Background Field Method Research Articles

Lattice determination of the Batalin-Vilkovisky function and the strong running interaction

The Batalin-Vilkovisky function is a central component in the modern formulation of the background field method and the physical applications derived from it. In the present work we report on novel lattice results for this particular quantity, obtained by capitalizing on its equality with the Kugo-Ojima function in the Landau gauge. The results of the lattice simulation are in very good agreement with the predictions derived from a continuum analysis based on the corresponding Schwinger-Dyson equations. In addition, we show that an important relation connecting this function with the ghost propagator is fulfilled rather accurately. With the aid of these results, we carry out the first completely lattice-based determination of the process-independent strong running interaction, employed in a variety of phenomenological studies.

Gravitational scattering and beyond from extreme mass ratio effective field theory

We explore a recently proposed effective field theory describing electromagnetically or gravitationally interacting massive particles in an expansion about their mass ratio, also known as the self-force (SF) expansion. By integrating out the deviation of the heavy particle about its inertial trajectory, we obtain an effective action whose only degrees of freedom are the lighter particle together with the photon or graviton, all propagating in a Coulomb or Schwarzschild background. The 0SF dynamics are described by the usual background field method, which at 1SF is supplemented by a “recoil operator” that encodes the wobble of the heavy particle, and similarly computable corrections appearing at 2SF and higher. Our formalism exploits the fact that the analytic expressions for classical backgrounds and particle trajectories encode dynamical information to all orders in the couplings, and from them we extract multiloop integrands for perturbative scattering. As a check, we study the two-loop classical scattering of scalar particles in electromagnetism and gravity, verifying known results. We then present new calculations for the two-loop classical scattering of dyons, and of particles interacting with an additional scalar or vector field coupling directly to the lighter particle but only gravitationally to the heavier particle.

Neutral pion polarizabilities from four-point functions in lattice QCD

We report a proof-of-principle lattice QCD simulation of the electric and magnetic polarizabilities for a neutral pion in the four-point function method. The results are based on the same quenched Wilson ensembles on a 243×48 lattice at β=6.0 with pion mass from 1100 to 370 MeV previously used for a charged pion. For electric polarizability, the results are largely consistent with those from the background field method and ChPT. In contrast, there are significant differences for magnetic polarizability among the four-point function method, the background field method, and ChPT. The situation points to the important role of disconnected diagrams for a neutral pion. We elucidate a transparent quark decomposition in the four-point function method that can be used to shed light on the issue.

Three-loop renormalization of the quantum action for a five-dimensional scalar cubic model with the usage of the background field method and a cutoff regularization

Three-loop renormalization of the quantum action for a five-dimensional scalar cubic model with the usage of the background field method and a cutoff regularization

Renormalization of the Einstein–Cartan theory in first-order form

We examine the Einstein–Cartan (EC) theory in first-order form, which has a diffeomorphism as well as a local Lorentz invariance. We study the renormalizability of this theory in the framework of the Batalin–Vilkovisky formalism, which allows for a gauge invariant renormalization. Using the background field method, we discuss the gauge invariance of the background effective action and analyze the Ward identities which reflect the symmetries of the EC theory. As an application, we compute, in a general background gauge, the self-energy of the tetrad field at one-loop order.

Three-loop renormalization of the quantum action for a four-dimensional scalar model with quartic interaction with the usage of the background field method and a cutoff regularization

The paper studies the quantum action for the four-dimensional real ϕ4-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization R-operation on the example of separate three-loop diagrams are also discussed. The explicit forms of the first three coefficients for the renormalization constants and for the β-function are presented. Consistency with previously known results is shown.

