Holographic D-brane constructions, governed by the Dirac-Born-Infeld (DBI) action, play a central role in the AdS/CFT correspondence, particularly in applications to quantum chromodynamics and condensed matter systems. In this work, we demonstrate how to equip these bottom-up holographic models with dynamical boundary gauge fields, thereby introducing electromagnetic interactions into their dual field theory descriptions. As a direct application of this formalism, we compute the dispersion relations of the lowest quasinormal modes around both equilibrium and nonequilibrium steady states, and show that their behavior matches the predictions from hydrodynamics with dynamical U(1) symmetry.

China

Yong Ahn

Matteo Baggioli

Spain

Hyun-Sik Jeong

Japan

Masataka Matsumoto

Holographic D-brane constructions with dynamical gauge fields

The Belle Collaboration recently reported a potential candidate for the spin-2 $$D^*\bar{D}^*$$ partner of the $\textit{X}$(3872), denoted as $$X_2$$ , with a mass of $$(4014.3\pm 4.0\pm 1.5)$$ MeV and a width of $$(4\pm 11\pm 6)$$ MeV. Assuming the $$X_2$$ as a $$D^*\bar{D}^*$$ molecular state, we calculate its hidden charm decay $$X_2 \rightarrow \pi ^+ \pi ^- \eta _{c}$$ via box loop diagram using an effective Lagrangian approach. To investigate the dependence of the decay width on the molecular structure, we consider four different phase angles: $$\theta =0$$ , $$\pi /6$$ , $$\pi /4$$ , and $$\pi /2$$ . The decay width is found to be sensitive to the mass of the $$X_2$$ . The decay partial widths are predicted to be on the order of 1 keV. We expect that the present calculations, performed in the molecular framework, would provide useful guidance for future experimental studies.

Hao-Dong Cai

Gang Li

Shi-Dong Liu

Hidden charmed decays of spin-2 partner of $\textit{X}$(3872) within $$D^*\bar{D}^*$$ molecular framework

The energy-momentum tensor form factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> is finite and negative in hadronic models and lattice QCD when only strong forces are included. However, when electromagnetic forces are considered, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> of charged hadrons undergoes a dramatic change: at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>, it changes sign and diverges like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><msqrt><mrow><mo>−</mo><mi>t</mi></mrow></msqrt></math> as shown for the proton in the classical model by Białynicki-Birula based on residual nuclear forces which can be understood as a mean field approach. We construct an analogous neutron model and show that this framework accurately explains the electromagnetic proton-neutron mass difference. We demonstrate that, after appropriately rescaling the residual nuclear forces, the model can reproduce lattice data on the nucleon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≲</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></mrow></math> as well as QED effects. Based on this realistic model description, we show that the proton and neutron <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> form factors are practically indistinguishable down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≈</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></mrow></math>, far below what can currently be accessed experimentally. We conclude that in the foreseeable future the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> form factors of the proton and neutron will practically look the same in experiments and phenomenology.

United States

Andrea Mejia

Peter Schweitzer

Energy-momentum tensor form factor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> of the proton and neutron

We present a neural network wave function framework for solving non-Abelian lattice gauge theories in a continuous group representation. Using a combination of SU(2) equivariant neural networks alongside an SU(2) invariant, physics-inspired ansatz, we learn a parametrization of the ground state wave function of SU(2) lattice gauge theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions. Our method, performed in the Hamiltonian formulation, has a straightforward generalization to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math>. We benchmark our approach against a solely invariant ansatz by computing the ground state energy, demonstrating the need for bespoke gauge equivariant transformations. We evaluate the Creutz ratio and average Wilson loop, and obtain results in strong agreement with perturbative expansions. Our method opens up an avenue for studying lattice gauge theories beyond one dimension, with efficient scaling to larger systems, and in a way that avoids both the sign problem and any discretization of the gauge group.

