In view of difficulty to reproduce observables in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup></math> mixing via the operator product expansion, we discuss the Dyson-Schwinger approach to this process. Formulated by the parametrization of quark propagators, SU(3) breaking relevant to charm mixing is evaluated in such a way that properly takes account of dynamical chiral symmetry breaking. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup><mo stretchy="false">→</mo><msup><mi>D</mi><mn>0</mn></msup></math> transition is discussed in the vacuum-insertion approximation with locality of the light valence-quark field, represented by the decay constant of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mn>0</mn></msup></math> meson as well as relevant momentum integrals. It is found that dimensionless mass-difference observable in this approach leads to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1.3</mn><mtext> </mtext><mi>–</mi><mtext> </mtext><mn>2.9</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, the order of magnitude comparable to the HFLAV data, and thereby offering a certain improvement as a theoretical framework.

China

Xiaotong Xie

Hiroyuki Umeeda

Jinglong Zhu

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>−</mo><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> mixing in the Dyson-Schwinger approach

In this work, nontrivial analytic structure of the quark propagator is discussed for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi></mrow></math>-meson inclusive decays. Attributed to invalidity of the standard Källén-Lehman spectral representation, complex conjugate poles alter the evaluation of decay rates, which lead to violation of quark-hadron duality. As phenomenological observables, widths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>X</mi><mi>c</mi></msub><mo>ℓ</mo><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math> decay as well as the lifetime of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>d</mi><mn>0</mn></msubsup></math> meson are discussed. In the presence of the mentioned nonanalytical contributions, the possibility for resolving the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> puzzle is addressed. It is demonstrated that there exists a parameter region, which explains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>d</mi><mn>0</mn></msubsup></math>-meson lifetime simultaneously within <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>1</mn><mi>σ</mi></mrow></math> via the complex conjugate poles from a charm quark.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>X</mi><mi>c</mi></msub><mo>ℓ</mo><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math> decay as a probe of complex conjugate poles

In light of the recent data for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msubsup><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msub><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub><mi>V</mi></math> decays, we perform a model-independent phenomenological analysis in the presence of quasielastic rescattering. With the Wilson coefficients including contributions beyond the standard model, the lifetimes of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> meson and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>d</mi><mn>0</mn></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mi>d</mi><mn>0</mn></msubsup></math> mixing are investigated to clarify correlations among the observables. We show that parameter regions for quasielastic rescattering, the size of color-suppressed tree amplitudes, and new physics are constrained due to the lifetime data. As a consequence, it is revealed that this scenario can be tested by the future LHCb measurement of the width difference in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>d</mi><mn>0</mn></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mi>d</mi><mn>0</mn></msubsup></math> mixing and semileptonic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetry.

Indonesia

Japan

Albertus Hariwangsa Panuluh

Satoshi Tanaka

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msubsup><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><mi>M</mi></math> decays in the presence of final-state interactions

We study violation of quark-hadron duality for <math><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>→</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi></mrow></msub><mi>l</mi><mi>ν</mi></mrow></math> and <math><mrow><mi>D</mi><mo>→</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>l</mi><mi>ν</mi></mrow></math> decays in the presence of the right-handed current operator. For this case, we show that duality violation has some aspects different from the standard model due to the existence of an instanton. In particular, the fermionic zero mode of an intermediate light quark can give a nonvanishing contribution owing to its chirality structure. A duality-violating component that arises from a single instanton is then analytically extracted as a finite-distance singularity, leading to an oscillatory correction to the observable in Minkowski space. Finally, we show that the size of duality violation for the lepton energy distribution in <math><mover><mi>B</mi><mo>¯</mo></mover></math> decays can be comparable to the chirally suppressed perturbative contribution, although the absolute size depends on the detail of the QCD vacuum.

Is the right-handed current contribution to <math><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>→</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>u</mi></mrow></msub><mi>l</mi><mi>ν</mi></mrow></math> decays corrected by the nontrivial vacuum in QCD?

