Tests of the Atomki anomaly in lepton pair decays of heavy mesons

G. López Castro ∗ and Néstor Quintero 3, † Departamento de F́ısica, Centro de Investigación y de Estudios Avanzados, Apartado Postal 14-740, 07000 México D.F., México Facultad de Ciencias Básicas, Universidad Santiago de Cali, Campus Pampalinda, Calle 5 No. 62-00, Código Postal 76001, Santiago de Cali, Colombia Departamento de F́ısica, Universidad del Tolima, Código Postal 730006299, Ibagué, Colombia The anomalies recently reported in lepton pair transitions of 8Be∗ and He nuclei may be attributed to the existence of a feeble interacting light vector boson X17. We study the effects of this hypothetic particle in the semileptonic H∗ → He+e− decays (H a Qq̄ meson) in the framework of the HQET+VMD model. Using current bounds on the X17 boson to quarks, we find that decays of D∗+ and D∗ s mesons can be enhanced with up to O(15%) corrections relative to the photon-mediated contributions. Dedicated experimental searches at current heavy meson factories may confirm the existence of this light boson or set stronger bounds of their couplings to ordinary matter.


Introduction
The existence of a light vector boson weakly coupled to Standard Model (SM) fermions, has been suggested as a solution to the observed discrepancy between the SM prediction and the experimental measurement of the muon g − 2 magnetic moment anomaly (see for example [1,2]). It may be also a good candidate as a mediator of dark and ordinary matter interactions [1,2]. Several strategies aiming their detection in different collider and fixed target experiments have not found any signal so far [3], but have excluded different regions in the mass and coupling strenghts of parameter space. Theoretically, different models can accomodate a light vector boson and its required interactions through dimension-four kinetic mixing with SM neutral gauge bosons and their interactions with fermionic currents of SM or dark matter particles [1,2].
The anomalies recently reported in the invariant-mass spectrum and angular distribution of lepton pairs produced in 8 Be * transitions to its ground state [4], reinforces the interest in searches of light vector bosons. The observed anomalies seems to require the existence of a spin-1 boson named X17 [4][5][6] with mass m X = (16.7 ± 0.35 ± 0.50) MeV and a relative ratio B( 8 Be * → 8 BeX)/B( 8 Be * → 8 Beγ) = 5.8×10 −6 [6]. Couplings to standard model first-generation fermions of O(10 −3 ) (in units of the the electron charge) required to explain data, do not appear to be discarded by other data. More recently, the same group seems to confirm the X17 particle in studies of the 0 − → 0 + transitions of 4 He [7].
The almost isosinglet nature and the small mass difference of nuclei involved in 8 Be * decay provides an ideal place to observe this light boson, in case it exists. The effects of isospin nuclear mixing that may partially explain the observed anomaly has been widely discussed [6][7][8][9]. Further studies in analogous systems will be very important in order to establish or discard this light boson. In the present letter, we propose the study of H * → He + e − decays, where H(H * ) is a heavy Qq spin-0 (spin-1) meson 1 . This system seems to be ideal to further test the Atomki anomaly: on the one hand, the mass-splitting in heavy mesons is large enough (see Table I) to produce the X17 boson on-shell; on the other hand, strong decays of H * are either very suppressed of forbidden by kinematics, leaving electromagnetic decays as dominant . Furthermore, the large amount of data produced at heavy meson factories would allow the test the proposed channels in the near future.
The Lagrangian describing the interaction of quark and lepton flavors f with the photon and the X boson is couplings strenghts e f and ε f given in units of the electron charge e. The photon and X boson couplings to hadrons are described each by a single vector form factor which takes into account their structure in the momentum transfer region 4m 2 e ≤ q 2 ≤ (m H * − m H ) 2 , with q = p e + + p e − . The form factors describing the couplings of the off-shell vector particles (V = γ, X) in H * (p H * , H * ) → H(p H )V (q) are defined from the hadronic amplitude For on-shell vector particles, this Lorentz-vector amplitude must be contracted with the vector polarization µ (q). The case of lepton pair production is discussed below.
