Precise Critical Exponents of the O(N)-Symmetric Quantum field Model using Hypergeometric-Meijer Resummation

In this work, we show that one can select different types of Hypergeometric approximants for the resummation of divergent series with different large-order growth factors. Being of $n!$ growth factor, the divergent series for the $\varepsilon$-expansion of the critical exponents of the $O(N)$-symmetric model is approximated by the Hypergeometric functions $_{k+1}F_{k-1}$. The divergent $_{k+1}F_{k-1}$ functions are then resummed using their equivalent Meijer-G function representation. The convergence of the resummation results for the exponents $\nu$,\ $\eta$ and $\omega$ has been shown to improve systematically in going from low order to the highest known six-loops order. Our six-loops resummation results are very competitive to the recent six-loops Borel with conformal mapping predictions and to recent Monte Carlo simulation results. To show that precise results extend for high $N$ values, we listed the five-loops results for $\nu$ which are very accurate as well. The recent seven-loops order ($g$-series) for the renormalization group functions $\beta,\gamma_{\phi^2}$ and $\gamma_{m^2}$ have been resummed too. Accurate predictions for the critical coupling and the exponents $\nu$, $\eta$ and $\omega$ have been extracted from $\beta$,$\gamma_{\phi^2}$ and $\gamma_{m^2}$ approximants.


