Self-dual Background Research Articles

Infrared structures of scattering on self-dual radiative backgrounds

The scattering of gluons and gravitons in trivial backgrounds is endowed with many surprising infrared features which have interesting conformal interpretations on the two-dimensional celestial sphere. However, the fate of these structures in more general asymptotically flat backgrounds is far from clear. In this paper, we consider holomorphic infrared structures in the presence of non-perturbative, self-dual background gauge and gravitational fields which are determined by freely specified radiative data. We make use of explicit formulae for tree-level gluon and graviton scattering in these self-dual radiative backgrounds, as well as chiral twistor sigma model descriptions of the classical dynamics. Remarkably, we find that the leading holomorphic part of tree-level collinear splitting functions — or celestial OPEs — and infinite-dimensional chiral soft algebras are undeformed by the background. We also compute all-order holomorphic celestial OPEs in the MHV sectors of gauge theory and gravity.

Self-dual fields on self-dual backgrounds and the double copy

We explore the double copy for self-dual gauge and gravitational fields on self-dual background spacetimes. We consider backgrounds associated with solutions of the second Plebanski equation and describe results with different gauge-fixing conditions. Finally we discuss the kinematic and w-algebras and the double copy, identifying modified Poisson structures and kinematic structure constants in the presence of the self-dual background. The self-dual plane wave and Eguchi-Hanson spacetimes are studied as examples and their respective w-algebras derived. Published by the American Physical Society 2024

Yang-Mills form factors on self-dual backgrounds

The construction of perturbative quantities on non-linear backgrounds leads to the possibility of incorporating strong field effects in perturbation theory. We continue a programme to construct QFT observables on self-dual backgrounds in Yang-Mills theory. The approach works with asymptotic data for fields defined at null infinity mathcal{I} , extending earlier work on Yang-Mills amplitudes on self-dual backgrounds to form factors and further incorporating supersymmetry. Since our analysis is based on reconstruction from data at null infinity, it naturally ties into work on celestial and twisted holography. We study form factors both in pure Yang-Mills and their supersymmetric counterparts in mathcal{N} = 4 SYM, giving a full treatment of mathcal{N} = 4 super-Yang-Mills at null infinity and their self-dual nonlinear backgrounds. We obtain tree-level MHV form factors around these backgrounds using new formulae for lifting operators to twistor space leading to simple dressings of the corresponding form factors around the vacuum. We give brief indications on how to go beyond the MHV sector by introducing dressed versions of the MHV diagram propagator. We discuss generating functionals of the MHV all plus 1-loop amplitude in this context together with its various dual conformal representations.

Zero modes of fermions trapped by giant vortices

Zero-energy solutions of the Dirac equation for the fermions bound to giant vortices of large winding number n are studied in the abelian Higgs and Chern-Simons Higgs models. The case of Jackiw-Rossi theory of the Majorana states in topological superconductors is discussed in detail. By expanding in inverse powers of n we find an analytic result for asymptotically all n solutions required by the index theorem. In the abelian Higgs model the zero modes fill the vortex core and reveal a universal structure independent of fine details of the gauge and scalar field interactions which, in particular, determines the general properties of the large-n superconducting cosmic strings. On the contrary, for the Chern-Simons Higgs vortices the zero modes are localized on the core boundary and the explicit solution is obtained for the supersymmetric couplings in a self-dual background.

Gluon Scattering on Self-Dual Radiative Gauge Fields

We present all-multiplicity formulae, derived from first principles in the MHV sector and motivated by twistor string theory for general helicities, for the tree-level S-matrix of gluon scattering on self-dual radiative backgrounds. These backgrounds are chiral, asymptotically flat gauge fields characterised by their free radiative data, and their underlying integrability is captured by twistor theory. Tree-level gluon scattering scattering amplitudes are expressed as integrals over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map related to the helicity configuration of the external gluons. In the MHV sector, our formula is derived from the Yang–Mills action; for general helicities the formulae are obtained using a background-coupled twistor string theory and pass several consistency tests. Unlike amplitudes on a trivial vacuum, there are residual integrals due to the functional freedom in the self-dual background, but for scattering of momentum eigenstates we are able to do many of these explicitly and even more is possible in the special case of plane wave backgrounds. In general, the number of these integrals is always less than expected from standard perturbation theory, but matches the number associated with space-time MHV rules in a self-dual background field, which we develop for self-dual plane waves.

One-loop multicollinear limits from 2-point amplitudes on self-dual backgrounds

For scattering amplitudes in strong background fields, it is — at least in principle — possible to perturbatively expand the background to obtain higher-point vacuum amplitudes. In the case of self-dual plane wave backgrounds we consider this expansion for two-point, one-loop amplitudes in pure Yang-Mills, QED and QCD. This enables us to obtain multicollinear limits of 1-loop vacuum amplitudes; the resulting helicity configurations are surprisingly restricted, with only the all-plus helicity amplitude surviving. These results are shown to be consistent with well-known vacuum amplitudes. We also show that for both abelian and non-abelian supersymmetric gauge theories, there is no helicity flip (and hence no vacuum birefringence) on any plane wave background, generalising a result previously known in the Euler-Heisenberg limit of super-QED.