We study lifetimes of heavy mesons in the 't Hooft model, a large-<math><msub><mrow><mi>N</mi></mrow><mrow><mi>c</mi></mrow></msub></math> theory of strong interaction in two-dimensional spacetime. Since this model is solvable, one can evaluate the total decay widths through hadronic amplitudes that are determined unambiguously within the formalism. We investigate quark-hadron duality by numerically comparing the exclusive result and one from the heavy quark expansion, with the contribution of Pauli interference being consistently incorporated. Within certain numerical accuracy, we find that the maximal difference between inclusive and exclusive rates for <math><mi>τ</mi><mo>[</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>]</mo><mo>/</mo><mi>τ</mi><mo>[</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>]</mo></math> <math><mo>(</mo><mi>τ</mi><mo>[</mo><msup><mrow><mi>B</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo><mo>/</mo><mi>τ</mi><mo>[</mo><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>]</mo><mo>)</mo></math> is as large as the current experimental (theoretical) error in four-dimensions unlike <math><mi>τ</mi><mo>[</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>]</mo><mo>/</mo><mi>τ</mi><mo>[</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>]</mo></math>, where an excellent agreement between the rates is seen.

Revisiting quark-hadron duality for heavy meson non-leptonic decays in two-dimensional QCD

We study local quark-hadron duality and its violation for the <math> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>−</mo> <msup> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </math> $$ {D}^0-{\overline{D}}^0 $$ , <math> <msubsup> <mi>B</mi> <mi>d</mi> <mn>0</mn> </msubsup> <mo>−</mo> <msubsup> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mi>d</mi> <mn>0</mn> </msubsup> </math> $$ {B}_d^0-{\overline{B}}_d^0 $$ and <math> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>−</mo> <msubsup> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mi>s</mi> <mn>0</mn> </msubsup> </math> $$ {B}_s^0-{\overline{B}}_s^0 $$ mixings in the ’t Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-N c limit. With the ’t Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the <math> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>−</mo> <msup> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </math> $$ {D}^0-{\overline{D}}^0 $$ mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the <math> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>−</mo> <msup> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mn>0</mn> </msup> </math> $$ {D}^0-{\overline{D}}^0 $$ mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the <math> <msubsup> <mi>B</mi> <mi>d</mi> <mn>0</mn> </msubsup> <mo>−</mo> <msubsup> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mi>d</mi> <mn>0</mn> </msubsup> </math> $$ {B}_d^0-{\overline{B}}_d^0 $$ and <math> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>−</mo> <msubsup> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mi>s</mi> <mn>0</mn> </msubsup> </math> $$ {B}_s^0-{\overline{B}}_s^0 $$ mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.

Quark-hadron duality for heavy meson mixings in the ’t Hooft model

We construct QCD sum rules for nonperturbative studies without assuming the quark-hadron duality for the spectral density at low energy on the hadron side. Instead, both resonance and continuum contributions to the spectral density are solved with the operator-product-expansion input on the quark side by treating sum rules as an inverse problem. This new formalism does not involve the continuum threshold, does not require the Borel transformation and stability analysis, and can be extended to extract properties of excited states. Taking the two-current correlator as an example, we demonstrate that the series of <math><mi>ρ</mi></math> resonances can emerge in our formalism, and the decay constants <math><msub><mi>f</mi><mrow><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></mrow></msub><mo>(</mo><msub><mi>f</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1450</mn><mo>)</mo></mrow></msub><mo>,</mo><msub><mi>f</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1700</mn><mo>)</mo></mrow></msub><mo>,</mo><msub><mi>f</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1900</mn><mo>)</mo></mrow></msub><mo>)</mo><mo>≈</mo><mn>0.22</mn></math> (0.19, 0.14, 0.14) GeV for the masses <math><msub><mi>m</mi><mrow><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></mrow></msub><mo>(</mo><msub><mi>m</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1450</mn><mo>)</mo></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1700</mn><mo>)</mo></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow><mi>ρ</mi><mo>(</mo><mn>1900</mn><mo>)</mo></mrow></msub><mo>)</mo><mo>≈</mo><mn>0.78</mn></math> (1.46, 1.70, 1.90) GeV are determined. We also show that the decay width <math><msub><mi>Γ</mi><mrow><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></mrow></msub><mo>≈</mo><mn>0.17</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> can be obtained by substituting a Breit-Wigner parametrization for the <math><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></math> pole on the hadron side. It is observed that quark condensates of dimension-six on the quark side are crucial for establishing those <math><mi>ρ</mi></math> resonances. Handling the conventional sum rules with the duality assumption as an inverse problem, we find that the multiple pole sum rules widely adopted in the literature do not describe the <math><mi>ρ</mi></math> excitations reasonably. The precision of our theoretical outcomes can be improved systematically by including higher-order and higher-power corrections on the quark side. Broad applications of this formalism to abundant low energy QCD observables are expected.

