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Abouzeid M. Shalaby

Publications and source records attributed to Abouzeid M. Shalaby.

At least 19 recordsLinked to original sources
arXiv2026

Dependence of Critical Exponents Accuracy on the Number of Fields in the O(N) -Invariant ϕ^4 Model

The study of large-$N$ cases of the $O(N)$-symmetric model at criticality has attracted significant attention in recent years. In particular, a recent high-precision Monte Carlo study ( Physical Review B 105, 054428 (2022)) reported the most accurate estimates to date for the critical exponents $ν$, $η$, and $ω$ for $N \ge4$. Within the framework of the renormalization group (RG)---the cornerstone of the modern theory of critical phenomena---powerful computational approaches such as the $\varepsilon$-expansion and non-perturbative RG are available. Since the effective expansion parameter in the $\varepsilon$-expansion, $σ= \frac{3}{N+8}$, decreases with increasing N, one expects resummation techniques to become progressively more accurate in the large-N regime. To test this expectation, we apply the entire-hypergeometric resummation algorithm developed by Shalaby et al. to the recently obtained seven-loop divergent $\varepsilon$-series for $N \ge 4$, yielding high-precision estimates for $ν$, $η$, and $ω$ . Our analysis confirms the anticipated improvement in accuracy with increasing N. To assess the significance of these results, we note that at the same seven-loop order the $O(2)$ case exhibits errors that are an order of magnitude larger than those obtained from experiment, Monte Carlo simulations, and conformal-field-theory analysis. In contrast, for sufficiently large N, the errors in the present work are of the same order of magnitude as those from Monte Carlo and non-perturbative RG methods, demonstrating the strong predictive power of our resummation approach in the large-N regime.

hep-th↗
arXiv2024

Accurate Critical Exponents from the Optimal Truncation of the epsilon- Expansion within the O(N)-symmetric Field Theory for large N

The perturbation series for the renormalization group functions of the $O(N)-$symmetric $ϕ^4$ field theory are divergent but asymptotic. They are usually followed by Resummation calculations to extract reliable results. Although the same features exist for QED series, their partial sums can return accurate results because the perturbation parameter is small. In this work, however, we show that, for $N\ge4$, the partial sum ( according to optimal truncation) of the series for the exponents $ν$ and $η$ gives results that are very competitive to the recent Monte Carlo and Conformal field calculations. The series can be written in terms of the perturbation parameter $α=σ\varepsilon=\frac{3}{N+8}\varepsilon$ which is smaller for larger $N$ and thus as $N$ increases one expects accurate perturbative results like the QED case. Such optimal truncation, however, doesn't work for the series of the critical exponent $ω$ ( for intermediate values of $N$) as the truncated series includes only the first order. Nevertheless, for $N\ge4$, the large-order parameter $σ=\frac{3}{N+8}$ is getting smaller which rationalizes for the use of the Pad$\acute{e}$ approximation. For that aim, we first obtain the seven-loop $\varepsilon$-series from the recent available corresponding $g$-expansion. The seven-loop Pad$\acute{e}$ approximation gives accurate results for the three exponents. Besides, for all the seven orders in the series, the large-$N$ limit leads to the exact result predicted by the non-perturbative $1/N$-Expansion.

hep-th↗
arXiv2024

Toward Exact Critical Exponents from the low-order loop expansion of the Effective Potential in Quantum Field Theory

The asymptotic strong-coupling behavior as well as the exact critical exponents from scalar field theory even in the simplest case of $1+1$ dimensions have not been obtained yet. Hagen Kleinert has linked both critical exponents and strong coupling parameters to each other. He used a clevert variational technique ( back to kleinert and Feynman) to extract accurate values for the strong coupling parameters from which he was able to extract precise critical exponents. In this work, we suggest a simple method of using the effective potential (low-order) to obtain exact values for the strong-coupling parameters for the $ϕ^{4}$ scalar field theory in $0+1$ and $1+1$ space-time dimensions. For the $0+1$ case, our results coincide with the well-known exact values from literature while for the $1+1$ case we test the results by obtaining the corresponding exact critical exponent. As the effective potential is a well-established tool in quantum field theory, we expect that the results can be easily extended to the most important three dimensional case and then the dream of getting exact critical exponents is made possible.

