We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey-Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) 3-vertices can be extracted from the same integrand. We analyze three distinct limits of the PT 3-vertex, recovering the unrefined topological vertex, the refined topological vertex, and the Macdonald refined topological vertex. Higher-rank extensions of PT counting and the DT/PT correspondence are also explored. From a quantum algebraic perspective, we construct an operator version of the equivariant PT 3-vertex and term it the Pandharipande-Thomas $\textit{qq}$-character. We then discuss its connection with the quantum toroidal $${\mathfrak{g}\mathfrak{l}}_{1}$$ .

France

Taro Kimura

Japan

Go Noshita

Gauge origami and quiver W-algebras. Part IV. Pandharipande-Thomas $\textit{qq}$-characters

We further develop the BPS/CFT correspondence between quiver W-algebras/ -characters and partition functions of gauge origami. We introduce -characters associated with multi-dimensional partitions with nontrivial boundary conditions which we call Donaldson-Thomas (DT) -characters. They are operator versions of the equivariant DT vertices of toric Calabi-Yau three and four-folds. Moreover, we revisit the construction of the D8 -characters with no boundary conditions and give a quantum algebraic derivation of the sign rules of the magnificent four partition function. We also show that under the proper sign rules, the D6 and D8 -characters with no boundary conditions all commute with each other and discuss its physical interpretation.

Gauge origami and quiver W-algebras. Part III. Donaldson-Thomas

We analyze the angular momentum balance for a particle undergoing Thomas precession. The relationships among relativistic torque, the center of mass, and the center of inertia for a spinning particle are clarified. We show that spin precession is accompanied by orbital angular momentum precession, and present examples of the resulting out-of-plane motion.

Canada

Andrzej Czarnecki

Andrei Zelnikov

Thomas precession, relativistic torque, and non-planar orbits

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas et al. to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant.

United States

Tyler Grover

Kory Stiffler

Patrick Vecera

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparametrization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow></math>, is directly related to the coadjoint elements of the Virasoro algebra. TW gravity exploits projective Gauss-Bonnet terms in the action functional which allows the theory to collapse to Einstein’s theory of general relativity in the limit that <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> vanishes. In this paper we develop the graded extension of TW gravity, super-TW gravity, in the framework of a DeWitt supermanifold. We construct the Lagrangian for super-TW gravity, give a detailed derivation of the classical field equations, and discuss the graded extension of the projective connection as a prelude to a future understanding of TW-supergravity (which has manifest supersymmetry) and its relationship to the super-Virasoro algebra.

Calvin Mera Sánchez

Vincent G. J. Rodgers

Graded extension of Thomas-Whitehead gravity

We have studied the dominant radiative transitions of the charmonium <math><mi>S</mi></math>- and <math><mi>P</mi></math>-wave states within the covariant confined quark model. The gauge invariant leading-order transition amplitudes have been expressed by using either the conventional Lorentz structures, or the helicity amplitudes, where it was effective. The renormalization couplings of the charmonium states have been strictly fixed by the compositeness conditions that excludes the constituent degrees of freedom from the space of physical states. We use the basic model parameters for the constituent c-quark mass <math><msub><mi>m</mi><mi>c</mi></msub><mo>=</mo><mn>1.80</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and the global infrared cutoff <math><mi>λ</mi><mo>=</mo><mn>0.181</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. We additionally introduce only one adjustable parameter <math><mi>ϱ</mi><mo>&gt;</mo><mn>0</mn></math> common for the charmonium states <math><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mrow><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub></mrow><mo>)</mo></mrow></math>, <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>(</mo><mrow><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow></math>, <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math>, <math><msub><mi>h</mi><mi>c</mi></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math>, and <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> to describe the quark distribution inside the hadron. This parameter describes the ratio between the charmonium “size” and its physical mass. The optimal value <math><mi>ϱ</mi><mo>=</mo><mn>0.485</mn></math> has been fixed by fitting the latest data for the partial widths of the one-photon radiative decays of the triplet <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo><mo>,</mo><mi>J</mi><mo>=</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></math>. Then, we calculate corresponding fractional widths for states <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>(</mo><mrow><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>)</mo></math> and <math><msub><mi>h</mi><mi>c</mi></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math>. Estimated results are in good agreement with the latest data. By using the fraction data from PDG2020 and our estimated partial decay width for <math><msub><mi>h</mi><mi>c</mi></msub><mo>(</mo><msub><mrow><mmultiscripts><mrow><mi>P</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math> we recalculate the “theoretical full width” <math><msubsup><mi>Γ</mi><msub><mi>h</mi><mi>c</mi></msub><mtext>theor</mtext></msubsup><mo>≃</mo><mo>(</mo><mn>0.57</mn><mo>±</mo><mn>0.12</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> in comparison with latest data <math><msubsup><mi>Γ</mi><msub><mi>h</mi><mi>c</mi></msub><mi>exp</mi></msubsup><mo>≃</mo><mo>(</mo><mn>0.7</mn><mo>±</mo><mn>0.4</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. We also repeated our calculations by gradually decreasing the global cutoff parameter and revealed that the results do not change for any <math><mi>λ</mi><mo>&lt;</mo><mn>0.181</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> up to the “deconfinement” limit.

