Finite-order Perturbation Theory Research Articles

Soft photon radiation and entanglement

We study the entanglement between soft and hard particles produced in generic scattering processes in QED. The reduced density matrix for the hard particles, obtained via tracing over the entire spectrum of soft photons, is shown to have a large eigenvalue, which governs the behavior of the Renyi entropies and of the non-analytic part of the entanglement entropy at low orders in perturbation theory. The leading perturbative entanglement entropy is logarithmically IR divergent. The coefficient of the IR divergence exhibits certain universality properties, irrespectively of the dressing of the asymptotic charged particles and the detailed properties of the initial state. In a certain kinematical limit, the coefficient is proportional to the cusp anomalous dimension in QED. For Fock basis computations associated with two-electron scattering, we derive an exact expression for the large eigenvalue of the density matrix in terms of hard scattering amplitudes, which is valid at any finite order in perturbation theory. As a result, the IR logarithmic divergences appearing in the expressions for the Renyi and entanglement entropies persist at any finite order of the perturbative expansion. To all orders, however, the IR logarithmic divergences exponentiate, rendering the large eigenvalue of the density matrix IR finite. The all-orders Renyi entropies (per unit time, per particle flux), which are shown to be proportional to the total inclusive cross-section in the initial state, are also free of IR divergences. The entanglement entropy, on the other hand, retains non-analytic, logarithmic behavior with respect to the size of the box (which provides the IR cutoff) even to all orders in perturbation theory.

Comment on “Improved many-body expansions from eigenvector continuation”

In a recent paper [Phys. Rev. C 101, 041302(R) (2020)] Demol et al. published an interesting method to improve results from finite-order perturbation theory. Their applications on nuclear many-body problems show that the numerical results are very accurate, capable of converting divergent series to fast-converging sequences. We argue that one reason for this impressive accuracy is connected to those of related methods which proved to be useful in many-electron theory.

Spurious poles in the scattering of electric and magnetic charges

Theories with both electric and magnetic charges (“mutually non-local” theories) have several major obstacles to calculating scattering amplitudes. Even when the interaction arises through the kinetic mixing of two, otherwise independent, U(1)’s, so that all low-energy interactions are perturbative, difficulties remain: using a self-dual, local formalism leads to spurious poles at any finite order in perturbation theory. Correct calculations must show how the spurious poles cancel in observable scattering amplitudes. Consistency requires that one type of charge is confined as a result of one of the U(1)’s being broken. Here we show how the constraints of confinement and parity conservation on observable processes manages to cancel the spurious poles in scattering and pair production amplitudes, paving the way for systematic studies of the experimental signatures of “dark” electric-magnetic processes. Along the way we demonstrate some novel effects in electric-magnetic interactions, including that the amplitude for single photon production of magnetic particles by electric particles vanishes.

Conformal mapping of the Borel plane: Going beyond perturbative QCD

The power corrections in the Operator Product Expansion (OPE) of QCD correlators can be viewed mathematically as an illustration of the transseries concept, which allows to recover a function from its asymptotic divergent expansion. Alternatively, starting from the divergent behavior of the perturbative QCD encoded in the singularities in the Borel plane, a modified expansion can be defined by means of the conformal mapping of this plane. A comparison of the two approaches concerning their ability to recover nonperturbative properties of the true correlator was not explored up to now. In the present paper, we make a first attempt to investigate this problem. We use for illustration the Adler function and observables expressed as integrals of this function along contours in the complex energy plane. We show that the expansions based on the conformal mapping of the Borel plane go beyond finite-order perturbation theory, containing an infinite number of terms when reexpanded in powers of the coupling. Moreover, the expansion functions exhibit nonperturbative features of the true function, while the expansions have a tamed behavior at large orders and are expected even to be convergent. Using these properties, we argue that there are no mathematical reasons for supplementing the expansions based on the conformal mapping of the Borel plane by additional arbitrary power corrections. Therefore, we make the conjecture that they provide an alternative to the standard OPE in approximating the QCD correlator. This conjecture allows to slightly improve the accuracy of the strong coupling extracted from the hadronic $\tau$ decay width. Using the optimal expansions based on conformal mapping and the contour-improved prescription of renormalization-group resummation, we obtain $\alpha_s(m_\tau^2)=0.314 \pm 0.006$, which implies $\alpha_s(m_Z^2)=0.1179 \pm 0.0008$.

