<p>It is a long-standing puzzle why the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> violation (CPV) in the baryon system has not yet been definitively established as in the meson one. We demonstrate that individual partial-wave CPV in the <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>p</mi><msup><mi>π</mi><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><msup><mi>K</mi><mo>−</mo></msup></math> decays can exceed 10%, but the destruction between the partial waves (the suppression by the small partial-wave weight) results in a small net direct CPV in the former (the latter) as measured currently. Our finding highlights the different dynamics responsible for CPVs in baryon and meson decays. We propose to probe the CPV observables associated with the angular distributions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>b</mi></mrow></msub><mo stretchy="false">→</mo><mi>p</mi><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1260</mn><mo stretchy="false">)</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>p</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></mrow></math> decay products, which are large enough for being identified experimentally.</p>

China

Jia-Jie Han

Ji-Xin Yu

Ya Li

Hsiang-nan Li

Jian-Peng Wang

Zhen-Jun Xiao

Fu-Sheng Yu

Establishing <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"><mi>b</mi></math>-Baryon Decays

<p>Within the framework of nonrelativistic quantum chromodynamics factorization, we calculate the next-to-leading order (NLO) perturbative QCD corrections to the form factors for the semileptonic decays of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> via (axial-)vector and (axial-)tensor currents. We obtain the complete analytical results for the form factors up to NLO, and provide their asymptotic expressions in the hierarchical heavy quark limit. The NLO corrections are found to be both significant and convergent in the relatively small squared transfer momentum (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math>) region, while also reducing the dependence on the renormalization scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math>. Finally, the theoretical predictions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> form factors over the full <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> range are provided.</p>

Qin Chang

Wei Tao

Ruilin Zhu

Next-to-leading order QCD corrections to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> form factors

We improve the perturbative QCD (PQCD) formalism for multi-body charmless hadronic meson decays, such as $$B\rightarrow VP_3 \rightarrow P_1P_2P_3$$ , by resolving the possible discrepancy in parametrizing the contribution of the -wave resonance to the two-meson distribution amplitudes (DAs) associated with the pairs $$P_1P_2=\pi \pi , K\pi , KK$$ . The determination of the Gegenbauer moments in the two-meson DAs is then updated in the global fit of the improved PQCD factorization formulas at leading order in the strong coupling $$\alpha _s$$ to available data for branching ratios and polarization fractions of three- and four-body decays. The convergence of the Gegenbauer expansion of the resultant two-meson DAs is manifest. The satisfactory quality of the fit implies the consistency of the PQCD framework for multi-body decays and the universality of the nonperturbative two-meson DAs. In particular, the predicted longitudinal polarization fraction $$f_0(B_s^0\rightarrow K^{*0} {\bar{K}}^{*0})=28.2^{+8.8}_{-9.5} \%$$ with the updated Gegenbauer moments matches well the measurement. The observable $$L_{K^{*0}{\bar{K}}^{*0}}=7.7^{+4.9}_{-3.8}$$ , defined as the ratio of the longitudinal amplitudes of the two -spin related channels $$B_s^0\rightarrow K^{*0} {\bar{K}}^{*0}$$ and $$B^0\rightarrow K^{*0} {\bar{K}}^{*0}$$ , accommodates the current data within errors. It is found that the direct asymmetries $${{\mathcal {A}}}^{0,||,\bot }_{\textrm{CP}}$$ in the polarization states of some four-body decays $$B\rightarrow V_1V_2\rightarrow (P_1P_2)(P_3P_4)$$ might be large, but the destruction among them result in small net violation. Our predictions can be confronted with LHCb and Belle-II data in the future.

Da-Cheng Yan

Zhou Rui

Improved global determination of two-meson distribution amplitudes from multi-body

We calculate the time-like $$\rho $$ electromagnetic (EM) form factor in the $$k_T$$ factorization formalism by including the next-to-leading-order (NLO) corrections of the leading-twist and sub-leading twist contributions. It’s observed that the NLO correction to the magnitude of the LO leading-twist form factor is lower than $$30\%$$ at large invariant mass squared $$Q^2 > 30 \,\text {GeV}^2$$ . It is found that the $$\rho $$ meson EM form factor is dominated by twist-3 contribution instead of by twist-2 one because of the end-point enhancement. The theoretical predictions of the moduli of three helicity amplitudes and total cross section are analyzed at $$\sqrt{s}=10.58$$ GeV, which are consistent with measurements from BABAR Collaboration.

