A Visible Coupling Benchmark for Massive Dark Photon Searches in D⁰→γA^′→γe⁺e^-

1. Introduction

A massive dark photon $A^{\prime }$ provides a well-motivated low-energy window onto hidden vector interactions. If the new boson $A^{\prime }$ has a nonzero mass, it can be produced on shell and searched for as a localized structure in a visible invariant-mass spectrum. In the minimal kinetic-mixing picture, constraints are commonly displayed in the $ϵ$-$m_{A^{\prime }}$ plane, and extensive fixed-target, $e^{+}{e}^{-}$, and hadron-collider programs have explored both visible and invisible $A^{\prime }$ decays [1,2,3,4,5,6]. This organization is most transparent when production and decay are governed by flavor-universal kinetic mixing[7,8]. It becomes less direct when the production vertex is flavor changing, because a visible final-state limit then constrains not only a microscopic coupling but also short-distance flavor dynamics, non-perturbative hadronic input, and the visible decay probability of the massive $A^{\prime }$ [9,10].

Rare charm decays are a natural setting for such a study. Standard Model flavor changing neutral current amplitudes in the up-quark sector are strongly suppressed, while exclusive charm modes are strongly affected by long distance hadronic effects and vector-meson resonances [11,12,13,14,15,16,17]. Consequently, inclusive rates alone are often difficult to interpret, whereas narrow structures and differential null tests can provide cleaner sensitivity to short-distance contributions [18,19,20,21,22,23]. Existing measurements and searches in radiative and dileptonic charm channels, including $D^0\to \varphi \gamma$, $D^0\to K^{*0}\gamma$, $D^0\to {\rho }^0\gamma$, $D^0\to \gamma \gamma$, and $D^0\to h^{+}{h}^{-}{\ell }^{+}{\ell }^{-}$, demonstrate the feasibility of such final states while emphasizing the need for controlled continuum and $\rho /\omega$, $\varphi$-region treatments [24,25,26,27,28,29]. The recent BESIII search for a massless dark photon in $c\to u{\gamma }^{\prime }$ decays is therefore closely related, but the massive visible case considered here is experimentally and interpretively distinct because the $A^{\prime }$ is reconstructed through $A^{\prime }\to e^{+}{e}^{-}$ [30].

The study focuses on $D^0\to \gamma A^{\prime }\to \gamma e^{+}{e}^{-}$ with $2m_e<m_{A^{\prime }}<m_{D^0}$. The signal topology is illustrated in Figure 1. The channel combines a two-body radiative production process with a narrow dielectron resonance search, but its rate cannot be mapped to a single microscopic parameter without additional assumptions. It depends simultaneously on the short-distance $c\to u{A}^{\prime }$ transition, the $D^0\to \gamma$ hadronic matrix element at $q^2=m_{A^{\prime }}^2$, and the branching fraction for $A^{\prime }\to e^{+}{e}^{-}$. The experimentally useful mass range is further shaped by photon and electron reconstruction, bremsstrahlung recovery, conversion backgrounds, endpoint phase-space suppression, and resonance vetoes or templates near the $\rho /\omega$ and $\varphi$ bands. A direct bound on $ϵ$ would therefore mix detector sensitivity with hadronic and model-dependent inputs.

To keep these ingredients separate, we parameterize the flavor-changing $c\to u{A}^{\prime }$ transition by an effective dipole operator and retain the $D^0\to \gamma$ hadronic transition as an explicit factor $F_{D^0\gamma }\left(q^2\right)$. We then combine this factor and the visible $A^{\prime }$ branching fraction with the effective dipole strength into a visible coupling $g_{\mathrm{vis}}$. In this normalization, $\mathcal{B}(D^0\to \gamma A^{\prime }\to \gamma e^{+}{e}^{-})$ is proportional to $g_{\mathrm{vis}}^2$ multiplied by the cubic two-body phase-space factor, so an experimental branching-fraction sensitivity can be converted into a $g_{\mathrm{vis}}$ reach without fixing a particular ultraviolet completion, form-factor model, or $A^{\prime }$ decay scenario. Detector-level inputs from BESIII-like, Belle II-like, LHCb-like, or future samples enter through the effective exposure, efficiency, mass resolution, background level, and resonance treatment [31,32,33,34]. The resulting framework is therefore a portable normalization for searches for a massive visible $A^{\prime }$ in $D^0\to \gamma e^{+}{e}^{-}$, with later translations to $g_{\mathrm{eff}}$ or $ϵ$ left to model-specific input.

