Hyperon-antihyperon pairs produced in high-energy electron-positron annihilation constitute a naturally spin-entangled system in the high-energy regime. Recently, a probabilistic amplification of entanglement, termed autodistillation, has been found in the daughter baryon-antibaryon pairs from hyperon decay and is constrained by an upper boundary. This work demonstrates that the quantum entanglement in this process may be accompanied by a decrease, constrained by a lower boundary, but it will not be completely lost. Thus, the entanglement of these systems undergoes redistribution within the phase space during the sequential decays of hyperons, highlighting the important role of hyperon polarization. By using the explicit spin density matrix of baryon pairs, it is also found that the quantumness of the system, characterized by quantum discord, always has the possibility to increase during decay processes, even when entanglement evaluated by concurrence and negativity does not increase.

China

Cong Li

Xu Cao

Ai-Qiang Guo

Chun-Xu Yu

Hong-Wei Zhang

Zhe Zhang

Entanglement redistribution of hyperon-antihyperon pair via sequential decay * Supported by the National Natural Science Foundation of China (12547111, 12447132, 12575089) and the National Key R&D Program of China (2023YFA1606703)

Using a sample of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation data corresponding to an integrated luminosity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4.4</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> collected with the BESIII detector at the BEPCII collider and produced at center-of-mass energies from 4600 to 4698 MeV, an amplitude analysis is performed of the singly Cabibbo-suppressed decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>p</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math>. The branching fractions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>p</mi><mi>ϕ</mi><mo stretchy="false">(</mo><mn>1020</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo><msup><mi>K</mi><mo>+</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1670</mn><mo stretchy="false">)</mo><msup><mi>K</mi><mo>+</mo></msup></math> are measured, where the latter two modes are decays that are observed for the first time. At the same time, with the detection efficiency based on the results of the amplitude analysis, the branching fraction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>p</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math> is updated to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>9.94</mn><mo>±</mo><mn>0.6</mn><msub><mn>5</mn><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>0.5</mn><msub><mn>0</mn><mrow><mi>syst</mi></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, which is consistent with the current world average value within one standard deviation. The result supersedes the previous BESIII measurement with precision improved by approximately a factor of 1.5.

M. Ablikim

Russia

M.N. Achasov

Sweden

P. Adlarson

X.C. Ai

Germany

R. Aliberti

Italy

A. Amoroso

Q. An

Y. Bai

JINR

O. Bakina

Y. Ban

H.R. Bao

Poland

V. Batozskaya

Mongolia

K. Begzsuren

N. Berger

M. Berlowski

M.B. Bertani

D. Bettoni

F. Bianchi

E. Bianco

A. Bortone

I. Boyko

United States

R.A. Briere

A. Brueggemann

H. Cai

M.H. Cai

X. Cai

A. Calcaterra

G.F. Cao

N. Cao

Türkiye

S.A. Cetin

X.Y. Chai

J.F. Chang

T.T. Chang

G.R. Che

Y.Z. Che

C.H. Chen

Chao Chen

G. Chen

H.S. Chen

H.Y. Chen

M.L. Chen

S.J. Chen

S.M. Chen

T. Chen

X.R. Chen

X.T. Chen

X.Y. Chen

Y.B. Chen

Y.Q. Chen

Z.K. Chen

J.C. Cheng

L.N. Cheng

South Korea

S.K. Choi

X. Chu

G. Cibinetto

F. Cossio

United Kingdom

J. Cottee-Meldrum

H.L. Dai

J.P. Dai

X.C. Dai

A. Dbeyssi

R.E. de Boer

D. Dedovich

C.Q. Deng

Z.Y. Deng

A. Denig

I. Denisenko

M. Destefanis

F. De Mori

X.X. Ding

Y. Ding

Y.X. Ding

J. Dong

L.Y. Dong

M.Y. Dong

X. Dong

M.C. Du

S.X. Du

X.L. Du

Y.Y. Duan

Z.H. Duan

P. Egorov

G.F. Fan

J.J. Fan

Y.H. Fan

J. Fang

S.S. Fang

W.X. Fang

Y.Q. Fang

L. Fava

F. Feldbauer

G. Felici

C.Q. Feng

J.H. Feng

L. Feng

Q.X. Feng

Y.T. Feng

M. Fritsch

Amplitude analysis of singly Cabibbo-suppressed decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><mi>p</mi><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>−</mo></msup></math>

