PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/nm8j-l3kyARTICLESPhenomenological aspects of field theory, general methodsRadiative decays of vector mesons with light-cone sum rulesRADIATIVE DECAYS OF VECTOR MESONS WITH LIGHT-CONE …WANG, GAO, ZHANG, MA, AND SUNWangZhi Jun1,*https://orcid.org/0009-0005-6050-6665GaoDi1,2,3,†ZhangKai Kai1,‡MaYuan Yuan1,§https://orcid.org/0000-0002-2059-7049SunYan Jun1,∥Institute of Theoretical Physics, College of Physics and Electronic Engineering, Northwest Normal Universityhttps://ror.org/00gx3j908, Lanzhou 730070, ChinaInstitute of Modern Physicshttps://ror.org/03x8rhq63, Chinese Academy of Sciences, Lanzhou 730000, ChinaLebedev Physical Institute, Russian Academy of Scienceshttps://ror.org/05qrfxd25, Leninsky Prospekt 53, 119991 Moscow, Russia
Contact author: 2023212251@nwnu.edu.cn
Contact author: digao@impcas.ac.cn
Contact author: zhangkk314@outlook.com
Contact author: Mayuanyuan917@163.com
Contact author: sunyanjun@nwnu.edu.cn
2June20261June20261131111600325November20256May2026Published by the American Physical Society2026authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Hadronic electromagnetic form factors and radiative decay properties offer a crucial window into the nonperturbative dynamics of quantum chromodynamics (QCD). In this work, we employ the light-cone sum rules (LCSR) method to systematically investigate the M1 radiative decay of vector mesons. Our study covers processes including K*-→K-γ, D*→Dγ, B*→Bγ, Ds*+→Ds+γ, and Bs*→Bsγ, and further extends to the excited charmonium state ψ(2S). Our calculations yield decay widths for K* and ψ(2S) that are in excellent agreement with experimental data. For the charm and bottom meson decays, where precise measurements are lacking, we provide theoretical predictions and compare them with other theoretical approaches. Most notably, our analysis reveals a universal linear dependence of the decay width on a function A(x) in the logarithmic coordinate system, which originates from the two-body decay dynamics and the ratio of the initial and final state decay constants. This relationship holds for the ground state V→Pγ processes here and suggests a broader applicability to radiative decays of ground-state vector mesons.
China Scholarship Council10.13039/501100004543202510530001INTRODUCTION
Quantum chromodynamics (QCD) is the fundamental theory describing the strong interaction. The property of asymptotic freedom enables the application of perturbative QCD in the high energy regime, leading to remarkable successes in describing both inclusive and exclusive processes at large momentum transfer. Nevertheless, investigating the properties of QCD in the nonperturbative domain represents a central challenge in modern particle physics. Electromagnetic transitions provide a crucial window into this domain. Among these, magnetic dipole (M1) transitions, where a spin-1 vector meson decays into a spin-0 pseudoscalar meson via photon emission, serve as powerful probes for investigating hadron structure. The corresponding transition form factors are pivotal for deepening our insight into nonperturbative QCD dynamics [1–8]. At present, the Particle Data Group (PDG) [9] has compiled extensive experimental data on hadron radiative decays. This highlights the demand for robust theoretical frameworks capable of interpreting these results and predicting new decay channels.
Currently, M1 radiative decays have been experimentally observed for numerous mesons. For example, for the ground state magnetic dipole (M1) radiative transition K*-→K-γ, the decay width is 50.37 keV [9]. For the charm-containing processes D*→Dγ and Ds*+→Ds+γ, only upper limits are available for the M1 radiative decay widths specifically 2.1 and 1.9 MeV, respectively [10–13]. Recently, the latest data from the Beijing Spectrometer III (BESIII) Collaboration reported measurements of the branching fractions for D*→Dγ and Ds*+→Ds+γ as (34.5±0.8±0.5)% and (93.57±0.38±0.22)%, respectively [14,15]. However, although the M1 radiative decays of the B* and Bs* mesons have been experimentally observed, their decay widths still lack precise experimental measurements [9]. For the excited state magnetic dipole radiative transition ψ(2S)→ηc(2S)γ, this transition poses a particular challenge due to the narrow phase space of ηc(2S). Subsequent studies have confirmed that ηc(2S) can be produced via radiative decay of ψ(2S). The Crystal Ball group [16] reported an excess of Eγ≃91MeV gamma rays from inclusive ψ(2S)→Xγ decays and interpreted this as possible evidence for the ηc(2S) with mass 3594±5MeV/c2. The Belle Collaboration later observed the pseudoscalar meson ηc(2S) in B meson decays [17]. Recently, the BESIII Collaboration investigated the M1 transition in the decay ψ(2S)→ηc(2S)γ, measuring the corresponding branching fraction as B(ψ(3686)→ηc(2S)γ)=(5.2±0.3±0.5-1.4+1.9)×10-4, and the decay width as Γ(ψ(3686)→ηc(2S)γ)=0.15-0.04+0.06keV[18]. Theoretically, a wide array of theoretical frameworks has been developed for M1 radiative decays, such as Godfrey-Isgur model [19–21], relativistic quark model [22], light-front quark model (LFQM) [23], light-cone quark model [24], meson loop corrections [25,26], QCD sum rules (QCDSR) [27–30], lattice quantum chromodynamics [31–37], and light-cone sum rules (LCSR) [38–41]. Nevertheless, significant challenges persist in the study of radiative decays and related research areas, as exemplified by the ρ-π puzzle [42–44] and several anomalous decay phenomena. Resolving these puzzles and advancing our fundamental understanding of hadronic structure and its underlying interaction mechanisms will necessitate sustained and concerted investigation on both experimental and theoretical fronts.
