2024 Analysis of the electromagnetic form factors and the radiative decays of the vector heavy-light mesons

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I. INTRODUCTION
As one of the most interesting research areas in particle physics, the experimental and theoretical investigations about the heavy flavor hadrons have been developing rapidly.This is due to their key roles in both understanding of QCD longdistance dynamics, and the determination of the fundamental parameters of the standard model (SM).Especially, the decay behavior of heavy flavor hadrons is an important and interesting research topic for testing the SM and finding new physics beyond the SM.These decay processes can be classified into leptonic, semileptonic and nonleptonic decays which involve both electroweak and strong interactions.However, theoretical investigations about the heavy hadrons is very difficult because the QCD is non-perturbative in low energy regions.Therefore, many phenomenological approaches  have been employed to analyze the decay processes of the heavy flavor hadrons.
The experimental data for the radiative decay of the vector meson D * + in Particle Data Group(PDG) [71] is Γ T otal (D * + ) = 83.4± 1.8 keV, and Γ(D * + →D + γ) Γ T otal (D * + ) ≈ 1.6 ± 0.4%.The decay widths for D * 0 and D * s are given as Γ T otal (D * 0 ) < 2.1 MeV, Γ(D * 0 →D 0 γ) Γ T otal (D * 0 ) ≈ 35.3 ± 0.9%, and Γ T otal (D * s ) < 1.9 MeV, Γ(D * s →D s γ) Γ T otal (D * s ) ≈ (93.5 ± 0.7)%, respectively.However, more exact values about these decays have not been determined yet.For the mesons B * ±(0) and B * s , the radiative decays as their main decay channels have not been determined in experiment.In general, more exact experimental results and theoretical predictions need to be provided, which can help us to understand the nature of the mesons and test the theoretical models.
The layout of this paper is as follows, After introduction in Sec.I, the radiative decays of the vector heavy-light mesons are analyzed in the framework of SM in Sec.II, and the electromagnetic form factor is introduced.In Sec.III, we systematically analyze the electromagnetic form factors of vector heavy-light meson to pseudoscalar heavy-light meson by the three-point QCDSR, where the contributions of perturbative part and vacuum condensate including qq , qg s σGq , g 2 s G 2 , f 3 G 3 and qq g 2 s G 2 are considered in QCD side.Sec.IV is employed to present the numerical results and discussions.Sec.V is reserved as conclusions.Some important figures are shown in Appendix.

II. THE RADIATIVE DECAY OF VECTOR HEAVY-LIGHT MESONS
To analyze the radiative decay of vector heavy-light mesons(V) to pseudoscalar heavy-light mesons(P) ( V → Pγ), we firstly write the following electromagnetic lagrangian where A µ is the electromagnetic field, q and Q denote the light (u, d or s) and heavy (c, b) quarks.e q and e Q are electric charges which are taken as − 1 3 e for d, s and b quarks, and 2 3 e for u and c quarks.The assignments of P, V, Q and q for different decay modes are shown in Table I.The assignments of P, V, Q and q for different decay modes. Mode From this lagrangian(Eq.( 1)), the decay amplitude can be expressed as the following form, where V(q 2 ) is electromagnetic form factor, p α and p ′β are four momentum of the vector and pseudoscalar heavy-light mesons, and ζ σ is the polarization vector of vector heavy-light mesons.According to Eqs. ( 2) and (3), the decay amplitude finally can be written as, where η µ is the polarization vector of the photon.This decay process can be explicitly illustrated by feynman diagrams in Fig. 1.

III. QCDSR FOR THE ELECTROMAGNETIC FORM FACTORS
To study the electromagnetic form factor of the vector heavy-light meson to the pseudoscalar heavy-light meson, we firstly write down the three-point correlation function, where T denotes the time ordered product, J P and J + ν are the interpolating currents of the pseudoscalar and vector heavylight mesons, and j µ is the electromagnetic current.These currents are taken as the following forms, In the framework of QCDSR, the correlation function will be calculated at two sides which are called the phenomenological side and the QCD side.Then, according to the quark hadron duality, the calculations of these two sides will be coordinated and the sum rules for electromagnetic form factor in the hadronic level can be obtained.