Energy-driven disorder in mean field QCD

An impact of the finite size effects on the vacuum free energy density of full QCD with $N_{\rm f}$ massless flavors in the presence of homogeneous (anti-)self-dual Abelian background gluon field is studied. The zero temperature free energy density of the four-dimensional spherical domain is computed as a function of the background field strength $B$ and domain radius $R$. Calculation is performed in the one-loop approximation improved by accounting for mixing of the quark and gluon quasi-zero modes with normal modes, with the use of the $\zeta$-function regularization. It is indicated that, under plausible assumption on the character of the mixing, the quantum correction to the free energy density has a minimum as a function of $B$ and $R$. Within the mean field approach to QCD vacuum based on domain wall network representation of the mean field, an existence of the minimum may prevent infinite growth of individual domain, thus protecting the vacuum from the long-range ordering, and, hence, serving as the origin of disorder in the statistical ensemble of domain wall networks, driven by the minimization of the overall free energy of the dominant gauge field configurations.

Maximal-Helicity-Violating Scattering of Gluons and Gravitons in Chiral Strong Fields.

We present all-multiplicity formulas for the tree-level scattering of gluons and gravitons in the maximal helicity violating (MHV) helicity configuration, calculated in certain chiral strong fields. The strong backgrounds we consider are self-dual plane waves in gauge theory and general relativity, which are treated exactly and admit a well-defined S matrix. The gauge theory background-coupled MHV amplitude is simply a dressed analog of the familiar Parke-Taylor formula, but the gravitational version has nontrivial new structures due to graviton tails. Both formulas have just one residual integral rather than the n-2 expected at n points from space-time perturbation theory; this simplification arises from the integrability of self-dual backgrounds and their corresponding twistor description. The resulting formulas pass several consistency checks and limit to the well-known expressions for MHV scattering of gluons and gravitons when the background becomes trivial.

Tensor Fields on Self-Dual Warped AdS3 Background

Considering rank s fields obey first order equation of motion, we study the dynamics of such fields in a 3-dimensional self-dual space-like warped AdS3 black hole background. We obtain a Klein–Gordon-like equation for tensor fields. By using gauge constraint and traceless condition, we will find the exact solutions of the equations of motion. Then, we will compute the quasi-normal modes by imposing appropriate boundary conditions at horizon and infinity.

On Painlevé/gauge theory correspondence

We elucidate the relation between Painlev\'e equations and four-dimensional rank one ${\cal N= 2}$ theories by identifying the connection associated to Painlev\'e isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlev\'e functions in terms of magnetic and dyonic Nekrasov partition functions for ${\cal N= 2}$ SQCD and Argyres-Douglas theories at self-dual Omega background $\epsilon_1+\epsilon_2= 0$, or equivalently in terms of $c= 1$ irregular conformal blocks.

SDYM equations on the self-dual background

We introduce the technique combining the features of integration schemes for self-dual Yang–Mills (SDYM) equations and multidimensional dispersionless integrable equations to get SDYM equations on the conformally self-dual background. Generating differential form is defined, and the dressing scheme is developed. Some special cases and reductions are considered.

Quasinormal frequencies of self-dual black holes

One simplified black hole model constructed from a semiclassical analysis of loop quantum gravity (LQG) is called self-dual black hole. This black hole solution depends on a free dimensionless parameter P known as the polymeric parameter and also on the $a_{0}$ area related to the minimum area gap of LQG. In the limit of P and $a_{0}$ going to zero, the usual Schwarzschild-solution is recovered. Here we investigate the quasinormal modes (QNMs) of massless scalar perturbations in the self-dual black hole background. We compute the QN frequencies using the sixth order WKB approximation method and compare them with numerical solutions of the Regge-Wheeler equation. Our results show that as the parameter P grows, the real part of the QN frequencies suffers an initial increase and then starts to decrease while the magnitude of the imaginary one decreases for fixed area gap $a_{0}$. This particular feature means that the damping of scalar perturbations in the self-dual black hole spacetimes are slower, and their oscillations are faster or slower according to the value of P.