Light-cone sum rules (LCSR) is a nonperturbative method based on the first principles of QCD. It combines the operator product expansion (OPE) with a description of nonperturbative dynamics in terms of light-cone distribution amplitude (LCDA), and has been proven highly effective in calculating key nonperturbative parameters for exclusive processes [45,46]. The method exhibits a comprehensive scope of applicability, enabling calculations of transition form factors [47–49], coupling constants [50], decay constants [51,52], magnetic moments of multiquark states and other physical quantities [53–57]. While the LCSR method itself is not new, its systematic application to several vector-to-pseudoscalar M1 transitions including ψ(2S)→ηc(2S)γ, supplements the existing literature. Unlike the photon wave functions employed in the literature [58], we adopt the light-cone distribution amplitudes of pseudoscalar mesons as the nonperturbative inputs.
In this work, we employ LCSR to carry out a systematic study of the M1 radiative decays of vector mesons (K*-, B*, Bs*, D*, Ds*+) and extend the formalism to the radiative transition of the excited charmonium state ψ(2S)→ηc(2S)γ. In addition to computing the transition form factors and decay widths and comparing them with existing data and other theoretical approaches, we have performed a comprehensive analysis of the systematics of these decays. Notably we attempt to analyze the behavior of the form factor across different transition channels with respect to the mass scale and a universal linear relationship between the decay width and a specific kinematic function A(x) is founded. This observed scaling may offer a fresh, unifying perspective on the systematics of radiative decays and could provide valuable insights into understanding broader hadronic anomalies, such as the ρ-π puzzle, by revealing underlying universal dynamics in M1 transitions.
This work is organized as follows. Section II outlines the theoretical framework and details the derivation of the transition form factors within the light-cone sum rules. Section III presents the numerical results, discusses their physical implications, and compares them with available experimental data and other theoretical approaches. Finally, Sec. IV summarizes our findings and provides an outlook for future research.
FORM FACTOR WITH LCSRForm factor for <inline-formula><mml:math display="inline"><mml:msup><mml:mi>K</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:math></inline-formula> transition
This section details the derivation of the transition form factors for the M1 radiative decays within the LCSR framework. We first present a comprehensive calculation for the process K*-→K-γ. The formalism for other heavy-light meson transitions (D*, B*, Ds*, Bs*) follows an analogous procedure, with the final results summarized for brevity. Subsequently, we extend the formalism to the charmonium sector to address the excited state transition ψ(2S)→ηc(2S)γ. For the K*-→K-γ decay channel, the following correlation function is employed: Πμν(Q,q)=i∫d4xeiq·x⟨K-(Q)|T{Jμ(x)Jν(0)}|0⟩,where Q is the four-momentum of the final-state K- meson, and q is the four-momentum of the outgoing photon. The currents are defined using two types of constructions: one with Jμ=s¯(x)γμs(x) and Jν=u¯(x)γνs(x), and the other with Jμ=u¯(x)γμu(x) and Jν=u¯(x)γνs(x). These correspond to the two ways in which M1 radiative decay occurs: the photon is emitted either from the s-quark line or from the u-quark line.