I. INTRODUCTION
Quantum field theory (QFT) represents an important tool to study critical phenomena for different physical systems. Critical phenomena is thus offering an indirect experimental test to the validity of QFT. The idea stems from the universal phenomena where a number of different systems can show up the same critical behavior in spite of their different microscopic details. A very clear example is the Ising model from magnetism and the one-component φ 4 model from QFT [1][2][3][4][5][6]. The more general example of the φ 4 scalar field theory with O(N)-symmetry can describe the critical phenomena in many physical systems that share the same respective symmetry. Regarding the N = 0, for example, the theory lies in the same universality class with polymers [7] while the N = 1 case describes the critical behavior of Ising-like models. For N = 2, the model describes a preferred orientation of a magnet in a plane while the case N = 3 can describe a rotationally invariant ferromagnet . Besides, the N = 4 case can mimic the phase transition in QCD at finite temperature with two light flavors [8].
The study of critical phenomena within quantum field theory has been reinforced by Wilson's introduction of the famous ε-expansion [9,10]. Wilson ideas made the renormalization group functions to take a place in the heart of predicting critical exponents from the study of QFT models [1,3,4]. However, the series generated by the ε-expansion is well known to be divergent [11] and thus resummation techniques are indispensable to extract reliable results from that series. In Ref. [12] (for instance), Borel transformation with conformal mapping technique has been used to resum divergent series of the critical exponents of the O(N)−symmetric model. Also in Ref. [13], the five-loops ε-expansion of the perturbation series for the critical exponents have been resummed using a strong-coupling resummation technique.
Resummation of the series generated by ε-expansion has been shown to be slightly less precise than the resummation of renormalization group functions at fixed dimensions [12]. This fact motivated the authors of the recent work in Ref. [14] to move one step forward toward the improvement of resummation predictions of the critical exponents from ε-expansion. In that reference, the six-loops perturbation series of the ε-expansion for the renormalization group functions of the O(N) model have been obtained and resummed using Borel with conformal mapping resummation algorithm. They obtained accurate results for the exponents ν, η and ω. However, this algorithm has three free parameters where their variations add to the uncertainty in the calculations. We will show in this work that a simple Hypergeometric-Meijer resummation algorithm [15], which has no free parameters, can result in competitive approximations for the critical exponents from the ε-expansion.
Methods that are using different approach (other than resummation) have been used in literature to extract accurate critical exponents of the O(N) model. Among these successful methods is Monte Carlo simulation which has been used to obtain accurate critical exponents of the O(N) model [16][17][18][19][20][21][22][23]. Besides, in recent years, researchers were able to extend the applicability of conformal bootstrap methods to three dimensions which in turn resulted in very accurate predictions for the critical exponents of the O(N) model too [24][25][26][27][28]. The results of these techniques besides the recent Borel resummation results will be used for comparison with our predictions from Hypergeometric-Meijer resummation of divergent series representing the critical exponents.
The divergence of perturbation series in QFT has been argued for the first time by Dyson [29]. From a mathematical point of view, singularities in the complex-plane are responsible for series divergence even for small argument [30]. The manifestation of divergence in a perturbation series appears in the form of large-order growth factors like n!, (2n)! and (3n)! (for instance). The appearance of such large-order behaviors stimulates the need for resummation of such type of perturbation series [31,32]. The most popular resummation technique is Borel and its different versions. In fact, the knowledge of the large-order behavior of a divergent series is needed not only to accelerate the convergence of resummation results but also to determine the type of the Borel transformation to be used. In our work, we will show that the large-order behavior is also important for our resummation (Hypergeometric-Meijer) algorithm [15] in order to select the suitable relation between the number of numerator and denominator parameters of the used Hypergeometric approximant.
Borel resummation and the Hypergeometric-Meijer algorithms share the need of the large-order behavior of a divergent series to select the suitable Borel-transform and the Hypergeometric approximant respectively. There exist, however, different features for both algorithms. One can get sufficient idea about the features of Borel resummation algorithm by going to its extensive use in literature. For the resummation of divergent series in QFT, one can visit some of past and recent successful studies that dealt with resummation of the divergent series of the renormalization group functions of the O(N)-symmetric model [1,4,5,12,14,[33][34][35][36]. Although resummation techniques used in literature like Borel and Borel-Padé can give reasonable results for the critical exponents of the O(N) model, these algorithms need a relatively high order of loop calculations which is not an easy task. To get an idea about how hard to have high orders of loops calculations, we assert that it took the researchers like 25 years to move forward from five-loops to six-loops calculations [14,37]. Even at the level of more simpler theories like the PT −symmetric iφ 3 field theory, the four loops renormalization group functions have been just recently obtained [33]. In going to more complicated theories that have fermionic as well as gauge boson sectors, the calculation of a relatively high loop orders is not an easy task. The Hypergeometric-Meijer algorithm, on the other hand, can give reasonable results even in using few orders from a perturbation series as input. It is thus very suitable for the study of non-perturbative features of a quantum field theory.
In Borel algorithms, results are always achieved via numerical calculations. This feature leads to the resummation of individual physical amplitudes one by one. The existence of a resummation algorithm that avoids this feature might help in getting other amplitudes without further resummation steps. Instead, we can obtain them from simple calculus. For instance, the vacuum energy or equivalently the effective potential is known to be the generating functional of the one-particle-irreducible amplitudes. Accordingly, getting a closed form resummation function for the effective potential enables one to get other amplitudes via functional differentiation [38,39]. The Hypergeometric-Meijer resummation as we will see can give accurate results as well as being simple and of closed form. Besides, it does not have any free parameters to fix like other resummation algorithms which use optimization tools to fix the introduced free parameters.
The Hypergeometric-Meijer resummation algorithm we use in this work is a development of the recently introduced simple Hypergeometric resummation algorithm [40]. In the Hypergeometric algorithm, the Hypergeometric approximant 2 F 1 (a, b; c; σz) has been suggested for the resummation of a divergent series. The four parameters a, b, c and σ are obtained by comparing the first four orders of the expansion of 2 F 1 (a, b; c; σz) in the variable z with the four available orders of the divergent series under consideration. To illustrate this more, consider a series representing a physical quantity Q (z) as: we have also the series expansion of c 0 2 F 1 (a, b; c; σz) as: For c 0 2 F 1 (a, b; c; σz) to serve as an approximant for Q (x) we have to set which can be solved to determine the unknown parameters a, b, c, d, σ in terms of the known coefficients c 1 , c 2 , c 3 and c 4 .
To accelerate the convergence of the algorithm, we suggested the employment of parameters from the asymptotic behavior of the perturbation series at large values of the argument z [41] or equivalently the strong coupling data. Our suggestion is based on the realization that when a − b is not an integer, the Hypergeometric function has the following asymptotic form [42]; Also the method has been generalized to accommodate higher orders from the perturbation series by using the generalized Hypergeometric function p F p−1 (a 1 , ...a p ; b 1 ....b p−1 ; σz) where the a i parameters are extracted from the asymptotic behavior of the perturbation series at large z value.
The Hypergeometric algorithm either the version in Ref. [40] or Ref. [41] cannot accommodate the large order data available for many perturbation series in physics. The point is that the series expansion of the Hypergeometric function 2 F 1 (a, b; c; σz) has a finite radius of convergence while it has been used for the resummation of a divergent series with zero radius of convergence. This means that the large order behavior of the expansion of the function 2 F 1 (a, b; c; σz) can not account explicitly for the n! growth factor characterizing a perturbation series with zero radius of convergence. In fact, in the Hypergeometric algorithm, the parameter σ ought to take large values to compensate for that [43,44] but itself cannot be considered as a large-order parameter. Indeed, employing parameters from large-order behavior is well known to accelerate the convergence of resummation algorithms (Borel for instance). Moreover, one can not apply the suitable Borel transform (divide by n! for instance ) unless we know the large order behavior of the perturbation series.
These facts led us to develop the Hypergeometric algorithm [15] by using the approximants Consider a divergent series that represents a physical amplitude Q(z) as where the first M + 1 orders are known. Assume that the large-order behavior of that series takes the from: In Ref. [15], we showed that when p = q + 2, the perturbative expansion of the Hypergeometric function p F q (a 1 , ...a p ; b 1 ....b q ; −σz) which has a zero-radius of convergence can be parametrized to give the same large-order behavior of the above perturbation series. Ac- is resummed using its representation in terms of Meijer-G function as follows [42]: Note that the authors in Ref. [44] used a Borel-Padé algorithm that leads to Meijer-G approximants parametrized by weak-coupling information.
One can generalize the idea of our previous work in Ref. [15] to other types of divergent series with growth factors other than n!. For instance, the divergent series of the ground state energy of the sixtic anharmonic oscillator has a zero radius of convergence but the growth factor is (2n)! while it is (3n)! for the octic anharmonic oscillator [45]. Knowing that the asymptotic form of the ratio of two Γ functions is given by [46]: one can easily conclude that either the Hypergeometric approximants [41] or p F p−2 (a 1 , ...a p ; b 1 ....b p−2 ; σz) used in Ref. [15] cannot account for the growth factors of the sixtic or octic ground state energies.
Accordingly, one can accept that there exists more than one type of Hypergeometric functions (different S = p − q) that are needed to approximate different divergent series in physics with different large-order growth factors.
Based on the idea that the large-order asymptotic behavior is responsible for the selection of the suitable Hypergeometric approximant for a perturbation series, one can list different p F q (a 1 , ...a p ; b 1 ....b q ; −σz) approximants for different growth factors as follows: 1. for divergent series that has the large-order behavior in Eq.(5) (n! growth factor), the 2. For a series that has a large-order behavior like γΓ 2n + 1 2 (−σ) n n b , n → ∞, the suitable one is p F p−3 (a 1 , ...a p ; b 1 ....b p−3 ; −σz). This is because one can easily show that for p = q + 3, one can get a similar large-order behavior. An example of such divergent series is the ground state energy of the sixtic anharmonic oscillator [45] 3. For the ground state energy of the octic anharmonic oscillator, the large order behavior is given by ∼ δ Γ 3n + 1 2 (−σ) n n b , n → ∞, which can be reproduced by the Based on this classification, knowing the large order behavior of a divergent series is essential not only to accelerate the convergence of the resummation algorithm but also to determine the suitable Hypergeometric approximant. A note to be mentioned is that, for p ≥ q + 2, the Hypergeometric function p F q (a 1 , ...a p ; b 1 ....b q ; σz) has a zero radius of convergence but it can be resumed using the closely related Meijer-G function (see Eq. (6)) which has the integral form [42]: The Hypergeometric-Meijer algorithm which will be used in this work to resum the divergent There exist some technical issues when applying the algorithm. The first issue is that for high orders, computer can take a relatively long-time to solve the set of equations like the one in Eq.(3). To overcome this problem, we generated the ratio R n = cn cn−1 and then solve the set of equations: For example, the approximant p F q (a 1 , ...a p ; b 1 ....b q ; σz) generates the following set of equations: . .
This trick decreases the degree of non-linearity in the set of equations and thus saves the computational time.
The other issue regarding the application of the Hypergeometric-Meijer algorithm is that at some orders one might find no solution for the set of equations defining the parameters in the Hypergeometric function. In this case, one resorts to a successive subtraction of the perturbation series. This trick is well known in resummation algorithms [4,44]. However, the subtracted series will have a different large-order b parameter where it increases by one per each subtraction ( see for instance sec.16.6 in Ref. [4]).