Parafermionic clock models and quantum resonance

We explore the $\mathbb{Z}_{N}$ parafermionic clock-model generalisations of the p-wave Majorana wire model. In particular we examine whether zero-mode operators analogous to Majorana zero-modes can be found in these models when one introduces chiral parameters to break time reversal symmetry. The existence of such zero-modes implies $N$-fold degeneracies throughout the energy spectrum. We address the question directly through these degeneracies by characterising the entire energy spectrum using perturbation theory and exact diagonalisation. We find that when $N$ is prime, and the length $L$ of the wire is finite, the spectrum exhibits degeneracies up to a splitting that decays exponentially with system size, for generic values of the chiral parameters. However, at particular parameter values (resonance points), band crossings appear in the unperturbed spectrum that typically result in a splitting of the degeneracy at finite order. We find strong evidence that these preclude the existence of strong zero-modes for generic values of the chiral parameters. In particular we show that in the thermodynamic limit, the resonance points become dense in the chiral parameter space. When $N$ is not prime, the situation is qualitatively different, and degeneracies in the energy spectrum are split at finite order in perturbation theory for generic parameter values, even when the length of the wire $L$ is finite. Exceptions to these general findings can occur at special "anti-resonant" points. Here the evidence points to the existence of strong zero modes and, in the case of the achiral point of the the $N=4$ model, we are able to construct these modes exactly.

Consistent Perturbative Fixed Point Calculations in QCD and Supersymmetric QCD.

We suggest how to consistently calculate the anomalous dimension γ_{*} of the ψ[over ¯]ψ operator in finite order perturbation theory at an infrared fixed point for asymptotically free theories. If the n+1 loop beta function and n loop anomalous dimension are known, then γ_{*} can be calculated exactly and fully scheme independently in a Banks-Zaks expansion through O(Δ_{f}^{n}), where Δ_{f}=N[over ¯]_{f}-N_{f}, N_{f} is the number of flavors, and N[over ¯]_{f} is the number of flavors above which asymptotic freedom is lost. For a supersymmetric theory, the calculation preserves supersymmetry order by order in Δ_{f}. We then compute γ_{*} through O(Δ_{f}^{2}) for supersymmetric QCD in the dimensional reduction scheme and find that it matches the exact known result. We find that γ_{*} is astonishingly well described in perturbation theory already at the few loops level throughout the entire conformal window. We finally compute γ_{*} through O(Δ_{f}^{3}) for QCD and a variety of other nonsupersymmetric fermionic gauge theories. Small values of γ_{*} are observed for a large range of flavors.

Complete normal ordering 1: Foundations

We introduce a new prescription for quantising scalar field theories (in generic spacetime dimension and background) perturbatively around a true minimum of the full quantum effective action, which is to ‘complete normal order’ the bare action of interest. When the true vacuum of the theory is located at zero field value, the key property of this prescription is the automatic cancellation, to any finite order in perturbation theory, of all tadpole and, more generally, all ‘cephalopod’ Feynman diagrams. The latter are connected diagrams that can be disconnected into two pieces by cutting one internal vertex, with either one or both pieces free from external lines. In addition, this procedure of ‘complete normal ordering’ (which is an extension of the standard field theory definition of normal ordering) reduces by a substantial factor the number of Feynman diagrams to be calculated at any given loop order. We illustrate explicitly the complete normal ordering procedure and the cancellation of cephalopod diagrams in scalar field theories with non-derivative interactions, and by using a point splitting ‘trick’ we extend this result to theories with derivative interactions, such as those appearing as non-linear σ-models in the world-sheet formulation of string theory. We focus here on theories with trivial vacua, generalising the discussion to non-trivial vacua in a follow-up paper.