2. H * H-Vector vertices. The form factors F H * HV (q) are evaluated in the framework of the heavy quark effective theory suplemented with vector dominance model (HQET+VDM) model [11,12] , which has shown to give a good description of H * → Hγ decays. Since we will normalize results for our observables to this radiative decay, we use the ratio of decay rates because they are rather insensitive to the specific q 2 -dependency of the form factor. This is due to the smallness of the H * − H mass splitting compared to typical hadronic scales (∼ 1 GeV 2 ).
The vector H * and pseudoscalar H heavy mesons are composed of a Qq pair, with Q = b, c and q = u, d, s. The hadronic matrix element of the electromagnetic current are given by [11]: with e Q (e q ) the electric charge of the heavy quark (light antiquark), and similarly, for the X boson current. A straightforward evaluation of the form factors in the HQET+VMD model [11] leads to with the effective light "quark mass" parameter The sum extends over light vector-meson resonances (V = ρ 0 , ω, φ) according to the light-quark content of heavy mesons [11]. For u and d quarks the sum extends over the ρ and ω mesons, and for the quark s only contributes the φ meson (we assume ideal mixing of vector mesons). Numerical inputs for couplings constants can be found in Ref [11] and are reproduced here for reference: g V = 5.8, λ = −0.289 ± 0.016 GeV −1 (updated from new experimental inputs) and f V (m V ) the decay constant (mass) of vector meson V . Using current experimental data for lepton-pair decays of vector mesons V → e − e + [13], one gets (f ρ , f ω , f φ ) = (0.171, 0.155, 0.232) GeV 2 , with very small uncertainties. In Table I we list values for the electromagnetic form factor predicted in the HQET+VMD model at q 2 = 0. The quoted uncertainty is dominated by the input on λ coupling, the H * HV strong coupling in this model. A comparison with the magnitude of the measured form factor (within square brackets), obtained from the measurement of the radiative decay D * + → D + γ branching fraction, give confidence on this model. Let us define the following ratio of two-body decay rates: where p V is the momentum of the final state boson in the rest frame of H * . This ratio exhibits two important differences with respect to the similar ratio defined in 8 Be * → 8 Be nuclear transitions [5]. First, since m H * m q (q 2 ) one would expect a suppression of the heavy quark contribution relative to the light quarks; this makes R X/γ (H * ) more sensitive to the Xqq couplings, which are relatively well bounded from other processes [5]. On the other hand, given the larger phase-space in heavy meson decays, this ratio is not suppressed by kinematics, as it happens for decay of 8 Be nucleus.
Predictions for the H * → HX decay fractions require an estimate of the ε Q,q couplings. Bounds for the couplings of the X17 boson to the quarks of the first generation needed to explain the 8 Be anomaly were obtained in Refs. [5,6]: ε u ±3.7 × 10 −3 , ε d ∓7.4 × 10 −3 , and 0.2 × 10 −3 |ε e | 1.4 × 10 −3 . Our study requires the knowledge of second-and third-generation couplings, namely strange ε s , charm ε c , and bottom ε b . A priori these parameters are independent [6], and need not be related to the first-generation couplings. Our simplest starting assumption is universality of down-and up-type quark ε f couplings, thus, we will take ε c = ε u and ε b = ε s = ε d ; henceforth, our results will be obtained under this assumption [5,6]. Values of the H * HX couplings and normalized Γ(H * → HX)/Γ(H * → Hγ) rates for these transitions are given in Table II. The ratios are enhanced with respect to the nuclear case owing to the unsuppressed kinematical ratio in R X/γ (H * ). Note that in the case of D * + s → D + s X transitions, this ratio can reach a value close to ten percent.  