Hsiang-nan Li

QCD sum rules with spectral densities solved in inverse problems

We analyze the electromagnetic current correlator at an arbitrary photon invariant mass <math><msup><mi>q</mi><mn>2</mn></msup></math> by exploiting its associated dispersion relation. The dispersion relation is turned into an inverse problem, by which the involved vacuum polarization function <math><mi>Π</mi><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> at low <math><msup><mi>q</mi><mn>2</mn></msup></math> is solved with the perturbative input of <math><mi>Π</mi><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> at large <math><msup><mi>q</mi><mn>2</mn></msup></math>. It is found that the result for <math><mi>Π</mi><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math>, including its first derivative <math><msup><mi>Π</mi><mo>′</mo></msup><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>)</mo></math>, agrees with those from lattice QCD, and its imaginary part accommodates the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math>-annihilation data. The corresponding hadronic vacuum polarization (HVP) contribution <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi></mrow></msubsup><mo>=</mo><mo>(</mo><mn>64</mn><msubsup><mn>1</mn><mrow><mo>−</mo><mn>63</mn></mrow><mrow><mo>+</mo><mn>65</mn></mrow></msubsup><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup></math> to the muon anomalous magnetic moment <math><mi>g</mi><mo>−</mo><mn>2</mn></math>, where the uncertainty arises from the variation of the perturbative input, also agrees with those obtained in other phenomenological and theoretical approaches. We point out that our formalism is equivalent to imposing the analyticity constraint to the phenomenological approach solely relying on experimental data and can improve the precision of the <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>HVP</mi></mrow></msubsup></math> determination in the Standard Model.

Vacuum polarization contribution to muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math> as an inverse problem

We calculate the parameters x and y for the D meson mixing in the Standard Model by considering a dispersion relation between them. The dispersion relation for a fictitious charm quark of arbitrary mass squared s is turned into an inverse problem, via which the mixing parameters at low s are solved with the perturbative inputs <math><mi>x</mi><mo>(</mo><mi>s</mi><mo>)</mo></math> and <math><mi>y</mi><mo>(</mo><mi>s</mi><mo>)</mo></math> from large s. It is shown that nontrivial solutions for x and y exist, whose values around the physical charm scale agree with the data in both CP-conserving and CP-violating cases. We then predict the observables <math><mo>|</mo><mi>q</mi><mo>/</mo><mi>p</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>≈</mo><mn>2</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></math> and <math><mi>A</mi><mi>r</mi><mi>g</mi><mo>(</mo><mi>q</mi><mo>/</mo><mi>p</mi><mo>)</mo><mo>≈</mo><mn>6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> degrees associated with the coefficient ratio for the D meson mixing, which can be confronted with more precise future measurements. Our work represents the first successful quantitative attempt to explain the D meson mixing parameters in the Standard Model.

Fanrong Xu

Fu-Sheng Yu

Meson Mixing as an Inverse Problem

We present predictions for CP violating phases of the Type-I seesaw model with four-zero textures on the Dirac mass matrix. For the four-zero textures, the effective low energy Majorana mass matrix is parametrized with seven parameters. They are three mass-dimensional parameters, two angles and two CP violating sources. The number of these parameters is less than that of the general description of the Majorana mass matrix with three neutrino masses, three mixing angles and three CP violating phases. In particular, only two independent CP violating sources give rise to three CP violating phases. The efficient and comprehensive method is proposed in this paper to investigate four-zero textures in Type-I seesaw. We numerically show the possible range of CP violating phases in the plane of a Dirac CP violating phase and one of Majorana phases. Some cases show the strong correlations among two phases. These correlations can be explained by hidden relations among the elements of Majorana matrix with the four-zero textures. The hidden relations are classified according to the position of one vanishing off-diagonal element of Majorana mass matrix. The Majorana mass matrix all of whose elements are non-vanishing also produces other hidden relations particularly in the case of four-zero textures. By applying the hidden relations, we describe the concrete correlations among CP violating phases.