Meson Distribution Amplitudes Research Articles

Exclusive production of double light neutral mesons at the e+e− colliders

In this work we investigate the exclusive production of a pair of light neutral mesons in e+e− annihilation, where the final state bears an even C-parity. The production processes can be initiated via the photon fragmentation or the nonfragmentation mechanism. While the fragmentation contribution can be rigorously accounted, the nonfragmentation contributions are calculated within the framework of collinear factorization, where only the leading-twist light-cone distribution amplitudes (LCDAs) of mesons are considered. Mediately solely by the nonfragmentation mechanism, the production rates of double light neutral pseudoscalar mesons are too small to be observed at the commissioning e+e− facilities. In contrast, the production rates of a pair of light neutral vector mesons are greatly amplified owing to the significant kinematic enhancement brought by the fragmentation mechanism. It is found that, at s=3.77 GeV, after including the destructive interference between the nonfragmentation and fragmentation contributions, the production rates for e+e−→ρ0ρ0 and ρ0ω can be lowered by about 10% and 30% relative to the fragmentation predictions. Future precise measurement of these exclusive double neutral vector meson production channels at BESIII experiment may provide useful constraints on the LCDAs of light vector mesons. Published by the American Physical Society 2024

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In this study, the form factors of B(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{(s)}$$\\end{document} to light D-wave tensor mesons with JPC=2--\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J^{PC} = 2^{--}$$\\end{document} (ρ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho _2$$\\end{document}, ω2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _2$$\\end{document}, K2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_2$$\\end{document} and ϕ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi _2$$\\end{document}) are calculated via the light cone sum rules (LCSR) in the framework of heavy quark effective eld theory (HQEFT). Firstly, the expressions of form factors in terms of the light cone distribution amplitudes (LCDAs) of tensor mesons are derived via the LCSR at the leading order of heavy quark expansion. Similar to the case of 2++\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{++}$$\\end{document} tensor mesons, the penguin type form factors can be obtained directly from the corresponding semileptonic ones. Considering the light tensor meson LCDAs to twist-3 and quark mass corrections, we give the numerical results of form factors systematically. Based on the form factors given here, we investigate the branching ratios, longitudinal polarization fractions and forward-backward asymmetries of relevant charged current induced semileptonic decays. Our results may be tested by more precise experiments in the future.

Distribution amplitude and decay constant of 1S and 2S state light mesons in the light-front quark model

Studying meson structures is essential for gaining insights into the nonperturbative nature of Quantum Chromodynamics (QCD). This study will focus on calculating the decay constant and distribution amplitudes (DAs) of unflavored light mesons (π and ρ) in the 1S and 2S states using the Light Front Quark Model. This study utilizes the QCD-motivated Hamiltonian, taking into account both contact and smeared spin-spin interactions. The two lowest harmonic oscillator bases are employed in this work to achieve improved results for the 2S states. The study found the optimal mixing parameter in basis expansion coefficients to be θ = 10°. Light meson properties, including the mass spectrum, decay constant, and twist-2 DAs, are then predicted using model parameters fixed through the variational principle. While the contact spin-spin interaction yields poor accuracy, the results from the smeared interaction generally agree well with experimental data and other theoretical models with $ f_{\pi}^{1S}=130 $ MeV and $ f_{\rho}^{1S}=210 $ MeV in the mixed state. Unlike the case for the 1S state, it should be noted that the properties of the 2S state are sensitive to the mixing parameter θ. In addition, we observe that the decay constant for ρ(2S) is $ f_{\rho}^{2S}=116 $ MeV. While for π(2S) the decay constant is extremely small with the value of $ f_{\pi}^{2S}=0.9 $ MeV, which is mainly due to the dynamical chiral symmetry breaking.

QCD light-cone distribution amplitudes of heavy mesons from boosted HQET

Light-cone distribution amplitudes (LCDAs) frequently arise in factorization theorems involving light and heavy mesons. The QCD LCDA for heavy mesons includes short-distance physics at energy scales of the heavy-quark mass. In this paper we achieve the separation of this perturbative scale from the purely hadronic effects by matching the QCD LCDA to the convolution of a perturbative function with the universal, quark-mass independent LCDA defined in heavy-quark effective theory. This factorization allows to resum potentially large logarithms between ΛQCD and mQ as well as between mQ and the scale Q of the hard process in the production of boosted heavy mesons at colliders. As an application we derive new theoretical predictions for the branching ratio of the decay W± → B±γ. Furthermore, we provide phenomenological models for the QCD LCDAs of both the B¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{B} $$\\end{document} and D mesons expressed as expansions in Gegenbauer polynomials.