A. The Phenomenological side
In the phenomenological side, a complete sets of hadronic states with the same quantum numbers as the interpolating currents J P and J + ν are inserted into the correlation function.By using the dispersion relation, the correlation function can be written as [72], where h.c.represents the contributions coming from higher resonances and continuum states.The hadronic matrix elements are defined by the following parameterized equations, with q = p − p ′ .In these above equations, f P and f V are the decay constants of the pseudoscalar and vector heavy-light mesons, ε µσαβ is the Levi-Civita tensor, and V(q 2 ) is the electromagnetic form factor. ζ ν is the polarization vector of vector heavy-light meson with the following properties, With these above equations, the correlation function in phenomenological side can be expressed as the following form, where Π phy (p, p ′ ) is called scalar invariant amplitude, ε µναβ p α p ′β is the corresponding tensor structure.

B. The QCD side
In this part, we firstly contract the quark filed with Wick's theorem, and then do the operator product expansion(OPE).After contraction of the quark filed, the correlation function in QCD side can be expressed as, where S mn q (x) and S mn Q (x) are the full propagators of light and heavy quarks, and they have the following forms [73,74], Here, λ a (a=1,...,8) are the Gell-Mann matrices, m and n are color indices, q denotes the light quarks(u, d, and s), Q represents the heavy quarks(c and b), σ αβ = i 2 [γ α , γ β ], and f λαβ , f αβµν have the following forms, 13) From these above equations, the correlation function in QCD side(Eq.( 11)) can also be transformed into the following form, where Π QCD is the scalar invariant amplitude in QCD side, and it can be divided into several parts according to different condensate terms, Here, Π pert represents the perturbative part, and Π qq , Π g 2 s G 2 , Π qg s σGq , Π f 3 G 3 and Π qq g 2 s G 2 are the vacuum condensate terms with dimension 3, 4, 5, 6 and 7, respectively.The perturbative part and the gluon condensate terms Π g 2 s G 2 and Π f 3 G 3 can be written as the following form according to the dispersion relation, where ρ(s, u, q 2 ) is the QCD spectral density and it can be represented as, with s = p 2 , u = p ′2 and q = p − p ′ .For the perturbative part, we firstly substitute the free propagator of heavy and light quarks in the momentum space in Eq. (11).Then, the correlation function can be expressed as the following form after performing the integrations in the coordinate and momentum space, Then, the QCD spectral density for the perturbative contribution can be obtained by putting all the quark lines on massshell using the Cutkoskys's rules(see Fig. 2), After performing integrations of these Dirac delta functions in Eq. ( 19), the spectral density can be expressed as, where, As for the gluon condensate terms g 2 s G 2 and f 3 G 3 , the following integral will be encountered, k (22) This integral can also be calculated by Cutkosky's rules according to the following transformations, Besides of these above contributions, the condensate terms of qq , qg s σGq and qq g 2 s G 2 are also taken into account in this work.The feynman diagrams of these condensate terms can be classified into two groups(see Fig. 3 and Fig. 4) which correspond to the correlation functions Π 1 and Π 2 in Eq. ( 11), respectively.

C. The electromagnetic form factor
We take the change of variables p 2 → −P 2 , p ′2 → −P ′2 and q 2 → −Q 2 and perform double Borel transforms [75,76] for the variables P 2 and P ′2 to both phenomenological and QCD sides.And then P 2 and P ′2 will be replaced by T 2 1 and T 2 2 which are called Borel parameters.After matching the phenomenological and QCD side by using quark-hadron duality, the sum rule for the electromagnetic form factor is obtained as, FIG. 3: Feynman diagrams of the vacuum condensate terms qq (a), g 2 s G 2 (b-g), f 3 G 3 (h-q), qg s σGq (r-t) and qq g s G 2 (u-x) for Π 1 in Eq. (11). where, In Eq. ( 24 FIG. 4: Feynman diagrams of the vacuum condensate terms g 2 s G 2 (a-f) and f 3 G 3 (g-p) for Π 2 in Eq. (11).