III. HYPERGEOMETRIC-MEIJER RESUMMATION FOR THE ε− EXPANSION OF CRITICAL EXPONENTS AND COUPLING UP TO FIVE LOOPS
The Lagrangian density of the O(N)-vector model is given by: At the fixed point, the β-function is zero which sets a critical coupling as a function of ε = 4 − d. Accordingly, one can obtain the renormalization group functions as power series in ε. In the following parts of this section, we list the resummation results (up to five loops) for the exponents ν, η and ω as well as the critical coupling of that model.
Two, three, four and five loops resummation for the exponent ν Up to five-loops, the power series for the reciprocal of the critical exponent ν is given by [4]: where The large-order parameters takes the form in Eq. (5) where [4] The suitable Hypergeometric approximant is thus can reproduce the large order behavior in Eq. (5). The number of unknown parameters in is 2p − 2 and thus we need an even number of equations to determine the unknown parameters. So we have two options: − Even number of loops as input: In this case we incorporate an even number (2p−2 ) of terms from the perturbation series to match with corresponding terms from the − Odd number of loops as input: in this case we take odd number (2p − 1) of loops to build odd number of equations and one equation from the large-order constraint: to determine the unknown numerator and denominator parameters.
So we list resummation results that involve odd or even number of perturbtive terms separately.

