arXiv subjects

J. Zinn-Justin

Publications and source records attributed to J. Zinn-Justin.

18 recordsLinked to original sources

arXiv2020

Two-Loop Corrections to the Large-Order Behavior of Correlation Functions in the One-Dimensional N-Vector Model

For a long time, the predictive limits of perturbative quantum field theory have been limited by our inability to carry out loop calculations to arbitrarily high order, which become increasingly complex as the order of perturbation theory is increased. This problem is exacerbated by the fact that perturbation series derived from loop diagram (Feynman diagram) calculations represent asymptotic (divergent) series which limits the predictive power of perturbative quantum field theory. Here, we discuss an ansatz which could overcome these limits, based on the observations that (i) for many phenomenologically relevant field theories, one can derive dispersion relations which relate the large-order growth (the asymptotic limit of "infinite loop order") with the imaginary part of arbitrary correlation functions, for negative coupling ("unstable vacuum"), and (ii) one can analyze the imaginary part for negative coupling in terms of classical field configurations (instantons). Unfortunately, the perturbation theory around instantons, which could lead to much more accurate predictions for the large-order behavior of Feynman diagrams, poses a number of technical as well as computational difficulties. Here, we study, to further the above mentioned ansatz, correlation functions in a one-dimensional (1D) field theory with a quartic self-interaction and an O(N) internal symmetry group, otherwise known as the 1D N-vector model. Our focus is on corrections to the large-order growth of perturbative coefficients, i.e., the limit of a large number of loops in the Feynman diagram expansion. We evaluate, in momentum space, the two-loop corrections for the two-point correlation function, and its derivative with respect to the momentum, as well as the two-point correlation function with a wigglet insertion. Also, we study the four-point function.

hep-th↗

arXiv2019

Charting the European Course to the High-Energy Frontier

We review the capabilities of two projects that have been proposed as the next major European facility, for consideration in the upcoming update of the European Strategy for Particle Physics: CLIC and FCC. We focus on their physics potentials and emphasise the key differences between the linear or circular approaches. We stress the uniqueness of the FCC-ee programme for precision electroweak physics at the $Z$ peak and the $WW$ threshold, as well as its unequalled statistics for Higgs physics and high accuracy for observing possible new phenomena in Higgs and $Z$ decays, whereas CLIC and FCC-ee offer similar capabilities near the $t \overline t$ threshold. Whilst CLIC offers the possibility of energy upgrades to 1500 and 3000 GeV, FCC-ee paves the way for FCC-hh. The latter offers unique capabilities for making direct or indirect discoveries in a new energy range, and has the highest sensitivity to the self-couplings of the Higgs boson and any anomalous couplings. We consider the FCC programme to be the best option to maintain Europe's place at the high-energy frontier during the coming decades.

hep-ex↗

arXiv2010

Imaginary Cubic Perturbation: Numerical and Analytic Study

The analytic properties of the ground state resonance energy E(g) of the cubic potential are investigated as a function of the complex coupling parameter g. We explicitly show that it is possible to analytically continue E(g) by means of a resummed strong coupling expansion, to the second sheet of the Riemann surface, and we observe a merging of resonance and antiresonance eigenvalues at a critical point along the line arg(g) = 5 pi/4. In addition, we investigate the convergence of the resummed weak-coupling expansion in the strong coupling regime, by means of various modifications of order-dependent mappings (ODM), that take special properties of the cubic potential into account. The various ODM are adapted to different regimes of the coupling constant. We also determine a large number of terms of the strong coupling expansion by resumming the weak-coupling expansion using the ODM, demonstrating the interpolation between the two regimes made possible by this summation method.

math-ph↗

arXiv2010

Order-dependent mappings: strong coupling behaviour from weak coupling expansions in non-Hermitian theories

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x^2+i v x^3, v real, has a real spectrum. This conjecture has been generalized to a class of so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v^2, and that some information about the location of level crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent "local" disk of convergence and analytic continuation by an order-dependent mapping of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.