On the phenomenological side, a complete set of hadronic states sharing the same quantum numbers as the final states is inserted between the two types of currents (Here, one of the two types of constructions is taken as an example for illustration, and the other yields analogous results on the phenomenological side.), yielding Πμν(Q,q)=i∫d4xeiq·x∑n∫d3P→1(2π)32E1θ(x0-0)eiQ·xe-iP1·x⟨K-(Q)|s¯(0)γμs(0)|n⟩⟨n|u¯(0)γνs(0)|0⟩+i∫d4xeiq·x∑n∫d3P→1(2π)32E1θ(0-x0)eiP1·x⟨K-(Q)|u¯(0)γνs(0)|n⟩⟨n|s¯(0)γμs(0)|0⟩.Isolating the ground state contributions gives Πμν(Q,q)=1mK*-2-(q+Q)2⟨K-(Q)|s¯(0)γμs(0)|K*-⟩⟨K*-|u¯(0)γνs(0)|0⟩+∫s0∞dsρ(Q2,s)s-(Q+q)2,where s0 is the threshold parameter that separates the ground state from excited states. Following Ref. [59], the matrix elements are parametrized as ⟨K-(Q)|s¯(0)γμs(0)|K*-⟩=ϵμαβγQβqαϵ*γFVP2mK*-+mK-,⟨K*-|u¯(0)γνs(0)|0⟩=mK*-fK*-ϵν,where mK*- is the mass of the K*- meson, mK- is the mass of the K- meson, fK*- is the decay constant, ϵν is the polarization vector, ϵμαβγ is a totally antisymmetric tensor, and FVP is the transition form factor. With these parametrizations, the correlator on the phenomenological side becomes Πμν(Q,q)=1mK*-2-(Q+q)22ϵμαβγQβqαϵ*γFVPmK*-fK*-ϵνmK*-+mK-+∫s0∞dsρ(Q2,s)s-(Q+q)2.
On the theoretical side of the correlator, the operator product expansion (OPE) is employed. After contracting the quark fields for the two types of currents, the corresponding propagators are substituted accordingly. By substituting the s-quark propagator S(0,x)=i∫d4k(2π)4e-ik·x{-ms-kαγαms2-k2} and the u-quark propagator S(x,0)=i∫d4k(2π)4e-ik·x{-mu+kαγαmu2-k2} into the correlation function, we obtain, corresponding to two types of current constructions, Πμν(Q,q)=-∫d4xeiqx∫d4k(2π)4e-ik·xkαms2-k2⟨K-(Q)|u¯(0)γνγαγμs(x)|0⟩,Πμν(Q,q)=∫d4xeiqx∫d4k(2π)4e-ik·xkαmu2-k2⟨K-(Q)|u¯(x)γμγαγνs(0)|0⟩.
The matrix elements in the above expression can be simplified using the following formula: γμγαγν=Sμανβγβ-iϵμανβγβγ5.
Since the light-cone distribution amplitude of K meson is defined as [60]⟨K-(Q)|u¯(0)γβγ5s(x)|0⟩=-iQβfK-∫01duei(1-u)Q·xϕK-(u)+highertwistterms,⟨K-(Q)|u¯(x)γβγ5s(0)|0⟩=-iQβfK-∫01dueiuQ·xϕK-(u)+highertwistterms.The above expression provides the parameterization of the matrix element of the K meson near the light cone. This matrix element can be described in terms of the decay constant fK- and the light-cone distribution amplitudes ϕK-(u). Here, u denotes the momentum fraction carried by the u quark, while u¯=1-u represents the momentum fraction carried by the s¯ antiquark. The correlation function is expressed in the following Eq. (9), corresponding to two types of constructions, respectively. Πμν(Q,q)=-ϵναμβfK-Qβqα∫01duϕK-(u)ms2-(q+u¯Q)2,Πμν(Q,q)=-ϵμανβfK-Qβqα∫01duϕK-(u)mu2-(q+uQ)2.
Applying the “quark-hadron duality” assumption to the hadronic spectral density in Eq. (6), ∫0∞dsρh(s)s-q2≃1π∫0∞dsImΠpert(s)s-q2,and matching the theoretical side with the phenomenological side yields FVPmK*-2-(q+Q)2=12fK-fK*-mK*-+mK-mK*-[∫0Δ1duϕK-(u)ms2-(q+u¯Q)2+∫Δ21duϕK-(u)mu2-(q+uQ)2],where the upper limit Δ1 and the lower limit Δ2 are given respectively as Δ1=12mK-2[-(s0+mK-2+P′2)2-4mK-2(s0-ms2)+(s0+mK-2-q2)],Δ2=12mK-2[(s0-mK-2+P′2)2+4mK-2(mu2+P′2)-(s0-mK-2-q2)],where the four-momenta are specified as Q2=mK-2 and P′2=-q2.
To suppress the contributions from higher excited states and the continuum, we apply the Borel transformation [45,60,61] to the four-momenta (Q+q), (uQ+q) and (u¯Q+q), BM2[1mK*-2-(Q+q)2]=1M2e-mK*-2/M2,BM2[1ms2-(uQ+q)2]=1uM2e-1uM2[ms2+u(1-u)Q2-(1-u)q2],BM2[1ms2-(u¯Q+q)2]=1u¯M2e-1u¯M2[ms2+u(1-u)Q2-uq2].Finally, the transition form factor is obtained FVP=12fK-fK*-mK*-+mK-mK*-[∫0Δ1duϕK-(u)u¯e-1u¯M2(ms2+uu¯mK-2-uq2)+mK*-2M2+∫Δ21duϕK-(u)ue-1uM2(mu2+uu¯mK-2-(1-u)q2)+mK*-2M2].