2. Effective Description and Visible Normalization

In this benchmark, the signal topology can be organized into two main ingredients that are useful to treat separately at the outset. The first is the short-distance flavour changing transition $c\to u{A}^{\prime }$, which is described by an effective operator at the charm scale. The second is the long-distance hadronic transition in which the $D^0$ meson emits the photon. Keeping these two ingredients separate makes the normalization transparent and allows the hadronic input to be updated independently.

At the lowest non-trivial order, the short distance flavor changing $c\to u{A}^{\prime }$ transition is represented by the dipole interaction,

$${\mathcal{L}}_{\mathrm{eff}}={\displaystyle \frac{1}{\Lambda }}\overline{u}{\sigma }_{\mu \nu }\left(C+C_5{\gamma }_5\right)c{F}^{\prime \mu \nu }+\mathrm{h}.\mathrm{c}.$$

(1)

where $F^{\prime \mu \nu }$ is the field strength tensor of the vector state $A^{\prime }$, ${\sigma }_{\mu \nu }=i[{\gamma }_{\mu },{\gamma }_{\nu }]/2$, and C and $C_5$ are dimensionless Wilson coefficients. The scale $\Lambda$ represents the heavy mass scale that suppresses the dimension-five operator.

In this benchmark, the term dark photon is used in a broad phenomenological sense: it denotes a massive visible vector state that can decay into $e^{+}{e}^{-}$. The flavor changing production vertex $c\to u{A}^{\prime }$ is not assumed to arise from minimal kinetic mixing alone, but is parameterized directly by the effective dipole operator in Equation (1). Consequently, translating the effective normalization introduced below into the conventional kinetic mixing parameter $ϵ$ requires additional theoretical input, as discussed in Section 4.

For an unpolarized benchmark rate, and under the common form factor normalization introduced below, the two dipole structures enter through the quadratic combination

$$g_{\mathrm{eff}}={\displaystyle \frac{\sqrt{{\left|C\right|}^2+{\left|C_5\right|}^2}}{\Lambda }},\left[g_{\mathrm{eff}}\right]={\mathrm{GeV}}^{-1}.$$

(2)

This definition should be understood as a convenient low energy coupling strength. It does not assume a specific ultraviolet model, nor does it fix possible CP phases that would require a more differential analysis.

The remaining part of the amplitude is the non-perturbative hadronic matrix element for the $D^0\to \gamma$ transition. Rather than assigning a model-dependent numerical value to this matrix element, we absorb it into a dimensionless transition factor $F_{D^0\gamma }\left(q^2\right)$, evaluated at $q^2=m_{A^{\prime }}^2$. To make this convention explicit, we fix the normalization of $F_{D^0\gamma }$ through the polarization summed amplitude,

$$\sum _{\mathrm{pol}}{\left|\mathcal{M}\left(D^0\to \gamma A^{\prime }\right)\right|}^2=2m_{D^0}^4{\left|F_{D^0\gamma }\left(m_{A^{\prime }}^2\right)\right|}^2{g}_{\mathrm{eff}}^2{\left(1-{\displaystyle \frac{m_{A^{\prime }}^2}{m_{D^0}^2}}\right)}^2.$$

(3)

With this definition, Equation (3) should be regarded as a normalization convention rather than as an independent dynamical prediction. The factor ${(1-m_{A^{\prime }}^2/m_{D^0}^2)}^2$ displays the momentum dependence generated by the radiative dipole structure, while the unknown strong-interaction normalization is kept in $F_{D^0\gamma }$. With this convention, lattice QCD calculations, light-cone sum rule estimates, vector-meson dominance inputs, or data driven normalizations can be incorporated simply by replacing $F_{D^0\gamma }\left(q^2\right)$, without changing the kinematic conversion used later.