math-ph↗

arXiv2010

Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators

This is the third article in a series of three papers on the resonance energy levels of anharmonic oscillators. Whereas the first two papers mainly dealt with double-well potentials and modifications thereof [see J. Zinn-Justin and U. D. Jentschura, Ann. Phys. (N.Y.) 313 (2004), pp. 197 and 269], we here focus on simple even and odd anharmonic oscillators for arbitrary magnitude and complex phase of the coupling parameter. A unification is achieved by the use of PT-symmetry inspired dispersion relations and generalized quantization conditions that include instanton configurations. Higher-order formulas are provided for the oscillators of degrees 3 to 8, which lead to subleading corrections to the leading factorial growth of the perturbative coefficients describing the resonance energies. Numerical results are provided, and higher-order terms are found to be numerically significant. The resonances are described by generalized expansions involving intertwined non-analytic exponentials, logarithmic terms and power series. Finally, we summarize spectral properties and dispersion relations of anharmonic oscillators, and their interconnections. The purpose is to look at one of the classic problems of quantum theory from a new perspective, through which we gain systematic access to the phenomenologically significant higher-order terms.

math-ph↗

arXiv2009

Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree

We present a unified treatment, including higher-order corrections, of anharmonic oscillators of arbitrary even and odd degree. Our approach is based on a dispersion relation which takes advantage of the PT-symmetry of odd potentials for imaginary coupling parameter, and of generalized quantization conditions which take into account instanton contributions. We find a number of explicit new results, including the general behaviour of large-order perturbation theory for arbitrary levels of odd anharmonic oscillators, and subleading corrections to the decay width of excited states for odd potentials, which are numerically significant.

math-ph↗

arXiv2008

Structure, Time Propagation and Dissipative Terms for Resonances

For odd anharmonic oscillators, it is well known that complex scaling can be used to determine resonance energy eigenvalues and the corresponding eigenvectors in complex rotated space. We briefly review and discuss various methods for the numerical determination of such eigenvalues, and also discuss the connection to the case of purely imaginary coupling, which is PT-symmetric. Moreover, we show that a suitable generalization of the complex scaling method leads to an algorithm for the time propagation of wave packets in potentials which give rise to unstable resonances. This leads to a certain unification of the structure and the dynamics. Our time propagation results agree with known quantum dynamics solvers and allow for a natural incorporation of structural perturbations (e.g., due to dissipative processes) into the quantum dynamics.

math-ph↗

arXiv2006

Quantum Dot Potentials: Symanzik Scaling, Resurgent Expansions and Quantum Dynamics

This article is concerned with a special class of the ``double-well-like'' potentials that occur naturally in the analysis of finite quantum systems. Special attention is paid, in particular, to the so-called Fokker-Planck potential, which has a particular property: the perturbation series for the ground-state energy vanishes to all orders in the coupling parameter, but the actual ground-state energy is positive and dominated by instanton configurations of the form exp(-a/g), where a is the instanton action. The instanton effects are most naturally taken into account within the modified Bohr-Sommerfeld quantization conditions whose expansion leads to the generalized perturbative expansions (so-called resurgent expansions) for the energy values of the Fokker-Planck potential. Until now, these resurgent expansions have been mainly applied for small values of coupling parameter g, while much less attention has been paid to the strong-coupling regime. In this contribution, we compare the energy values, obtained by directly resumming generalized Bohr-Sommerfeld quantization conditions, to the strong-coupling expansion, for which we determine the first few expansion coefficients in powers of g^(-2/3). Detailed calculations are performed for a wide range of coupling parameters g and indicate a considerable overlap between the regions of validity of the weak-coupling resurgent series and of the strong-coupling expansion. Apart from the analysis of the energy spectrum of the Fokker-Planck Hamiltonian, we also briefly discuss the computation of its eigenfunctions. These eigenfunctions may be utilized for the numerical integration of the (single-particle) time-dependent Schroedinger equation and, hence, for studying the dynamical evolution of the wavepackets in the double-well-like potentials.