III.1.1. Two-loops Resummation for ν
For p = q + 2, the lowest order Hypergeometric approximant for ν −1 is thus: For this resummation function, one needs to determine the two parameters a 1 and a 2 by matching the perturbative expansion of 2 2 F 0 (a 1 , a 2 ; ; − 3 N +8 ε) with the first two terms in the perturbation series in Eq. (12). In this case we get: from which we obtain the results: To test the accuracy of this two -loops resummation function, let us note that for N = 1, the recent Monte Carlo calculation [16] gives υ = 0.63002(10). Our two-loops Hypergeometric-Meijer resummation gives the result υ = 0.66209. This result is very reasonable in taking into account that the algorithm is fed with only the first two orders from the perturbation series as input. For N = 0, the a recent accurate prediction is listed in Ref. [19] as ν = 0.5875970(4) while our two loops resummation gives ν = 0.60890. For N = 2, Monte Carlo calculations gives υ = 0.6690 [16] while the two-loops gives ν = 0.711526. So it seems that the simple Hypergeometric-Meijer resummation algorithm we follow in this work gives reasonable results even with very low orders of perturbation series as input. It is expected that the resummation of higher orders will improve the accuracy of the results which we will do in the following subsections.

III.1.2. Three-loops resummation for ν
For more accurate results, one can go to the higher three-loops order of Hypergeometric- . Although it is parametrized by four parameters (a 1 , a 2 , a 3 and b 1 ), the use of the large order constraint [15]: leads to the need of three terms only from perturbation series to determine the parameters.
So to determine them (a 1 , a 2 , a 3 and b 1 ), we solve the set of equations: The predictions of this order are given in table-I for different N values and compared to two, four and five loops resummation results and to the Janke-Kleinert resummation (up to five-loops) in Ref. [4] and the Borel-with conformal mapping in Refs. [12,14]. One can easily realize that the convergence has been greatly improved when moved from two-loops to the three-loops resummation. N This Work JK [4] BCM [12], [14] 2 F 0 : ε 2 3 F 1 : ε 3 3 F 1 : ε 4 4 F 2 : ε 5 The obvious acceleration of the convergence of the algorithm from two to three loops is strongly recommending the Hypergeometric-Meijer resummation algorithm to take a place among the preferred algorithms to resum divergent series with large order behavior of the form in Eq. (5). Other features that recommend it for resummation of divergent series is that it does not include any free parameters and of closed form as well.
can also be used to resum the perturbation series up to four loops but in this case we have to solve the set of equations: The prediction of this order of resummation is also listed in table-I where it shows that the accuracy is improving in a systematic way when moving to higher orders.