The experimentally reconstructed final state is $\gamma e^{+}{e}^{-}$, rather than the intermediate state $A^{\prime }$ alone. The observable rate must therefore include the probability for $A^{\prime }$ to decay into an electron and positron pair. We combine the coupling associated with the short distance transition, the hadronic normalization, and the visible branching fraction into a coupling used at the search level,

$$g_{\mathrm{vis}}\left(m_{A^{\prime }}\right)\equiv \left|F_{D^0\gamma }\left(m_{A^{\prime }}^2\right)\right|\sqrt{\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})}{g}_{\mathrm{eff}}.$$

(4)

The quantity $g_{\mathrm{vis}}$ has the same mass dimension as $g_{\mathrm{eff}}$, namely ${\mathrm{GeV}}^{-1}$. All projected rates in the following analysis are expressed in terms of $g_{\mathrm{vis}}$. If a particular model later specifies both $F_{D^0\gamma }\left(q^2\right)$ and $\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})$, the same results can be translated back to $g_{\mathrm{eff}}$, and then to the microscopic parameters of that model.

3. Signal Branching Fraction from Two-Body Kinematics

After the visible normalization has been specified, the signal rate can be obtained from the two body decay $D^0\to \gamma A^{\prime }$. In the rest frame of the $D^0$ meson, and treating the photon as massless, the common momentum of the two final state particles is

$$|{\overrightarrow{p}}_{\gamma }|=|{\overrightarrow{p}}_{A^{\prime }}|={\displaystyle \frac{m_{D^0}^2-m_{A^{\prime }}^2}{2m_{D^0}}}={\displaystyle \frac{m_{D^0}}{2}}\left(1-{\displaystyle \frac{m_{A^{\prime }}^2}{m_{D^0}^2}}\right).$$

(5)

This expression is purely kinematic. It shows that the available momentum vanishes when the mass of $A^{\prime }$ approaches the $D^0$ mass.

Throughout this benchmark we consider the visible massive-$A^{\prime }$ range

$$2m_e<m_{A^{\prime }}<m_{D^0}.$$

(6)

The lower bound follows from the requirement that the intermediate state can decay into an $e^{+}{e}^{-}$ pair, while the upper bound is set by the two-body kinematics of $D^0\to \gamma A^{\prime }$. In practice, the experimentally useful region is further restricted by reconstruction thresholds, conversion backgrounds, resonance regions, and endpoint phase-space loss, as discussed below.

The partial width is obtained by multiplying the squared amplitude by the two body phase space. Since the parent $D^0$ meson has spin zero, no average over initial spin states is required. After summing over the final state polarizations, one obtains

$$\Gamma (D^0\to \gamma A^{\prime })={\displaystyle \frac{|{\overrightarrow{p}}_{\gamma }|}{8\pi m_{D^0}^2}}\sum _{\mathrm{pol}}{\left|\mathcal{M}(D^0\to \gamma A^{\prime })\right|}^2.$$

(7)

Substituting the amplitude normalization introduced in Equation (3) and the momentum in Equation (5) gives

$$\Gamma (D^0\to \gamma A^{\prime })={\displaystyle \frac{m_{D^0}^3}{8\pi }}{\left|F_{D^0\gamma }\left(m_{A^{\prime }}^2\right)\right|}^2{g}_{\mathrm{eff}}^2{\left(1-{\displaystyle \frac{m_{A^{\prime }}^2}{m_{D^0}^2}}\right)}^3.$$

(8)

Here the transition factor is evaluated at $q^2=m_{A^{\prime }}^2$. The cubic dependence has a clear kinematic origin: one power arises from the two body phase space, while the remaining two powers are supplied by the radiative dipole structure of the amplitude.