Strange-quark mass effects in the Bs meson’s light-cone distribution amplitude

We investigate the differences between the light-cone distribution amplitudes (LCDAs) of Bs mesons and Bq mesons (with q = u, d) induced by a non-vanishing strange-quark mass ms ≠ 0 (compared to mq ≃ 0). To this end, we consider the so-called “radiative tail” which is related to the short-distance expansion of the relevant light-ray operators in heavy-quark effective theory. We extend the calculation of the according matching coefficients, including operators linear in ms for both the leading and sub-leading 2-particle LCDAs. Based on a generic parameterization for the leading LCDA, we discuss the effect on its shape on a quantitative level, and compare our findings with recent results on the inverse moments of the Bq and Bs LCDAs from QCD sum rules.

Study pion distribution amplitudes from the electromagnetic form factor by data-driven dispersion relation

We study the pion electromagnetic form factor in the modulus squared dispersion relation, and do the model independent extraction of the chiral mass embedded in the sub-leading twist light-cone distribution amplitude. The motivation of this work is the recent measurement of timelike form factor in the resonant regions, which makes up the piece lacking solid QCD-based calculation. With the perturbative QCD calculation up to next-to-leading-order of strong coupling and twist four level of meson distribution amplitudes, we obtain the chiral mass of pion meson as m_0^pi (1 , textrm{GeV}) = 1.31^{+0.27}_{-0.30} , textrm{GeV}. Due to the chiral enhancement contribution from the sub-leading twist distribution amplitudes in the perturbative QCD calculation, we can not extract the lowest Gegenbauer moments of pion from the study of electromagnetic form factor. This problem could be settled down with the foresee Belle-II and BESIII measurement of the transition form factor especially in the large momentum transfers regions.

Contribution of the electromagnetic dipole operator O7 to the B¯s→μ+μ− decay amplitude

We construct a factorization theorem that allows one to systematically include QCD corrections to the contribution of the electromagnetic dipole operator in the effective weak Hamiltonian to the ${\overline{B}}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decay amplitude. We first rederive the known result for the leading-order QED box diagram, which features a double-logarithmic enhancement associated with the different rapidities of the light quark in the ${\overline{B}}_{s}$ meson and the energetic muons in the final state. We provide a detailed analysis of the cancellation of the related endpoint divergences appearing in individual momentum regions, and show how the rapidity logarithms can be isolated by suitable subtractions applied to the corresponding bare factorization theorem. This allows us to include in a straightforward manner the QCD corrections arising from the renormalization-group running of the hard matching coefficient of the electromagnetic dipole operator in soft-collinear effective theory, the hard-collinear scattering kernel, and the ${B}_{s}$-meson distribution amplitude. Focusing on the contribution from the double endpoint logarithms, we derive a compact formula that resums the leading-logarithmic QCD corrections.

Phenomenological studies on neutral B-meson decays into J/psi f_1 and eta _c f_1

The axial-vector mesons f_1(1285) and f_1(1420) are particularly viewed as the mixtures of flavor states f_n and f_s with mixing angle varphi . In order to determine this angle, we study the B_{d,s}^0rightarrow J/psi f_1(1285,1420) and B_{d,s}^0rightarrow eta _c f_1(1285,1420) decays in perturbative QCD (PQCD) approach, including the effects of vertex corrections, nonfactorizable diagrams and penguin operators. Not only the branching fractions, but also the direct CP asymmetries and the polarization fractions are calculated. It is found that the branching fractions of these decays are large enough to be measured in the running LHCb and Belle-II experiments. Moreover, in comparison with the observed {mathcal {B}}(B_{d,s}^0 rightarrow J/psi f_1(1285)), B_s^0 rightarrow (J/psi , eta _c) f_1(1420) decays have large branching fractions, which could be measured promisingly through f_1(1420) rightarrow K_S^0 K^pm pi ^mp in experiments. We also propose several ratios that could be used to further constrain the absolute value of the mixing angle varphi , but its sign cannot be determined yet in these decays. The direct CP asymmetries of these decays indicate the penguin pollution in the B_d^0 rightarrow (J/psi , eta _c) f_1 decays cannot be neglected. We acknowledge that there are large theoretical uncertainties arising from the distribution amplitudes of axial-vector mesons and charmonium states, and more precise nonperturbative parameters are called. The comparisons between our results and future experimental data would help us to understand the nature of f_1 states and to test the PQCD approach.

