769(11)

V. TWO-DIMENSIONAL HYPERGEOMETRIC-MEIJER RESUMMATION
In two dimensions or equivalently ε = 2, there are two main differences from the three dimensional case. The first is that for N ≥ 2, there is no broken-symmetry phase [33].
For the other difference, since ε = 2 is a large-value and the strong-coupling asymptotic behavior of the O(N) symmetric model is not known yet, one expects a slower convergence of the resummation of the perturbation series. For the g expansion, it has been argued that the β function is not analytic at the fixed point [33,34,41] which in turn slows the convergence down too. The effect of the non-analiticity of the β function is higher in two dimensions. This leads to inaccurate predictions for critical exponents from the g expansion in two dimensional case [34]. Accordingly, testing the resummation algorithm for the ε = 2 case offers an interesting point about the capability of the ε-expansion to predict reliable results for that case. Apart from inaccurate resummation results from the g-expansion as  [36]. Results from conformal bootstrap calculations [24,29] for ν and ω , while η from Ref [22] are listed. Besides, MC simulations for ω is taken from Ref. [17] while ν and η are from Ref. [13]. The six-loop BCM resummation (ε 6 ) is taken from Ref. [29] and five-loops (ε 5 ) from same reference. NPRG results up to O(∂ 4 ) [27] are shown in the last row.  (9) 0.817 (30) 0.765 (30) 0.794 (9) 0.795 (6) 0.761 (12) well as prvious results of the ε-expansion that needs more improvement, exact values for the two dimensional critical exponents are known and thus can be used to test the reliability of any approximating method.
For N = 0, our resummation result for the critical exponent ν is 0.75148 while the exact result is assumed to be 3 4 [42] and the recent Borel with conformal mapping resummation for six-loop yields the result ν = 0.741(4) [29]. For the critical exponent ω, our prediction is 1.9554 while the exact value is 2 [42,43] and the recent six-loop resummation in Ref. [29] gives the result 1.90 (25). For η, we get the value 0.18955 while the exact result is For Ising-like case (N = 1), we obtained the result ν = 0.96964 compared to the well known exact result ν = 1 [44] while BCM result for six loops gives the value ν = 0.952 (14).
For ω we get the result 1.7202 while the exact value is ω = 1.75 [45] and BCM result is 1.71 (9). Our prediction for η is 0.22277 while the exact value is 0.25 [42] and the six-loop BCM resummation result is 0.237 (27).
From the calculations above and those listed in  The seven-loop (ε 7 ) hypergeometric-Meijer resummation for the exponents ν , η and ω for the Self-avoiding walks (N = 0) and the Ising-like model (N = 1) in two dimensions (ε = 2). For comparison with other predictions, we list the results from the same algorithm but for six loops, BCM results [29] for six loops too. Exact results for N = 1 for ν and η are obtained in the seminal article in Ref. [44] and ω from Ref. [45]. For N = 0, exact values for ν, η and ω are conjectured in Ref. [42].  [29,36]. In this work, we used our hypergeometric-Meijer algorithm [36,37] to resum the up to ε 7 series for the critical exponents from the O(N)-symmetric φ 4 theory for N = 0, 2, 3, 4.
The resummation results has shown clear improvement for the previous six-loop results.
The most reflecting quantity for the improvement of the six-loop results is the specific heat critical exponent of the XY model. Taking into account that the result α = −0.0127 (3) from zero-gravity experiment in Ref. [38], the BCM six-loop result from ref. [29] which is While the predictions of the renormalization group at fixed dimensions gives accurate results in three-dimensions [4,28], the story is different for the two dimensional cases. In two dimensions, the renormalization group at fixed dimensions gives inaccurate results especially for the critical exponents of small values [32,34]. The reason behind this is the nonanaliticity of the β-function at the fixed point [29,33,34]. The ε-expansion on the other hand might not suffer from this problem [35]. We tested our resummation results in two dimensions and found an overall improvements to our six-loop resummation results in Ref. [36].
Our algorithm while simple gives astonishing results for the critical exponents which are competitive to the results from more sophisticated resummation algorithms, numerical methods as will as conformal field theory. This puts it among the preferred resummation algorithms applied to different problems in physics. A note to be mentioned here is that this work (up to the best of knowledge) represents the first resummation results for the ε 7