arXiv subjects

M. Stingl

Publications and source records attributed to M. Stingl.

7 recordsLinked to original sources

arXiv2022

Scalars from Gauge Fields

In an Euclidean SU(2) $\otimes$ U(1) gauge theory without fermions, we identify scalar-field variables, functionals of the gauge fields and coming in different representations of isospin, which (i) are of mass dimension one in $d=4$, (ii) couple to their parent gauge fields through suitable gauge-covariant derivatives, and (iii) can be endowed with a hypercharge despite their parents having none. They can be interpreted as projections of the gauge vectors onto an orthonormal basis that is defined by the fields themselves. We inquire as to whether these scalars can perform the usual tasks, normally fulfilled by external scalar fields, of spontaneous symmetry breaking and mass generation through vacuum expectation values. The gauge Lagrangian, expressed in terms of these scalars, automatically has quartic and cubic terms; no extra coupling constant for quartic scalar self-interactions is needed. VEV formation takes place in one of four scalar fields populating the classical potential-energy minimum. There are nine massive Higgs particles, a neutral triplet at a mass of $m_Z \sqrt{2}$, and three conjugate pairs of charged ones at $m_W \sqrt{2}$. Seven quasi-Goldstone scalars remain massless. This results in a qualitatively correct pattern of heavy-vector masses and mixing, with the analog of the mixing angle determined by theory. Higgs-type hypercharge and charge assignments emerge naturally.

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arXiv2006

Variational Worldline Approximation for the Relativistic Two-Body Bound State in a Scalar Model

We use the worldline representation of field theory together with a variational approximation to determine the lowest bound state in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. Self-energy and vertex corrections are included approximately in a consistent way as well as crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. In a path integral description of an appropriate current-current correlator an effective, retarded action is obtained by integrating out the meson field. As in the polaron problem we employ a quadratic trial action with variational functions to describe retardation and binding effects through multiple meson exchange.The variational equations for these functions are derived, discussed qualitatively and solved numerically. We compare our results with the ones from traditional approaches based on the Bethe-Salpeter equation and find an enhanced binding contrary to some claims in the literature. For weak coupling this is worked out analytically and compared with results from effective field theories. However, the well-known instability of the model, which usually is ignored, now appears at smaller coupling constants than in the one-body case and even when self-energy and vertex corrections are turned off. This induced instability is investigated analytically and the width of the bound state above the critical coupling is estimated.

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arXiv2005

Worldline Variational Approximation: A New Approach to the Relativistic Binding Problem

We determine the lowest bound-state pole of the density-density correlator in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. This is done by employing the worldline representation of field theory together with a variational approximation as in Feynman's treatment of the polaron. Unlike traditional methods based on the Bethe-Salpeter equation, self-energy and vertex corrections are (approximately) included as are crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. The well-known instability of the model due to self-energy effects leads to large qualitative and quantitative changes compared to traditional approaches which neglect them. We determine numerically the critical coupling constant above which no real solutions of the variational equations exist anymore and show that it is smaller than in the one-body case due to an induced instability. The width of the bound state above the critical coupling is estimated analytically.

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arXiv2002

Field-Theory Amplitudes as Resurgent Functions

A series of informal seminars at graduate-student level on the subject of coupling dependence in quantum field theory, with an elementary introduction to the notion of resurgent function that forms the appropriate framework for the coupling dependence of strictly renormalizable theories. While most of the discussion is pedagogical, there are also a few things for the expert: we demonstrate, by studying a model spectral integral, that an amplitude may possess both the t'Hooft set of singularities in the coupling-constant plane, and Borel-plane singularities of the infrared-renormalon type in its perturbative part, and yet be uniquely reconstructible from its resurgent symbol, the appropriate generalization of the semiconvergent perturbation expansion. In the same model we demonstrate the virtues of a quasi-perturbative expansion, obtained by resummation of the resurgent symbol in its nonperturbative direction, and which in contrast to the perturbative one is Borel summable. On the basis of this expansion, we discuss a systematic approximation method for the reconstruction of correlation functions with the resurgent coupling dependence typical of strictly renormalizable and asymptotically free theories.

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arXiv1998

Extended Iterative Scheme for QCD: Three-point Vertices

In the framework of a generalized iterative scheme introduced previously to account for the non-analytic coupling dependence associated with the renormalization-group invariant mass scale Lambda, we establish the self-consistency equations of the extended Feynman rules (Lambda-modified vertices of zeroth perturbative order) for the three-gluon vertex, the two ghost vertices, and the two vertices of massless quarks. Calculations are performed to one-loop-order, in Landau gauge, and at the lowest approximation level (r=1) of interest for QCD. We discuss the phenomenon of compensating poles inherent in these equations, by which the formalism automatically cancels unphysical poles on internal lines, and the role of composite-operator information in the form of equation-of-motion condensate conditions. The observed near decoupling of the four-gluon conditions permits a solution to the 2-and-3-point conditions within an effective one-parameter freedom. There exists a parameter range in which one solution has all vertex coefficients real, as required for a physical solution, and a narrower range in which the transverse-gluon and massless-quark propagators both exhibit complex-conjugate pole pairs.

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arXiv1998

Extended Iterative Scheme for QCD: the Four-Gluon Vertex

We study the self-consistency problem of the generalized Feynman rule (nonperturbatively modified vertex of zeroth perturbative order) for the 4-gluon vertex function in the framework of an extended perturbation scheme accounting for non-analytic coupling dependence through the Lambda scale. Tensorial structure is restricted to a minimal dynamically closed basis set. The self-consistency conditions are obtained at one loop, in Landau gauge, and at the lowest approximation level (r=1) of interest for QCD. At this level, they are found to be linear in the nonperturbative 4-gluon coefficients, but strongly overdetermined due to the lack of manifest Bose symmetry in the relevant Dyson-Schwinger equation. The observed near decoupling from the 2-and-3-point conditions permits least-squares quasisolutions for given 2-and-3-point input within an effective one-parameter freedom. We present such solutions for N_F=2 massless quarks and for the pure gluon theory, adapted to the 2-and-3-point coefficients determined previously.

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arXiv1996

A Systematic Extended Iterative Solution for QCD

An outline is given of an extended perturbative solution of Euclidean QCD which systematically accounts for a class of nonperturbative effects, while allowing renormalization by the perturbative counterterms. Proper vertices Gamma are approximated by a double sequence Gamma[r,p], with r the degree of rational approximation w.r.t. the QCD mass scale Lambda, nonanalytic in the coupling g, and p the order of perturbative corrections in g-squared, calculated from Gamma[r,0] - rather than from the perturbative Feynman rules Gamma(0)(pert) - as a starting point. The mechanism allowing the nonperturbative terms to reproduce themselves in the Dyson-Schwinger equations preserves perturbative renormalizability and is tied to the divergence structure of the theory. As a result, it restricts the self-consistency problem for the Gamma[r,0] rigorously - i.e. without decoupling approximations - to the superficially divergent vertices. An interesting aspect of the scheme is that rational-function sequences for the propagators allow subsequences describing short-lived excitations. The method is calculational, in that it allows known techniques of loop computation to be used while dealing with integrands of truly nonperturbative content.

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