Reemergence of rapidity-scale uncertainty in soft-collinear effective theory

The artificial separation of a full-theory mode into distinct collinear and soft modes in soft-collinear effective theory leads to divergent integrals over rapidity, which are not present in the full theory. Rapidity divergence introduces an additional scale into the problem, giving rise to its own renormalization group with respect to this new scale. Two contradicting claims exist in the literature concerning rapidity scale uncertainty. One camp has shown that the results of perturbative calculations depend on the precise choice of rapidity scale. The other has derived an all-order factorization formula with no dependence on rapidity scale, by using a form of analytic regulator to regulate rapidity divergences. We deliver a simple resolution to this controversy by deriving an alternative form of the all-order factorization formula with an analytic regulator that, despite being formally rapidity scale independent, reveals how rapidity scale dependence arises when it is truncated at a finite order in perturbation theory. With our results, one can continue to take advantage of the technical ease and simplicity of the analytic regulator approach while correctly taking into account rapidity scale dependence. As an application, we update our earlier study of $WW$ production with jet veto by including rapidity scale uncertainty. While the central values of the predictions are unchanged, the scale uncertainties are increased and consistency between the next-to-leading logarithm and next-to-next-to-leading logarithm calculations is improved.

Perturbative renormalization of lattice $ \mathcal{N} = 4 $ super Yang-Mills theory

We consider N=4 super Yang-Mills theory on a four-dimensional lattice. The lattice formulation under consideration retains one exact supersymmetry at non-zero lattice spacing. We show that this feature combined with gauge invariance and the large point group symmetry of the lattice theory ensures that the only counterterms that appear at any order in perturbation theory correspond to renormalizations of existing terms in the bare lattice action. In particular we find that no mass terms are generated at any finite order of perturbation theory. We calculate these renormalizations by examining the fermion and auxiliary boson self energies at one loop and find that they all exhibit a common logarithmic divergence which can be absorbed by a single wavefunction renormalization. This finding implies that at one loop only a fine tuning of the finite parts is required to regain full supersymmetry in the continuum limit.

Dynamical structure factor at smallq for theXXZ spin-1/2 chain

We combine Bethe ansatz and field theory methods to study the longitudinal dynamical structure factorSzz(q,ω) for theanisotropic spin-1/2 chain in the gapless regime. Using bosonization, we derive a low-energy effectivemodel, including the leading irrelevant operators (band curvature terms) whichaccount for boson decay processes. The coupling constants of the effectivemodel for finite anisotropy and finite magnetic field are determined exactly bycomparison with corrections to thermodynamic quantities calculated by Betheansatz. We show that a good approximation for the shape of the on-shell peak ofSzz(q,ω) inthe interacting case is obtained by rescaling the result for free fermions by certain coefficients extractedfrom the effective Hamiltonian. In particular, the width of the on-shell peak is argued to scale likeδωq∼q2 and this prediction is shown to agree with the width of the two-particle continuumat finite fields calculated from the Bethe ansatz equations. An exception to theq2 scaling is found at finite field and large anisotropy parameter (near theisotropic point). We also present the calculation of the high-frequency tail ofSzz(q,ω) in the region using finite-order perturbation theory in the band curvature terms. Both thewidth of the on-shell peak and the high-frequency tail are compared withSzz(q,ω) calculated by Bethe ansatz for finite chains using determinant expressions for the formfactors and excellent agreement is obtained. Finally, the accuracy of the form factors ischecked against the exact first moment sum rule and the static structure factor calculatedby density matrix renormalization group (DMRG).