The experimentally visible final state also requires the decay of $A^{\prime }$ into an electron pair. Therefore, the partial width must be multiplied by ${\tau }_{D^0}/\hslash$ and by the visible branching fraction. Using the definition of $g_{\mathrm{vis}}$, the visible signal branching fraction becomes

$${\mathcal{B}}_{\mathrm{sig}}\equiv \mathcal{B}(D^0\to \gamma A^{\prime }\to \gamma e^{+}{e}^{-})={\displaystyle \frac{{\tau }_{D^0}}{\hslash }}{\displaystyle \frac{m_{D^0}^3}{8\pi }}{g}_{\mathrm{vis}}^2\left(m_{A^{\prime }}\right){\left(1-{\displaystyle \frac{m_{A^{\prime }}^2}{m_{D^0}^2}}\right)}^3.$$

(9)

For this benchmark, Equation (9) is the main conversion between the coupling used at the search level and the visible branching fraction. It shows that the rate is proportional to $g_{\mathrm{vis}}^2$ and that the rate is suppressed when the available two-body momentum becomes small near the endpoint. The corresponding benchmark branching fraction curves, which visualize the $g_{\mathrm{vis}}^2$ scaling and the endpoint suppression implied by Equation (9), are shown in Appendix A.

The same branching fraction fixes the area of the peak in the reconstructed dielectron mass spectrum. If the intrinsic width of $A^{\prime }$ is much smaller than the detector mass resolution, the observed signal is controlled by the detector response. For a normalized resolution function G,

$${\displaystyle \frac{d\mathcal{B}}{d{m}_{e^{+}{e}^{-}}}}={\mathcal{B}}_{\mathrm{sig}}G(m_{e^{+}{e}^{-}};m_{A^{\prime }},{\sigma }_m),\int G(m_{e^{+}{e}^{-}};m_{A^{\prime }},{\sigma }_m)d{m}_{e^{+}{e}^{-}}=1.$$

(10)

where the normalization of G ensures that the integral of the peak is ${\mathcal{B}}_{\mathrm{sig}}$. If the intrinsic width is not negligible compared with the mass resolution, the physical line shape should first be described by a normalized Breit Wigner function and then smeared with the detector response,

$${\displaystyle \frac{d\mathcal{B}}{d{m}_{e^{+}{e}^{-}}}}={\mathcal{B}}_{\mathrm{sig}}\left[BW(m_{e^{+}{e}^{-}};m_{A^{\prime }},{\Gamma }_{A^{\prime }})\otimes G(m_{e^{+}{e}^{-}};{\sigma }_m)\right].$$

(11)

The narrow peak expression in Equation (10) is used as a transparent benchmark for a weakly coupled visible state. It fixes the signal yield and the local peak shape, but it does not replace the background model of a real analysis. In an experimental study, the smooth continuum, sidebands, reconstruction efficiency, and the $\rho /\omega$, $\varphi$ resonance regions should be treated with experiment-specific templates or vetoes.

4. Coupling Reach and Interpretation

The preceding section gives a direct relation between the visible branching fraction and the coupling used at the search level. This relation can be inverted to express the coupling reach in terms of a detector level sensitivity to the visible branching fraction. Denoting this sensitivity by ${\mathcal{B}}_{\mathrm{sens}}$, and solving Equation (9) for $g_{\mathrm{vis}}$, one obtains

$$g_{\mathrm{vis}}^{\lim }(m_{A^{\prime }};{\mathcal{B}}_{\mathrm{sens}})={\left[{\displaystyle \frac{8\pi \hslash {\mathcal{B}}_{\mathrm{sens}}}{{\tau }_{D^0}{m}_{D^0}^3}}\right]}^{1/2}{\left(1-{\displaystyle \frac{m_{A^{\prime }}^2}{m_{D^0}^2}}\right)}^{-3/2}.$$

(12)

This expression is the conversion used to draw the coupling reach curves. The square root dependence reflects the fact that rates are proportional to squared amplitudes. As a result, an improvement of the branching fraction sensitivity by a factor of ten improves the coupling reach by a factor of $\sqrt{10}$. The last factor shows the loss of reach near the kinematic endpoint, where the two body phase space becomes small.

Figure 2 shows the reach in $g_{\mathrm{vis}}$ obtained from representative values of ${\mathcal{B}}_{\mathrm{sens}}$. The curves depend only on the visible branching fraction sensitivity and on the kinematic factor in Equation (12). The rise at large $m_{A^{\prime }}$ is therefore not a consequence of an assumed background model, but follows from the closing of the available phase space.