IV. NUMERICAL RESULTS AND DISCUSSIONS
The masses of the heavy and light quarks, and the values of vacuum condensate terms are energy-scale dependent, which can be expressed as the following forms according to the renormalization group equation(RGE), where , Λ QCD = 210 MeV, 292 MeV and 332 MeV for the flavors n f =5, 4 and 3, respectively [71].The MS masses of the heavy and light quarks are adopted from the PDG [71], where m c (m c ) = 1.275 ± 0.025 GeV, m b (m b ) = 4.18 ± 0.03 GeV, m u(d) (µ = 1GeV) = 0.006 ± 0.001 GeV and m s (µ = 2GeV) = 0.095 ± 0.005 GeV.As for the values of other input parameters, they are all listed in Table II, where the quark and quark-gluon condensate parameters take their values in the energy scale µ = 1 GeV.Based on our previous researches about the heavy-light mesons [77], the values of decay constants are selected by following energy-scale, where µ = 1 GeV for D * → D electromagnetic form factor, µ = 2.5 GeV for B * → B, µ = 1.1 GeV for D * s → D s and µ = 2.6 GeV for B * s → B s .[77] In the framework of QCDSR, two criteria should be satisfied, which are the pole dominance and convergence of operator product expansion (OPE).The pole and continuum contribution can be defined as [61], with 2 dsdu(28) It can be seen from Eqs. ( 24) and ( 27) that the results of QCDSR depend on input parameters such as the Borel parameters T 2 1 and T 2 2 , the continuum threshold parameters s 0 and u 0 , and the square momentum Q 2 .In this article, we take T 2 1 = T 2 and T 2 2 = kT 2 1 = kT 2 .k is a constant which is related to the ratio of vector and pseudoscalar heavy-light mesons and can be expressed as k = m 2 V m 2

