We present the first calculation of the connected scalar matrix element and the momentum fraction of the charm quark within the ${3}/{2}^{+}$ and ${3}/{2}^{-}$ triply charmed baryons on lattice QCD. The results are based on overlap valence fermions on two ensembles of $ N_f=2+1 $ domain wall fermion configurations with two lattice spacings. The corresponding sea quark pion masses are 300 MeV and 278 MeV. The separated contributions to the triply charmed baryon mass are derived through the decomposition of the QCD energy-momentum tensor. The contribution of the connected charm quark matrix element to the triply charmed baryon is about 3/2 times that of the charmonium, and it is almost 70% of the total mass. The mass splitting of ${3}/{2}^{+}$ and ${3}/{2}^{-}$ triply charmed baryons is mainly from the $ \langle H_{E}\rangle $ of the QCD energy-momentum tensor. A mass decomposition based on the quark model is also studied for comparison.

China

Jin-Bo Li

Long-Cheng Gui

Wen Qin

Wei Sun

Jian Liang

Triply charmed baryons mass decomposition from lattice QCD * Supported by the National Natural Science Foundation of China (12175036, 11935017, 12175073, 12205311, 12222503), the Natural Science Foundation of Hunan Province, China (2023JJ30380, 2024JJ6300), the Department of Education Foundation of Hunan Province, China (20A310, 22B0044), and the Major Project of Basic and Applied Basic Research of Guangdong Province, China (2020B0301030008)

<p>We calculate the leading-twist light cone distribution amplitudes of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>s</mi><mo stretchy="false">)</mo></math> baryon using lattice QCD methods within the framework of large-momentum effective theory. Our numerical computations are carried out with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> stout smeared clover fermions and a Symanzik gauge action on a lattice with spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.077</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and a pion mass of 303 MeV. To approach the large-momentum region, we simulate the equal-time correlations with the hadron momentum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>P</mi></mrow><mrow><mi>z</mi></mrow></msup><mo>=</mo><mo stretchy="false">{</mo><mn>2.52</mn><mo>,</mo><mn>3.02</mn><mo>,</mo><mn>3.52</mn><mo stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. We investigate the analytic properties of the baryon quasidistribution amplitude in coordinate space and validate them through our lattice calculations. After renormalization and extrapolation, we present results for the three-dimensional distribution in momentum fractions of the two light quarks. Based on these findings, we briefly discuss the phenomenological impact on weak decays of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Λ</mi><mi>b</mi></msub></math> and outline potential systematic uncertainties that can be improved in the future. This work lays the theoretical foundation for accessing baryon light cone distribution amplitudes from lattice QCD.</p>

Poland

Min-Huan Chu

Haoyang Bai

Jun Hua

United States

Xiangdong Ji

Germany

Andreas Schäfer

Yushan Su

Wei Wang

Xiaonu Xiong

Yi-Bo Yang

Jun Zeng

Jian-Hui Zhang

Qi-An Zhang

Light cone distribution amplitude for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi></math> baryon from lattice QCD

We compute the overlap Dirac spectrum on three gauge ensembles generated using <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor domain wall fermions. The three ensembles have different lattice spacings, and two of them have quark masses tuned to the physical point. The spectral density is determined up to <math><mi>λ</mi><mo>∼</mo><mn>100</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> with subpercentage statistical uncertainty. We find that the density is close to a constant below <math><mi>λ</mi><mo>∼</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> as predicted by chiral perturbative theory (<math><mi>χ</mi><mi>PT</mi></math>) and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order <math><mi>χ</mi><mi>PT</mi></math> form and using the nonperturbative renormalization using the regularization independent momentum subtraction scheme, the SU(2) (keeping the strange quark mass at the physical point) and SU(3) chiral condensates at <math><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> 2 GeV are determined to be <math><mi>Σ</mi><mo>=</mo><mo>(</mo><mn>265.4</mn><mo>(</mo><mn>0.5</mn><mo>)</mo><mo>(</mo><mn>4.2</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi><msup><mo>)</mo><mn>3</mn></msup></math> and <math><msub><mi>Σ</mi><mn>0</mn></msub><mo>=</mo><mo>(</mo><mn>234.3</mn><mo>(</mo><mn>0.5</mn><mo>)</mo><mo>(</mo><mn>25.8</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi><msup><mo>)</mo><mn>3</mn></msup></math>, respectively. The pion decay constants are also determined to be <math><mi>F</mi><mo>=</mo><mn>84.1</mn><mo>(</mo><mn>1.9</mn><mo>)</mo><mo>(</mo><mn>8.0</mn><mo>)</mo></math> and <math><msub><mi>F</mi><mn>0</mn></msub><mo>=</mo><mn>58.6</mn><mo>(</mo><mn>0.5</mn><mo>)</mo><mo>(</mo><mn>10.0</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant <math><msub><mi>L</mi><mn>6</mn></msub></math>. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with <math><mrow><mo>∼</mo><mn>10</mn><mo>%</mo><mi>–</mi><mn>20</mn><mo>%</mo></mrow></math> statistical uncertainties using the Dirac spectral density.

