arXiv subjects

Jan Fischer

Publications and source records attributed to Jan Fischer.

34 records · Page 2Linked to original sources

arXiv2008

Exponential decay in a spin bath

We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a well-defined purely exponential decay for special (`narrowed') bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spin-bath dynamics have been investigated in the non-Markovian limit, giving super-exponential or power-law decay of correlation functions. We calculate the relevant decoherence time T_2 explicitly for free-induction decay and find a simple expression with dependence on bath polarization, magnetic field, the shape of the electron wave function, dimensionality, total nuclear spin I, and isotopic concentration for experimentally relevant heteronuclear spin systems.

cond-mat.mes-hall↗

arXiv2007

Correlated projection operator approach to non-Markovian dynamics in spin baths

The dynamics of an open quantum system is usually studied by performing a weak-coupling and weak-correlation expansion in the system-bath interaction. For systems exhibiting strong couplings and highly non-Markovian behavior this approach is not justified. We apply a recently proposed correlated projection superoperator technique to the model of a central spin coupled to a spin bath via full Heisenberg interaction. Analytical solutions to both the Nakajima-Zwanzig and the time-convolutionless master equation are determined and compared with the results of the exact solution. The correlated projection operator technique significantly improves the standard methods and can be applied to many physical problems such as the hyperfine interaction in a quantum dot.

quant-ph↗

arXiv2007

Comment on "Infrared freezing of Euclidean QCD observables"

Recently, P. M. Brooks and C.J. Maxwell [Phys. Rev. D{\bf 74} 065012 (2006)] claimed that the Landau pole of the one-loop coupling at $Q^2=Λ^2$ is absent from the leading one-chain term in a skeleton expansion of the Euclidean Adler ${\cal D}$ function. Moreover, in this approximation one has continuity along the Euclidean axis and a smooth infrared freezing, properties known to be satisfied by the "true" Adler function. We show that crucial in the derivation of these results is the use of a modified Borel summation, which leads simultaneously to the loss of another fundamental property of the true Adler function: the analyticity implied by the Källen-Lehmann representation.

hep-ph↗

arXiv2006

Infrared freezing of Euclidean observables and analyticity in perturbative QCD

The renormalization-group improved finite order expansions of the QCD observables have an unphysical singularity in the Euclidean region, due to the Landau pole of the running coupling. Recently it was claimed that, by using a modified Borel representation, the leading one-chain term in a skeleton expansion of the Euclidean QCD observables is finite and continuous across the Landau pole, and then exhibits an infrared freezing behaviour, vanishing at $Q^2=0$. In the present paper we show, using for illustration the Adler-${\cal D}$ function, that the above Borel prescription violates the causality properties expressed by energy-plane analyticity: the function ${\cal D}(Q^2)$ thus defined is the boundary value of a piecewise analytic function in the complex plane, instead of being a standard analytic function. So, the price to be paid for the infrared freezing of Euclidean QCD observables is the loss of a fundamental property of local quantum field theory.

hep-ph↗

arXiv2005

On the infrared freezing of perturbative QCD in the Minkowskian region

The infrared freezing of observables is known to hold at fixed orders of perturbative QCD if the Minkowskian quantities are defined through the analytic continuation from the Euclidean region. In a recent paper [1] it is claimed that infrared freezing can be proved also for Borel resummed all-orders quantities in perturbative QCD. In the present paper we obtain the Minkowskian quantities by the analytic continuation of the all-orders Euclidean amplitudes expressed in terms of the inverse Mellin transform of the corresponding Borel functions [2]. Our result shows that if the principle of analytic continuation is preserved in Borel-type resummations, the Minkowskian quantities exhibit a divergent increase in the infrared regime, which contradicts the claim made in [1]. We discuss the arguments given in [1] and show that the special redefinition of Borel summation at low energies adopted there does not reproduce the lowest order result obtained by analytic continuation.

hep-ph↗

arXiv2003

Analytic structure in the coupling constant plane in perturbative QCD

We investigate the analytic structure of the Borel-summed perturbative QCD amplitudes in the complex plane of the coupling constant. Using the method of inverse Mellin transform, we show that the prescription dependent Borel-Laplace integral can be cast, under some conditions, into the form of a dispersion relation in the a-plane. We also discuss some recent works relating resummation prescriptions, renormalons and nonperturbative effects, and show that a method proposed recently for obtaining QCD nonperturbative condensates from perturbation theory is based on special assumptions about the analytic structure in the coupling plane that are not valid in QCD.

hep-ph↗

arXiv2002

Renormalization scheme dependence in the case of a QCD non-power perturbative expansion

A novel, non-power, expansion of QCD quantities replacing the standard perturbative expansion in powers of the renormalized couplant a has recently been introduced and examined by two of us. Being obtained by analytic continuation in the Borel plane, the new expansion functions W_n(a) share the basic analyticity properties with the expanded quantity. In this note we investigate the renormalization scale dependence of finite order sums of this new expansion for the phenomenologically interesting case of the tau-lepton decay rate.