cond-mat.other↗

arXiv2006

Effective Action and Phase Structure of Multi-Layer Sine-Gordon Type Models

We analyze the effective action and the phase structure of N-layer sine-Gordon type models, generalizing the results obtained for the two-layer sine-Gordon model found in [I. Nandori, S. Nagy, K. Sailer and U. D. Jentschura, Nucl. Phys. B725, 467-492 (2005)]. Besides the obvious field theoretical interest, the layered sine-Gordon model has been used to describe the vortex properties of high transition temperature superconductors, and the extension of the previous analysis to a general N-layer model is necessary for a description of the critical behaviour of vortices in realistic multi-layer systems. The distinction of the Lagrangians in terms of mass eigenvalues is found to be the decisive parameter with respect to the phase structure of the N-layer models, with neighbouring layers being coupled by quadratic terms in the field variables. By a suitable rotation of the field variables, we identify the periodic modes (without explicit mass terms) in the N-layer structure, calculate the effective action and determine their Kosterlitz-Thouless type phase transitions to occur at a coupling parameter β^2_{c} = 8 N π, where N is the number of layers (or flavours in terms of the multi-flavour Schwinger model).

hep-th↗

arXiv2002

Chiral Anomalies and Topology

When a quantum field theory has a symmetry, global or local like in gauge theories, in the tree or classical approximation formal manipulations lead to believe that the symmetry can also be implemented in the full quantum theory, provided one uses the proper quantization rules. While this is often true, it is not a general property and therefore requires a proof because simple formal manipulations ignore the unavoidable divergences of perturbation theory. The existence of invariant regularizations allows solving the problem in most cases but the combination of gauge symmetry and chiral fermions leads to subtle issues. Depending on the specific group and field content, anomalies are found: obstructions to the quantization of chiral gauge symmetries.Because anomalies take the form of local polynomials in the fields, are linked to local group transformations, but vanish for global (rigid) transformations they have a topological character.In these notes we review various perturbative and non-perturbative regularization techniques, and show that they leave room for possible anomalies when both gauge fields and chiral fermions are present. We determine the form of anomalies in simple examples. We relate anomalies to the index of the Dirac operator in a gauge background. We exhibit gauge instantons that contribute to the anomaly in the example of the CP(N-1) models and SU(2) gauge theories. We briefly mentioned a few physical consequences. For many years the problem of anomalies had been discussed only within the framework of perturbation theory. New non-perturbative solutions based on lattice regularization have recently been proposed. We describe the so-called overlap and domain wall fermion formulations.

hep-th↗

arXiv2001

Higher-Order Corrections to Instantons

The energy levels of the double-well potential receive, beyond perturbation theory, contributions which are non-analytic in the coupling strength; these are related to instanton effects. For example, the separation between the energies of odd- and even-parity states is given at leading order by the one-instanton contribution. However to determine the energies more accurately multi-instanton configurations have also to be taken into account. We investigate here the two-instanton contributions. First we calculate analytically higher-order corrections to multi-instanton effects. We then verify that the difference betweeen numerically determined energy eigenvalues, and the generalized Borel sum of the perturbation series can be described to very high accuracy by two-instanton contributions. We also calculate higher-order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two-instanton effect and with exact data for the first 200 perturbative coefficients.

math-ph↗

arXiv2000

Quantum Field Theory at Finite Temperature: An Introduction

In these notes we review some properties of Statistical Quantum Field Theory at equilibrium, i.e Quantum Field Theory at finite temperature. We explain the relation between finite temperature quantum field theory in (d,1) dimensions and statistical classical field theory in d+1 dimensions. This identification allows to analyze the finite temperature QFT in terms of the renormalization group and the theory of finite size effects of the classical theory. We discuss in particular the limit of high temperature (HT) or the situation of finite temperature phase transitions. There the concept of dimensional reduction plays an essential role. Dimensional reduction in some sense reflects the known property that quantum effects are not important at high temperature. We illustrate these ideas with several standard examples, phi^4 field theory, the non-linear sigma model and the Gross-Neveu model, gauge theories. We construct the corresponding effective reduced theories at one-loop order, using the technique of mode expansion of fields in the imaginary time variable. In models where the field is a vector with N components, the large N expansion provides another specially convenient tool to study dimensional reduction.

hep-ph↗

arXiv2000

Precise determination of critical exponents and equation of state by field theory methods

Renormalization group, and in particular its Quantum Field Theory implementation has provided us with essential tools for the description of the phase transitions and critical phenomena beyond mean field theory. We therefore review the methods, based on renormalized phi^4_3 quantum field theory and renormalization group, which have led to a precise determination of critical exponents of the N-vector model (R. Guida and J. Zinn-Justin, J. Phys. A31 (1998) 8103. cond-mat/9803240). and of the equation of state of the 3D Ising model (R. Guida and J. Zinn-Justin, Nucl. Phys. B489 [FS] (1997) 626, hep-th/9610223.). These results are among the most precise available probing field theory in a non-perturbative regime.