P
. The threshold parameters are defined as s 0 = (m V +∆ V ) 2 and u 0 = (m P +∆ P ) 2 .The values of ∆ V and ∆ P should be smaller than the experimental value of the distance between the ground and first excited state.Taking the electromagnetic form factor V 1 of D * → D as an example, we simply discuss how to select s 0 and u 0 .Fixing Q 2 = 1GeV 2 in Eq. ( 28), we firstly plot the electromagnetic form factor V 1 on Borel parameter T 2 in Fig. 5, where different values of s 0 and u 0 are adopted.It is indicated that the results have good stability when s 0 is taken to be 6.30GeV 2 (∆ V =0.5 GeV) and u 0 is 5.57 GeV 2 (∆ P =0.5 GeV).Thus, the final results will be obtained by taking ∆ V = ∆ P = 0.4, 0.5 and 0.6 GeV, where 0.5 GeV is used to determine the central values of the electromagnetic form factors, and 0.4, 0.6 GeV are for the lower and upper bounds of the final results.
An appropriate region of the Borel parameter T 2 needs to be chosen to make the final results stable and reliable.This region is commonly called 'Borel platform'.After repeated trial and contrast, we finally determine the Borel platforms which are shown in Fig. 8 in Appendix.From Fig. 8, we can see that the values of the form factors have good stability in the Borel regions, which means the convergence of OPE is well satisfied.In addition, the pole and continuum contributions in the Borel platforms are plotted in Fig. 9.We can see from Fig. 9 that the central values of pole contribution in the Borel platform satisfy the condition of pole dominance (> 50%).That is to say, two criteria of pole dominance and convergence of OPE are both satisfied.The Borel platform, pole contributions in the Borel platform, and the values of electromagnetic form factors in Q 2 = 1 GeV 2 are listed in Table III.By taking different values of Q 2 , the electromagnetic form factors V i (Q 2 ) in space-like regions (Q 2 > 0) can be obtained, where Q 2 is in the range of 0.5 ∼ 5.5 GeV 2 for V 1 (Q 2 ), and . For the values of the electromagnetic form factors at Q 2 = 0, which are used to analyze the radiative decay of the vector heavy-light mesons, they can be obtained by extrapolating the results V i (Q 2 ) into Q 2 = 0.This process is realized by selecting appropriate analytical functions to fit the results V i (Q 2 ) in space-like region.In general, the single pole fitting or exponential fitting are usually used to roughly parameterize the electroweak form factors.More precise and physical method is the z− series expansion which is used to extrapolate the heavy-to-light electroweak form factors to large momentum transfer regions(q 2 = −Q 2 >> 0) [82][83][84].For the electromagnetic form factors of vector heavylight mesons to pseudoscalar heavy-light mesons, both the initial and final states are heavy mesons, and we only take care of the values at Q 2 = 0. Thus, we do not directly employ the z− series expansion method to fit the electromagnetic form factors.After repeated trial and error, we find that the results in space-like regions (Q 2 > 0) can be well fitted by the combinations of the single pole, exponential and polynomial functions.In addition, we know that the series expansion of the exponential function and z− series expansion are essentially consistent with each other in their mathematical form.Thus, the simple parameterizations method is adopted to fit the form factors in this work, and it has the following form, where A, B, C and D are the fitting parameters and their values are listed in Table IV.The fitting results are also explicitly shown in Figs. 6 and 7.By setting Q 2 = 0 in the analytical functions(Eq.( 29)), the values of electromagnetic form factors in Q 2 = 0(V i (0)) are obtained, which are also listed in the last column of Table IV.According to the heavy-hadron chiral perturbation theory, the relation of the form factors V 1 (0) V 2 (0) < 1 should be satisfied [85].From Table IV, we find the values of V 1 (0) V 2 (0) for D * s → D s , B * → B and B * s → B s are all lower than 1, which means the results are consistent well with the conclusion of heavy-hadron chiral perturbation theory.For the form factor of D * → D, the central value of V 1 (0) is slightly larger than that of V 2 (0), which looks as if these results are contradict with the conclusion of chiral perturbation theory.Actually, if we consider the error bar of the results, the relation V 1 (0) V 2 (0) < 1 is still roughly satisfied.It is known that the next-to-leading order contribution is important in small Q 2 regions.However, the calculation of this diagram is very complicated for three-point QCDSR, thus we do not consider this contribution, which may lead to errors in the final results.In addition, the parametrization of the form factors is model dependent, it may also lead to some errors in the results.According to these above discussions, we think there are no contradiction between the results of QCDSR and that of the heavy-hadron chiral perturbation theory.Basing on these above analysis about the electromagnetic form factors, we can now discuss the radiative decays of vector heavy-light mesons.The standard form of two body decay width can be written as, where M i and M f represent the masses of initial and final mesons, J is the total angular momentum of the initial meson, Σ denotes the summation of all the polarization vectors, and T is the scattering amplitude.With Eqs. ( 4) and (30), the decay width of vector heavylight meson to pseudoscalar heavy-light meson and photon can be expressed as, Here, Q Q[q] denote the electric charges of quarks, which are taken as 2  3 for u and c quarks, − 1 3 for d, s and b quarks.α is the electromagnetic fine structure constant whose value is 1  137 .Using the values about electromagnetic form factors V i (0) in Table IV, we can obtain the decay widths of radiative decay for vector heavy-light mesons.The results of present work and those of other collaboration's are all listed in Table V.
According to Particle Data Group(PDG) [71], the experimental data of the total decay width for D * + is 83.4 ± 1.8 keV, and the ratio of the decay width of D * + → D + γ to the total width is about (1.6 ± 0.4)%.The results of present work about this radiative decay is 0.17 +0.08 −0.07 keV, which is about 0.20% of the total width 83.4 ± 1.8 keV and is smaller than the experi-mental data.In Ref [51], the radiative decay for D * + was also analyzed by QCDSR, and their result is compatible with ours, which is also smaller than the experimental data.The total widths for D * 0 and D * s are given as Γ(D * 0 ) < 2.1 MeV and Γ(D * s ) < 1.9 MeV, and the ratios of the radiative decays are about (35.3 ± 0.9)% and (93.5 ± 0.7)% of the total widths.However, the exact values of these decay widths have not be determined yet.Our results for D * 0 and D * s are 1.74 +0.40 −0.37 keV and 0.029 +0.009 −0.008 keV, which are compatible with the experiment data.In addition, the electromagnetic form factor which has the same expression as ours was firstly analyzed by Lattice QCD [87], their radiative decay width about D * s → D s γ is given as 0.066 ± 0.026 keV.Considering the uncertainties, our result is in agreement well with theirs.We also notice that the radiative decays of the heavy-light mesons were analyzed in the framework of light-cone QCD sum rules(LCSR) by consedering the subleading power corrections in recent years [88,89].In Ref [88], the electromagnetic coupling constants of the heavy-light mesons were obtained by considering the contributions from twist-two photon distribution amplitude at next-to-leading-order in α s .For the convenience of comparing the results, we use their electromagnetic coupling constants to get the corresponding decay widths which are shown in Table .V. As for the results of mesons B * ±(0) and B * s , it can be seen from Table V that the theoretical results of different collaborations are not consistent well with each other, which needs to be further testified by experiments in the future.