Dynamical field theory for glass-forming liquids, self-consistent resummations andtime-reversal symmetry

We analyse the symmetries and the self-consistent perturbative approaches of dynamicalfield theories for glass-forming liquids. In particular, we focus on the time-reversalsymmetry, which is crucial to obtain fluctuation–dissipation relations (FDRs). Previousfield theoretical treatment violated this symmetry, whereas others pointed out thatconstructing symmetry-preserving perturbation theories is a crucial and open issue. In thiswork we solve this problem and then apply our results to the mode-coupling theory of theglass transition (MCT).We show that in the context of dynamical field theories for glass-forming liquidstime-reversal symmetry is expressed as a nonlinear field transformation that leaves theaction invariant. Because of this nonlinearity, standard perturbation theories generically donot preserve time-reversal symmetry and in particular fluctuation–dissipationrelations. We show how one can cure this problem and set up symmetry preservingperturbation theories by introducing some auxiliary fields. As an outcome we obtainSchwinger–Dyson dynamical equations that automatically preserve FDR and that serveas a basis for carrying out symmetry-preserving approximations. We apply ourresults to the mode-coupling theory of the glass transition, revisiting previousfield theory derivations of MCT equations and showing that they genericallyviolate FDR. We obtain symmetry-preserving mode-coupling equations and discusstheir advantages and drawbacks. Furthermore, we show, contrary to previousworks, that the structure of the dynamic equations is such that the ideal glasstransition is not cut off at any finite order of perturbation theory, even in thepresence of coupling between current and density. The opposite results found inprevious field theoretical works, such as the ones based on nonlinear fluctuatinghydrodynamics, were only due to an incorrect treatment of time-reversal symmetry.

AN APPLICATION OF THIRD FORMALISM OF QUANTUM STATISTICS AND A DIAGONALIZATION THEOREM IN ASYMPTOTIC EXACT THEORY OF SUPERCONDUCTIVITY

Starting from an electron-phonon interacting model, by means of the third formalism of quantum statistics, and with the aid of a diagonalization theorem, the Tc of superconductors from weak coupling to strong coupling cases are studied in a unified way. Our results are comparable with those of McMillan's theory, (which is valid up to the electron-phonon (e-ph) coupling parameter λ~1, a good agreement is shown for λ<1.3) and the Allen and Dynes's theory. Especially our results are very close to those of Hg spectrum with λ~2.0 and some strong coupling compounds. The theory is exact in the thermodynamic limit, without introducing variation method, the compensation of the dangerous diagrams in finite order in perturbation theory, abnormal green's function etc.

Supersymmetry on the lattice and the Leibniz rule

The major obstacle to a supersymmetric theory on the lattice is the failure of the Leibniz rule. We analyze this issue by using the Wess–Zumino model and a general Ginsparg–Wilson operator, which is local and free of species doublers. We point out that the Leibniz rule could be maintained on the lattice if the generic momentum k μ carried by any field variable satisfies | ak μ |< δ in the limit a→0 for arbitrarily small but finite δ. This condition is expected to be satisfied generally if the theory is finite perturbatively, provided that discretization does not induce further symmetry breaking. We thus first render the continuum Wess–Zumino model finite by applying the higher derivative regularization which preserves supersymmetry. We then put this theory on the lattice, which preserves supersymmetry except for a breaking in interaction terms by the failure of the Leibniz rule. By this way, we define a lattice Wess–Zumino model which maintains the basic properties such as U(1)× U(1) R symmetry and holomorphicity. We show that this model reproduces continuum theory in the limit a→0 up to any finite order in perturbation theory; in this sense all the supersymmetry breaking terms induced by the failure of the Leibniz rule are irrelevant. We then suggest that this discretization may work to define a low energy effective theory in a non-perturbative way.