Through phenomenological and theoretical treatments of the correlation function, followed by matching procedures, the form factor for the K*-→K-γ process has been obtained. This methodology is universally applicable to other processes, and for D*→Dγ, B*→Bγ, Ds*+→Ds+γ and Bs*→Bsγ, the form factors are FVP=12fDfD*mD*+mDmD*[∫0Δ1duϕD(u)u¯e-1u¯M2(mc2+uu¯mD2-uq2)+mD*2M2+∫Δ21duϕD(u)ue-1uM2(mu2+uu¯mD2-(1-u)q2)+mD*2M2],FVP=12fDs+fDs*+mDs*++mDs+mDs*+[∫0Δ1duϕDs+(u)u¯e-1u¯M2(mc2+uu¯mDs+2-uq2)+mDs*+2M2+∫Δ21duϕDs+(u)ue-1uM2(ms2+uu¯mDs+2-(1-u)q2)+mDs*+2M2],FVP=12fBfB*mB*+mBmB*[∫0Δ1duϕB(u)u¯e-1u¯M2(mb2+uu¯mB2-uq2)+mB*2M2+∫Δ21duϕB(u)ue-1uM2(md2+uu¯mB2-(1-u)q2)+mB*2M2],FVP=12fBsfBs*mBs*+mBsmBs*[∫0Δ1duϕBs(u)u¯e-1u¯M2(mb2+uu¯mBs2-uq2)+mBs*2M2+∫Δ21duϕBs(u)ue-1uM2(ms2+uu¯mBs2-(1-u)q2)+mBs*2M2],where M2 is Borel parameter, mD* (mB*, mDs*+, mBs*) is the mass of the initial state meson, mD (mB, mDs+, mBs) is the mass of the final state meson, and fD* (fB*, fDs*+, fBs*) and fD (fB, fDs+, fBs) are the decay constants of the initial and final state mesons, respectively. Additionally, mc and mb are the masses of the c quark and the b quark, respectively.
Form factor for <inline-formula><mml:math display="inline"><mml:mi>ψ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>S</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> transition
The LCSR method can also be applied to transitions involving excited states. We consider the M1 radiative decay ψ(2S)→ηc(2S)γ. The correlation function is defined as Πμν(Q,q)=i∫d4xeiq·x⟨ηc(2S)(Q)|T{Jμc(x)Jνc(0)}|0⟩,where the electromagnetic currents of the c quark are given by Jμc=c¯(x)γμc(x) and Jνc=c¯(x)γνc(x).
On the phenomenological side of the correlation function, we insert the hadron state ∑n|n⟩⟨n|, which includes J/ψ, ψ(2S), higher excited states, and the hadronic continuum. Since the mass of state J/ψ is lower than that of state ηc(2S), the transition J/ψ→ηc(2S)γ cannot occur. Therefore, the transition ψ(2S)→ηc(2S)γ is isolated, Πμν(Q,q)=1mψ(2S)2-(q+Q)2⟨ηc(2S)(Q)|c¯(0)γμc(0)|ψ(2S)⟩⟨ψ(2S)|c¯(0)γνc(0)|0⟩+∫s0∞dsρ(Q2,s)s-(Q+q)2,where ρ(Q2,s) is defined similarly to Eq. (3).
The decay matrix elements in Eq. (17) are parametrized as follows: ⟨ηc(2S)(Q)|c¯(0)γμc(0)|ψ(2S)⟩=ϵμαβγQβqαϵγ*FVP2mψ(2S)+mηc(2S),⟨ψ(2S)|c¯(0)γνc(0)|0⟩=mψ(2S)fψ(2S)ϵν,where mψ(2S) is the mass of the ψ(2S) meson, mηc(2S) is the mass of the ηc(2S) meson, fψ(2S) is the decay constant, and FVP is the transition form factor. Through the substitution of Eqs. (18) and (19) into Eq. (17), the phenomenological result is obtained Πμν(Q,q)=1mψ(2S)2-(Q+q)22ϵμαβγQβqαϵγ*FVPmψ(2S)fψ(2S)ϵνmψ(2S)+mηc(2S)+∫s0∞ds1πImΠ(Q,q)θ(s-s0)s-(Q+q)2.