V. CONCLUSIONS
In this article, the electromagnetic form factors in spacelike regions (Q 2 > 0) for vector heavy-light meson to pseudoscalar heavy-light meson are firstly analyzed within the framework of three-point QCDSR.Then, the electromagnetic form factors in Q 2 = 0 are obtained by fitting the results into analytical functions in space-like regions.Based on these results, the radiative decays of the vector heavy-light mesons are systematically analyzed.We hope the results about electromagnetic form factors and radiative decays can help to shed more light on the nature of the hadrons.Appendix: The pole and continuum contributions, and the contributions of different vacuum condensate terms.
), BB[ ] denote the double Borel transforms.It needs to be explained that the vacuum condensate terms qq , qg s σGq and qq g 2 s G 2 for the electromagnetic form fac-tor V 2 (Q 2 ) in Eq. (24) will vanish after taking double Borel transforms.The threshold parameters s 0 and u 0 are introduced in dispersion integral to eliminate the contributions of higher resonances and continuum states.They should satisfy the relations m 2 V ≤ s 0 ≤ m ′2 V and m 2 P ≤ u 0 ≤ m ′2 P , where m V[P] and m ′ V[P] are the masses of the ground and the first excited state of the vector (pseudoscalar) heavy-light mesons.The masses of the ground and excited states commonly fulfill the relation m ′ V[P] = m V[P] + ∆, where ∆ is usually taken as the value of 0.4 ∼ 0.6 GeV[61].

FIG. 5 :
FIG. 5: The D * → D electromagnetic form factor V 1 on Borel parameter T 2 with different s 0 (a) and u 0 (b).

FIG. 6 :
FIG. 6: The fitting results for D * 0(±) → D 0(±) (a,b) and D * ± s → D ± s (c,d).The blue circles are the form factors in space-like regions(Q 2 > 0), the red solid lines are the fitting curves, and the red bands represent the error regions.

FIG. 7 :
FIG. 7:The fitting results for B * 0(±) → B 0(±) (a,b) and B * 0 s → B 0 s (c,d).The blue circles are the form factors in space-like regions(Q 2 > 0), the red solid lines are the fitting curves, and the red bands represent the error regions.

TABLE II :
Input parameters (IP) in this work.

TABLE III :
The Borel platform, pole contributions (Pole) and the values of electromagnetic form-factors in Q 2 = 1 GeV 2 for different radiative decay modes.
Mode Form factor Borel platform Pole(%)

TABLE IV :
The parameters for the analysis function and the values of electromagnetic form factors in Q 2 = 0 for different radiative decay modes.