RENORMALIZATION GROUP APPROACHES TO STRONGLY CORRELATED ELECTRON SYSTEMS

In recent years the numerical renormalization group (NRG) method has been extended to the calculation of dynamic response functions and transport properties of magnetic impurity models. The approach can now be applied more widely to lattice models of strongly correlated electron systems by the use of dynamical mean field theory (DMFT), in which the lattice problem is transformed into one for an effective impurity with an additional self-consistency constraint. We review these developments and assess the potential for further applications of this approach. We also discuss an alternative approach to renormalization, renormalized perturbation theory, in which the leading asymptotically exact results for the low temperature regime for a number of magnetic impurity models can be obtained within finite order perturbation theory.

Scale-independent calculation of sin 2θefflept

We present a calculation of the electroweak mixing parameter sin 2 θ eff lept that incorporates known higher order effects, shares the desirable convergence properties of the MS scheme, and has the important theoretical advantage of being strictly independent of the electroweak scale in finite orders of perturbation theory. We also show how this formulation can be extended to the calculation of the W mass M W . The results provide accurate, scale-independent evaluations of these important parameters, as functions of the Higgs boson mass M H , and are compared with previous calculations in order to analyze the scheme and scale dependence of the electroweak corrections.

QCD from a five-dimensional point of view

We propose a 5-dimensional definition for the physical 4D-Yang-Mills theory. The fifth dimension corresponds to the Monte-Carlo time of numerical simulations of QCD_4. The 5-dimensional theory is a well-defined topological quantum field theory that can be renormalized at any given finite order of perturbation theory. The relation to non-perturbative physics is obtained by expressing the theory on a lattice, a la Wilson. The new fields that must be introduced in the context of a topological Yang-Mills theory have a simple lattice expression. We present a 5-dimensional critical limit for physical correlation functions and for dynamical auto-correlations, which allows new Monte-Carlo algorithm based on the time-step in lattice units given by $\e = g_0^{-13/11}$ in pure gluodynamics. The gauge-fixing in five dimensions is such that no Gribov ambiguity occurs. The weight is strictly positive, because all ghost fields have parabolic propagators and yield trivial determinants. We indicate how our 5-dimensional description of the Yang-Mills theory may be extended to fermions.

Asymptotic structure of perturbative series for $\tau$ lepton decay observables: $m_s^2$ corrections

In a previous paper [1] we performed an analysis of the asymptotic structure of the perturbation theory series for semileptonic $\tau$ lepton decays in the massless limit. We extend our analysis to the Cabibbo suppressed $\Delta S=1$ decay modes of the $\tau$ lepton. In particular we address the problem of $m_s^2$ corrections to the theoretical formulas. The properties of the asymptotic behavior of the finite order perturbation theory series for the coefficient functions of the $m_s^2$ corrections are studied.

Asymptotic structure of perturbative series for $\tau$ lepton observables

We analyze tau lepton decay observables, namely moments of the hadronic spectral density in the finite energy interval (0,M_\tau), within finite order perturbation theory including \alpha_s^4 corrections. The start of asymptotic growth of perturbation theory series is found at this order in a scheme invariant manner. We establish the ultimate accuracy of finite order perturbation theory predictions and discuss the construction of optimal observables.

Asymptotic structure of perturbative series for $\tau$ lepton observables

We analyze \(\tau\) lepton decay observables, namely the moments of the hadronic spectral density in the finite energy interval \((0,M_\tau)\), within finite order perturbation theory including \(\alpha_{\mathrm{s}}^4\) corrections. The start of the asymptotic growth of the perturbation theory series is found at this order in a scheme-invariant manner. We establish the ultimate accuracy of finite order perturbation theory predictions and discuss the construction of optimal observables.

Asymptotic structure of perturbative series for

In a previous paper we performed an analysis of asymptotic structure of perturbation theory series for semileptonic $\tau$-lepton decays in massless limit. We extend our analysis to the Cabibbo suppressed $\Delta S=1$ decay modes of the $\tau$ lepton. In particular we address the problem of $m_s^2$ corrections to theoretical formulas. The properties of the asymptotic behavior of the finite order perturbation theory series for the coefficient functions of the $m_s^2$ corrections are studied.