On the other hand, the theoretical result for the correlation function is given by Πμν(Q,q)=2ϵμανβfηc(2S)Qβqα∫01duϕηc(2S)(u)mc2-(q+uQ)2,where ϕηc(2S)(u) is defined analogously to Eq. (8). Matching the phenomenological side with the theoretical side and applying the Borel transformation to both sides, the analytical expression of the form factor for the process ψ(2S)→ηc(2S)γ is obtained FVP=mψ(2S)+mηc(2S)mψ(2S)fηc(2S)fψ(2S)∫Δ1duϕηc(2S)(u)ue-1uM2[mc2+u(1-u)mηc(2S)2+(1-u)P′2]+mψ(2S)2M2,where Δ=12mηc(2S)2[(s0-mηc(2S)2+P′2)2+4mηc(2S)2(mc2+P′2)-(s0-mηc(2S)2-q2)].
Light-cone distribution amplitudes
The LCDAs serve as essential nonperturbative input parameters. The typical functional forms of LCDAs employed in this work, along with the required parameters, are provided below.
For the process K*-→K-γ, the LCDA of K- meson is chosen as [62]ϕK(u,μ)=6u(1-u)[1+∑n=14anK(μ)Cn3/2(2u-1)],where Cn3/2 denotes the Gegenbauer polynomials and anK represents the corresponding Gegenbauer moments. In our calculation, the adopted Gegenbauer coefficients are a1K=0.09±0.13 and a2K=0.03±0.03[63].
For the decays B*→Bγ, Bs*→Bsγ, D*→Dγ and Ds*+→Ds+γ, the LCDAs of the heavy-light mesons B, D, Ds+ and Bs are parametrized exponentially as [64]ϕH(u,μ)=N(α,β)4uu¯e4αuu¯+β(u-u¯),where N(α,β)=16α5/2[4α(βsinh(β)+2αcosh(β))+eα+β24α(-2α+4α2-β2)×{Erf(α-β2α)+Erf(α+β2α)}]-1.
The parameters α and β are presented in Table I, and Erf here denotes the error function.
I10.1103/nm8j-l3ky.t1
The parameters α and β in the LCDAs of D, B, Ds+, and Bs mesons at the scale μ=2GeV[64].
For the process ψ(2S)→ηc(2S)γ, the BHL approach [61,65,66] is used to define the LCDA of ηc(2S) meson, ϕηc(2S)(u,μ0)=26fηc(2S)∫|k→⊥|2<μ02d2k→⊥16π3Ψηc(2S)(u,k→⊥),Ψηc(2S)λ1λ2(u,k→⊥)=ϕBHL(u,k→⊥)χλ1λ2(u,k→⊥)=Ae-b2k→⊥2+mc*2u(1-u)χλ1λ2(u,k→⊥),where Ψηc(2S)λ1λ2(u,k→⊥) is the wave function of ηc(2S) meson, and χλ1λ2(u,k→⊥)[67–69] is the spin wave function, as presented in Table II. The parameters A and b2 are determined by the following two constraints:
II10.1103/nm8j-l3ky.t2
The spin wave function for ηc(2S). Here, mc*=1.8GeV[70] is the constituent quark mass of the c quark, and λ1 and λ2 denote the helicities of the c quark and the anti-c quark, respectively.
26fηc(2S)∫01dx∫|k→⊥|2<μ02d2k→⊥16π3∑λ1λ2Ψηc(2S)λ1λ2(ui,k→⊥)=1,∫01dx∫d2k→⊥16π3|ϕBHL(u,k→⊥)|2=Pηc(2S).The first equation in (28) is the normalization condition, and the second equation corresponds to the probability normalization condition, where Pηc(2S) denotes the proportion of the Fock state. Here, we select the leading order in the Fock state expansion with Pηc(2S)=0.5, and assume that the initial scale of the distribution amplitude of ηc(2S) meson is μ0=mc*. Using these inputs, the two undetermined parameters are obtained as A=21.7713GeV-1 and b2=0.06787GeV-2.
In the above, the distribution amplitudes of each process were determined. Below, the images of the selected distribution amplitudes are presented as shown in Fig. 1, from which it is evident that the distribution amplitude of ηc(2S) has no nodes.
110.1103/nm8j-l3ky.f1
The distribution amplitudes of the K, ηc(2S), D, Ds, B, and Bs mesons.
Decay width
Once the form factors are determined, the M1 radiative decay widths for the processes K*-→K-γ, D*→Dγ, B*→Bγ, Ds*+→Ds+γ, and Bs*→Bsγ, can be obtained [59,71]Γ=α3|FVP|2(mi+mf)2kγ3,where α is the fine structure constant, kγ=(Mi2-Mf2)/2Mi, and the coupling constant FVP(q2) for a real photon is equal to the form factor FVP(q2) in the limit q2→0.
NUMERICAL RESULTS AND DISCUSSIONInput parameters
In the preceding sections, analytic expressions for the form factors and decay widths of the six decay processes have been derived. In the subsequent discussion, numerical results will be provided and compared with experimental measurements along with predictions from other theoretical approaches.