We analyze the radiative decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>P</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>D</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>S</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>P</mi><mtext> </mtext><mtext> </mtext><msub><mi mathvariant="normal">Λ</mi><mi>c</mi></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>D</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>S</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>P</mi><mtext> </mtext><mtext> </mtext><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub></mrow></math> charmed baryons, belonging to the flavor antitriplet (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mn mathvariant="bold">3</mn><mo stretchy="false">¯</mo></mover><mi mathvariant="normal">F</mi></msub></math>), using the constituent quark model. We compute electromagnetic transitions from ground and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave states to ground states, as well as from second-shell states to both ground and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave final states. Electromagnetic decay widths are especially valuable for identifying resonances when multiple states share similar mass and total decay width. We give branching ratios of several electromagnetic decay widths that can confirm the assignment of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3055</mn><mo stretchy="false">)</mo></math> reported by LHCb. Likewise, we provide branching-ratio predictions that can help guide the assignment of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>3080</mn><mo stretchy="false">)</mo></math> states, and discuss their possible interpretation either as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mi>D</mi></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="bold">J</mi><mi mathvariant="bold">P</mi></msup><mo>=</mo><mn>5</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></math> or as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>S</mi></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="bold">J</mi><mi mathvariant="bold">P</mi></msup><mo>=</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>+</mo></msup></math>. For the first time, this work provides calculations of electromagnetic decays for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>ρ</mi></msub></math>-wave states, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi><mo>−</mo><mi>λ</mi></math> mixed configurations, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ρ</mi></math>-mode radially excited states in singly charmed baryons of the flavor antitriplet. Both experimental and model-dependent uncertainties are taken into account throughout our analysis.

Mexico

R. Gamboa-Goni

H. García-Tecocoatzi

Ailier Rivero-Acosta

A. Gutierrez-Rodriguez

A. Ramirez-Morales

E. Santopinto

Carlos Alberto Vaquera-Araujo

Radiative decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>P</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>D</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>S</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>P</mi><mtext> </mtext><mtext> </mtext><msub><mi mathvariant="normal">Λ</mi><mi>c</mi></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>D</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>S</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mi>P</mi><mtext> </mtext><mtext> </mtext><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub></mrow></math> charmed baryons

Using a dataset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>7.33</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation data collected with the BESIII detector at center-of-mass energies from 4.128 to 4.226 GeV, we report an updated measurement of the branching fraction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>τ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></math> via four <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>τ</mi><mo>+</mo></msup></math> decay modes: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>τ</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>τ</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>τ</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>τ</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mn>0</mn></msup><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>τ</mi></msub></math>. The branching fraction is determined to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>τ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>5.37</mn><mo>±</mo><mn>0.0</mn><msub><mn>8</mn><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>0.0</mn><msub><mn>6</mn><mrow><mi>syst</mi></mrow></msub><mo stretchy="false">)</mo><mo>%</mo></math>. The product of the modulus of the Cabibbo-Kobayashi-Maskawa matrix element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> decay constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></msub></math> is measured to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>f</mi></mrow><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mo stretchy="false">(</mo><mn>248.2</mn><mo>±</mo><msub><mrow><mn>1.9</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><msub><mrow><mn>1.4</mn></mrow><mrow><mi>syst</mi></mrow></msub><mo>±</mo><msub><mrow><mn>0.6</mn></mrow><mrow><mtext>input</mtext></mrow></msub><mo>±</mo><msub><mrow><mn>0.8</mn></mrow><mrow><mi>EM</mi></mrow></msub><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. Both the branching fraction and the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></msub><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo></math> are the most precise results yet obtained. Then, taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></msub></math> from lattice quantum chromodynamics calculations results in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mn>0.993</mn><mo>±</mo><mn>0.00</mn><msub><mn>8</mn><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>0.00</mn><msub><mn>6</mn><mrow><mi>syst</mi></mrow></msub><mo>±</mo><mn>0.00</mn><msub><mn>3</mn><mtext>input</mtext></msub><mo>±</mo><mn>0.00</mn><msub><mn>3</mn><mrow><mi>EM</mi></mrow></msub></math>. Conversely, one finds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></msub><mo>=</mo><mo stretchy="false">(</mo><mn>255.0</mn><mo>±</mo><msub><mn>1.9</mn><mrow><mi>stat</mi></mrow></msub><mo>±</mo><msub><mn>1.4</mn><mrow><mi>syst</mi></mrow></msub><mo>±</mo><msub><mn>0.6</mn><mtext>input</mtext></msub><mo>±</mo><msub><mn>0.8</mn><mrow><mi>EM</mi></mrow></msub><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> when taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo stretchy="false">|</mo></math> from the CKMfitter group as an input. Combining with the world average value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub></math>, the ratio of the branching fractions between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>τ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub></math> is estimated to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>10.04</mn><mo>±</mo><mn>0.29</mn></math>, which is consistent with the Standard Model prediction of lepton flavor universality.