hep-ph↗

arXiv2001

Analytic continuation and perturbative expansions in QCD

Starting from the divergence pattern of perturbative quantum chromodynamics, we propose a novel, non-power series replacing the standard expansion in powers of the renormalized coupling constant $a$. The coefficients of the new expansion are calculable at each finite order from the Feynman diagrams, while the expansion functions, denoted as $W_n(a)$, are defined by analytic continuation in the Borel complex plane. The infrared ambiguity of perturbation theory is manifest in the prescription dependence of the $W_n(a)$. We prove that the functions $W_n(a)$ have branch point and essential singularities at the origin $a=0$ of the complex $a$-plane and their perturbative expansions in powers of $a$ are divergent, while the expansion of the correlators in terms of the $W_n(a)$ set is convergent under quite loose conditions

hep-ph↗

arXiv2001

Dispersion relations in differential form

Various forms of derivative dispersion relations, in which the dispersion integral is replaced by a series of derivatives of the imaginary part of a scattering amplitude, are reviewed. Conditions of their validity and practical applicability as well as their relevance to high-energy small-angle hadron-hadron scattering are discussed.

hep-th↗

arXiv2000

Convergence of the expansion of the Laplace-Borel integral in perturbative QCD improved by conformal mapping

The optimal conformal mapping of the Borel plane was recently used to accelerate the convergence of the perturbation expansions in QCD. In this work we discuss the relevance of the method for the calculation of the Laplace-Borel integral expressing formally the QCD Green functions. We define an optimal expansion of the Laplace-Borel integral in the principal value prescription and establish conditions under which the expansion is convergent.

hep-ph↗

arXiv1999

Operator product expansion and analyticity

We discuss the current use of the operator product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the euclidean region, we observe how the bound varies with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions chosen. The results obtained are discussed in connetcion with calculations of the coupling constant α_{s} from the τdecay.

hep-ph↗

arXiv1998

Accelerated convergence of perturbative QCD by optimal conformal mapping of the Borel plane

The technique of conformal mappings is applied to enlarge the convergence of the Borel series and to accelerate the convergence of Borel-summed Green functions in perturbative QCD. We use the optimal mapping, which takes into account the location of all the singularities of the Borel transform as well as the present knowledge about its behaviour near the first branch points. The determination of α_{s}(m_τ) from the hadronic decay rate of the τ-lepton is discussed as an illustration of the method.

hep-ph↗

arXiv1998

The operator-product expansion away from euclidean region

The role of the operator-product expansion in QCD calculations is discussed. Approximating the two-point correlation function by several terms and assuming an upper bound on the truncation error along the euclidean ray, we consider two model situations to examine how the bound develops with increasing deflection from the euclidean ray towards the cut. We obtain explicit bounds on the truncation error and show how they worsen with the increasing deflection. The result does not support the believe that the remainder is constant for all angles in the complex energy plane. Further refinements of the formalism are dicussed.

hep-ph↗

arXiv1997

On the role of power expansions in quantum field theory

Methods of summation of power series relevant to applications in quantum theory are reviewed, with particular attention to expansions in powers of the coupling constant and in inverse powers of an energy variable. Alternatives to the Borel summation method are considered and their relevance to different physical situations is discussed. Emphasis is placed on quantum chromodynamics. Applications of the renormalon language to perturbation expansions (resummation of bubble chains) in various QCD processes are reported and the importance of observing the full renormalization-group invariance in predicting observables is emphasized. News in applications of the Borel-plane formalism to phenomenology are conveyed. The properties of the operator-product expansion along different rays in the complex plane are examined and the problem is studied how the remainder after subtraction of the first $n$ terms depends on the distance from euclidean region. Estimates of the remainder are obtained and their strong dependence on the nature of the discontinuity along the cut is shown. Relevance of this subject to calculations of various QCD effects is discussed.

hep-ph↗

arXiv1996

A Surprise in Sum Rules - Modulating Factors

A generic physical situation is considered where Im $Π$, the imaginary part of polarization operator (generalized susceptibility), can be measured on a finite interval and the high frequency asymptotics (up to a few orders) of $Π$ can be calculated theoretically. In such a case, it is desirable to derive an equivalent form of the Kramers-Kronig dispersion relation, the so-called sum rule, in which both the high-frequency part of Im $Π$ in the dispersion integral and the high-order contribution to $Π$ are suppressed. We provide a general framework for derivation of such sum rules, without any recourse to an infinite-order differential operator. We derive sum rules for a wide set of weight functions and show that any departure from the $e^{-t}$ behaviour of the weight function in sum rules leads to modulating factors on the theoretical side of sum rules, providing its low frequency regularization. We argue that by including modulating factors one can extend the domain of validity of sum rules further to an intermediate region of frequencies and can account for ``bumps" which were observed numerically on the phenomenological side of sum rules at ``intermediate'' frequencies.

hep-ph↗

arXiv1995

High-order behaviour and summation methods in perturbative QCD

After reviewing basic facts about large-order behaviour of perturbation expansions in various fields of physics, I consider several alternatives to the Borel summation method and discuss their relevance to different physical situations. Then I convey news about the singularities in the Borel plane, and discuss the topical subject of the resummation of renormalon chains and its application in various QCD processes.

hep-ph↗