The input parameters consist of two parts: some are taken from the PDG [9] (e.g., quark masses), compiled in Table III and others are required by the LCSR analysis, including the thresholds and the Borel parameters.
III10.1103/nm8j-l3ky.t3
Parameters required for all processes include quark masses, decay constants, and meson masses.
Within the LCSR framework, the threshold and the Borel parameter are crucial input parameters. As a common practice, the threshold is taken to be the squared mass of the first excited state meson, although its precise value depends on the specific decay channel and must be fixed separately. For the decay K*-→K-γ, the first excited state of the K* meson, K*(1410), has a mass of 1.414 GeV, so the threshold s0 is approximately 1.9GeV2. The Borel parameter is typically determined by imposing two criteria:
The combined contribution from higher resonances and the continuum is less than 30%;
The physical observables exhibit only mild dependence on the Borel parameter.
Transition form factors
Based on the criteria and constraints established above, the dependence of the form factors on the threshold s0 and Borel parameter M2 is discussed as follows.
The form factor for the K*-→K-γ process is shown in Fig. 2(a), with the threshold parameter chosen as s0≈1.9GeV2. As illustrated in Fig. 2(a), our results exhibit high stability against variations in the threshold parameter and also remain highly stable under changes in the Borel parameter, indicating that our choice of the Borel region is reasonable.
210.1103/nm8j-l3ky.f2
The dependence of the form factors for (a) K*-→K-γ and (b) ψ(2S)→ηc(2S)γ processes on the threshold s0 and Borel parameter M2.
Building on the transition form factors we have obtained for ground state vector mesons [60], we extend our calculation to excited state M1 radiative decay ψ(2S)→ηc(2S)γ. As shown in Fig. 2(b), the results of the form factor exhibits a certain sensitivity to the Borel parameter, while maintaining only weak dependence on the thresholds.
Experimental data on the D*→Dγ transition remain imprecise at present. In Fig. 3(a), we choose the threshold near the square of the mass of D1*0(2600)[9]. Within the LCSR framework, the form factor is remarkably stable against variations of the thresholds. By contrast, the results exhibit a certain dependence on the Borel parameter.
310.1103/nm8j-l3ky.f3
The dependence of the form factors for (a) D*→Dγ and (b) Ds*+→Ds+γ processes on the threshold s0 and Borel parameter M2.
Similar to the D*→Dγ process, we select the threshold near the square of the Ds1*(2700)± meson mass [9], s0=7.29GeV2. As shown in Fig. 3(b), the transition form factor for Ds*+→Ds+γ exhibits excellent stability under variations of the thresholds, and the results of the form factor display only a mild dependence on the Borel parameter.
In experiments, the total decay widths of the B* and Bs* mesons and the branching fractions for their M1 radiative decays have not yet been precisely determined. As shown in Fig. 4, the transition form factors for the B*→Bγ and Bs*→Bsγ processes exhibit a high degree of stability under variations of the Borel parameter, with almost negligible impact. Further, the form factors display weak dependence on the threshold. This minor variation stems primarily from the insufficient knowledge of the first excited state in these decay channels, which directly affects the choice of the threshold parameter and constitute the main source of uncertainty.
410.1103/nm8j-l3ky.f4
The dependence of the form factors for (a) B*→Bγ and (b) Bs*→Bsγ processes on the threshold s0 and Borel parameter M2.
Numerical results for the form factors <inline-formula><mml:math display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>V</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>
Based on the above sensitivity analysis, we present the numerical values of the transition form factors at fixed Borel parameters and thresholds in Table IV at the limit q2=0. The total uncertainty incorporates contributions from various sources, including the Borel parameter M2, the threshold parameter, quark masses, decay constants, and higher-twist distribution amplitudes. In the extraction of the form factors, the contributions from excited states are all below 20%, while those from higher-twist distribution amplitudes are less than 3% and the radiative correction a few per mille. In this work, with the parameters choices s0=1.96GeV2 and M2=5GeV2, we present numerical results for the K*-→K-γ transition form factor. The parameters adopted for the other channels are D*→Dγ (s0=6.76GeV2, M2=4GeV2), B*→Bγ (s0=32GeV2, M2=9GeV2), Ds*+→Ds+γ (s0=7.29GeV2, M2=7GeV2), Bs*→Bsγ (s0=34GeV2, M2=14GeV2), and ψ(2S)→ηc(2S)γ (s0=16GeV2, M2=15GeV2).
IV10.1103/nm8j-l3ky.t4
Numerical results of the form factors FVP(q2) at the limit q2→0.
In this work, we define q2 as the four-momentum squared of the photon, i.e., the photon virtuality. For the radiative decays V→P+γ considered here, the photon is real, which corresponds to the physical point q2=0. Our light-cone sum rule calculations directly yield the form factors at this point, denoted as FVP(0) and listed in Table IV. These values can be directly inserted into the decay width formula Eq. (29), enabling a direct comparison with experimental measurements.