First-order formalism for β functions in bosonic sigma models from supersymmetry breaking

We consider the renormalization group flow equation for the two-dimensional sigma models with the Kähler target space. The first-order formulation allows us to treat perturbations in these models as current-current deformations. We demonstrate, however, that the conventional first-order formalism misses certain anomalies in the measure, and should be amended. We reconcile beta functions obtained within the conformal perturbation theory for the current-current deformations with traditional “geometric” results obtained in the background field methods, in this way resolving the peculiarities pointed out in O. Gamayun [Peculiarities of beta functions in sigma models, ]. The result is achieved by the supersymmetric completion of the first-order sigma model. Published by the American Physical Society 2024

Neutrino charge radius and electromagnetic dipole moments via scalar and vector leptoquarks

The one-loop contribution of scalar and vector leptoquarks (LQs) to the electromagnetic properties (NEPs) of massive Dirac neutrinos is presented via an effective Lagrangian approach. For the contribution of gauge LQs to the effective neutrino charge radius defined in Bernabeu et al. (Phys Rev Lett 89:101802, 2002. https://doi.org/10.1103/PhysRevLett.89.101802 [Erratum: Phys Rev Lett 89:229902 (2002)]), we considered a Yang–Mills scenario and used the background field method for the calculation. Analytical results for nonzero neutrino mass are presented, which can be useful to obtain the NEPs of heavy neutrinos, out of which approximate expressions are obtained for light neutrinos. For the numerical analysis we concentrate on the only renormalizable scalar and vector LQ representations that do not need extra symmetries to forbid tree-level proton decay. Constraints on the parameter space consistent with current experimental data are then discussed and it is found that the scalar R~2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\widetilde{R}}_2$$\\end{document} and the vector U1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_1$$\\end{document} representations could yield the largest LQ contributions to the NEPs: for LQ couplings to both left- and right handed neutrinos of the order of O(1) and a LQ mass of 1.5-2 TeV, the magnetic dipole moment (MDM) of a Dirac neutrino with a mass in the eV scale can be of the order of 10-9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-9}$$\\end{document}–10-10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-10}$$\\end{document}μB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _B$$\\end{document}, whereas its neutrino electric dipole moment (EDM) can reach values as high as 10-20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-20}$$\\end{document}–10-19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-19}$$\\end{document} ecm. On the other hand, the effective NCR can reach values up to 10-35\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-35}$$\\end{document} cm2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} even if LQs do not couple to right-handed neutrinos, in which case the EDM vanishes and the contribution to the MDM would be negligible.

Revisiting renormalization group equations of the SMEFT dimension-seven operators

In this work, we revisit the renormalization group equations (RGEs) of dimension-seven (dim-7) operators in the Standard Model effective field theory (SMEFT) resulting from mixing among dim-7 operators themselves by means of the background field method. Adopting a recently proposed physical basis for dim-7 operators, we achieve the explicit RGEs of all non-redundant dim-7 operators in the SMEFT for the first time. Together with those originating from the dim-5 and dim-6 operators, these results constitute the complete RGEs of dim-7 operators, and hence can be exploited to study full RG-running effects on some lepton- or baryon-number-violating processes involving dim-7 operators in the SMEFT, such as neutrino masses, neutrinoless double beta decay, meson and nucleon decays. We perform an analysis of the structure and perturbative power counting of the obtained one-loop anomalous dimension matrix, which is consistent with a non-renormalization theorem and the naive dimension analysis. Additionally, a partial check on some results is carried out by means of different tools and quantum field gauges.

Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries

We consider two-dimensional Rayleigh–Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function h. We prove rigorous upper bounds on the Nusselt number Nu which capture the dependence on the curvature of the boundary κ and the (non-constant) friction coefficient α explicitly. If h∈W2,∞ and κ satisfies a smallness condition with respect to α, we find Nu≲Ra12+∥κ∥∞, where Ra is the Rayleigh number, which agrees with the predicted Spiegel–Kraichnan scaling when κ = 0. This bound is obtained via local regularity estimates in a small strip at the boundary. When h∈W3,∞ , the functions κ and α are sufficiently small in L∞ and the Prandtl number Pr is sufficiently large, we prove upper bounds using the background field method, which interpolate between Ra12 and Ra512 with non-trivial dependence on α and κ. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime Pr⩾Ra57 , we improve the Ra12 -upper bound, showing Nu≲α,κRa37, where ≲α,κ hides an additional dependency of the implicit constant on α and κ.