Netherlands

Thomas Spriggs

Eliska Greplova

Switzerland

Juan Carrasquilla

Jannes Nys

Accurate Ground States of SU(2) Lattice Gauge Theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">D</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>1</mn><mi mathvariant="normal">D</mi></mrow></math>

In this work, we construct the <italic>D</italic>-wave isoscalar <math altimg='si1.svg'><mrow><mi>&#960;</mi><mi>&#960;</mi><mo linebreak='goodbreak'>/</mo><mi>K</mi><mover accent='true'><mi>K</mi><mo>&#175;</mo></mover></mrow></math> coupled-channel Omn&#232;s matrix, formulated to satisfy unitarity, analyticity, and the appropriate asymptotic behavior. We employ a two-channel <italic>K</italic>-matrix model containing poles associated with the <italic>f</italic> <inf>2</inf>(1270) and <math altimg='si7.svg'><mrow><msubsup><mi>f</mi><mrow><mn>2</mn></mrow><mo>&#8242;</mo></msubsup><mrow><mo>(</mo><mn>1525</mn><mo>)</mo></mrow></mrow></math> resonances. The resulting unitary scattering matrix, which reproduces the experimental <italic>&#960;&#960;</italic>&#8239;&#8594;&#8239; <italic>&#960;&#960;</italic> and <math altimg='si27.svg'><mrow><mi>&#960;</mi><mi>&#960;</mi><mo>&#8594;</mo><mi>K</mi><mover accent='true'><mi>K</mi><mo>&#175;</mo></mover></mrow></math> data and PDG information, serves as input to the homogeneous two-channel Muskhelishvili-Omn&#232;s equation. We compare our Omn&#232;s matrix with previous constructions based on <math altimg='si27.svg'><mrow><mi>&#960;</mi><mi>&#960;</mi><mo>&#8594;</mo><mi>K</mi><mover accent='true'><mi>K</mi><mo>&#175;</mo></mover></mrow></math> phases extracted from sums of Breit-Wigner amplitudes. The Omn&#232;s matrix developed here provides a reliable dispersive input for form-factor calculations and resonance studies in the tensor-meson sector. As an application, we show that it enables a simultaneous and accurate description of the BESIII <italic>J</italic>/ <italic>&#968;</italic>&#8239;&#8594;&#8239; <italic>&#960;</italic> 0 <italic>&#960;</italic> 0 <italic>&#947;</italic> and <italic>J</italic>/ <italic>&#968;</italic>&#8239;&#8594;&#8239; <italic>K<inf>S</inf>K<inf>S</inf>&#947;</italic> spectra in the <math altimg='si2.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math> electric-dipole (E1) partial wave.

Germany

Igor Danilkin

Oleksandra Deineka

Emilie Passemar

Marc Vanderhaeghen

Coupled-channel Omnès matrix for the <italic>D</italic>-wave isoscalar <math altimg='si1.svg'><mrow><mi>&#960;</mi><mi>&#960;</mi><mo linebreak='goodbreak'>/</mo><mi>K</mi><mover accent='true'><mi>K</mi><mo>&#175;</mo></mover></mrow></math> system and its application to <italic>J</italic>/ <italic>&#968;</italic>&#8239;&#8594;&#8239; <italic>&#960;</italic> 0 <italic>&#960;</italic> 0 <italic>&#947;</italic>,&#8201; <italic>K<inf>S</inf>K<inf>S</inf>&#947;</italic>

We consider supersymmetric solutions of $\textit{D}$ = 5 Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets. We consider solutions that admit both the R-symmetry Killing vector, $$ \mathcal{K} $$ , constructed as a bilinear in the Killing spinor, as well as an additional Killing vector $\textit{ℓ}$. Using $\textit{ℓ}$ to perform a dimensional reduction to $\textit{D}$ = 4, $$ \mathcal{N} $$ = 2 gauged supergravity, we show how the $\textit{D}$ = 5 on-shell action can be computed using equivariant localization. We illustrate the formalism with some examples, computing the supersymmetric Casimir energy and the supersymmetric index of the dual SCFT without using the explicit supergravity solutions.