Concerning the q2 dependence, we note that the form factor FVP(q2) as a function of photon virtuality is primarily relevant for processes involving virtual photons, such as V→P+l++l-. In such cases, a single-pole parametrization motivated by the vector meson dominance (VMD) model is commonly adopted to describe this dependence. For the real photon decay V→P+γ, however, the physical kinematic point is q2=0, which is consistent with 0≤q2≤(MV-MP)2 in the LCSR, and therefore, no extrapolation is required.
Numerical results for the decay widths
In the aforementioned section, we discussed the transition form factors, an important nonperturbative physical parameter. In this section, we will present the results of the M1 radiative decay widths using the transition form factors. We visually illustrate the dependence of M1 radiative decay widths on the thresholds s0 and Borel parameters M2.
The Fig. 5 shows the dependence of the decay widths for the processes K*-→K-γ and ψ(2S)→ηc(2S)γ on the thresholds s0 and the Borel parameters M2. It is evident from the figure that the M1 radiative decay width remains remarkably stable under variations of both the threshold parameter and the Borel parameter.
510.1103/nm8j-l3ky.f5
The dependence of the decay widths for (a) K*-→K-γ and (b) ψ(2S)→ηc(2S)γ processes on the threshold s0 and Borel parameter M2.
The M1 radiative decay width for the K*-→K-γ process, as quoted in the Particle Data Group (PDG) review, is 50.37 keV [9]. The result obtained in this work using the LCSR method is 48.10±15.05keV, with a relative deviation of approximately 4% compared to the PDG value. For the excited state ψ(2S)→ηc(2S)γ process, the calculated result in this work is 0.20±0.07keV, and the relative deviation from the experimental value of 0.15 keV [18] is approximately 33%. This discrepancy could be attributed to several factors, including the neglect of higher-order corrections, the sensitivity of the excited-state calculation to the LCSR parameters and the challenges in modeling the LCDA for the ηc(2S) meson.
Theoretical predictions for the M1 radiative decay widths of four processes (D*→Dγ, B*→Bγ, Ds*+→Ds+γ and Bs*→Bsγ) are presented in Figs. 6 and 7. In Table V, we compare our results of the decay widths with those obtained from experiments and from other theoretical approaches, with the errors in our results coming from the Borel parameter, the threshold parameter, quark masses, decay constants, and higher-twist distribution amplitudes.
610.1103/nm8j-l3ky.f6
The dependence of the decay widths for (a) D*→Dγ and (b) Ds*+→Ds+γ processes on the threshold s0 and Borel parameter M2.
710.1103/nm8j-l3ky.f7
The dependence of the decay widths for (a) B*→Bγ and (b) Bs*→Bsγ processes on the threshold s0 and Borel parameter M2.
V10.1103/nm8j-l3ky.t5
The M1 radiative decay widths for all processes are presented (in units of keV). The first row shows our theoretical predictions, and the second row displays the corresponding experimental measurements. Furthermore, we systematically compare our theoretical results with predictions from several models, including QCD sum rules (QCDSR), vector confining potentials (VCP), relativistic quark model (RQM), nonrelativistic potential model (PM), effective mass scheme (EMS), scalar confining potentials (SCP), light-front quark model (LFQM), and nonrelativistic quark model (NRQM).
We observe that, as we move from K* to the D*,Ds*,ψ(2S),B*, and Bs* mesons, the meson masses increase successively, while the decay widths exhibit an overall decreasing trend. This behavior reflects the mass suppression effect characteristic of heavy quarks, which is physically expected and consistent with the predictions of heavy quark effective theory [80,81]. Furthermore, this trend is in good agreement with the patterns shown by other theoretical approaches, such as RIQ [75], LFQM [23,24,71], PM [77,78], and EMS [77], as presented in Table V. The radiative decay widths for D* and B* mesons predicted by our LCSR approach are in good agreement with those from other LCSR calculations within uncertainties. For the charmonium excited state, we present the first LCSR calculation and find that our result is consistent within uncertainty with the prediction from the nonrelativistic quark model.
Linear relationship
An intriguing systematic behavior emerges from our analysis of the ground-state V→Pγ decays. We find that the decay widths exhibit a universal scaling relation when plotted against a specific kinematic function. Motivated by the two-body decay dynamics and the role of decay constant, we define the function A(x) as A(x)=(fffi1(mf+mi)2)kγ3(FVP)x,where, x is a free power to be determined, fi and ff are the decay constants of the initial and final states, mi and mf are the masses corresponding to the initial and final state mesons and kγ is the photon momentum.