Table 1 gives representative numerical values of the converted coupling reach. These values are consistent with Equation (12), using the $D^0$ mass, the $D^0$ lifetime, and ℏ in units matched to GeV. The table is intended as a benchmark for rescaling rather than as a substitute for an experiment specific limit.

Expressing the reach in terms of $g_{\mathrm{vis}}$ keeps the detector-level sensitivity separated from the hadronic normalization and from the visible decay probability of the intermediate state. The quantity $g_{\mathrm{vis}}$ should therefore not be read as a microscopic coupling by itself. It already contains the hadronic transition factor and the visible branching fraction of $A^{\prime }$. Once a specific model or a data driven input provides these quantities, the corresponding effective coupling is obtained from

$$g_{\mathrm{eff}}^{\lim }\left(m_{A^{\prime }}\right)={\displaystyle \frac{g_{\mathrm{vis}}^{\lim }(m_{A^{\prime }};{\mathcal{B}}_{\mathrm{sens}})}{\left|F_{D^0\gamma }\left(m_{A^{\prime }}^2\right)\right|\sqrt{\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})}}}.$$

(13)

Here the transition factor is evaluated at $q^2=m_{A^{\prime }}^2$. This equation separates the experimental result from the external physics inputs. The experimental analysis constrains $g_{\mathrm{vis}}$, while the conversion to $g_{\mathrm{eff}}$ requires information on the hadronic normalization and on the decay probability of $A^{\prime }$ into the electron pair. In this sense, $g_{\mathrm{vis}}$ is the most model-independent quantity to quote, whereas $g_{\mathrm{eff}}$ should be reported only after specifying $F_{D^0\gamma }\left(q^2\right)$ and $\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})$. Different choices of these two inputs therefore rescale the same $g_{\mathrm{vis}}$ reach multiplicatively, without changing the detector-level conversion in Equation (12).

Figure 3 visualizes the reinterpretation step encoded in Equation (13). For a fixed detector level reach on the search level coupling $g_{\mathrm{vis}}$, the inferred constraint on the underlying effective flavor changing coupling $g_{\mathrm{eff}}$ depends on two external inputs: the hadronic transition normalization $|F_{D^0\gamma }\left(m_{A^{\prime }}^2\right)|$ and the visible decay probability $\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})$. The color scale therefore represents the multiplicative factor that converts a quoted $g_{\mathrm{vis}}$ limit into an input-dependent $g_{\mathrm{eff}}$ limit. Smaller values of either input weaken the inferred constraint on $g_{\mathrm{eff}}$, even when the experimental sensitivity to $g_{\mathrm{vis}}$ is unchanged. Thus, Figure 3 should not be interpreted as imposing a particular form factor model or dark sector decay scenario. Instead, it provides a rescaling map that allows future theoretical or phenomenological inputs to be applied to the same experimental search level result. This makes explicit the central separation of the present framework: detector level sensitivity is encoded in $g_{\mathrm{vis}}$, whereas the translation to $g_{\mathrm{eff}}$ is left to external theoretical input.

With this separation made explicit, the $g_{\mathrm{vis}}$ representation provides a useful way to position the present framework relative to other research perspectives. Unlike the conventional $ϵ$ versus $m_{A^{\prime }}$ plane used in ordinary dark photon searches, the present benchmark is organized around a charm transition with changing flavor and a visible intermediate state. Table 2 summarizes this distinction and compares the role of the $g_{\mathrm{vis}}$ framework with related approaches.

This comparison clarifies the scope of the benchmark. Ordinary dark photon searches provide important external context, but they do not directly constrain the $c\to u{A}^{\prime }$ amplitude without additional assumptions. Rare charm studies are closer in motivation, although they are often sensitive to long distance hadronic effects. Standard Model studies of the visible $D^0$ decay spectrum are essential for modeling the continuum, sidebands, and resonance regions. The $g_{\mathrm{vis}}$ formulation is therefore best understood as a reusable normalization that connects visible sensitivity at the detector level to a charm transition with changing flavor, while the translation to $g_{\mathrm{eff}}$, microscopic couplings, or the kinetic mixing parameter $ϵ$ requires external hadronic and decay inputs.