Light-cone distribution amplitudes of heavy mesons with QED effects

We discuss the QED-generalized leading-twist light-cone distribution amplitudes of heavy mesons, that appear in QCD QED factorization theorems for exclusive two-body B decays. In the presence of electrically charged particles, these functions should be more appropriately regarded as soft functions for heavy-meson decays into two back-to-back particles. In this paper, we derive the one-loop anomalous dimension of these soft functions and study their behaviour under renormalization-scale evolution, obtaining an exact solution in Laplace space. In addition, we provide numerical solutions for the soft functions and analytical solutions to all orders in the strong and to first order in the electromagnetic coupling. For the inverse (and inverse-logarithmic) moments, we obtain an all-order solution in both couplings. We further provide numerical estimates for QED corrections to the inverse moments.

A_1(1260)-meson longitudinal twist-2 distribution amplitude and the Drightarrow a_1(1260)ell ^+nu _ell decay processes

In the paper, we investigate the moments langle xi _{2;a_1}^{Vert ;n}rangle of the axial-vector a_1(1260)-meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale mu _0=1~{mathrm{GeV}} are langle xi _{2;a_1}^{Vert ;2}rangle |_{mu _0} = 0.223 pm 0.029, langle xi _{2;a_1}^{Vert ;4}rangle |_{mu _0} = 0.098 pm 0.008, langle xi _{2;a_1}^{Vert ;6}rangle |_{mu _0} = 0.056 pm 0.006, langle xi _{2;a_1}^{Vert ;8}rangle |_{mu _0} = 0.039 pm 0.004 and langle xi _{2;a_1}^{Vert ;10}rangle |_{mu _0} = 0.028 pm 0.003, respectively. We then construct a light-cone harmonic oscillator model for a_1(1260)-meson longitudinal twist-2 distribution amplitude phi _{2;a_1}^{Vert }(x,mu ), whose model parameters are fitted by using the least squares method. As an application of phi _{2;a_1}^{Vert }(x,mu ), we calculate the transition form factors (TFFs) of Drightarrow a_1(1260) in large and intermediate momentum transfers by using the QCD light-cone sum rules approach. At the largest recoil point (q^2=0), we obtain A(0) = 0.130_{ - 0.013}^{ + 0.015}, V_1(0) = 1.898_{-0.121}^{+0.128}, V_2(0) = 0.228_{-0.021}^{ + 0.020}, and V_0(0) = 0.217_{ - 0.025}^{ + 0.023}. By applying the extrapolated TFFs to the semi-leptonic decay D^{0(+)} rightarrow a_1^{-(0)}(1260)ell ^+nu _ell , we obtain {mathcal {B}}(D^0rightarrow a_1^-(1260) e^+nu _e) = (5.261_{-0.639}^{+0.745}) times 10^{-5}, {mathcal {B}}(D^+rightarrow a_1^0(1260) e^+nu _e) = (6.673_{-0.811}^{+0.947}) times 10^{-5}, {mathcal {B}}(D^0rightarrow a_1^-(1260) mu ^+ nu _mu )=(4.732_{-0.590}^{+0.685}) times 10^{-5}, {mathcal {B}}(D^+ rightarrow a_1^0(1260) mu ^+ nu _mu )=(6.002_{-0.748}^{+0.796}) times 10^{-5}.

Exclusive production of light vector mesons at next-to-leading order in the dipole picture

Exclusive production of light vector mesons in deep inelastic scattering is calculated at next-to-leading order in the dipole picture in the limit of high photon virtuality. The resulting expression is free of any divergences and suitable for numerical evaluations. The higher-order corrections are found to be numerically important, but they can be mostly captured by the nonperturbative fit parameters describing the initial condition for the small-$x$ evolution of the dipole scattering amplitude. The vector meson production cross section is shown to depend only weakly on the meson distribution amplitude and the factorization scale. We also present phenomenological comparisons of our result to the existing exclusive $\phi$ and $\rho$ production data from HERA and find an excellent agreement at high virtualities.