Ya-Lan Zhang

Si-Yu Wang

Chao Wang

Next-to-leading-order time-like rho electromagnetic form factors in $${k_{\textrm{T}}}$$ factorization

Within the nonrelativistic QCD (NRQCD) framework, we complete the three-loop calculations of the NRQCD renormalization constants and the matching coefficients, for the heavy flavor-changing temporal vector, spatial-spatial tensor, spatial-temporal axial-tensor currents, which are coupled to the P-wave   <math> <mi>c</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ c\overline{b} $$  mesons. We further study the ten decay constants for the   <math> <mi>c</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ c\overline{b} $$  mesons (B  c ,   <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>∗</mo> </msubsup> </math>  $$ {B}_c^{\ast } $$ ,   <math> <msubsup> <mi>B</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> <mo>∗</mo> </msubsup> </math>  $$ {B}_{c0}^{\ast } $$ , B  c1) coupled with the ten heavy flavor-changing currents involving (pseudo-)scalar, (axial-)vector and (axial-)tensor up to the next-to-next-to-next-to-leading order (N3LO) of α  s . We obtain the six ratios of decay constants by approximating them to the corresponding ratios of matching coefficients. We find the N3LO perturbative QCD results for the six ratios of decay constants have good convergence and weak scale-dependence. We finally predict the hierarchical relationship among the ten decay constants for the   <math> <mi>c</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ c\overline{b} $$  mesons coupled with the ten currents.

Decay constants of   <math> <mi>c</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ c\overline{b} $$  mesons involving the ten heavy flavor-changing currents at N3LO QCD

The scalars  $$f_0$$  <math> <msub> <mi>f</mi> <mn>0</mn> </msub> </math>   closest to 1.5 GeV contain the mesons  $$f_0(1370)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1370</mn> <mo>)</mo> </mrow> </mrow> </math>  ,  $$f_0(1500)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1500</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$f_0(1710)$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1710</mn> <mo>)</mo> </mrow> </mrow> </math>  , and the latter two ones are usually viewed as the potential candidates for the scalar glueballs. In this work, by including the important contributions from the vertex corrections, we study the decays  $$B^0 \rightarrow J/\psi f_0$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </math>   within the improved perturbative QCD approach and analyze the possible scalar glueball hunting. Together with the two mixing models, namely,  $$f_0(1500) (f_0(1710))$$  <math> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1500</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1710</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>   being the primary scalar glueball in model I (II), and two classification scenarios, namely,  $$f_0$$  <math> <msub> <mi>f</mi> <mn>0</mn> </msub> </math>   being the  $$q\bar{q}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   excited (ground) states in scenario 1 (2), the branching fractions associated with their ratios for  $$B^0 \rightarrow J/\psi f_0$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </math>   are evaluated comprehensively. The predictions with still large uncertainties in the considered two mixing models are roughly consistent with currently limited data, which indicates that both more rich data and more precise predictions are urgently demanded to figure out the scalar glueball clearly in the future. Moreover, several interesting ratios between the branching fractions of  $$B^0 \rightarrow J/\psi f_0(\rightarrow \pi ^+ \pi ^-/K^+ K^-)$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> <mo>/</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>K</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> </mrow> </math>   and  $$B^0 \rightarrow J/\psi \rho ^0/\phi (\rightarrow \pi ^+ \pi ^-/ K^+ K^-)$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <msup> <mi>ρ</mi> <mn>0</mn> </msup> <mo>/</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> <mo>/</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>K</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> </mrow> </math>   that could help us to understand the nature of scalar  $$f_0$$  <math> <msub> <mi>f</mi> <mn>0</mn> </msub> </math>   are defined and predicted theoretically. These ratios should be examined in future experiments.