5. Yield, Sensitivity, and Resonance Treatment

The visible branching fraction obtained above can be converted into an expected event yield once the effective exposure is specified. We define the effective exposure as the number of reconstructed $D^0$ candidates multiplied by the total signal efficiency. With this definition,

$$E_{\mathrm{eff}}=N_{D^0}{ϵ}_{\mathrm{rec}},N_{\mathrm{sig}}=E_{\mathrm{eff}}{\mathcal{B}}_{\mathrm{sig}}.$$

(14)

where ${ϵ}_{\mathrm{rec}}$ includes the relevant reconstruction and selection effects. If both $D^0$ and ${\overline{D}}^0$ samples are used, $N_{D^0}$ should denote the total number of neutral charm mesons included in the analysis. This relation gives the basic scaling of the signal yield with exposure and branching fraction.

For a search at a fixed value of $m_{A^{\prime }}$, the signal is counted in a mass window W centered on the tested mass. The expected signal and background yields in this window can be written as

$$S=E_{\mathrm{eff}}{\mathcal{B}}_{\mathrm{sig}}{ϵ}_W,B={\int }_W{N}_{D^0}{ϵ}_{\mathrm{bkg}}{\displaystyle \frac{d{\mathcal{B}}_{\mathrm{bkg}}}{d{m}_{e^{+}{e}^{-}}}}d{m}_{e^{+}{e}^{-}}.$$

(15)

Here ${ϵ}_W$ denotes the fraction of the signal line shape contained in the chosen window. The function ${ϵ}_{\mathrm{bkg}}$ describes this mass-dependent background efficiency, and $d{\mathcal{B}}_{\mathrm{bkg}}/d{m}_{e^{+}{e}^{-}}$ denotes the corresponding differential background branching fraction before detector effects.

For each tested value of $m_{A^{\prime }}$, the window yields S and B are converted into a local sensitivity according to an experiment-specific statistical criterion. Since this work is intended as a normalization benchmark, we do not prescribe a unique discovery or exclusion test. Instead, we define $S_{min}\left(m_{A^{\prime }}\right)$ as the minimum signal yield required by the adopted counting or likelihood procedure. This quantity may depend on the expected background, systematic uncertainties, the mass window definition, and, in a mass scan, the treatment of the look-elsewhere effect.

The same counting setup also determines the branching fraction sensitivity that enters Equation (12). Solving Equation (15) for the signal branching fraction and replacing S by $S_{\min }$ gives

$$B_{\mathrm{sens}}\left(m_{A^{\prime }}\right)={\displaystyle \frac{S_{\min }\left(m_{A^{\prime }}\right)}{E_{\mathrm{eff}}{ϵ}_W\left(m_{A^{\prime }}\right)}}.$$

(16)

In the background-free 90% C.L. limit, one may use $S_{\min }\simeq 2.3$. In the background-dominated case, $S_{\min }$ is obtained by solving the chosen local significance or exclusion condition using the statistical model specified above. The resulting $B_{\mathrm{sens}}$ is therefore a sensitivity at the tested mass point. In a mass scan, this local sensitivity should be distinguished from a global exclusion or discovery criterion, since scanning many tested masses increases the probability of finding a background fluctuation somewhere in the spectrum.

A schematic mass spectrum can then be written as a sum of normalized templates,

$${\displaystyle \frac{dN}{d{m}_{e^{+}{e}^{-}}}}=N_{\mathrm{cont}}{f}_{\mathrm{cont}}\left(m_{e^{+}{e}^{-}}\right)+\sum _{V=\rho /\omega ,\varphi }{N}_V{T}_V\left(m_{e^{+}{e}^{-}}\right)+N_{A^{\prime }}G(m_{e^{+}{e}^{-}};m_{A^{\prime }},{\sigma }_m).$$

(17)

where $f_{\mathrm{cont}}$ describes the smooth continuum, $T_V$ denotes the vector meson templates for $V=\rho /\omega$, $\varphi$, and G denotes the detector resolution template for the $A^{\prime }$ signal. The coefficients $N_{\mathrm{cont}}$, $N_V$, and $N_{A^{\prime }}$ are the corresponding event yields. The structure of Equation (17) is illustrated schematically in Figure 4.