C.S. Akondi

Y.H. An

X.L. Bao

W. Chen

J. Cheng

Y.Q. Du

Updated measurement of the branching fraction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>τ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></math>

In the present work, we investigate the molecular properties of the hidden charm pentaquark states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> with a coupled-channel framework that combines heavy quark spin symmetry and the local hidden gauge formalism. By solving the Bethe-Salpeter equation with the cutoff method, we obtain the pole trajectories, wave functions, and root-mean-square radii. For the hidden charm system, the full coupled-channel interactions respecting the heavy quark spin symmetry are essential to generate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub></math> states, which significantly affect the poles’ widths. The dominant bound channels are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><msub><mi mathvariant="normal">Σ</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><msub><mi mathvariant="normal">Σ</mi><mi>c</mi></msub></math>, which couple strongly to the lower decay channels. In contrast, for the hidden charm strange system, the full heavy quark spin symmetry treatment is not necessary, where the splitting pseudoscalar meson-baryon (PB) and vector meson-baryon (VB) sectors yield similar results. The main bound channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub></math> couple strongly to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></msub><msub><mi mathvariant="normal">Λ</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mi>s</mi><mo>*</mo></msubsup><msub><mi mathvariant="normal">Λ</mi><mi>c</mi></msub></math>, respectively, but weakly to the lower decay channels, different from the hidden charm case. The trajectories of the pole widths for the loosely bound channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup></math> exhibit distinct behaviors. Notably, all the primary bound channels have similar binding energies in the single-channel interactions due to equally attractive potentials. Furthermore, we also calculate the wave functions and root-mean-square radii of the corresponding poles. The wave functions are localized within <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>0</mn><mo>∼</mo><mn>6</mn><mtext> </mtext><mi>fm</mi></mrow></math> and vanish fast beyond 4 fm. The root-mean-square radii, evaluated by two consistent methods, typically lie between 0.5 and 2 fm, comparable to the characteristic scale of molecular states. The root-mean-square radii depend on the pole trajectories and differ among the full coupled-channel case, the split PB and VB sectors, and the single-channel interactions.

Jing-Zhi Cao

Huan-Yu Wei

Jiao-Xue Yang

Jian Sun

Chu-Wen Xiao

Study of the molecular properties of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> states