hep-th↗
arXiv2022

High-Order Parametrization of the Hypergeometric-Meijer Approximants

In previous articles, we showed that, based on large-order asymptotic behavior, one can approximate a divergent series via the parametrization of a specific hypergeometric approximant. The analytical continuation is then carried out through a Mellin-Barnes integral representation of the hypergeometric approximant or equivalently using an equivalent form of the Meijer G-Function. The parametrization process involves the solution of a non-linear set of coupled equations which is hard to achieve (might be impossible) for high orders using normal PCs. In this work, we extend the approximation algorithm to accommodate any order (high or low) of the given series in a short time. The extension also allows us to employ non-perturbative information like strong-coupling and large-order asymptotic data which are always used to accelerate the convergence. We applied the algorithm for different orders (up to O($29$)) of the ground state energy of the $x^4$ anharmonic oscillator with and without the non-perturbative information. We also considered the available $20$ orders for the ground sate energy of the $\mathcal{PT}-$symmetric $ix^{3}$ anharmonic oscillator as well as the given $20$ orders of its strong-coupling expansion or equivalently the Yang-Lee model. For high order weak-coupling parametrization, accurate results have been obtained for the ground state energy and the non-perturbative parameters describing strong-coupling and large-order asymptotic behaviors. The employment of the non-perturbative data accelerated the convergence very clearly. The High temperature expansion for the susceptibility within the $SQ$ lattice has been also considered and led to accurate prediction for the critical exponent and critical temperature.

hep-th↗
arXiv2021

Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions

For theories that exhibit second order phase transition, we conjecture that the large-order asymptotic behavior of the strong-coupling ( High-Temperature) series expansion takes the form $σ^{n} n^{b}$ where $b$ is a universal parameter. The associated critical exponent is then given by $b+1$. The series itself can be approximated by the hypergeometric approximants $_{p}F_{p-1}$ which can mimic the same large-order behavior of the given series. Near the tip of the branch cut, the hypergeometric function $_{p}F_{p-1}$ has a power-law behavior from which the critical exponent and critical coupling can be extracted. The conjecture has been tested in this work for the perturbation series of the ground state energy of the Yang-Lee model as a strong-coupling form of the $\mathcal{PT}$-symmetric $iϕ^3$ theory and the High-Temperature expansion within the Ising model. From the known $b$ parameter for the Yang-Lee model, we obtained the exact critical exponents which reflects the universality of $b$. Very accurate prediction for $b$ has been obtained from the many orders available for the High-Temperature series expansion of the Ising model which in turn predicts accurate critical exponent. Apart from critical exponents, the hypergeometric approximants for the Yang-Lee model show almost exact predictions for the ground state energy from low orders of perturbation series as input.

cond-mat.stat-mech↗
arXiv2020

The $λ$-point anomaly in view of the seven-loop Hypergeometric resummation for the critical exponent $ν$ of the $O(2)$ $ϕ^4$ 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 $ν$ for the $O(2)$-symmetric $ϕ^4$ model. Our prediction gives the result $ν=0.6711(7)$ which is compatible with the value $ν=0.6709(1)$ 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 ( $ν=0.6716(6)$), the most precise result from Monte Carlo simulations ($ν=0.67169(7)$) as well as the recent conformal bootstrap calculations ($ν=0.67175(10)$). 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 $ν$ exponent in a way that may favor the theoretical predictions.

hep-th↗
arXiv2020

Critical Exponents of the O(N)-symmetric $ϕ^4$ Model from the \boldmath{\large{ $\varepsilon^7$}} 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 $ν,\ ω$ and $η$ 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 $α$ for the $XY$ model. We found the result $-0.0123(11)$ which is compatible with the famous experimental result of -0.0127(3) from the specific heat of zero gravity liquid helium superfluid transition 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 a significant improvement to the previous six-loop resummation predictions.

hep-th↗
arXiv2020

Weak-Coupling, Strong-Coupling and Large-Order Parametrization of the Hypergeometric-Meijer Approximants