III.1.4. Five-loops resummation for ν
In this case we use the approximants 4 F 2 (a 1 , ..., a 4 ; b 1 ...b 4 ; − 3 N +8 ε) where the unknown parameters are determined from the set of equations: For this order, we get even more precise results for the ν-exponent which are also presented in table-I and compared to the five-loops resummation from other algorithms in Refs. [4,12].
Also to compare with other recent theoretical predictions, for N = 0, we get the result ν = 0.587142 compared to the recent accurate Monte Carlo simulation prediction from Ref.

Resummation of Four and Five-loops series for η exponent
For the critical exponent η of the O(N) model, the ε-expansion up to five loops is given by [4] where and the large-order for η of this model takes the form in Eq. (5) where [4] σ = 3 N + 8 and b = 3 + N 2 .
Note that the factored series ( has the large-order parameters The lowest order approximant is thus 2 F 0 which in this case is a four-loops approximant.

III.2.1. Four-loops resummation for η
The Hypergeometric-Meijer approximant is then: The resummation results of that order are shown in table-III. The results are reasonable but since the Hypergeometric approximant 2 F 0 has few number of parameters, it is expected that the improvement of the results needs higher loops to be incorporated.

III.2.2. The η five-loop resummation
In this case the Hypergeometric approximant is To determine the four unknown parameters we use the equations: Accordingly, the Hypergeometric-Meijer approximant for this order is given by: Our predictions that incorporate the fourth and fifth orders of divergent series of the ηexponent are listed in table-III . It is very clear that the simple algorithm we follow gives accurate results for few terms from the perturbation series as input. This can be more elaborated by looking at the large number of estimates for critical exponents in Ref. [47] too.
In fact, for the same order of perturbation series involved, the precision of resummation results for η are always less than that in ν or ω because the lowest order in the perturbation series of η is ε 2 and thus always approximated by Hypergeometric approximants of fewer parameters than that for ν or ω.

III.3. Resummation of the exponent ω
For the exponent ω we have the five-loops perturbation series as: where [2]  N This work JK [4] BCM [12], [14] 2 F 0 : ε 4 3 F 1 : ε 5 and the large-order parameters for that exponent are The two-loops resummation gives reasonable but not precise results so in the following, we shall list the resummation of three, four and five loops.

III.3.2. The ω four-loops Resummation
In this case also we use the approximant 3 F 1 (a 1 , a 2 , a 3 ; b 1 ; −σε) but we replace the fourth equation in the set in Eqs. (30) by:

III.3.3. ω five-loops approximant
The Hypergeometric function that can accommodate five-loops is Accordingly, the fifth order resummation for ω is In table-IV, we compared our results to predictions from the Janke-Kleinert Resummation for five-loops ε-expansion in Ref. [4] and Borel with conformal mapping in Refs. [12,14] for N = 0, 1, 2, 3 and 4. Again, the comparison shows that the algorithm we follow gives very accurate results from few orders of the perturbation series as input. In the way to get the ε-expansion for the critical exponents one has to obtain the dependance of the critical coupling on ε first. The expansion for the critical coupling g c up to fifth order is given by [4]: while the large order parameters are σ = 3 N +8 and b = 4+ N 2 . The third order approximation takes the form 3 F 1 (a 1 , a 2 , a 3 ; b 1 ; −σε) − 1 while the fourth order takes the same form except in the equations determining the parameters we use the large order constraint a 1 + a 2 + a 3 − b 1 − 2 = b. For the five-loops resummation we resummed the series for N = 1, 2, 3 and 4 using the Hypergeometric approximant f 1 3 F 1 (a 1 , a 2 , a 3 ; b 1 ; σε). For N = 0, however, we resummed the subtracted series gc(ε)−f 1 ε the Hypergeometric approximant: with the constraint a 1 + a 2 + a 3 − b 1 − 2 = b + 2. Such technical steps are well known in resummation techniques [4,44] which can be used in case no solution has been found for the equations defining the parameters. The prediction of these orders are shown in table-V and compared with other resummation results from Refs. [4,12,35,50]. In Ref. [14], the six-loops order of the renormalization group functions has been obtained and resummed using Borel with conformal mapping algorithm. The work led to the improvement of the previous resummation predictions of the five-loops order in Refs. [4,12]. This six-loops order of perturbation series represents a good test for the accuracy and stability of our resummation algorithm. We shall thus extend our work in the previous section to incorporate the six-loops weak-coupling data to compare with the recent results of Borel resummation and numerical predictions.  (6) 0.03627 (10) 0.82929 0.820 (7) 0.832 (6) This work [14] [16] This work [14] [17] A different ε has been used in Ref. [14] as the space-time dimension has been set as d − 2ε.
Accordingly, the n th coefficients in each perturbation series has to be divided by 2 n to keep the definition used in our work ( d − ε). For the critical exponent ν we then have where the first five coefficients ( c i ) are given by Eq.(13) while the sixth coefficients are given in table-VII. Accordingly we use the approximant 2 4 F 2 (a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 ; −σε) for the resummation of the ν −1 series above. In table VI, one can realize that our six-loop resummation for the critical exponent ν is very competitive either to the six-loops Borel with conformal mapping algorithm in Ref. [14] or Monte Carlo calculations ( ours are closer to numerical results).
For the critical exponent η, we have the series up to fifth order in Eq.(21) and we add the sixth coefficient from Ref. [14] as shown in table-VII. The Hypergeometric approximant 3 F 1 has been used for the resummation of the six-loops perturbation series of η and its resummation results are presented in table VI too.
For the critical exponent ω, the sixth coefficients e 6 are listed in Table- In the minimal subtraction scheme, Oliver Schnetz has obtained the seven-loops order of the renormalization group functions β, γ m 2 and γ φ for the O(N)-symmetric model [51]. Here γ m 2 is the mass anomalous dimension while γ φ represents the field anomalous dimension. In the following we list our resummation results for N = 0, 1, 2, 3 and 4 while the results are compared to recent calculations from different techniques in tables VIII, IX,X, XI and XII.

V.1. Resummation results for self-avoiding walks (N = 0)
For N = 0 and in three dimensions, the seven-loops order for the β-function is given by: We resummed this series using the approximant ( 5 F 3 (a 1 , a 2 , a 3 , a 4 , a 5 which resulted in the Meijer-G approximant of the form: The critical coupling is obtained from the zero of the β-function where we found g c = 0.53430.
The series for correction to scaling critical exponent ω is obtained from differentiating the above series with respect to g and it has been resummed using the approximant where the large-order constraint has been employed and we found the result ω = 0.85650. This result can be compared with the recent Monte Carlo simulations calculations in Ref. [19] that predicts the result ω = ∆ 1 ν = 0.899(12) (see table-VIII for comparison with different methods). The field anomalous dimension is also given by: The suitable Hypergeometric approximant used is The critical exponent η is obtained from the relation η = 2γ φ (g c ) where we get the result η = 0.03129. In a recent conformal bootstrap calculation the result η = 2∆ φ − 1 = 0.0282 (4) has been obtained [52] while the Monte Carlo result is η = 0.031043(3) in Refs. [14,18].
The seven-loops perturbation series for the anomalous mass dimension γ m 2 has been obtained in the same reference [51] where: We used ( 5 F 3 (a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 ; −g) − 1) too for the resummation of this series. The ν-exponent is then The recent Monte Carlo prediction gives the value ν = 0.63002 (10) in Ref. [16] while in Ref. [26] one can find the result ν = 0.62999(5) using conformal bootstrap calculations.
Likewise, the field anomalous dimension up to seven loops is given by: which is approximated by g( 4 F 2 (a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 ; −g)−1) and gives the result η = 0.03740.
Again the Monte Carlo simulations in Ref. [17] gives the values η = 0.0365(3). Also Monte Carlo simulations and finite-size scaling of 3D Potts Models in Ref. [23] gives the result η = 5 − 2y h = 0.036 (6) and the conformal bootstrap calculations is 0.0378(32) [25].  [14,27] for ν and ω, while η from Ref. [25]. MC simulations for ω is taken from Ref. [21] while ν and η are from Ref. [17]. The six-loop BCM resummation (ε 6 ) is taken from Ref. [14] and five-loops (ε 5 ) from same reference. NPRG results up to O(∂ 4 ) [54] are shown in the last row.  (30) 0.794 (9) 0.795 (6) 0.761 (12) A note to be mentioned is that one should not judge the convergence of the seven-loops resummation results by comparing with six-loops resummation or lower order resummation in this work. The point is that the seven-loops resummation in this work applied for the g-series but for the other orders we resummed the ε-series. Our aim behind resumming both available series is to test our algorithm using different types of perturbation series. To have an idea about the good convergence of our algorithm for the resummation of the g-series one should look at different orders of resummation of the g-series itself. For instance, for N = 4, we get ω = 0.77963 from five-loop resummation of the g-series, ω = 0.78162 from six loops compared to the seven-loops result in table-XII as ω = 0.80325.