Particle in a self-dual dyon background: Hidden free nature and exotic superconformal symmetry

We show that a nonrelativistic particle in a combined field of a magnetic monopole and $1/{r}^{2}$ potential reveals a hidden, partially free dynamics when the strength of the central potential and the charge-monopole coupling constant are mutually fitted to each other. In this case the system admits both a conserved Laplace-Runge-Lenz vector and a dynamical conformal symmetry. The supersymmetrically extended system corresponds then to a background of a self-dual or anti-self-dual dyon. It is described by a quadratically extended Lie superalgebra $D(2,1;\ensuremath{\alpha})$ with $\ensuremath{\alpha}=1/2$, in which the bosonic set of generators is enlarged by a generalized Laplace-Runge-Lenz vector and its dynamical integral counterpart related to Galilei symmetry, as well as by the chiral ${\mathbb{Z}}_{2}$-grading operator. The odd part of the nonlinear superalgebra comprises a complete set of $24=2\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ fermionic generators. Here a usual duplication comes from the ${\mathbb{Z}}_{2}$-grading structure; the second factor can be associated with a triad of scalar integrals---the Hamiltonian, the generator of special conformal transformations, and the squared total angular momentum vector, while the quadruplication is generated by a chiral spin vector integral which exits due to the (anti-)self-dual nature of the electromagnetic background.

Erratum: Gluon condensation and scaling exponents for the propagators in Yang-Mills theory [Phys. Rev. D83, 045014 (2011)

We investigate the infrared (strong-coupling) regime of SU(N)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field strength invariant F_{\mu {\nu}}F^{\mu {\nu}} from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the beta function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared imposes a new upper bound on the infrared exponents of the propagators. For the scaling solution, this bound reads kappa_c < 23/38 which, together with Zwanziger's horizon condition kappa_c> 1/2, defines a rather narrow window for this critical exponent. Current estimates from functional methods indeed satisfy these bounds.

Self-duality of Green-Schwarz sigma-models

We study fermionic T-duality symmetries of integrable Green-Schwarz sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes, constructed as \( {\mathbb{Z}_4} \) supercosets of superconformal algebras. We find three algebraic conditions that guarantee self-duality of the backgrounds under fermionic T-duality, we classify those that satisfy them and construct the map of the monodromy matrix. We introduce new T-duality directions, where some of them contain no bosonic directions, along which the backgrounds are self-dual. We find that the only self-dual backgrounds are AdS n × Sn for n = 2, 3, 5. In addition we find that the backgrounds AdS n × S1 for n = 2, 3, 5, AdS4 × S2 and AdS2 × S4 are self-dual at the level of the classical action, but have a non-trivial transformation of the dilaton.

Gluon condensation and scaling exponents for the propagators in Yang-Mills theory

We investigate the infrared (strong-coupling) regime of SU($N$)-Yang-Mills theory on a self-dual background. We present an evaluation of the full effective potential for the field-strength invariant ${F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}$ from nonperturbative gauge correlation functions and find a nontrivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the $\ensuremath{\beta}$ function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators. Consistency between both gauges in the infrared imposes a new upper bound on the infrared exponents of the propagators. For the scaling solution, this bound reads ${\ensuremath{\kappa}}_{c}l23/38$ which, together with Zwanziger's horizon condition ${\ensuremath{\kappa}}_{c}g1/2$, defines a rather narrow window for this critical exponent. Current estimates from functional methods indeed satisfy these bounds.

Some chirality-related properties of the 4D massive Dirac propagator and determinant in an arbitrary gauge field

For a 4-D massive Dirac field in the background of arbitrary gauge fields, we show that the Dirac propagator and functional determinant are completely determined by knowledge of the corresponding quantities for just one of the chirality sectors of the second-order Dirac operator. This generalizes the related, previously known, statements in (anti-)self-dual background gauge fields. The logarithms of the (renormalized) functional determinants from the two chirality sectors are shown to be different only by a term reflecting the integrated chiral anomaly.

NEW RELATIONS BETWEEN SPINOR AND SCALAR ONE-LOOP EFFECTIVE LAGRANGIANS IN CONSTANT BACKGROUND FIELDS

Simple new relations are presented between the one-loop effective Lagrangians of spinor and scalar particles in constant curvature background fields, both electromagnetic and gravitational. These relations go beyond the well-known cases for self-dual background fields.

Self-duality and supersymmetry

We observe that the Hamiltonian H=D̸2, where D̸ is the flat 4d Dirac operator in a self-dual gauge background, is supersymmetric, admitting 4 different real supercharges. A generalization of this model to the motion on a curved conformally flat 4d manifold exists. For an Abelian self-dual background, the corresponding Lagrangian can be derived from known harmonic superspace expressions.

Gravity, Twistors and the MHV Formalism

We give a self-contained derivation of the MHV amplitudes for gravity and use the associated twistor generating function to define a twistor action for the MHV diagram approach to gravity. Starting from a background field calculation on a spacetime with anti self-dual curvature, we obtain a simple spacetime formula for the scattering of a single, positive helicity linearized graviton into one of negative helicity. Re-expressing our integral in terms of twistor data allows us to consider a spacetime that is asymptotic to a superposition of plane waves. Expanding these out perturbatively yields the gravitational MHV amplitudes of Berends, Giele & Kuijf. We go on to take the twistor generating function off-shell at the perturbative level. Combining this with a twistor action for the anti self-dual background, we obtain a twistor action for the MHV diagram approach to perturbative gravity. We finish by extending these results to supergravity, in particular N=4 and N=8.