More accurate σGG→h,Γh→GGAAΨ¯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sigma \\left(\\mathcal{GG}\ o h\\right),\\Gamma \\left(h\ o \\mathcal{GG},\\mathcal{AA},\\overline{\\Psi}\\Psi \\right) $$\\end{document} and Higgs width results via the geoSMEFT

We develop Standard Model Effective Field Theory (SMEFT) predictions of σ(GG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{GG} $$\\end{document} → h), Γ(h → GG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{GG} $$\\end{document}), Γ(h → AA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{AA} $$\\end{document}) to incorporate full two loop Standard Model results at the amplitude level, in conjunction with dimension eight SMEFT corrections. We simultaneously report consistent Γ(h → Ψ¯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}\\Psi $$\\end{document}) results including leading QCD corrections and dimension eight SMEFT corrections. This extends the predictions of the former processes Γ, σ to a full set of corrections at Ov¯T2/Λ216π22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^2/{\\varLambda}^2{\\left(16{\\pi}^2\\right)}^2\\right) $$\\end{document} and Ov¯T4/Λ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^4/{\\Lambda}^4\\right) $$\\end{document}, where v¯T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\overline{v}}_T $$\\end{document} is the electroweak scale vacuum expectation value and Λ is the cut off scale of the SMEFT. Throughout, cross consistency between the operator and loop expansions is maintained by the use of the geometric SMEFT formalism. For Γ(h → Ψ¯Ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\\Psi}\\Psi $$\\end{document}), we include results at Ov¯T2/Λ216π2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\overline{v}}_T^2/{\\Lambda}^2\\left(16{\\pi}^2\\right)\\right) $$\\end{document} in the limit where subleading mΨ → 0 corrections are neglected. We clarify how gauge invariant SMEFT renormalization counterterms combine with the Standard Model counter terms in higher order SMEFT calculations when the Background Field Method is used. We also update the prediction of the total Higgs width in the SMEFT to consistently include some of these higher order perturbative effects.

Quasi-de sitter solution, negative norm state and Krein space in cosmology

Recent observational data have shown two important results about cosmic inflation: First, the primordial power spectrum is approximately scale invariant. Secondly, the space–time geometry during inflation is quasi-de Sitter. In this paper, we study the background field method and the concept of negative norm states in the framework of Krein space to use in early universe cosmology. For the inflaton field equation, based on the observations and taking into account our studies, we suggest an alternative proposal. We will take the quasi-de Sitter solutions as the physical geometry with the positive norm and the pure de Sitter solutions as the background geometry with the negative norm. In the following, we review the consequences of applying the proposal to calculate some cosmological quantities.

On two-loop divergences of effective action in 6D, mathcal{N} = (1, 1) SYM theory

We study the off-shell structure of the two-loop effective action in 6D, mathcal{N} = (1, 1) supersymmetric gauge theories formulated in mathcal{N} = (1, 0) harmonic superspace. The off-shell effective action involving all fields of 6D, mathcal{N} = (1, 1) supermultiplet is constructed by the harmonic superfield background field method, which ensures both manifest gauge covariance and manifest mathcal{N} = (1, 0) supersymmetry. We analyze the off-shell divergences dependent on both gauge and hypermultiplet superfields and argue that the gauge invariance of the divergences is consistent with the non-locality in harmonics. The two-loop contributions to the effective action are given by harmonic supergraphs with the background gauge and hypermultiplet superfields. The procedure is developed to operate with the harmonic-dependent superpropagators in the two-loop supergraphs within the superfield dimensional regularization. We explicitly calculate the gauge and the hypermultiplet-mixed divergences as the coefficients of frac{1}{varepsilon^2} and demonstrate that the corresponding expressions are non-local in harmonics.

New covariant Feynman rules for effective field theories

We provide a new and completely general formalism to compute the effective field theory matching contributions from integrating out massive fields in a manifestly gauge covariant way, at any desired loop order. The formalism is based on old ideas such as the background field method and the heat kernel, however we add some crucial new ingredients that greatly improve the simplicity and general applicability of the approach. We formulate our method in terms of Feynman rules, the resulting effective action is expressed in terms of local heat kernel coefficients. We also provide as supplementary material a mathematica code that facilitates the computation of these coefficients.

