Dirac Structure Research Articles

Renormalization and mixing of staple-shaped Wilson line operators on the lattice revisited

Transverse-momentum-dependent parton distribution functions and wave functions (TMDPDFs/TMDWFs) can be extracted from lattice calculations of appropriate Euclidean matrix elements of staple-shaped Wilson line operators. We investigate the mixing pattern of such operators under lattice renormalization using symmetry considerations. We perform an analysis for operators with all Dirac structures, which reveals mixings that are not present in one-loop lattice perturbation theory calculations. We also present the relevant one-loop matching in a renormalization scheme that does not introduce extra non-perturbative effects at large distances, both for the TMDPDFs and for the TMDWFs. Our results have the potential to greatly facilitate numerical calculations of TMDPDFs and TMDWFs on the lattice.

Probing new physics in class-I B-meson decays into heavy-light final states

With updated experimental data and improved theoretical calculations, several significant deviations are being observed between the Standard Model predictions and the experimental measurements of the branching ratios of {overline{B}}_{(s)}^0to {D}_{(s)}^{left(ast right)+}{L}^{-} decays, where L is a light meson from the set {π, ρ, K(∗)}. Especially for the two channels {overline{B}}^0to {D}^{+}{K}^{-} and {overline{B}}_s^0to {D}_s^{+}{pi}^{-} , both of which are free of the weak annihilation contribution, the deviations observed can even reach 4–5σ. Here we exploit possible new-physics effects in these class-I non-leptonic B-meson decays within the framework of QCD factorization. Firstly, we perform a model-independent analysis of the effects from twenty linearly independent four-quark operators that can contribute, either directly or through operator mixing, to the quark-level b → coverline{u}d(s) transitions. It is found that, under the combined constraints from the current experimental data, the deviations observed could be well explained at the 1σ level by the new-physics four-quark operators with γμ(1 − γ5) ⨂ γμ(1 − γ5) structure, and also at the 2σ level by the operators with (1 + γ5) ⨂ (1 − γ5) and (1 + γ5) ⨂ (1 + γ5) structures. However, the new-physics four-quark operators with other Dirac structures fail to provide a consistent interpretation, even at the 2σ level. Then, as two specific examples of model-dependent considerations, we discuss the case where the new-physics four-quark operators are generated by either a colorless charged gauge boson or a colorless charged scalar, with their masses fixed both at the 1 TeV. Constraints on the effective coefficients describing the couplings of these mediators to the relevant quarks are obtained by fitting to the current experimental data.

Structural identifiability of linear Port Hamiltonian systems

This paper, puts in discussion the structural identifiability of LTI Port-Controlled Hamiltonian (PCH) systems, in order to develop a specific identification and control theory. This is due to their remarkable properties of power conservation and stability under power preserving interconnection. The main part of the paper, presents a power based identifiability approach, with specific propositions and definitions. It is based on the power knowledge associated with the system ports, interconnected by a Dirac structure, for selected input signals. In a preliminary section, corresponding transfer functions, system outputs, Markov parameters, observability conditions, port-observability or infinite Grammians are defined for each port. Beside this, a port-identifiability concept is introduced for the identifiability analysis of one port. It is proved that between the input and system ports, a specific model can be determined for identification analysis, preserving in the same time the PCH structure. As examples to demonstrate the theory, a controlled LC circuit and a DC motor are selected for the lossless and lossy cases, respectively.

SuperTracer: a calculator of functional supertraces for one-loop EFT matching

We present SuperTracer, a Mathematica package aimed at facilitating the functional matching procedure for generic UV models. This package automates the most tedious parts of one-loop functional matching computations. Namely, the determination and evaluation of all relevant supertraces, including loop integration and Dirac algebra manipulations. The current version of SuperTracer also contains a limited set of output simplifications. However, a further reduction of the output to a minimal basis using Fierz identities, integration by parts, simplification of Dirac structures, and/or light field redefinitions might still be necessary. The code and example notebooks are publicly available at .1

Integrability of quotients in Poisson and Dirac geometry

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids integrating two interesting families of geometric structures: (i) a special class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson groups and (ii) Dirac homogeneous spaces.

Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow

Part I of this paper presented a systematic derivation of the Stokes–Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes–Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved.In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.

On Lie bialgebroid crossed modules

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples [Formula: see text] is established, where [Formula: see text] is a co-quadratic Lie algebroid and [Formula: see text] is a pair of transverse Dirac structures in [Formula: see text].

Port-Hamiltonian formulation of two-phase flow models

Abstract Two-phase flows are frequently modelled and simulated using the Two-Fluid Model (TFM) and the Drift Flux Model (DFM). This paper proposes Stokes–Dirac structures with respect to which port-Hamiltonian representations for such two-phase flow models can be obtained. We introduce a non-quadratic candidate Hamiltonian function and present dissipative Hamiltonian representations for both models. We then use the structure of the corresponding formally skew-adjoint operator to derive a Stokes–Dirac structure for the two variants of multi-phase flow models. Moreover, we discuss the difficulties in deriving a port-Hamiltonian formulation of the DFM with general slip conditions, and argue why this model may not be energy-consistent.

On Curvature and Torsion in Courant Algebroids

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.

Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids

In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-algebroids and split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac structure of a CLWX 2-algebroid and show that the graph of a Maurer–Cartan element of the homotopy Poisson algebra of degree 3 associated to a split Lie 2-bialgebroid is a weak Dirac structure. Various examples including the string Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions and left-symmetric algebroids are given.

Port-Hamiltonian modelling of fluid dynamics models with variable cross-section

Many single- and multi-phase fluid dynamical systems are governed by non-linear evolutionary equations. A key aspect of these systems is that the fluid typically flows across spatially and temporally varying cross-sections. We, first, show that not any choice of state-variables may be apt for obtaining a port-Hamiltonian realization under spatially varying cross-section. We propose a modified choice of the state-variables and then represent fluid dynamical systems in port-Hamiltonian representations. We define these port-Hamiltonian representations under spatial variation in the cross-section with respect to a new proposed state-dependent and extended Stokes- Dirac structure. Finally, we account for temporal variations in the cross-section and obtain a suitable structure that respects key properties, such as, for instance, the property of dissipation inequality.

Differential operator Dirac structures

Abstract As shown before, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended to pairs of linear differential operators defining a formally skew-adjoint relation between flows and efforts. Furthermore it is shown how the underlying repeated integration by parts operation is streamlined by the use of two-variable polynomial calculus. Dirac structures defined by formally skew adjoint operators together with differential operator effort constraints are treated within the same framework. Finally it is sketched how the approach can be also used for Lagrangian subspaces on bounded domains.

A port-Dirac formulation for thermodynamics of non-simple systems

The paper presents a port-Dirac formulation for thermodynamics of non-simple systems, in which we consider a non-simple system whose thermodynamic states may be represented by several entropy variables. Here we regard such a non-simple system as an interconnected system with ports that can be represented by a port-Dirac system in the context of Dirac structures. We first show some extension of the Lagrange-d’Alembert principle for obtaining the evolution equations of such non-simple systems. Then, a Dirac structure is constructed on the Pontryagin bundle over a thermodynamic configuration manifold. Further, a port-Dirac dynamical formulation for such non-simple systems is demonstrated, where the developed evolution equations are to be equivalent with the generalized Lagrange-d’Alembert equations. The validity of the proposed approach is finally illustrated by two examples of an adiabatic piston and a resistive circuit with entropy production.

Some notes on port-Hamiltonian systems on Banach spaces

Abstract We consider port-Hamiltonian systems from a functional analytic perspective. Dirac structures and Hamiltonians on Banach spaces are introduced, and an energy balance is proven. Further, we consider port-Hamiltonian systems on Banach manifolds, and we present some physical examples that fit into the presented theory.

Wave function of perturbed Hamiltonian in graphene

In this paper, we use the generalized Dirac structure beyond the linear regime of graphene. This is probed using the a deformation of the Dirac structure in graphene by the generalized uncertainty principle. Here, the Planck length is replaced by the graphene lattice spacing. As the graphene sheet is bounded by two boundaries, we analyze this system with suitable boundary conditions. We solve the perturbed Hamiltonian and derive the wave function for this system. We observe that the energy of this system gets corrected due to this deformation. We explicitly calculate these corrections to the energy of this system.

The geometry of graded cotangent bundles

Given a vector bundle A→M we study the geometry of the graded manifolds T∗[k]A[1], including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, such as higher Courant algebroids on A⊕⋀k−1A∗ and higher Dirac structures therein, semi-direct products of Lie algebroid structures on A with their coadjoint representations up to homotopy, and branes on certain AKSZ σ-models.

Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

Charmed baryon spectrum from lattice QCD near the physical point

We calculate the low-lying spectrum of charmed baryons in lattice QCD on the $32^3\times64$, $N_f=2+1$ PACS-CS gauge configurations at the almost physical pion mass of $\sim 156$ MeV/c$^2$. By employing a set of interpolating operators with different Dirac structures and quark-field smearings for the variational analysis, we extract the ground and first few excited states of the spin-$1/2$ and spin-$3/2$, singly-, doubly-, and triply-charmed baryons. Additionally, we study the $\Xi_c$-$\Xi_c^\prime$ mixing and the operator dependence of the excited states in a variational approach. We identify several states that lie close to the experimentally observed excited states of the $\Sigma_c$, $\Xi_c$ and $\Omega_c$ baryons, including some of the $\Xi_c$ states recently reported by LHCb. Our results for the doubly- and triply-charmed baryons are suggestive for future experiments.

Dirac structures in nonequilibrium thermodynamics for simple open systems

Dirac structures are geometric objects that generalize Poisson structures and presymplectic structures on manifolds. They naturally appear in the formulation of constrained mechanical systems and play an essential role in structuring a dynamical system through the energy flow between its subsystems and elements. In this paper, we show that the evolution equations for open thermodynamic systems, i.e., systems exchanging heat and matter with the exterior, admit an intrinsic formulation in terms of Dirac structures. We focus on simple systems in which the thermodynamic state is described by a single entropy variable. A main difficulty compared to the case of closed systems lies in the explicit time dependence of the constraint associated with entropy production. We overcome this issue by working with the geometric setting of time-dependent nonholonomic mechanics. We define two types of Dirac dynamical systems for the nonequilibrium thermodynamics of open systems, based either on the generalized energy or the Lagrangian. The variational formulations associated with the Dirac dynamical systems are also presented.

LOCAL FORMULAS FOR MULTIPLICATIVE FORMS

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis–Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.