VI. SUMMARY AND CONCLUSIONS
We show that divergent series with different large-order behaviors can be approximated by different generalized Hypergeometric functions p F q (a 1 , ...a p ; b 1 ....b q ; σz). The relation between the number of numerator and denominator parameters (p and q) is determined from the growth factor in the large-order behavior of the divergent series. For a divergent series with a growth factor n!, the series expansion of the Hypergeometric function p F q (a 1 , ...a p ; b 1 ....b q ; σz) where p = q + 2 can reproduce a large-order behavior with same growth factor. Accordingly, the Hypergeometric function p F p−2 (a 1 , .. The parametrization of the Hypergeometric function is then followed by the resummation step of using a representation in terms of Meijer-G function. We applied the algorithm to resum the divergent series representing critical exponents ν (ν −1 ), η and ω as well as the critical coupling up to ε 5 order as input. For N equals 0, 1, 2, 3 and 4, the results ought to be reasonable even for very low order of perturbation used to parametrize the Hypergeometric approximant. The results are greatly improved in using third order and being more precise in going to fourth order while the fifth order offers very competitive predictions when compared to other resummation algorithms in literature.
To show that the precise results extends to higher N values, we resummed the perturbation series for the exponent ν for N = 6, 8, 10 and 12. The precision of the results can be seen from table-II where we listed the 5 th order resummation results for the exponent ν and compared it with other methods.
All the Hypergeometric functions p F p−2 (a 1 , ...a p ; b 1 ....b p−2 ; σz) share the same analytic behavior. Accordingly, one expects no surprises in going to higher orders of resummation.
To test this clear fact as well as to seek more improved results, we resummed the six-loops order for the perturbation series for the exponents ν, η and ω for N = 1, 2, 3 and 4. The results are showing improved predictions for those exponents. When compared to other calculations, our results for the critical exponents are compatible with the recent six-loops BC resummation method in Ref [14], MC simulations calculations [16,17,[20][21][22][23]53] and conformal bootstrap methods [24,24,25,27,28,52].
The very recent seven-loops order (coupling-series) for the renormalization group functions β, γ φ and γ m 2 has been resummed too. Up to the best of our knowledge, no other resummation algorithm has been used to resum this order. Very accurate results for the critical coupling and the exponent ν have been extracted from the resummed functions.
In all of our calculations, we used weak-coupling and large-order data as input. The a i parameters in the Hypergeometric functions p F q (a 1 , ...a p ; b 1 ....b p−2 ; σz) are well known to represent the strong-coupling data [15]. However, the strong coupling expansion for the series under consideration has not been obtained yet ( up to the best of our knowledge ).
Accordingly, we cannot get benefited from this fact in further acceleration of the convergence of the resummation algorithm. However, the expansion coefficients of the Hypergeometric function depend on the strong-coupling parameters and they in turn constrained to mach the weak-coupling and large-order data. Accordingly, this algorithm is linking the unknown strong-coupling parameters to the known weak-coupling and large-order data. Thus the algorithm has the ability to predict the non-perturbative asymptotic strong-coupling behavior of a quantum field theory from knowing the weak coupling and large-order data. In other algorithms, this asymptotic behavior is predicted from optimization techniques and different optimizations can even lead to different results for the same theory.