3. Lepton pair production. The decay amplitude for lepton pair production H * (P H * ) → H(P H )e + (p + )e − (p − ) is the coherent sum of the photon and X-boson mediated amplitudes M(H * → He + e − ) = M γ + M X , where (V = γ, X): where µ =ū(p − )γ µ v(p + ) is the leptonic current and G H * Hγ (q 2 ) = −F H * Hγ (q 2 )/q 2 , G H * HX (q 2 ) = ε e F H * HX (q 2 )/(q 2 − m 2 X + im X Γ X ). As in Ref. [5,6], we assume negligible decays of the X17 boson into neutrino channels, such that its full width is given by Γ X ≡ Γ(X → e + e − ) = (α em ε 2 e m X /3)(1 + 2r e ) √ 1 − 4r e = 8.0 × 10 −8 MeV (r e = m 2 e /m 2 X ), corresponding to the maximum value for ε e . Given this very narrow width of X17, the lepton pair invariant mass distribution, normalized to the radiative decay width, becomes the sum of the non-interfering photon and X-boson mediated distributions, namely (α em is the fine structure constant, and λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz): Table III displays the decay rates of the lepton-pair production in H * → H transitions for different heavy mesons. These rates are normalized to the radiative width Γ(H * → Hγ) = (α em /3)|F H * Hγ (0)| 2 | p γ | 3 in order to cancel remaining model-dependent terms in the form factors (all other lepton-pair and angular distributions in the following are normalized to this radiative width). The largest contribution of the X17 boson is observed for the D * + and D * + s , making these channels the most sensitive for the observation of this light boson effects. Our prediction for the D s channel normalized to the radiative decay is B(D * s → D s e + e − ) = (7.8 ± 0.6) × 10 −3 , in good agreeement with the corresponding ratio measured by the CLEO collaboration (7.2 +1.8 −1.6 ) × 10 −3 [16]. A previous estimate of this ratio B(D * s → D s e + e − ) = 6.5 × 10 −3 was estimated in Ref. [16] based on the model proposed in [17] which includes only the electromagnetic contribution.
The lepton-pair invariant mass distributions due to photon (solid-red) and X17-boson (dashed-blue) exchange are shown separately in Figure 1 for the six different decay channels under consideration. The shaded bands around each curve represents the theoretical error. The peak due to the production of the X17 boson in each channel is not located very close to the end of the lepton-pair spectrum as it happens in the nuclear case, avoiding in this way possible end-point kinematical effects. In contradistinction to the on-shell X17 production, the effect of this boson is noticeably only for the D * + (D * s ) → D + (D s )e + e − decay. The corresponding peaks of this boson contribution is suppressed by one or two orders of magnitude in all other cases, relative to the photon contribution. Note that we are assuming universality bounds on heavier quark ε c,s,b couplings; since this is a conservative assumption, the experimental study of heavy mesons transitions involving lepton pairs may serve to set bounds on these unknown couplings of the hypothetical X17 boson.
Finally, let us comment that the angular distribution of the e + e − pair, in the rest frame of the decaying particle, will be peaked closer to the collinear configuration compared to the nuclear case of 8 Be * transitions, where θ(e + e − ) ∼ 140 0 . This happens because the X17 boson is produced with a larger velocity, while in nuclear transitions this boson is produced almost at rest.
4. Conclusions. The hypothetical light vector boson X17, proposed as a solution for the anomaly observed in lepton-pair production of 8 Be * and 4 He transitions, can be studied in the clean environment provided by vector to pseudoscalar heavy mesons transitions in Belle, Belle II and BESIII factories. These H * (Qq) → H(Qq)e + e − decays are free from theoretical uncertainties associated to nuclear isospin mixing. We have used the HQET+VMD dominate to model the hadronic form factors of 1 − → 0 − meson transitions, although our results are little-dependent on the model's uncertainties because the rates are normalized to the dominant H * → Hγ electromagnetic decays. Although all branching fractions of the considered heavy meson channels are sensitive to the effects of the X17 boson, decays of D * + and D * s mesons turn out to be the most sensitive ones. Improved measurements of these leptonic decay channels can set additional powerful constraints on the X17 boson couplings to ordinary fermions or, eventually, confirm the existence of this light boson.