hep-th↗

arXiv1996

The O(N) vector model in the large N limit revisited: multicritical points and double scaling limit

The multicritical points of the $O(N)$ invariant $N$ vector model in the large $N$ limit are reexamined. Of particular interest are the subtleties involved in the stability of the phase structure at critical dimensions. In the limit $N \to \infty$ while the coupling $g \to g_c$ in a correlated manner (the double scaling limit) a massless bound state $O(N)$ singlet is formed and powers of $1/N$ are compensated by IR singularities. The persistence of the $N \to \infty$ results beyond the leading order is then studied with particular interest in the possible existence of a phase with propagating small mass vector fields and a massless singlet bound state. We point out that under certain conditions the double scaled theory of the singlet field is non-interacting in critical dimensions.

hep-th↗

arXiv1994

2D Gravity and Random Matrices

We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising--like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

hep-th↗

arXiv1993

Large Order Behaviour of 2D Gravity Coupled to $d<1$ Matter

We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general $(p,q)$ conformal matter. In a previous work it was proven that at large order $k$ the string susceptibility had a generic $a^kΓ(2k-\ud)$ behaviour. Moreover the constant $a$, relevant for the problem of Borel summability, was determined for all one-matrix models. We here obtain a set of equations for this constant in the general $(p,q)$ model. String equations can be derived from the construction of two differential operators $P,Q$ satisfying canonical commutation relations $[P,Q]=1$. We show that the equation for $a$ is determined by the form of the operators $P,Q$ in the spherical or semiclassical limits. The results for the general one-matrix models are then easily recovered. Moreover, since for the $(p,q)$ string models such $p=(2m+1)q\pm1$ the semiclassical forms of $P,Q$ are explicitly known, the large order behaviour is completely determined. This class contains all unitary $(q+1,q)$ models for which the answer is specially simple. As expected we find that the topological expansion for unitary models is not Borel summable. \preprint{SPhT/92-163}, Plain-TeX, macro harvmac

hep-th↗

arXiv1992

The $O(n)$ model on a random surface: critical points and large order behaviour

In this article we report a preliminary investigation of the large $N$ limit of a generalized one-matrix model which represents an $O(n)$ symmetric model on a random lattice. The model on a regular lattice is known to be critical only for $-2\le n\le 2$. This is the situation we shall discuss also here, using steepest descent. We first determine the critical and multicritical points, recovering in particular results previously obtained by Kostov. We then calculate the scaling behaviour in the critical region when the cosmological constant is close to its critical value. Like for the multi-matrix models, all critical points can be classified in terms of two relatively prime integers $p,q$. In the parametrization $p=(2m+1)q \pm l$, $m,l$ integers such that $0<l<q$, the string susceptibility exponent is found to be $γ_{\rm string}=-2l/(p+q-l)$. When $l=1$ we find that all results agree with those of the corresponding $(p,q)$ string models, otherwise they are different.\par We finally explain how to derive the large order behaviour of the corresponding topological expansion in the double scaling limit.

hep-th↗

arXiv1991

$O(N)$ Vector Field Theories in the Double Scaling Limit

$O(N)$ invariant vector models have been shown to possess non-trivial scaling large $N$ limits, at least perturbatively within the loop expansion, a property they share with matrix models of 2D quantum gravity. In contrast with matrix models, however, vector models can be solved in arbitrary dimensions. We present here the analysis of field theory vector models in $d$ dimensions and discuss the nature and form of the critical behaviour. The double scaling limit corresponds for $d>1$ to a situation where a bound state of the $N$-component fundamental vector field $ϕ$, associated with the $ϕ^2$ composite operator, becomes massless, while the field $ϕ$ itself remains massive. The limiting model can be described by an effective local interaction for the corresponding $O(N)$ invariant field. It has a physical interpretation as describing the statistical properties of a class of branched polymers.\par It is hoped that the $O(N)$ vector models, which can be investigated in their most general form, can serve as a test ground for new ideas about the behaviour of 2D quantum gravity coupled with $d>1$ matter.

hep-th↗