Without Borel or Pad$\acute{e}$ techniques, we show that for a divergent series with $n!$ large-order growth factor, the set of Hypergeometric series $_{k+1}F_{k-1}$ represents suitable approximants for which there exist no free parameters. The divergent $_{k+1}F_{k-1}$ series are then resummed via their representation in terms of the Meijer-G function. The choice of $_{k+1}F_{k-1}$ accelerates the convergence even with only weak-coupling information as input. For more acceleration of the convergence, we employ the strong-coupling and large-order information. We obtained a new constraint that relates the difference of numerator and denominator parameters in the Hypergeometric approximant to one of the large-order parameters. To test the validity of that constraint, we employed it to obtain the exact partition function of the zero-dimensional $ϕ^4$ scalar field theory. The algorithm is also applied for the resummation of the ground state energies of $ϕ_{0+1}^{4}$ and $iϕ_{0+1}^{3}$ scalar field theories. We get accurate results for the whole coupling space and the precision is improved systematically in using higher orders. Precise results for the critical exponents of the $O(4)$-symmetric field model in three dimensions have been obtained from resummation of the recent six-loops order of the corresponding perturbation series. The recent seven-loops order for the $β$-function of the $ϕ^{4}_{3+1}$ field theory has been resummed which shows non-existence of fixed points. The first resummation result of the seven-loop series representing the fractal dimension of the two-dimensional self-avoiding polymer is presented here where we get a very accurate value of $d_f=1.3307$ compared to its exact value ($4/3\approx1.3333$).

hep-th↗
arXiv2020

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 $ν$,\ $η$ and $ω$ 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 $ν$ which are very accurate as well. The recent seven-loops order ($g$-series) for the renormalization group functions $β,γ_{ϕ^2}$ and $γ_{m^2}$ have been resummed too. Accurate predictions for the critical coupling and the exponents $ν$, $η$ and $ω$ have been extracted from $β$,$γ_{ϕ^2}$ and $γ_{m^2}$ approximants.

hep-th↗
arXiv2019

Extrapolating the precision of the Hypergeometric Resummation to Strong couplings with application to the $\mathcal{PT-}$Symmetric $iϕ^{3}$ Field Theory

In PRL 115, 143001 (2015), H. Mera et al. developed a new simple but precise Hypergeometric Resummation technique. In this work, we suggest to obtain half of the parameters of the Hypergeometric function from the strong coupling expansion of the physical quantity. Since these parameters are taking now their exact values they can improve the precision of the technique for the whole range of the coupling values. The second order approximant $\,_{2}F_{1}$ of the algorithm is applied to resum the perturbation series of the ground state energy of the $\mathcal{PT-}$symmetric $(iϕ^{3})_{0+1}$ field theory. It gives accurate results compared to exact calculations from the literature specially for very large coupling values. The $\mathcal{PT-}$ symmetry breaking of the Yang-Lee model has been investigated where third, fourth and fifth orders were able to get very accurate results when compared to other resummation methods involving $150$ orders. The critical exponent $ν$ of the $O(4)$-symmetric model in three dimensions has been precisely obtained using only first order of perturbation series as input. The algorithm can be extended easily to accommodate any order of perturbation series in using the generalized Hypergeometric function $\,_{k+1}F_{k}$ as it shares the same analytic properties with $\,_{2}F_{1}$.

hep-th↗
arXiv2018

Effective Action study of $\mathcal{PT}$-Symmetry Breaking for the non-Hermitian $\left( iϕ^{3}\right) _{6-ε}$ Theory and The Yang-Lee Edge Singularity

We use the effective potential method to study the $\mathcal{PT}$-symmetry breaking of the non-Hermitian $iϕ^{3}$ field theory in $6-ε$ space-time dimensions. The critical exponents so obtained coincide with the exact values listed in the literature. We showed that at the point of $\mathcal{PT}$-symmetry breaking, the vacuum-vacuum amplitude is certainly zero and the fugacity is one which mimics a Yang-Lee edge singularity in magnetic systems. What makes this work interesting is that it takes into account problems which are always overlooked in the literature for the Yang-Lee model like stability, unitarity and generation of Stokes wedges at space-time dimensions for which divergences occur in the theory . Besides, here we make direct calculation of critical exponents from the dependance of the order parameter on external magnetic field not from the density of zeros of the partition function.

hep-th↗
arXiv2014

Vacuum Stability of the $\mathcal{PT}$-Symmetric $\left( -ϕ^{4}\right) $ Scalar Field Theory

In this work, we study the vacuum stability of the classical unstable $\left( -ϕ^{4}\right) $ scalar field potential. Regarding this, we obtained the effective potential, up to second order in the coupling, for the theory in $1+1$ and $2+1$ space-time dimensions. We found that the obtained effective potential is bounded from below, which proves the vacuum stability of the theory in space-time dimensions higher than the previously studied $0+1$ case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the $\left( -ϕ^{4}\right) $ theory at the quantum mechanical level while our work extends the argument to the level of field quantization.