Pietro Genolini

United Kingdom

Jerome Gauntlett

Yusheng Jiao

Jaeha Park

James Sparks

Equivariant localization for $\textit{D}$ = 5 gauged supergravity

Motivated by the interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></mrow></math> as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mtext> </mtext></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> molecular state, heavy-quark spin symmetry predicts a spin-2 partner, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math>, which can be regarded as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mtext> </mtext></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> molecule with quantum numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>. Its experimental confirmation, however, remains elusive. In this work, we investigate the production mechanisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math> in the reactions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><mi>p</mi><mo stretchy="false">→</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi><mo stretchy="false">→</mo><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math> within an effective Lagrangian framework. The production processes are modeled via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>-channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>/</mo><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mtext> </mtext></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> meson exchanges, while initial-state interactions (ISIs) mediated by Pomeron and Reggeon exchanges are also taken into account. Our calculations indicate that the total cross sections can reach the pb level, suggesting that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math> may be accessible at current and future experiments such as AMBER@CERN and LHCb. Inclusion of ISIs enhances the cross sections by nearly 1 order of magnitude. The differential distributions show distinct angular behaviors for the two reactions: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>−</mo></msup><mi>p</mi></math> reaction exhibits a forward-peaked distribution, whereas the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi></math> reaction shows a dip near central angles. This study provides a quantitative theoretical benchmark for future experimental searches of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math> and highlights the importance of ISIs in high-energy particle investigations.

Min Yuan

Bo Nan Zhang

Yin Huang

Search for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mtext> </mtext></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> molecular state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>4013</mn><mo stretchy="false">)</mo></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>−</mo></msup><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi></math> collisions

A potential quark model is used to search for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>s</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>u</mi><mi>d</mi><mi>s</mi><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>0</mn></mrow></msup></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math> pentaquark state that has recently been observed experimentally by LHCb Collaboration at 4338.2 MeV, with a width of 7.0 MeV and high statistical significance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo form="prefix">&gt;</mo><mn>15</mn><mi>σ</mi></math>. Our model Hamiltonian reproduces the masses of the low-lying charmed and strange hadrons. We use the Gaussian expansion method to solve the five-body Schrödinger equation. Employing the real scaling method, including the relevant meson-baryon thresholds explicitly, sharp resonances are distinguished from the meson-baryon scattering states. We incorporate new color states of the color-octet meson and baryon configurations as well as the color-singlet configurations for the five-quark states. We find no <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math> resonance close to the observed state, and also none in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math> state. This increases the likelihood that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>P</mi><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></mrow><mn>0</mn></msubsup></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math> hadronic molecule rather than a compact state.

Emiko Hiyama

Atsushi Hosaka

Makoto Oka

Georg Wolschin

Quark-model search for compact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>c</mi><mover accent="true"><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>u</mi><mi>d</mi><mi>s</mi></mrow></math> pentaquark states

