Symmetric Φ4 Theory Research Articles

Self-similar sequence transformation for critical exponents

Self-similar sequence transformation is an original type of nonlinear sequence transformations allowing for defining effective limits of asymptotic sequences. The method of self-similar factor transformations is shown to be regular. This method is applied for calculating the critical exponents of the O(N)-symmetric φ4 theory in three dimensions by summing asymptotic ε expansions. It is shown that this method is straightforward and essentially simpler than other summation techniques involving complicated numerical calculations, while enjoying comparable accuracy.

Critical exponents from the weak-coupling, strong-coupling and large-order parametrization of the hypergeometric ([formula omitted]) approximants

In this work, we suggest a new parametrization for the hypergeometric (k+1Fk) approximants introduced by Mera et al. (2015). The new parametrization enables the approximants to accommodate all perturbative and non-perturbative information of the divergent series as input. Also, the parametrization has been shown to account for the n! growth factor of the given perturbation series provided that one of the denominator parameters of the hypergeometric approximant takes large values. The algorithm with the new parametrization has been tested using two quantum mechanical problems where one can incorporate the weak-coupling, strong-coupling and large-order information. Accurate results have been obtained in using a relatively low order from the perturbation series. Since strong-coupling behavior is not yet known for the renormalization group functions of the O(N)-symmetric ϕ4 theory, we used weak-coupling and large-order parametrization to resum the seven-loop critical exponents ν,η and ω for N=0,1,2,3,4. In view of the recent results from six-loop resummation as well as Monte Carlo simulations and conformal bootstrap calculations, our results show a clear improvement to the six-loop results.

Field theories for loop-erased random walks

Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the n→0 limit of ϕ4-theory with O(n)-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the O(n)-symmetric ϕ4 theory at n=−2 whose link to LERWs was known in two dimensions due to conformal field theory. Here it is established in arbitrary dimension via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. We explicitly show that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. This allows us to compute the fractal dimension of LERWs to order ε5 where ε=4−d. In particular, in d=3 our theory gives zLERW(d=3)=1.6243±0.001, in excellent agreement with the estimate z=1.62400±0.00005 of numerical simulations.

Critical mass renormalization in renormalized ϕ4 theories in two and three dimensions

We consider the O(N)-symmetric ϕ4 theory in two and three dimensions and determine the nonperturbative mass renormalization needed to obtain the ϕ4 continuum theory. The required nonperturbative information is obtained by resumming high-order perturbative series in the massive renormalization scheme, taking into account their Borel summability and the known large-order behavior of the coefficients. The results are in good agreement with those obtained in lattice calculations.

Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the (4−∈)-Dimensionally Regularized O(n)-Symmetric φ4 Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the O(n)-symmetric φ4 theory are found based on the instanton approach in the minimal subtraction scheme for the (4−∈)-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.

Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4−e)-expansion for the scaling function of the pair correlator of the O(n)-symmetric φ4-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the e3 order of the expansion of the scaling function in the small-τ domain.

Minimal renormalization without ϵ-expansion: Three-loop amplitude functions of the 0( n) symmetric Φ4 theory in three dimensions below Tc

We present an analytic three-loop calculation for thermodynamic quantities of the O(n) symmetric Φ 4 theory below T c within the minimal subtraction scheme at fixed dimension d = 3. Goldstone singularities arising at an intermediate stage in the calculation of O( n) symmetric quantities cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the three-loop terms of the amplitude functions ƒ Φ, F + and F − of the order parameter and the specific heat above and below Tc, respectively, without using the e = 4- d expansion. A Borel resummation for the case n = 2 yields resummed amplitude functions f Φ and F − that are slightly larger than the one-loop results. Accurate knowledge of these functions is needed for testing the renormalization-group prediction of critical-point universality along the λ −line of superfluid 4He. Combining the three-loop result for F − with a recent five-loop calculation of the additive renormalization constant of the specific heat yields excellent agreement between the calculated and measured universal amplitude ratio A +/ A - of the specific heat of 4He. In addition we use our result for f Φ to calculate the universal combination Rc of the amplitudes of the order parameter, the susceptibility and the specific heat for n = 2 and n = 3. Our Borel-resummed three-loop result for R c is significantly more accurate than the previous result obtained from the ϵ-expansion up to O( ϵ 2.

Higgs mechanism in the Gaussian functional approximation

The Higgs mechanism is explicity verified and its gauge invariance confirmed in the Gaussian ground-state functional approximation to theO(2) symmetric ϕ4 theory coupled to an Abelian gauge field. We demonstrate this by using the recent proof of the Goldstone theorem in the said approximation and the Englert-Brout-Schwinger method. The mass of the vector gauge boson is evaluated and expressed in terms of the dressed properties of the scalar fields and the gauge field coupling constant.

Scaling analysis of the O(4)-symmetric φ4-theory in the broken phase

We study the O(4)-symmetric φ 4-theory in the scaling region of the broken phase using the standard and a Symanzik improved action with infinite bare self-coupling λ. A high precision Monte Carlo simulation is performed by applying the reflection cluster algorithm. Employing the histogram method we analytically continue to a sequence of values of the hopping parameter κ neighbouring the actually simulated ones. In the investigated vicinity of the critical point κ c finite volume effects effecting, e.g., the determination of the field expectation value Σ and the mass m σ of the σ-particle, are very well described by one-loop renormalized perturbation theory. We carry out a detailed scaling analysis on a high level of precision. Finally we discuss the upper bound on the Higgs mass for both kinds of actions.

O(N) symmetric finite-temperatureϕ 4 theory in 2+1 dimensions

Starting from the grand partition function, the symmetry breaking of a BE gas in 2+1 dimension is investigated both analytically and numerically. For this the Gaussian effective potential method is extended toμ (chemical potential)-dependentO(N) finite-temperature symmetricϕ4 theory. The relevant integrals involved are evaluated in a simple and straightforward manner. The results are compared with those of others obtained by the loop expansion method.

Computation of phase shift in four-dimensional O(4) φ4 theory by Monte Carlo simulation

Abstract The phase shift of the O(4) symmetric φ4 theory in the symmetric phase is calculated numerically using the relation between phase shift and energy level of two-particle state recently derived by Luscher. The result is checked against the perturbative prediction for the phase shift, and the practical feasibility of the method is discussed.

Scaling analysis of the O(4)-symmetric Φ4-theory in the broken phase

We study the Φ 4 -theory deep in the scaling region of the broken phase using the standard and a Symanzik improved action with infinite bare self-coupling λ. In the investigated vicinity of the critical point κ c finite volume effects are very well described by one-loop renormalized perturbation theory. We carry out a detailed scaling analysis of our results on a level of precision which is actually higher than that of Lüscher and Weisz.

Five-loop renormalization group functions of O(n)-symmetric φ4-theory and ϵ-expansions of critical exponents up to ϵ5

Motivated by the discovery of errors in six of the 135 diagrams in the published five-loop expansions of the β-function and the anomalous dimensions of the O(n)-symmetric φ4-theory in D=4-ϵ dimensions we present the results of a full analytic reevaluation of all diagrams. The divergences are removed by minimal subtraction and ϵ-expansions are given for the critical exponents η,ν and ω up to order ϵ5.