Jia-Le Ren

Min-Qi Li

Xin Liu

Zhi-Tian Zou

Ying Li

The  $$B^0 \rightarrow J/\psi f_0(1370,1500,1710)$$  <math> <mrow> <msup> <mi>B</mi> <mn>0</mn> </msup> <mo>→</mo> <mi>J</mi> <mo>/</mo> <mi>ψ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1370</mn> <mo>,</mo> <mn>1500</mn> <mo>,</mo> <mn>1710</mn> <mo>)</mo> </mrow> </mrow> </math>   decays: an opportunity for scalar glueball hunting

We present the three-loop calculations of the nonrelativistic QCD (NRQCD) current renormalization constants and corresponding anomalous dimensions, and the matching coefficients for the spatial-temporal tensor and spatial-spatial axial-tensor currents with two different heavy quark masses. We obtain the convergent decay constant ratio up to the next-to-next-to-next-to-leading order (N3LO) for the S-wave vector meson   <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>∗</mo> </msubsup> </math>  $$ {B}_c^{\ast } $$  involving the tensor and axial-tensor currents. We obtain the three-loop finite (ϵ 0) term in the ratio of the QCD heavy flavor-changing tensor current renormalization constant in the on-shell (OS) scheme to that in the modified-minimal-subtraction (  <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$ ) scheme, which is helpful to obtain the three-loop matching coefficients for all heavy flavor-changing (axial-)tensor currents.

Three-loop matching of heavy flavor-changing (axial-)tensor currents

Within the framework of non-relativistic QCD (NRQCD) factorization, we compute the matching coefficients between full Quantum Chromodynamics (QCD) and NRQCD for the heavy flavor-changing vector, axial-vector, scalar and pseudo-scalar currents up to next-to-next-to-next-to-leading order (N3LO). We accomplish the analytical expressions for the three-loop renormalization constants and the corresponding anomalous dimensions for all of the four NRQCD currents with two different heavy flavors. The three-loop QCD corrections to the matching coefficients turn out to be significantly larger than lower order corrections. By employing the scale relation, we approximately simulate the N3LO correction to the wave functions at the origin for the vector   <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>∗</mo> </msubsup> </math>  $$ {B}_c^{\ast } $$  meson and the pseudo-scalar B  c  meson from the known result for the equal-mass heavy quarkonium in potential NRQCD (pNRQCD). We find large cancellations at the third order between the matching coefficients and the wave functions at the origin, and obtain the convergent decay constants of   <math> <msubsup> <mi>B</mi> <mi>c</mi> <mo>∗</mo> </msubsup> </math>  $$ {B}_c^{\ast } $$  and B  c  up to N3LO. We present the complete perturbative QCD predictions for the decay constants, leptonic decay widths, and branching ratios of the beauty-charmed mesons.

Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications

We compute the matching coefficient between the quantum chromodynamics (QCD) and the non-relativistic QCD (NRQCD) for the flavor-changing scalar current involving the heavy charm and bottom quark, up to the three-loop order within the NRQCD factorization. For the first time, we obtain the analytical expressions for the three-loop renormalization constant  $$\tilde{Z}_s(x,R_f)$$  <math> <mrow> <msub> <mover> <mi>Z</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>R</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>   and the corresponding anomalous dimension  $$\tilde{\gamma }_s(x,R_f)$$  <math> <mrow> <msub> <mover> <mi>γ</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>R</mi> <mi>f</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>   for the NRQCD scalar current with the two heavy bottom and charm quark. We present the precise numerical results for those relevant coefficients  $$(C_{FF}(x_0), \ldots , C_{FBB}(x_0))$$  <math> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mrow> <mi>FF</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mi>FBB</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </math>   with an accuracy of about thirty digits. The three-loop QCD correction turns out to be significantly large. The obtained matching coefficient  $$C_s(\mu _f,\mu ,m_b,m_c)$$  <math> <mrow> <msub> <mi>C</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>μ</mi> <mi>f</mi> </msub> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <msub> <mi>m</mi> <mi>b</mi> </msub> <mo>,</mo> <msub> <mi>m</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> </mrow> </math>   is helpful to analyze the threshold behaviours when two different heavy quarks are close to each other and form the double heavy  $$B_c$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   meson family.

Three-loop QCD matching of the flavor-changing scalar current involving the heavy charm and bottom quark

Within the framework of perturbative QCD factorization, we investigate the nonfactorizable contributions to the factorization-forbidden quasi-two-body decays   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   with   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $  . We compare our predicted branching ratios for the   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   decay with available experiment data as well as predictions by other theoretical studies. The branching ratios of these decays are consistent with data and other theoretical predictions. However, in the Cabibbo-suppressed decays   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   with   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $  , the values of the branching ratios are of the order of   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   and   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $  . The ratio   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   between the decays   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $     $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   and   $ B_{(s)}\rightarrow h\chi_{c0}\rightarrow h\pi^+\pi^-(K^+K^-) $   and the distribution of branching ratios for different decay modes in invariant mass are considered in this study.