Worldline Formalism Research Articles

Semi-transparent boundary conditions in the worldline formalism

The interaction of a quantum field with a background containing a Dirac delta function with support on a surface of codimension 1 represents a particular kind of matching conditions on that surface for the field. In this paper, we show that the worldline formalism can be applied to this model. We obtain the asymptotic expansion of the heat kernel corresponding to a scalar field on in the presence of an arbitrary regular potential and subject to this kind of matching conditions on a flat surface. We also consider two such surfaces and compute their Casimir attraction due to the vacuum fluctuations of a massive scalar field weakly coupled to the corresponding Dirac deltas.

The conformal window from the worldline formalism

We use the worldline formalism to derive a universal relation for the lower boundary of the conformal window in non-supersymmetric QCD-like theories. The derivation relies on the convergence of the expansion of the fermionic determinant in terms of Wilson loops. The expansion shares a similarity with the lattice strong coupling expansion and the genus expansion in string theory. Our result relates the lower boundary of the conformal window in theories with different representations and different gauge groups. Finally, we use SQCD to estimate the boundary of the conformal window in QCD-like theories and compare it with other approaches.

Interplay between geometry and temperature for inclined Casimir plates

We provide further evidence for the nontrivial interplay between geometry and temperature in the Casimir effect. We investigate the temperature dependence of the Casimir force between an inclined semi-infinite plate above an infinite plate in D dimensions using the worldline formalism. Whereas the high-temperature behavior is always found to be linear in T in accordance with dimensional-reduction arguments, different power-law behaviors at small temperatures emerge. Unlike the case of infinite parallel plates, which shows the well-known T{sup D} behavior of the force, we find a T{sup D-1} behavior for inclined plates, and a {approx}T{sup D-0.3} behavior for the edge effect in the limit where the plates become parallel. The strongest temperature dependence {approx}T{sup D-2} occurs for the Casimir torque of inclined plates. Numerical as well as analytical worldline results are presented.

On the Yang–Mills two-loop effective action with worldline methods

We derive the two-loop effective action for covariantly constant field strength of pure Yang–Mills theory in the presence of an infrared scale. The computation is done in the framework of the worldline formalism, based on a generalization procedure of constructing multiloop effective actions in terms of the bosonic worldline path integral. The two-loop β-function is correctly reproduced. This is the first derivation in the worldline formulation, and serves as a non-trivial check on the consistency of the multiloop generalization procedure in the worldline formalism.

Detours and paths: BRST complexes and worldline formalism

We construct detour complexes from the BRST quantization of worldline diffeomorphism invariant systems. This yields a method to efficiently extract physical quantum field theories from particle models with first class constraint algebras. As an example, we show how to obtain the Maxwell detour complex by gauging N=2 supersymmetric quantum mechanics in curved space. Then we concentrate on first class algebras belonging to a class of recently introduced orthosymplectic quantum mechanical models and give generating functions for detour complexes describing higher spins of arbitrary symmetry types. The first quantized approach facilitates quantum calculations and we employ it to compute the number of physical degrees of freedom associated to the second quantized, field theoretical actions.

Gravitational corrections to the Euler-Heisenberg Lagrangian

We use the worldline formalism for calculating the one-loop effective action for the Einstein-Maxwell background induced by charged scalars or spinors, in the limit of low energy and weak gravitational field but treating the electromagnetic field nonperturbatively. The effective action is obtained in a form which generalizes the standard proper-time representation of the Euler-Heisenberg Lagrangian. We compare with previous work and discuss possible applications.

Simplicity in the structure of QED and gravity amplitudes

We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude.

Dependence of the Direction of the Casimir Force on the Shape of the Boundary

The Casimir force due to a massless scalar field satisfying Dirichlet boundary conditions may draw a piston into the neck or toward the bulb of a flasklike container. In the worldline formalism this is due to competing contributions to the interaction energy from two types of Brownian bridges. Geometrical subtractions define a finite interaction energy. The procedure does not require regularization and appears well suited for numerical studies. The worldline analysis is robust with respect to variations in the shape of both piston and flask.

Scalar heat kernel with boundary in the worldline formalism

The worldline formalism has in recent years emerged as a powerful tool for the computation of effective actions and heat kernels. However, implementing nontrivial boundary conditions in this formalism has turned out to be a difficult problem. Recently, such a generalization was developed for the case of a scalar field on the half-space + × D−1, based on an extension of the associated worldline path integral to the full D using image charges. We present here an improved version of this formalism which allows us to write down non-recursive master formulas for the n-point contribution to the heat kernel trace of a scalar field on the half-space with Dirichlet or Neumann boundary conditions. These master formulas are suitable to computerization. We demonstrate the efficiency of the formalism by a calculation of two new heat-kernel coefficients for the half-space, a4 and a9/2.

Beyond the quenched/probe-brane approximation in lattice/holographic QCD

We propose a method to improve the quenched approximation. The method, based on the worldline formalism, takes into account effects of quark loops. The idea is mostly useful for AdS/CFT (holographic) calculations. To demonstrate the method we estimate screening (string breaking) effects by a simple holographic calculation.

AdS/CFT CORRESPONDENCE VIA R-CURRENT CORRELATION FUNCTIONS REVISITED

Motivated by realizing open/closed string duality in the work by Gopakumar [Phys. Rev. D70, 025009 (2004)], we study two- and three-point correlation functions of R-current vector fields in [Formula: see text] super-Yang–Mills theory. These correlation functions in free field limit can be derived from the worldline formalism and are written as heat kernel integrals in the position space. We show that reparametrizing these integrals converts them to the expected AdS supergravity results which are known in terms of bulk to boundary propagators. We expect that this reparametrization corresponds to transforming open string moduli parametrization to the closed string ones.