hep-th↗
arXiv2010

On the $iϕ^{3}$ $\mathcal{PT}$-symmetric Scalar Field Theory

In this work, we show that, for the $iϕ^{3}$ scalar field theory, their exists a contradiction between the assumption that the field is real and the fact that the quantized as well as the classical fields have to satisfy the Klein-Gordon equation. In solving the Klein-Gordon equation for the theory under investigation, we realized that the field is a pure imaginary solitary wave which spoils out the non-Hermiticity of the theory. Thus, instead of being non-Hermitian, the $iϕ^{3}$ scalar field theory is a kind of a Hermitian-Lee-Wick theory which suffers from the existence of the famous ghost states and instability problems. We applied a Canonical transformation to obtain a Non-Hermitian and non-$\mathcal{PT}$-symmetric representation which leads to the invalidity of the previous trials in the literature to cure the ghost states problem. Moreover, the solitonic solution is a non-topological one which is a very strange result to appear for a one component field theory. To account for this strange result, we conjecture that the $iϕ^{3}$ scalar field theory has an equivalent Hermitian and non-Lee wick theory that have a conserved Noether current.

hep-th↗
arXiv2009

Reply to "Comment on 'New ansatz for metric operator calculation in pseudo-Hermitian field theory '", arXiv:0906.4293

In this report, we reply to a recent comment by Carl M. Bender, Gregorio Benincasa and Hugh F. Jones on our work 'New ansatz for metric operator calculation in pseudo-Hermitian field theory (Phys. Rev. D. 79, 107702 (2009)). In fact, they figured out that there exist sign errors in our work which leaded to the conclusion of the invalidity of the ansatz introduced in our work. Here, we show that, the ansatz is valid in d+1 space-time dimensions, which by itself is a new and very important result. The importance of the work comes from the fact that it is the first time to have a metric operator for a quantum field theory which is local in the fields as well as valid in 3+1 space-time dimensions. Moreover, it is composed of the operators in the Hamiltonian itself which makes the Feynmann diagram calculations for the physical amplitudes go the same way as in conventional theories.

hep-th↗
arXiv2009

New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory

In this work, a new ansatz is introduced to make the calculations of the metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume that the metric operator is not only a functional of the field operators $ϕ$ and its conjugate field $π$ but also on the field gradient $\nablaϕ$. Rather than the locality of the metric operator obtained, the ansatz enables one to calculate the metric operator just once for all dimensions of the space-time. We calculated the metric operator of the $iϕ^{3}$ scalar field theory up to first order in the coupling. The higher orders can be conjectured from their corresponding operators in the quantum mechanical case available in the literature. We assert that, the calculations existing in literature for the metric operator in field theory are cumbersome and are done case by case concerning the dimension of space-time in which the theory is investigated. Moreover, while the resulted metric operator in this work is local, the existing calculations for the metric operator leads to a non-local one. Indeed, we expect that the new results introduced in this work will greatly lead to the progress of the studies in Pseudo-Hermitian field theories where there exist a lack of such kind of studies in the literature. In fact, with the aid of this work a rigorous study of a $\mathcal{PT}$-symmetric Higgs mechanism can be reached.

hep-th↗
arXiv2009

Effective Field calculations of the Energy Spectrum of the $\mathcal{PT}% $-Symmetric ($-x^{4}$) Potential

In this work, we show that the traditional effective field approach can be applied to the $\mathcal{PT}$-symmetric wrong sign ($-x^{4}$) quartic potential. The importance of this work lies in the possibility of its extension to the more important $\mathcal{PT}$-symmetric quantum field theory while the other approaches which use complex contours are not willing to be applicable. We calculated the effective potential of the massless $-x^{4}$ theory as well as the full spectrum of the theory. Although the calculations are carried out up to first order in the coupling, the predicted spectrum is very close to the exact one taken from other works. The most important result of this work is that the effective potential obtained, which is equivalent to the Gaussian effective potential, is bounded from below while the classical potential is bounded from above. This explains the stability of the vacuum of the theory. The obtained quasi-particle Hamiltonian is non-Hermitian but $\mathcal{PT}$-symmetric and we showed that the calculation of the metric operator can go perturbatively. In fact, the calculation of the metric operator can be done even for higher dimensions (quantum field theory) which, up till now, can not be calculated in the other approaches either perturbatively or in a closed form due to the possible appearance of field radicals. Moreover, we argued that the effective theory is perturbative for the whole range of the coupling constant and the perturbation series is expected to converge rapidly (the effective coupling $g_{eff}={1/6}$).

hep-th↗