D(s) to light tensor meson form factors and semileptonic decays via LCSR in HQEFT

In this paper, the form factors of D(s) to light tensor mesons a2(1320), [Formula: see text], [Formula: see text], [Formula: see text] and corresponding semileptonic decays are investigated through the method of light cone sum rules (LCSR) in the framework of heavy quark effective field theory (HQEFT). Considering the light cone distribution amplitudes (DAs) of tensor mesons to twist-3, we give numerical results of form factors systematically, which are necessary inputs for the analysis of relevant D(s) decays. Based on the form factor results, the relevant semileptonic decays induced by charged current (CC) as well as flavor changing neutral current (FCNC) are studied. Our predictions for the branching ratios, longitudinal polarization fractions and forward–backward asymmetries can be tested by more precise experiments such as LHCb, BESIII and BelleII in the future.

Direct ρL meson production in proton–proton and proton–antiproton collisions

We have investigated the contribution of the Higher Twist (HT) Feynman diagrams to the large-[Formula: see text] inclusive [Formula: see text] meson production cross-section in proton–proton and proton–antiproton collisions and we discuss the phenomenological consequences of possible HT contributions to cross-sections. To extract HT subprocesses from Leading Twist (LT) background, we use various [Formula: see text] meson distribution amplitudes (DAs). In the numerical calculations, the dependencies of the HT contribution on the transverse momentum and the rapidity are discussed with special emphasis on DAs. Analysis of our results shows that HT contributions decrease more rapidly than LT contributions with increasing [Formula: see text]. The preceding results demonstrate that HT contributions must be considered especially in the region of low [Formula: see text] and the HT contribution to the cross-section depends on the choice of the meson DAs.

Eta ^{(prime )}-meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay D_s^+ rightarrow eta ^{(prime )}ell ^+ nu _ell

In this paper, we make a detailed discussion on the eta and eta ^prime -meson leading-twist light-cone distribution amplitude phi _{2;eta ^{(prime )}}(u,mu ) by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments langle xi ^n_{2;eta ^{(prime )}}rangle |_{mu _0} with n = (2, 4, 6) can be determined, where the initial scale mu _0 is set as the usual choice of 1 GeV. Numerically, we obtain langle xi _{2;eta }^2rangle |_{mu _0} =0.231_{-0.013}^{+0.010}, langle xi _{2;eta }^4 rangle |_{mu _0} =0.109_{ - 0.007}^{ + 0.007}, and langle xi _{2;eta }^6 rangle |_{mu _0} =0.066_{-0.006}^{+0.006} for eta -meson, langle xi _{2;eta '}^2rangle |_{mu _0} =0.211_{-0.017}^{+0.015}, langle xi _{2;eta '}^4 rangle |_{mu _0} =0.093_{ - 0.009}^{ + 0.009}, and langle xi _{2;eta '}^6 rangle |_{mu _0} =0.054_{-0.008}^{+0.008} for eta '-meson. Next, we calculate the D_srightarrow eta ^{(prime )} transition form factors (TFFs) f^{eta ^{(prime )}}_{+}(q^2) within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are f^{eta }_+(0) = 0.476_{-0.036}^{+0.040} and f^{eta '}_+(0) = 0.544_{-0.042}^{+0.046}. After extrapolating TFFs to the allowable physical regions within the series expansion, we obtain the branching fractions of the semi-leptonic decay, i.e. D_s^+rightarrow eta ^{(prime )}ell ^+ nu _ell , i.e. {{mathcal {B}}}(D_s^+ rightarrow eta ^{(prime )} e^+nu _e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})times 10^{-2} and {{mathcal {B}}}(D_s^+ rightarrow eta ^{(prime )} mu ^+nu _mu )=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})times 10^{-2} for ell = (e, mu ) channels respectively. And in addition to that, the mixing angle for eta -eta ' with varphi and ratio for the different decay channels {{mathcal {R}}}_{eta '/eta }^ell are given, which show good agreement with the recent BESIII measurements.

Cross-link relations between π and ρ -meson channels and the QCD vacuum

We discuss cross-link relations between the $\ensuremath{\pi}$ and $\ensuremath{\rho}$-meson channels emerging from two different descriptions of the QCD vacuum: instanton physics and QCD sum rules with nonlocal condensates (NLCs). We derive in both schemes an intriguing linear relation between the $\ensuremath{\pi}$ and the ${\ensuremath{\rho}}^{\ensuremath{\parallel}}$-meson distribution amplitudes in terms of their conformal coefficients and work out the specific impact of the scalar NLC in these two channels. Using a simple model with Gaussian decay of the scalar NLC, we are able to relate it to the moments of the pion nonsinglet parton distribution function measurable in experiment---a highly nontrivial result. The implications for the pion and the ${\ensuremath{\rho}}^{\ensuremath{\parallel}}$-meson distribution amplitudes entailed by the obtained cross-link relations are outlined in terms of two generic scenarios.