We discuss the production of <math altimg='si10.svg'><mrow><mi>D</mi><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover></mrow></math> pairs in <math altimg='si2.svg'><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>&#8722;</mo></msup></mrow></math> collisions, where <italic>D</italic> refers to either <italic>D</italic> 0 or <math altimg='si30.svg'><msup><mi>D</mi><mo>+</mo></msup></math>. The continuum mechanism with the <italic>t</italic>/ <italic>u</italic>-channel vector-meson <italic>D</italic>* exchanges are considered. The results of the calculation depend on the parameter of the off-shell form-factor for the virtual <italic>D</italic>* mesons. The <italic>D</italic>* <italic>D&#947;</italic> coupling constants are found from the <italic>D</italic>*&#8239;&#8594;&#8239; <italic>D&#947;</italic> decays. We find relatively large contribution for the <math altimg='si4.svg'><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mrow><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover></mrow><mn>0</mn></msup></mrow></math> channel and much smaller contribution in the <math altimg='si1.svg'><mrow><msup><mi>D</mi><mo>+</mo></msup><msup><mi>D</mi><mo>&#8722;</mo></msup></mrow></math> channel. In the second case we consider also the <italic>D</italic> &#8239;&#177;&#8239; exchanges. We conclude that the bump at <math altimg='si31.svg'><mrow><msub><mi>M</mi><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mrow><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover></mrow><mn>0</mn></msup></mrow></msub><mo linebreak='goodbreak'>=</mo><mn>3.8</mn></mrow></math> GeV observed by the Belle and BaBar Collaborations has rather continuum origin than it corresponds to the broad resonance <italic>&#967;</italic> <inf><italic>c</italic>0</inf>(3860). We discuss also production of the <italic>&#967;</italic> <inf><italic>c</italic>2</inf>(3930) resonance which is a candidate for the <italic>&#967;</italic> <inf><italic>c</italic>2</inf>(2 <italic>P</italic>) state. This state can decay into both <math altimg='si10.svg'><mrow><mi>D</mi><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover></mrow></math> channels, however the branching fractions are not well known at present. From a comparison of our model results to the BaBar data we find <math altimg='si32.svg'><mrow><mi>B</mi><mo>(</mo><msub><mi>&#967;</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mrow><mo>(</mo><mn>3930</mn><mo>)</mo></mrow><mo>&#8594;</mo><mi>D</mi><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover><mo>)</mo><mo>=</mo><mn>0.58</mn><mo>&#177;</mo><mn>0.13</mn></mrow></math> using the two-photon width <math altimg='si33.svg'><mrow><msub><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle><mrow><mi>&#947;</mi><mi>&#947;</mi></mrow></msub><mo linebreak='goodbreak'>=</mo><mn>0.544</mn></mrow></math> keV (obtained for the Buchm&#252;ller-Tye potential) evaluated within the light-front approach (NRQCD limit). Our finding of <math altimg='si34.svg'><mrow><msub><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle><mrow><mi>&#947;</mi><mi>&#947;</mi></mrow></msub><mo linebreak='goodbreak'>&#215;</mo><mi>B</mi><mrow><mo>(</mo><msub><mi>&#967;</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mrow><mo>(</mo><mn>3930</mn><mo>)</mo></mrow><mo>&#8594;</mo><mi>D</mi><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><mn>0.32</mn><mo>&#177;</mo><mn>0.07</mn></mrow></math> keV is close to the Belle and BaBar results. Realistic predictions of the differential distributions in several variables and integrated cross-sections are given for the Belle II kinematics.

Poland

Izabela Babiarz

Piotr Lebiedowicz

Wolfgang Schäfer

Antoni Szczurek

Competition of <italic>&#967;<inf>c</inf></italic>(2 <italic>P</italic>) quarkonia and continuum in <math altimg='si9.svg'><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>&#8722;</mo></msup><mo>&#8594;</mo><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>&#8722;</mo></msup><mi>D</mi><mover accent='true'><mi>D</mi><mo>&#175;</mo></mover></mrow></math>

This paper presents a detailed analysis of the three-body final state interactions in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><msup><mi>K</mi><mo>+</mo></msup></math> process, whose phase space is sufficient small. To precisely extract the resonance parameters, for instance the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>3930</mn><mo stretchy="false">)</mo></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math> invariant mass distributions, in this process, one has to take into account final state interaction, especially the three-body final state interaction. We employ the dispersive Khuri-Treiman formalism, combined with a parametrization of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math> interaction based on heavy quark spin symmetry. By performing a simultaneous fit to the experimental data from LHCb, BABAR, and Belle collaborations, the scheme with three-body interaction successfully describes the invariant mass distributions of the three two-body subsystems. We precisely extract the pole structures of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>3930</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ψ</mi><mo stretchy="false">(</mo><mn>3770</mn><mo stretchy="false">)</mo></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3.913</mn><mo>−</mo><mn>0.016</mn><mi>i</mi><mtext> </mtext><mi>GeV</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>3.761</mn><mo>−</mo><mn>0.006</mn><mi>i</mi><mtext> </mtext><mi>GeV</mi></mrow></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup></math> decay. By performing the pole trajectory analysis on a uniformized complex plane, we find that both of them stem from the input bare state.

Xin-Yue Hu

Jiahao He

Pengyu Niu

Qian Wang

Thailand

Yupeng Yan

Three-body final state interactions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><msup><mi>K</mi><mo>+</mo></msup></math> decays