Scalar field with Robin boundary conditions in the worldline formalism

The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this paper we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A1, A3/2 and A2 of the heat kernel expansion of a scalar field on the positive real line.

Photon–graviton mixing in an electromagnetic field

Einstein–Maxwell theory implies the mixing of photons with gravitons in an external electromagnetic field. This process and its possible observable consequences have been studied at tree level for many years. We use the worldline formalism for obtaining an exact integral representation for the one-loop corrections to this amplitude due to scalars and fermions. We study the structure of this amplitude, and obtain exact expressions for various limiting cases.

D0-brane recoil revisited

One-loop string scattering amplitudes computed using the standard D0-brane conformal field theory (CFT) suffer from infrared divergences associated with recoil. A systematic framework to take recoil into account is the worldline formalism, where fixed boundary conditions are replaced by dynamical D0-brane worldlines. We show that, in the worldline formalism, the divergences that plague the CFT are automatically cancelled in a non-trivial way. The amplitudes derived in the worldline formalism can be reproduced by deforming the CFT with a specific "recoil operator", which is bilocal and different from the ones previously suggested in the literature.

Dynamics of theO(N)model in a strong magnetic background field as a modified noncommutative field theory

In the presence of a strong magnetic field, the effective action of a composite scalar field in a scalar $O(N)$ model is derived using two different methods. First, in the framework of world-line formalism, the 1PI $n$-point vertex function for the composites is determined in the limit of a strong magnetic field. Then, the $n$-point effective action of the composites is calculated in the regime of lowest Landau level dominance. It is shown that in the limit of a strong magnetic field, the results coincide and an effective field theory arises which is comparable with the conventional noncommutative field theory. In contrast to the ordinary case, however, the UV/IR mixing is absent in this modified noncommutative field theory.

Heavy-quark condensate at zero and finite temperatures for various forms of the short-distance potential

With the use of the world-line formalism, the heavy-quark condensate in the SU(N)-QCD is evaluated for the cases when the next-to-1/ r term in the quark-antiquark potential at short distances is either quadratic, or linear. In the former case, which takes place in the stochastic vacuum model, the standard QCD-sum-rules result is reproduced, while the latter result is a novel one. Explicitly, it is UV-finite only in less than four dimensions. This fact excludes a possibility to have, in four dimensions, very short strings, whose length is smaller than the vacuum correlation length. Accordingly, the linear term in the potential also vanishes at such distances, and OPE is not violated. In any number of dimensions, the obtained novel expression for the quark condensate depends on the string tension at short distances, rather than on the gluon condensate, and grows linearly with the number of colors in the same way as the standard QCD-sum-rules expression. The use of the world-line formalism enables one to generalize further both results to the case of finite temperatures. A generalization of the QCD-sum-rules expression to the case of an arbitrary number of space-time dimensions is also obtained and is shown to be UV-finite, provided this number is smaller than six.

Mixed heavy-quark–gluon condensate in the stochastic vacuum model and dual superconductor

The world-line formalism is used for the evaluation of the mixed heavy-quark-gluon condensate in two models of QCD - the stochastic vacuum model and the dual superconductor one. Calculations are performed for an arbitrary dimensionality of space-time $d\ge 2$. While in the stochastic vacuum model, the condensate is UV finite up to $d=8$, in the dual superconductor model it is UV divergent at any $d\ge 2$. A regularization of this divergence is proposed, which makes quantitative the condition of the type-II dual superconductor. The obtained results are generalized to the case of finite temperatures. Corrections to the both, mixed and standard, heavy-quark condensates, which appear due to the variation of the gauge field at the scale of the vacuum correlation length, are evaluated within the stochastic vacuum model. These corrections diminish the absolute values of the condensates, as well as the ratio of the mixed condensate to the standard one.

Geometry of spin-field coupling on the worldline

We derive a geometric representation of couplings between spin degrees of freedom and gauge fields within the worldline approach to quantum field theory. We combine the string-inspired methods of the worldline formalism with elements of the loop-space approach to gauge theory. In particular, we employ the loop (or area) derivative operator on the space of all holonomies which can immediately be applied to the worldline representation of the effective action. This results in a spin factor that associates the information about spin with "zigzag" motion of the fluctuating field. Concentrating on the case of quantum electrodynamics in external fields, we obtain a purely geometric representation of the Pauli term. To one-loop order, we confirm our formalism by rederiving the Heisenberg-Euler effective action. Furthermore, we give closed-form worldline representations for the all-loop order effective action to lowest nontrivial order in a small-N_f expansion.

Quantum effective actions from nonperturbative worldline dynamics

We demonstrate the feasibility of a nonperturbative analysis of quantum field theory in the worldline formalism with the help of an efficient numerical algorithm. In particular, we compute the effective action for a super-renormalizable field theory with cubic scalar interaction in four dimensions in quenched approximation (small-$N_f$ expansion) to all orders in the coupling. We observe that nonperturbative effects exert a strong influence on the infrared behavior, rendering the massless limit well defined in contrast to the perturbative expectation. Our numerical method is based on a direct use of probability distributions for worldline ensembles, preserves all Euclidean spacetime symmetries, and thus represents a new nonperturbative tool for an investigation of continuum quantum field theory.

One loop photon-graviton mixing in an electromagnetic field: Part 1

Photon-graviton mixing in an electromagnetic field is a process of potential interest for cosmology and astrophysics. At the tree level it has been studied by many authors. We consider the one-loop contribution to this amplitude involving a charged spin 0 or spin 1/2 particle in the loop and an arbitrary constant field. In the first part of this article, the worldline formalism is used to obtain a compact two-parameter integral representation for this amplitude, valid for arbitrary photon energies and background field strengths. The calculation is manifestly covariant througout.