Andrei Alexandru

Yu-Jiang Bi

Terrence Draper

Keh-Fei Liu

Detecting the flavor content of the vacuum using the Dirac operator spectrum

The decay <math><mi>η</mi><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup></math> is an ideal process in which to study flavor-conserving <math><mi>C</mi></math> and <math><mi>C</mi><mi>P</mi></math> violation beyond the Standard Model. We deduce the <math><mi>C</mi></math>- and <math><mi>C</mi><mi>P</mi></math>-odd quark operators that contribute to <math><mi>η</mi><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup></math> originating from the mass-dimension-six Standard Model effective field theory. The corresponding hadron-level operators that generate a nonvanishing <math><mi>I</mi><mo>=</mo><mn>0</mn></math> amplitude at order <math><msup><mi>p</mi><mn>6</mn></msup></math> in the chiral effective theory are presented for the first time, to the best of our knowledge, in addition to the leading-order operators ascribed to the <math><mi>I</mi><mo>=</mo><mn>2</mn></math> final state. By fitting the KLOE-2 and the most recent BESIII experimental data, we determine the coefficients of the lowest-order <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math><mi>I</mi><mo>=</mo><mn>2</mn></math> amplitudes and estimate the potential new physics energy scale. We also perform an impact study of the future <math><mi>η</mi><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>π</mi><mn>0</mn></msup></math> experiments.

Jun Shi

Susan Gardner

Revisiting <math><mrow><mi>C</mi></mrow></math> and <math><mi>C</mi><mi>P</mi></math> violation in <math><mrow><mi>η</mi><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mn>0</mn></msup></mrow></math> decay

We present a lattice QCD calculation of the transverse-momentum-dependent wave function (TMDWF) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of highly improved staggered quarks with lattice spacing <math><mi>a</mi><mo>=</mo><mn>0.121</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> from the MILC collaboration, and one with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavor clover fermions and tree-level Symanzik gauge action generated by the CLS collaboration with <math><mi>a</mi><mo>=</mo><mn>0.098</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWF simulated on the lattice, we extract the TMDWF. The results for both lattice ensembles are compatible and a comparison with a phenomenological parametrization is made. Our studies provide a first attempt of ab initio calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.

Jin-Chen He

Hai-Tao Shu

Ji-Hao Wang

Hong Kong

Transverse-momentum-dependent wave functions of the pion from lattice QCD

The hadron mass can be obtained through the calculation of the trace of the energy-momentum tensor in the hadron which includes the trace anomaly and sigma terms. The anomaly due to conformal symmetry breaking is believed to be an important ingredient for hadron mass generation and confinement. In this work, we will present the calculation of the glue part of the trace anomaly form factors of the pion up to <math><msup><mi>Q</mi><mn>2</mn></msup><mo>∼</mo><mn>4.3</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math> and the nucleon up to <math><msup><mi>Q</mi><mn>2</mn></msup><mo>∼</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></math>. The calculations are performed on a domain wall fermion ensemble with overlap valence quarks at seven valence pion masses varying from <math><mo>∼</mo><mn>250</mn></math> to <math><mo>∼</mo><mn>540</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, including the unitary point <math><mo>∼</mo><mn>340</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. We calculate the radius of the glue trace anomaly for the pion and the nucleon from the <math><mi>z</mi></math> expansion. By performing a two-dimensional Fourier transform on the glue trace anomaly form factors in the infinite momentum frame with no energy transfer, we also obtain their spatial distributions for several valence quark masses. The results are qualitatively extrapolated to the physical valence pion mass with systematic errors from the unphysical sea quark mass, discretization effects in the renormalization sum rule, and finite-volume effects to be addressed in the future. We find the pion’s form factor changes sign, as does its spatial distribution, for light quark masses. This explains how the trace anomaly contribution to the pion mass approaches zero toward the chiral limit.

Bigeng Wang

Fangcheng He

France

Gen Wang

Trace anomaly form factors from lattice QCD

We calculate the nucleon electric dipole moment (EDM) from the <math><mi>θ</mi></math> term with overlap fermions on three domain wall lattices with different sea pion masses at lattice spacing 0.11 fm. Due to the chiral symmetry conserved by the overlap fermions, we have well-defined topological charge and chiral limit for the EDM. Thus, the chiral extrapolation can be carried out reliably at nonzero lattice spacings. We use three to four different partially quenched valence pion masses for each sea pion mass and find that the EDM dependence on the valence and sea pion masses behaves oppositely, which can be described by partially quenched chiral perturbation theory. With the help of the cluster decomposition error reduction technique, we determine the neutron and proton EDM at the physical pion mass to be <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>−</mo><mn>0.00148</mn><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>31</mn><mo>)</mo></mrow><mover><mrow><mi>θ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mtext> </mtext><mtext> </mtext><mi>e</mi><mo>·</mo><mi>fm</mi></mrow></math> and <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mn>0.0038</mn><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow><mover><mrow><mi>θ</mi></mrow><mrow><mo>¯</mo></mrow></mover><mtext> </mtext><mtext> </mtext><mi>e</mi><mo>·</mo><mi>fm</mi></mrow></math>. This work is a clear demonstration of the advantages of using chiral fermions in the nucleon EDM calculation and paves the road to future precise studies of the strong <math><mi>C</mi><mi>P</mi></math> violation effects.