Figure 4 provides a visual representation of the template decomposition in Equation (17). The blue dashed curve represents the smooth continuum component $N_{\mathrm{cont}}{f}_{\mathrm{cont}}\left(m_{e^{+}{e}^{-}}\right)$, while the red dashed structures indicate the vector-meson contributions associated with the $\rho /\omega$ and $\varphi$ regions. The green narrow peaks illustrate potential $A^{\prime }$ signal contributions at possible mass values. The shaded vertical bands emphasize the resonance-dominated mass intervals, where the interpretation of a narrow visible $A^{\prime }$ signal is expected to be most analysis dependent.

Equation (17) is intended only as a schematic description of the mass spectrum. It should not be read as a unique Standard Model background prediction. In a real analysis, the continuum shape, sidebands, reconstruction efficiency, and resonance regions should be treated with experiment specific templates or vetoes. If interference among resonant and continuum amplitudes is relevant, the template sum should be replaced by an amplitude level model.

The $\rho /\omega$ and $\varphi$ regions are therefore the main limitations for a visible-spectrum interpretation. In the benchmark presentation, the intervals near 0.72 - 0.82 GeV and 0.99 - 1.04 GeV can be treated either as veto regions or as regions that require dedicated vector-meson templates. This treatment preserves the usefulness of the signal normalization while avoiding an overinterpretation of the schematic spectrum in regions dominated by known hadronic structures.

Taken together, the yield, significance, and spectrum relations provide a practical way to reinterpret a detector-level search in terms of the visible coupling defined above. Their cleanest use is in regions where the intermediate state is narrow compared with the experimental mass resolution and where the visible decay can be treated as prompt, so that the acceptance effects are contained in ${ϵ}_{\mathrm{rec}}$. Near the $\rho /\omega$ and $\varphi$ bands, the same normalization can still be used, but the spectral interpretation should be tied to experiment-specific vetoes or resonance templates. With these qualifications, the framework provides a reusable conversion between visible branching fraction sensitivity, $g_{\mathrm{vis}}$ reach, and exposure scaling, while translations to $g_{\mathrm{eff}}$ or to the ordinary kinetic-mixing parameter require additional hadronic or model input. Representative detector-level inputs for BESIII-like, Belle II-like, and LHCb-like massive $A^{\prime }$ searches are summarized in Appendix B. These inputs are used only for rescaling purposes and are not treated as existing limits on the specific decay chain.

6. Conclusions

Rare charm searches with visible narrow structures are difficult to interpret in a single microscopic parameter because the observed rate depends simultaneously on the short-distance transition, the non-perturbative $D^0\to \gamma$ matrix element, and the visible decay probability of the intermediate state. In this work, we have organized these ingredients into a visible coupling benchmark for the decay chain $D^0\to \gamma A^{\prime }\to \gamma e^{+}{e}^{-}$. The definition of $g_{\mathrm{vis}}$ keeps the detector level sensitivity separated from the hadronic and visible decay inputs, so that experimental branching fraction sensitivities can be compared without committing to a specific ultraviolet completion.

Within this normalization, the visible branching fraction is controlled by $g_{\mathrm{vis}}^2$ and by the cubic two-body phase space factor. The inverse relation then converts a representative sensitivity $B_{\mathrm{sens}}$ into a reach on $g_{\mathrm{vis}}$, with the expected degradation near the kinematic endpoint. The same framework also connects the branching fraction reach to event yields through the effective exposure $E_{\mathrm{eff}}$, making it straightforward to rescale the benchmark to different reconstructed charm samples, efficiencies, and mass resolutions.

The resulting formulation is intended as a rescaling tool rather than as a universal bound on the kinetic mixing parameter. Once experiment-specific efficiencies, background templates, resonance vetoes, and mass scan procedures are specified, the formulae can be used to compare BESIII-like, Belle II-like, LHCb-like, or future charm samples in a common visible coupling language. A further translation from $g_{\mathrm{vis}}$ to $g_{\mathrm{eff}}$ or to microscopic dark sector parameters requires additional input on $F_{D^0\gamma }\left(q^2\right)$ and $\mathcal{B}(A^{\prime }\to e^{+}{e}^{-})$.