Gauge invariance and finite counterterms in chiral gauge theories

We derive the finite one-loop counterterm required to restore the Ward Identities broken by the regularization scheme in chiral gauge theories. Our result is an analytic expression applicable to a wide class of regularizations satisfying a few general properties. We adopt the background field method, which ensures background gauge invariance in the quantized theory, and focus on renormalizable chiral theories with arbitrary gauge group and fermions in general representations. Our approach can be extended to theories involving scalars, such as the Standard Model, or to non-renormalizable theories, such as the SMEFT. As a concrete application, we work out the finite counterterm at one loop in the Standard Model, within dimensional regularization and the Breitenlohner-Maison-’t Hooft-Veltman prescription for γ5.

Patterns of gauge symmetry in the background field method

The correlation functions of Yang–Mills theories formulated in the background field method satisfy linear Slavnov–Taylor identities, which are naive generalizations of simple tree level relations, with no deformations originating from the ghost-sector of the theory. In recent years, a stronger version of these identities has been found to hold at the level of the background gluon self-energy, whose transversality is enforced separately for each special block of diagrams contributing to the gluon Schwinger–Dyson equation. In the present work we demonstrate by means of explicit calculations that the same distinct realization of the Slavnov–Taylor identity persists in the case of the background three-gluon vertex. The analysis is carried out at the level of the exact Schwinger–Dyson equation for this vertex, with no truncations or simplifying assumptions. The demonstration entails the contraction of individual vertex diagrams by the relevant momentum, which activates Slavnov–Taylor identities of vertices and multi-particle kernels nested inside these graphs; the final result emerges by virtue of a multitude of extensive cancellations, without the need of performing explicit integrations. In addition, we point out that background Ward identities amount to replacing derivatives of propagators by zero-momentum background-gluon insertions, in exact analogy to standard properties of Abelian gauge theories. Finally, certain potential applications of these results are briefly discussed.

Non-Abelian Gauge Theories with Composite Fields in the Background Field Method

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green’s functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green’s functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion.

Schwinger–Dyson truncations in the all-soft limit: a case study

We study a special Schwinger–Dyson equation in the context of a pure SU(3) Yang–Mills theory, formulated in the background field method. Specifically, we consider the corresponding equation for the vertex that governs the interaction of two background gluons with a ghost–antighost pair. By virtue of the background gauge invariance, this vertex satisfies a naive Slavnov–Taylor identity, which is not deformed by the ghost sector of the theory. In the all-soft limit, where all momenta vanish, the form of this vertex may be obtained exactly from the corresponding Ward identity. This special result is subsequently reproduced at the level of the Schwinger–Dyson equation, by making extensive use of Taylor’s theorem and exploiting a plethora of key relations, particular to the background field method. This information permits the determination of the error associated with two distinct truncation schemes, where the potential advantage from employing lattice data for the ghost dressing function is quantitatively assessed.

Chiral vortical catalysis

Gluon interaction introduces remarkable corrections to the magnetic polarization effects on the chiral fermions, which is known as the inverse magnetic catalysis. It is a natural speculation that the vorticity, which has many similar properties as magnetic field, would bring non-negligible contribution to the chiral rotational suppression. Using an intuitive semi-classical background field method we studied the rotation dependence of the effective strong interaction coupling constant. Contrary to the magnetic field case the rotation increases the effective coupling which would slow down the condensate melting with temperature. This could be named as the chiral vortical catalysis or inverse rotation suppression. Imposing such dependence on the 4-fermion coupling in the NJL model, we numerically checked this analysis qualitatively. The pseudo critical temperature is shown to rise with the rotation and approach saturation eventually which may be induced by the model cutoff.

Background field method and generalized field redefinitions in effective field theories

We study the non-linear background field redefinitions arising at the quantum level in a spontaneously broken effective gauge field theory. The non-linear field redefinitions are crucial for the symmetric (i.e. fulfilling all the relevant functional identities of the theory) renormalization of gauge-invariant operators. In a general R_\xiRξ-gauge the classical background-quantum splitting is also non-linearly deformed by radiative corrections. In the Landau gauge these deformations vanish to all orders in the loop expansion.