Five-loop Approximation Research Articles

Renormalized coupling constants for the three-dimensional scalar λ ϕ 4 field theory and pseudo-ε-expansion

Renormalized coupling constants g_{2k} that enter the critical equation of state and determine nonlinear susceptibilities of the system possess universal values g*_{2k} at the Curie point. They are calculated, along with the ratios R_{2k} = g_{2k}/g_4^{k-1}, for the three-dimensional scalar \lambda\phi^4 field theory within the pseudo-\epsilon-expansion approach. Pseudo-\epsilon-expansions for g*_6, g*_8, R*_6, and R*_8 are derived in the five-loop approximation, numerical estimates are presented for R*_6 and R*_8. The higher-order coefficients of the pseudo-\epsilon-expansions for the sextic coupling are so small that simple Pade approximants turn out to be sufficient to yield very good numerical results. Their use gives R*_6 = 1.650 while the most recent lattice estimate is R*_6 = 1.649(2). For the octic coupling pseudo-\epsilon-expansions are less favorable from the numerical point of view. Nevertheless, Pade-Borel resummation leads in this case to R*_8 = 0.890, the number differing only slightly from the values R*_8 = 0.871, R*_8 = 0.857 extracted from the lattice and field-theoretical calculations.

Higgs Decay, $Z$ Decay and the QCD Beta Function

Recent developments in perturbative QCD, leading to the beta function in five-loop approximation are presented. In a first step the two most important decay modes of the Higgs boson are discussed: decays into a pair of gluons and, alternatively, decays into a bottom-antibottom quark pair. Subsequently the quark mass anomalous dimension is presented which is important for predicting the value of the bottom quark mass at high scales and, consequently, the Higgs boson decay rate into pair of massive quarks, in particular into $b\bar b$. In the next section the $\alpha_s^4$ corrections to the vector- and axial-vector correlator are discussed. These are the essential ingredients for the evaluation of the QCD corrections to the cross section for electron-positron annihilation into hadrons at low and at high energies, to the hadronic decay rate of the $\tau$ lepton and for the $Z$-boson decay rate into hadrons. Finally we present the prediction for the QCD beta-function in five-loop approximation, discuss the analytic structure of the result and compare with experiment at low and at high energies.

Universal effective coupling constant ratios of 3D scalarϕ4field theory and pseudo-ϵexpansion

The ratios R 2 k = g 2 k / g k − 1 4 of renormalized coupling constants g 2 k entering the small-field equation of state approach universal values R * 2 k at criticality. They are calculated for the three-dimensional λ ϕ 4 field theory within the pseudo- ϵ expansion approach. Pseudo- ϵ expansions for R * 6 , R * 8 , R * 10 are derived in the five-loop approximation, numerical estimates are obtained with a help of the Pade–Borel–Leroy resummation technique. Its use gives R * 6 = 1.6488, the number which perfectly agrees with the most recent lattice result R * 6 = 1.649. For the octic coupling the pseudo- ϵ expansion is less favorable numerically. Nevertheless the Pade–Borel–Leroy resummation leads to the estimate R * 8 = 0.890 close to the values R * 8 = 0.87, R * 8 = 0.857 extracted from the lattice and field-theoretical calculations. The pseudo- ϵ expansion for R * 10 turns out to have big and rapidly increasing coefficients. This makes correspondent estimates strongly dependent on the Borel–Leroy shift parameter b and prevents proper evaluation of R * 10

Adler function, sum rules and Crewther relation of order [formula omitted]: The singlet case

The analytic result for the singlet part of the Adler function of the vector current in a general gauge theory is presented in five-loop approximation. Comparing this result with the corresponding singlet part of the Gross-Llewellyn Smith sum rule (Baikov et al., 2010 [1]), we successfully demonstrate the validity of the generalized Crewther relation for the singlet part. This provides a non-trivial test of both our calculations and the generalized Crewther relation. Combining the result with the already available non-singlet part of the Adler function (Baikov et al., 2008 [2], Baikov et al., 2010 [3]) we arrive at the complete O(αs4) expression for the Adler function and, as a direct consequence, at the complete O(αs4) correction to the e+e− annihilation into hadrons in a general gauge theory.

QCD decoupling at four loops

We present the matching condition for the strong coupling constant α s ( n f ) at a heavy quark threshold to four loops in the modified minimal subtraction scheme. Our results lead to further decrease of the theoretical uncertainty of the evolution of the strong coupling constant through heavy quark thresholds. Using a low energy theorem we furthermore derive the effective coupling of the Higgs boson to gluons (induced by a virtual heavy quark) in four- and (partially) through five-loop approximation.

Critical thermodynamics of two-dimensional N-vector cubic model in the five-loop approximation

The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.

Critical behavior of two-dimensional cubic andMNmodels in the five-loop renormalization group approximation

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.

Critical thermodynamics of two-dimensional systems in the five-loop renormalization-group approximation

The paper is devoted to the calculation of renormalization-group (RG) functions in the O(n)-symmetry two-dimensional model of the λϕ 4 type in the five-loop approximation and to an analysis of the critical behavior of systems described by this model. Five-loop expansions for the β function and the critical indices are determined in bulk theory. They are summed up using the Padé-Borel and Padé-Borel-Le Roy methods, making it possible to optimize the summation procedure and to estimate the accuracy of the obtained numerical values. It is shown that in the Ising (n=1) case, as well as in other cases, the inclusion of the five-loop contribution to the β function displaces the coordinate of the Wilson fixed point only insignificantly, leaving it outside the interval formed by the results of computations on lattices; even “spreads” of the error in the renormalization group and lattice estimates do not overlap. This discrepancy is attributed to the effect of the nonanalytic com-ponent of the β function, which cannot be determined in perturbation theory. A computation of critical indices proves that, although the inclusion of the five-loop terms in the corresponding RG expansion slightly improves the concordance with the exact results, the nonanalytic contributions are apparently also significant in this case.

Five-loop renormalization-group expansions for the three-dimensionaln-vector cubic model and critical exponents for impure Ising systems

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pade-Borel-Leroy resummation under n = 0. These exponents are found to be: \gamma = 1.325 +/- 0.003, \eta = 0.025 +/- 0.01, \nu = 0.671 +/- 0.005, \alpha = - 0.0125 +/- 0.008, \beta = 0.344 +/- 0.006. For the correction-to-scaling exponent, the less accurate estimate \omega = 0.32 +/- 0.06 is obtained.

Critical exponents for a three-dimensional impure Ising model in the five-loop approximation

The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Pade-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.

On the calculation of 2 + ε RG functions in the Gross-Neveu model from large- N expansions of critical exponents

Using knowledge of the explicit n-dependence of the RG functions (in a MS scheme) and expressions of the critical exponents in the framework of the large- N expansion in the Gross-Neveu model we derive RG functions in the four- and five-loop approximation up to some unknown coefficients.

ϵ-Expansion for critical exponents: The O(ϵ 5) approximation

The O(ϵ 5) approximation to critical exponents of phase transitions of the second kind is calculated using the five-loop approximation to renormalization group functions of the gϕ 4 model in four dimensions. A special resummation technique is applied to the series obtained.