Charged current electroproduction of a charmed meson at an electron-ion collider

We calculate the amplitude for exclusive electroweak production of a pseudoscalar ${D}_{s}$ or a vector ${D}_{s}^{*}$ charmed strange meson on an unpolarized nucleon, through a charged current, in leading order in ${\ensuremath{\alpha}}_{s}$. We work in the framework of the collinear QCD approach where generalized gluon distributions factorize from perturbatively calculable coefficient functions. We include both $O({m}_{c})$ terms in the coefficient functions and $O({M}_{D})$ mass term contributions in the heavy meson distribution amplitudes. We show that this process may be accessed at future electron-ion colliders.

Light-cone distribution amplitudes of light mesons with QED effects

We discuss the generalization of the leading-twist light-cone distribution amplitude for light mesons including QED effects. This generalization was introduced to describe virtual collinear photon exchanges at the strong-interaction scale ΛQCD in the factorization of QED effects in non-leptonic B-meson decays. In this paper we study the renormalization group evolution of this non-perturbative function. For charged mesons, in particular, this exhibits qualitative differences with respect to the well-known scale evolution in QCD only, especially regarding the endpoint-behaviour. We analytically solve the evolution equation to first order in the electromagnetic coupling αem, which resums large logarithms in QCD on top of a fixed-order expansion in αem. We further provide numerical estimates for QED corrections to Gegenbauer coefficients as well as inverse moments relevant to (QED-generalized) factorization theorems for hard exclusive processes.

S -wave contributions to the B(s)→χc1(ππ,Kπ,KK) decays

We make a detailed study of the three-body decays $B_{(s)}\rightarrow \chi_{c1} hh'$, where $h^{(')}$ is either a pion or kaon, by taking into account the $S$-wave states in the $hh'$ invariant mass distribution within the perturbative QCD approach. The two meson distribution amplitudes (DAs) are introduced to capture the strong interaction related to the production of the $hh'$ system. We calculate the branching ratios for the $S$-wave components and observe large values of order $10^{-4}$ for some Cabibbo-favored decays, which are accessible to the LHCb and Belle II experiments. The obtained branching ratio $\mathcal{B}(B\rightarrow \chi_{c1}K^*_{0}(1430)(\rightarrow K^+\pi^-))=(5.1^{+0.6}_{-0.8})\times 10^{-5}$ consistent with the data from Belle within errors. Moreover, we also predict the differential distributions in the $hh'$ invariant mass for the decays under consideration, which await the future experimental test. In addition, the corresponding $\chi_{c1}(2P)$ channels are also investigated, which are helpful to clarify the nature of the $X(3872)$ state.

Global analysis of hadronic two-body B decays in the perturbative QCD approach

Based on the flavor structure of four-quark effective operators, we develop an automatic computation program to calculate hadronic two-body $B$ meson decay amplitudes, and apply it to their global analysis in the perturbative QCD (PQCD) approach. Fitting the PQCD factorization formulas for $B\to PP,VP$ decays at leading order in the strong coupling $\alpha_s$ to measured branching ratios and direct CP asymmetries, we determine the Gegenbauer moments in light meson light-cone distribution amplitudes (LCDAs). It is found that most of the fitted Gegenbauer moments of the twist-2 and twist-3 LCDAs for the pseudoscalar meson $P$ ($P=\pi$, $K$) and vector meson $V$ ($V=\rho$, $K^*$) agree with those derived in QCD sum rules. The shape parameter for the $B_s$ meson distribution amplitude and the weak phase $\phi_3(\gamma)=(75.1\pm2.9)^\circ$ consistent with the value in Particle Data Group are also obtained. It is straightforward to extend our analysis to higher orders and higher powers in the PQCD approach, and to the global determination of LCDAs for other hadrons.

B -meson Ioffe-time distribution amplitude at short distances

We propose the approach for a lattice investigation of light cone distribution amplitudes (LCDA) of heavy-light mesons, such as the $B$ meson, using the formalism of parton pseudodistributions. A basic ingredient of the approach is the study of short-distance behavior of the $B$-meson Ioffe-time distribution amplitude (ITDA), which is a generalization of the $B$-meson LCDA in coordinate space. We construct a reduced ITDA for the $B$ meson, and derive the matching relation between the reduced ITDA and the LCDA. The reduced ITDA is ultraviolet finite, which guarantees that the continuum limit exists on the lattice.