We investigate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><msub><mi>η</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> scattering in the <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><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> channels up to a center-of-mass energy of 6.6 GeV. The calculations are carried out at two unphysical pion masses, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>420</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and 250 MeV in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn></math> lattice QCD. For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub></math>, we extract the finite-volume energy levels on two lattices with an identical lattice spacing (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>≃</mo><mn>0.136</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>) but different spatial volumes. Since the coupled-channel effects between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><msub><mi>η</mi><mi>c</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> channels are found to be negligible, we analyze the corresponding scattering properties using the single-channel L"uscher method. We find that the interactions in these dicharmonium systems are dominated by the quark rearrangement effect. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> channel, the near-threshold attraction in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> and repulsion in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><msub><mi>η</mi><mi>c</mi></msub></math> can be explained through the Fierz rearrangement. The attractive interaction in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mo>​</mo><mn>1</mn></msup><msub><mi>S</mi><mn>0</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> channel allows for the existence of a near-threshold scalar structure, which may correspond to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>6200</mn><mo stretchy="false">)</mo></math>. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> channel, while the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mo>​</mo><mn>5</mn></msup><msub><mi>S</mi><mn>2</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> system exhibits a repulsive interaction near threshold, the scattering amplitude has a Castillejo-Dalitz-Dyson zero at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>6.45</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> and a resonance pole at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msub></mrow></msqrt><mo>=</mo><mrow><mo stretchy="false">[</mo><mn>6.543</mn><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mn>0.548</mn><mo stretchy="false">(</mo><mn>34</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>420</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msub></mrow></msqrt><mo>=</mo><mrow><mo stretchy="false">[</mo><mn>6.538</mn><mo stretchy="false">(</mo><mn>13</mn><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mn>0.537</mn><mo stretchy="false">(</mo><mn>56</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mtext> </mtext><mi>GeV</mi></mrow></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>250</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, where the uncertainties are statistical. This resonance may correspond to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>6600</mn><mo stretchy="false">)</mo></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi><mo stretchy="false">(</mo><mn>6400</mn><mo stretchy="false">)</mo></math>) reported by the ATLAS and CMS Collaborations. Our result supports its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> assignment, in agreement with the latest spin-parity determination by CMS. Despite the observed dominance of the quark rearrangement effect, the light-hadron dynamics underlying dicharmonium scattering need to be explored at lighter pion masses. Furthermore, since our lattices are coarse and the lattice volumes are small, the associated systematic uncertainties should be controlled in future studies using more sophisticated lattice setups.

Geng Li

Chunjiang Shi

Ying Chen

Wei Sun

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>J</mi><mo>/</mo><mi>ψ</mi></math> scattering from lattice QCD

We investigate the elliptic-flow splitting between prompt <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> mesons within a heavy-quark transport framework with sequential hadronization, in which <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> forms at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1.2</mn><mspace width="0.16em"></mspace><msub><mi>T</mi><mi>c</mi></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math>. We present model calculations for the <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>T</mi></msub></math>-differential <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub></math> and predictions for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>/</mo><msup><mi>D</mi><mn>0</mn></msup></mrow></math> yield ratio in O-O <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>–</mo><mspace width="0.16em"></mspace><mn>20</mn><mo>%</mo></mrow></math> collisions at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV, where preliminary ALICE data are available. The sequential scenario reproduces the observed <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msup><mi>D</mi><mn>0</mn></msup><mo>)</mo></mrow><mo>&gt;</mo><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>)</mo></mrow></mrow></math> ordering and predicts an enhanced <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>/</mo><msup><mi>D</mi><mn>0</mn></msup></mrow></math> ratio at low <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>T</mi></msub></math>, whereas a simultaneous baseline yields the opposite ordering. Decomposing the hadronic splitting into its partonic components, we show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub></math> ordering is driven by the late-stage flow accumulated by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math>-parent charm quarks during the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1.2</mn><mspace width="0.16em"></mspace><msub><mi>T</mi><mi>c</mi></msub><mo>→</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></math> interval, with hadronic rescattering essential to preserve the signal in small systems. A systematic scan across nine collision configurations spanning O-O and Pb-Pb centralities reveals a universal linear scaling between the hadronic splitting and the partonic flow increment accumulated during this window. This establishes the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>–</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></mrow></math> flow splitting as a hadronization chronometer of the QGP at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV.

Hui Du

Xiao-Wei Hao

Wei Dai

Jiaxing Zhao

Ben-Wei Zhang

Enke Wang

System-size dependence of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>0</mn></msup><mtext>−</mtext><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></mrow></math> flow splitting from early <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> formation at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV

This paper proposes a novel mechanism based on the instantaneous Bethe-Salpeter (Salpeter) equation for investigating wave function mixing in vector mesons such as <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>. Conventional theories typically treat <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 a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>S</mi><mo>−</mo><mn>1</mn><mi>D</mi></math> mixed state; however, considering only tensor forces or relativistic corrections alone often leads to mixing angles that are too small and inconsistent with experimental data. Phenomenological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>S</mi><mo>−</mo><mn>1</mn><mi>D</mi></math> mixing requires experimental data as input to determine the mixing angles, resulting in limited theoretical studies on states like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>1</mn><mi>D</mi><mo>,</mo><mn>2</mn><mi>D</mi><mo stretchy="false">)</mo></math> in the absence of experimental data. To more accurately describe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>−</mo><mi>D</mi></math> mixing and its relativistic effects, this paper systematically compares four relativistic wave function representations (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>φ</mi><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>φ</mi><mn>2</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>φ</mi><mn>3</mn></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>φ</mi><mn>4</mn></msub></math>) by solving the Salpeter equation and calculates the mass spectra and dileptonic decay widths of charmonium and bottomonium. The study finds that the wave function representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>φ</mi><mn>2</mn></msub></math> can simultaneously reproduce the experimental data of both charmonium and bottomonium well. Further analysis reveals that, in addition to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>−</mo><mi>D</mi></math> mixing, the wave functions of vector mesons contain a non-negligible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave component, meaning they are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>−</mo><mi>P</mi><mo>−</mo><mi>D</mi></math> mixed states. We predict the mixing angles for bottomonium <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>1</mn><mi>D</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">ϒ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>D</mi><mo stretchy="false">)</mo></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mn>1.7</mn><msubsup><mrow><mn>8</mn></mrow><mrow><mo>−</mo><mn>0.25</mn></mrow><mrow><mo>+</mo><mn>0.32</mn></mrow></msubsup><mo stretchy="false">)</mo><mi>°</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>5.4</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>0.76</mn></mrow><mrow><mo>+</mo><mn>1.10</mn></mrow></msubsup><mo stretchy="false">)</mo><mi>°</mi></math>, with dileptonic decay widths of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>2.29</mn><mrow><mo>−</mo><mn>0.69</mn></mrow><mrow><mo>+</mo><mn>0.86</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>eV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mn>10.5</mn><mrow><mo>−</mo><mn>3.1</mn></mrow><mrow><mo>+</mo><mn>4.2</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>eV</mi></math>, respectively.

Wen-Yuan Ke

Qiang Li

Tianhong Wang

Tai-Fu Feng

Guo-Li Wang

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mo>−</mo><mi>P</mi><mo>−</mo><mi>D</mi></math> mixing in vector quarkonia from the Salpeter equation with optimized wave function representations

Motivated by the first observation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> violation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math>-baryon decays, the search for baryonic decays exhibiting large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> violation will be a primary focus in the coming years. We propose that significant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math>-violating effects exist in the decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>b</mi></msub><mo stretchy="false">→</mo><mi mathvariant="normal">Λ</mi><mi>D</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math> denotes a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> eigenstate of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mtext>−</mtext><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> system. The predicted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> asymmetries for both the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math>-even and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math>-odd modes can reach magnitudes as large as 50%, making these decays promising targets for measurement at the LHCb experiment. Additionally, we predict for the first time several nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math>-violating observables associated with angular distribution parameters, providing valuable complementary information in the search for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> violation in baryon decays. Furthermore, we propose a novel strategy to extract the Cabibbo-Kobayashi-Maskawa angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi></math> by combining data on angular distribution parameters and decay rates from the relevant channels. We emphasize that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>b</mi></msub><mo stretchy="false">→</mo><mi mathvariant="normal">Λ</mi><mi>D</mi></math> decays are among the most promising candidates for determining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi></math> in the baryon sector. Our findings may offer new insights for future theoretical and experimental investigations.

Zhou Rui

Zhi-Tian Zou

Ya Li

Ying Li

Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>b</mi></msub><mo stretchy="false">→</mo><mi mathvariant="normal">Λ</mi><mi>D</mi></math> decays and extraction of the Cabibbo-Kobayashi-Maskawa angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi></math>

We extend the results of Lappi et al. [<a href="http://dx.doi.org/10.1140/epjc/s10052-023-12365-2">Eur. Phys. J. C 84, 84 (2024)</a>] to show that it is possible to obtain expressions for the longitudinal, singlet, and gluon structure functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>L</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>S</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math> in deep inelastic scattering directly in terms of the measured functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>F</mi><mn>2</mn></msub><mo>/</mo><mi>ln</mi><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> modulo nonsinglet corrections expected to be small at very small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. The latter can be treated at low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> using existing quark distributions. Our results are presented consistently to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>, correcting and extending the mixed-order results of Lappi et al..

Iran

G.R. Boroun

Loyal Durand

Phuoc Ha

Direct determination of the structure functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>L</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>S</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>F</mi><mn>2</mn></msub><mo>/</mo><mi>d</mi><msup><mi>Q</mi><mn>2</mn></msup></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>