Nucleon electric dipole moment from the <math><mi>θ</mi></math> term with lattice chiral fermions

We calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudoscalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton.

Lisa Walter

Lattice calculation of the intrinsic soft function and the Collins-Soper kernel

The separation of the connected and disconnected sea partons, which were uncovered in the Euclidean path-integral formulation of the hadronic tensor, is accommodated with an extended parametrization of the nonperturbative parton distribution functions in the CT18 global analysis. This is achieved with the help of the distinct small <math><mi>x</mi></math> behaviors of these two sea partons and the constraint from the lattice calculation of the ratio of the strange momentum fraction to that of the <math><mover><mi>u</mi><mo>¯</mo></mover></math> or <math><mover><mi>d</mi><mo>¯</mo></mover></math> in the disconnected insertion. The whole dataset of CT18 is used in this CT18CS fit. The impact of the recent SeaQuest data on the <math><mover><mi>d</mi><mo>¯</mo></mover><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mover><mi>u</mi><mo>¯</mo></mover><mo>(</mo><mi>x</mi><mo>)</mo></math> distribution of CT18CS is also discussed. The separate momentum fractions for the valence, the connected sea and disconnected sea of <math><mi>u</mi></math> and <math><mi>d</mi></math>, the strange and the gluon partons are presented at the input scale <math><mi>μ</mi><mo>=</mo><mn>1.3</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> for the first time. They can be compared term-by-term with systematic error controlled lattice calculations.

Tie-Jiun Hou

Mengshi Yan

C.P. Yuan

Connected and disconnected sea partons from the CT18 parametrization of PDFs

We present a calculation of the proton momentum and angular momentum decompositions using overlap fermions on a <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor RBC/UKQCD domain-wall lattice at 0.143 fm with a pion mass of 171 MeV which is close to the physical one. A complete determination of the momentum and angular momentum fractions carried by up, down, strange, and glue inside the proton has been done with valence pion masses varying from 171 to 391 MeV. We have utilized fast Fourier transform on the stochastic-sandwich method for connected-insertion parts and the cluster-decomposition error reduction technique technique for disconnected-insertion parts has been used to reduce statistical errors. The full nonperturbative renormalization and mixing between the quark and glue operators are carried out. The final results are normalized with the momentum and angular momentum sum rules and reported at the physical valence pion mass at <math><mover><mi>MS</mi><mo>¯</mo></mover><mo>(</mo><mi>μ</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>)</mo></math>. The renormalized momentum fractions for the quarks and glue are <math><mo>⟨</mo><mi>x</mi><msup><mo>⟩</mo><mi>q</mi></msup><mo>=</mo><mn>0.491</mn><mo>(</mo><mn>20</mn><mo>)</mo><mo>(</mo><mn>23</mn><mo>)</mo></math> and <math><mo>⟨</mo><mi>x</mi><msup><mo>⟩</mo><mi>g</mi></msup><mo>=</mo><mn>0.509</mn><mo>(</mo><mn>20</mn><mo>)</mo><mo>(</mo><mn>23</mn><mo>)</mo></math>, respectively, and the renormalized total angular momentum fractions for quarks and glue are <math><mn>2</mn><msup><mi>J</mi><mi>q</mi></msup><mo>=</mo><mn>0.539</mn><mo>(</mo><mn>22</mn><mo>)</mo><mo>(</mo><mn>44</mn><mo>)</mo></math> and <math><mn>2</mn><msup><mi>J</mi><mi>g</mi></msup><mo>=</mo><mn>0.461</mn><mo>(</mo><mn>22</mn><mo>)</mo><mo>(</mo><mn>44</mn><mo>)</mo></math>, respectively. The quark spin fraction is <math><mi>Σ</mi><mo>=</mo><mn>0.405</mn><mo>(</mo><mn>25</mn><mo>)</mo><mo>(</mo><mn>37</mn><mo>)</mo></math> from our previous work and the quark orbital angular momentum fraction is deduced from <math><mn>2</mn><msup><mi>L</mi><mi>q</mi></msup><mo>=</mo><mn>2</mn><msup><mi>J</mi><mi>q</mi></msup><mo>−</mo><mi>Σ</mi></math> to be 0.134(22)(44).

Proton momentum and angular momentum decompositions with overlap fermions