Remarkably, when plotting the decay width Γ against A(x) in the logarithmic coordinate system, all data points for the ground-state transitions align along a straight line for an optimal power of x≈2.1, as shown in Fig. 8. This observed linear dependence, log(Γ) ∝ log(A(2.1)) suggests a universal pattern governing these M1 radiative decays. The fact that the optimal power is not exactly 2 [as a naive leading-order analysis might suggest in Eq. (29)] indicates the significance of contributions beyond the leading order, such as higher-order perturbative QCD corrections and higher-twist effects. Additionally, we have also collected the meson masses, decay constants, radiative decay widths, and transition form factors for the processes J/ψ→ηcγ and ρ+→π+γ[9,38,82], also fall neatly on the same fitted line. This observation could provide a fresh, unifying perspective on the systematics of radiative decays.
810.1103/nm8j-l3ky.f8
Linear dependence of the decay width on the function A in the logarithmic coordinate system.
SUMMARY
In this work, we study the magnetic dipole (M1) radiative decays of vector mesons within the framework of LCSR. Firmly grounded in the first principles of QCD, this method provides a computationally efficient tool for determining transition form factors with minimal model dependence. We apply it to a set of vector mesons systems spanning the strange, charm, and bottom sectors (K*, B*, Bs*, D*, Ds*+) and extend the formalism to the radiative transition between excited charmonium states, ψ(2S)→ηc(2S)γ. The core of this work lies in the computation of the transition form factors for these decay channels, from which the corresponding M1 radiative decay widths are derived.
Our numerical results of the decay widths are consistent with existing experimental data. For the ground state vector meson channel K*-→K-γ, we predict Γ(K*-→K-γ)=48.10±15.05keV, in excellent agreement with the measured value of 50.37 keV [9]. For the charmonium transition ψ(2S)→ηc(2S)γ, we obtain Γ(ψ(2S)→ηc(2S)γ)=0.20±0.07keV, which is consistent with the experimental value of 0.15 keV [18] within uncertainties. For channels not yet measured experimentally, we provide the following theoretical predictions: Γ(D*→Dγ)=7.96±3.31keV, Γ(Ds*+→Ds+γ)=1.36±0.08keV, Γ(B*→Bγ)=0.194±0.200keV, and Γ(Bs*→Bsγ)=0.05±0.06keV. A comprehensive comparison with a range of other theoretical approaches including quark models, lattice QCD, and alternative QCD sum rule calculations shows broad consistency, thereby further reinforcing the credibility of our predictions.
While this work provides individual predictions for decay widths, it also reveals a linear behavior governing these transition processes. For the vector mesons [K*, D*, Ds*+, ψ(2S), B*, Bs*], the decay width exhibits a monotonic decrease with increasing vector meson mass. This is consistent with the expectations of methods such as heavy quark effective theory. Moreover, when analyzed in logarithmic coordinates, the widths for all channels including charmonium systems reveal a striking linear scaling relation with an appropriate kinematic function derived from two-body decay dynamics. This finding suggests a potential universality underlying M1 transitions. Deviations from the fitted linear relation are primarily attributed to the leading order truncation in our computational framework, particularly the omission of higher order perturbative QCD corrections and higher twist contributions.
However, the physical origin of this scaling behavior remains to be clarified. Whether it reflects an underlying symmetry or common dynamical features across different flavor sectors requires further investigation. To elucidate its nature and test its generality, extending our analysis to other radiative transitions, such as electric dipole decays, is expected in the future. These extended studies will help determine whether the observed linear scaling is specific to M1 transitions or a more general manifestation of nonperturbative QCD dynamics.
In summary, the light-cone sum rules framework has been employed to compute radiative partial widths for a representative set of vector meson decays, yielding results consistent with experimental data while providing valuable physical insights. The observed linear behavior offers a new perspective on the systematics of hadronic radiative transitions. In subsequent work, we will focus on incorporating higher order corrections and related factors to enhance the precision of our results and reduce theoretical uncertainties. Furthermore, we plan to include processes with different spin configurations, thereby enabling more rigorous tests of the linear behavior observed in the present study. On the experimental front, high precision measurements of radiative widths for bottom and charmed mesons from ongoing and future facilities, particularly Belle II and LHCb, are anticipated to provide critical tests of these theoretical predictions and further constrain nonperturbative QCD models. Such coordinated efforts will undoubtedly advance our fundamental understanding of strong interaction dynamics in the nonperturbative regime.
ACKNOWLEDGMENTS
We would like to thank Wen-Nian Liu, Chen Wang, and Wen-Xuan Zhang for stimulating discussions. D. G. acknowledges the support of the China Scholarship Council program (Project No. 202510530001).
DATA AVAILABILITY
The data that support the findings of this article are not publicly available. The data are available from the authors upon reasonable request.
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