JINR

Mongolia

Gurjav Ganbold

Germany

Thomas Gutsche

Mikhail A. Ivanov

Chile

Russia

Valery E. Lyubovitskij

Radiative transitions of charmonium states in the covariant confined quark model

Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity over a d-dimensional manifold that is intimately related to string theory through reparametrization invariance. Unparametrized geodesics are the ubiquitous structure that ties together string theory and higher dimensional gravitation. This is realized through the projective geometry of Tracy Thomas. The projective connection, due to Thomas and later Whitehead, admits a component that in one dimension is in one-to-one correspondence with the coadjoint elements of the Virasoro algebra. This component is called the diffeomorphism field <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> in the literature. It also has been shown that in four dimensions, the TW action collapses to the Einstein-Hilbert action with cosmological constant when <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> is proportional to the Einstein metric. These previous results have been restricted to either particular metrics, such as the Polyakov 2D metric, or were restricted to coordinates that were volume preserving. In this paper, we review TW gravity and derive the gauge invariant TW action that is explicitly projectively invariant and general coordinate invariant. We derive the covariant field equations for the TW action and show how fermionic fields couple to the gauge invariant theory. The independent fields are the metric tensor <math><msub><mi>g</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math>, the fundamental projective invariant <math><msub><msup><mi>Π</mi><mi>a</mi></msup><mrow><mi>b</mi><mi>c</mi></mrow></msub></math>, and the diffeomorphism field <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow></math>.

Samuel Brensinger

Kenneth Heitritter

General structure of Thomas-Whitehead gravity

Motivated by the calculation of observables in the decays <math><msub><mi>Λ</mi><mi>b</mi></msub><mo>→</mo><msub><mi>Λ</mi><mi>c</mi></msub><mrow><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>±</mo></msup><mo>)</mo></mrow><mo>ℓ</mo><mover><mi>ν</mi><mo>¯</mo></mover></math>, we present a calculation of form factors in the quark model. Our scheme combines a spectroscopic model with the internal wave functions and the Bakamjian-Thomas (BT) relativistic formalism to get the wave functions in motion. In the heavy quark limit, the current matrix elements provide the Isgur-Wise (IW) function. This limit is covariant, satisfies a large set of sum rules, and has been successfully applied to mesons. On the other hand, for baryons, we meet difficulties using standard spectroscopic models. This leads us to propose a phenomenological model, a Q-pointlike-diquark model, nonrelativistic, with harmonic oscillator forces, giving both a reasonable low-lying spectrum and the expected slope of the IW function. To begin, we extract this slope from lattice QCD data and find it to be around <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup><mo>∼</mo><mn>2</mn></math>, that we use as a guideline. We are not able to reproduce the right <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup></math> using certain typical standard <math><mrow><mtext>linear</mtext><mo>+</mo><mtext>Coulomb</mtext></mrow></math> potential models, both with three quarks <math><mi>Q</mi><mi>q</mi><mi>q</mi></math> or in a Q-pointlike-diquark picture. These difficulties seem to derive from the high sensitivity of <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup></math> to the structure of the light quark subsystem in a relativistic scheme. Finally, we present our model, and fixing its parameters to yield the correct spectrum and <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup><mo>∼</mo><mn>2</mn></math>, we apply it to the calculation of observables. By studying Bjorken sum rule, we show that the inelastic IW function is large, and therefore, the transitions <math><msub><mi>Λ</mi><mi>b</mi></msub><mo>→</mo><msub><mi>Λ</mi><mi>c</mi></msub><mrow><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>,</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>)</mo></mrow><mo>ℓ</mo><mover><mi>ν</mi><mo>¯</mo></mover></math> could be studied at LHCb. Interestingly, some observables in the <math><mi>τ</mi></math> case present zeroes for specific values of <math><msup><mi>q</mi><mn>2</mn></msup></math> that could be tests of the Standard Model.

D. Bečirević

A. Le Yaouanc

V. Morénas

L. Oliver

Heavy baryon wave functions, Bakamjian-Thomas approach to form factors, and observables in <math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>→</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><msup><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo><mo>ℓ</mo><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> transitions

Recent lattice QCD calculations showed that a <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub></math> similar to the deuteron with baryon number <math><mi>B</mi><mo>=</mo><mn>2</mn></math> and with a small binding energy might exist. In this work we propose a hadronic molecular approach to study the dynamical properties of this exotic state. We employed a phenomenological Lagrangian approach to describe the coupling of the <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub></math> to its constituents and the strong decays into conventional hadrons, <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub><mo>→</mo><mi>Λ</mi><mi>Ξ</mi></math> and <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub><mo>→</mo><mi>Σ</mi><mi>Ξ</mi></math>. Predictions for the sum of the decay rates are in the range of a few hundred keV. In addition, we find that the <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub><mo>→</mo><mi>Λ</mi><mi>Ξ</mi></math> mode is dominant, preferably searched for in a future Relativistic Heavy Ion Collider (RHIC) experiment.

China

Cheng-Jian Xiao

Yu-Bing Dong

Dian-Yong Chen

Towards the decay properties of deuteron-like state <math><msub><mi>d</mi><mrow><mi>N</mi><mi>Ω</mi></mrow></msub></math>

We present a study of the <math><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1535</mn><mo>)</mo></math> resonance electroexcitation in a soft-wall AdS/QCD model. Both the transverse <math><msubsup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mi>p</mi></msubsup></math> and longitudinal <math><msubsup><mi>S</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mi>p</mi></msubsup></math> helicity amplitudes are calculated resulting in good agreement with data and with the MAID parametrization.

Ivan Schmidt

<math><mi>γ</mi><mi>N</mi><mo>→</mo><msup><mi>N</mi><mo>*</mo></msup><mo>(</mo><mn>1535</mn><mo>)</mo></math> transition in soft-wall AdS/QCD