New Signature of low mass Z ′ in J/ψ decays

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to predict possible events at both facilities.Notably, we foresee a substantial enhancement in the precision of the lower limit estimation of α NP as well as a reduction in statistical uncertainty with upcoming STCF experiments.Furthermore, it is essential to highlight that a null result in the measurement of α NP would impose stringent constraints, requiring the Z ′ − q − q couplings to be on the order of 10 −2 .
Searching for such a gauge boson helps us to gain more insights about the fundamental theory beyond the SM.Experimentally, direct searches for the Z ′ boson are conducted in various types of high energy colliders, including e + e − colliders like LEP, and hadron colliders such as Tevatron and LHC.Various mass ranges of Z ′ are scanned, and the couplings of Z ′ with both leptons and quarks are constrained.In the case of leptonic collider searches, the agreement between LEP-II measurements and the SM predictions regarding the cross-section of e + e − → f f implies that either M Z ′ > 209 GeV, or that the Z ′ couplings with leptons are smaller than 10 −2 [15][16][17].Stronger constraints have also been found through the dark photon searches in various experiments [18][19][20][21].The limit on the couplings between Z ′ and leptons is almost around 10 −4 .Besides, some indirect searches through neutrino-electron scatterings are also proposed and severe constraints are given various neutrino experiments [22,23].
These searches have led to the consideration of the leptophobic Z ′ boson, which interacts exclusively with quarks and is extensively searched on hadronic colliders.
Through extensive scanning of the dijet mass spectrum, upper limits on the Z ′ couplings have been established by the CMS collaboration in the mass range from several TeV down to 10 GeV [24][25][26].For Z ′ bosons with masses below 10 GeV, comprehensive explorations on the hadron colliders are limited due to significant background interferences.While some progress has been made through nonstandard quarkonium decays [27], there remains a pressing need for additional strategies to comprehensively investigate this specific low mass range.
In addressing this critical research gap, we propose to conduct the search of the Z ′ boson on lepton colliders, such as Beijing Spectrometer III (BESIII) and the forthcoming Super Tau Charm Factory (STCF), which have a very clean background as well as large volumn of data sample.The BESIII collaboration achieved a significant milestone, accumulating a staggering 10 10 J/ψ events by 2019, with considerable amount of events producing polarized baryon-antibaryon pairs [31].Utilizing the entanglement of final states has enabled the extraction of observables at an unprecedented level of accuracy, offering an excellent platform for probing new physics (NP) phenomena [32,33].Moreover, the future STCF is designed to take 1ab −1 data, corresponding to 3.4 × 10 12 J/ψ events per year [34], promising even higher precision in relevant processes.One of the Z ′ models has been proposed to relieve the tensions in the J/ψ → π + π − and ψ(2S) → π + π − branching fractions with fitted pion form factors [35].In this work, we focus on parity violation in J/ψ → ΛΣ 0 and its charge conjugate.Dominated by a single virtual photon exchange, the nonperturbative effects stemming from gluon exchanges in such decays are comparatively suppressed, allowing for a factorizable amplitude at the first order in theoretical calculations [36,37].Furthermore, the SM prediction for parity violation in J/ψ → ΛΣ 0 + c.c. is vanishingly small, leading to a clean background for the detection of Z ′ .The BESIII collaboration has very recently analyzed CP violation in J/ψ → ΛΣ 0 [38].They have measured the ratio between electric and magnetic form factors with high precision, demonstrating their capability to accurately reconstruct decay distributions.However, their work assumed spatial inverse parity symmetry, a constraint we have relaxed in our research, representing the main contribution of our study.
The parity violating effect is characterized by the polarization asymmetry parameter α NP for the decay of J/ψ → ΛΣ 0 .Experimentally, α NP is available from the angular distribution as follows: where α Λ = 0.748 (7) is the asymmetry parameter in Λ → pπ − [26], and θ p is the angle between ⃗ p Λ and ⃗ p p defined at the rest frames of Λ, respectively.Theoretically, α NP is defined as where h λλ are the helicity amplitudes of J/ψ → ΛΣ 0 with λ and λ the helicities of Λ and Σ 0 , respectively.When parity symmetry holds, we have Additionally, the angular distribution for the charge-conjugate process, namely, , is given by simply substituting (α NP , α Λ , α) for (α NP , α Λ , α) in Eq. ( 1).It is important to note that α Λ denotes the asymmetry parameter for Λ → pπ + , with a measured value of −0.757(4) according to the Particle Data Group (PDG) [26].
Under the assumption that CP symmetry is conserved in the decay of Λ → pπ − , we construct the CP-even and CP-odd observables as α ± = (α ± α)/2.It is worth highlighting that within the SM, both α + and α − remain below the threshold of 10 −3 .
Furthermore, by considering two fold cascade decays, such as the case depicted in Fig. 1, more observables can be extracted.
We adopt the general effective Lagrangian describing the Z ′ boson, as prescribed by the PDG [26].In the context of the J/ψ → ΛΣ 0 decay process, our analysis focuses exclusively on the isovector-axial vector current, denoted as (ūγ µ γ 5 u − dγ µ γ 5 d), and the vector current of cγ µ c.Consequently, the effective Lagrangian tailored for our investigation is as follows: where g A = (g R u − g R d )/4 and g V = (g R u + g L )/2 represent the pertinent coupling constants.The vector current cγ µ c is dictated by the annihilation of J/ψ, and the axial vector currents of u, d quarks are considered to introduce parity violation.Due to the requirement of Hermiticity, g A and g V must be real, thereby ensuring CP conservation and α − = 0. Other terms in the Lagrangian are collectively designated as C, and do not affect the detection of Z ′ .In the presence of such a Z ′ boson, parity violation arises from the interference between amplitudes associated with J/ψ → Z ′ * /γ * → ΛΣ 0 .
These amplitudes are labeled as A Z ′ /γ and, at the first order, they are given as: where is the propagator of Z ′ , and M Z ′ (Γ Z ′ ) corresponds to its mass (decay width).Here, f ψ and M ψ represent the decay constant and mass of J/ψ, respectively, while α em corresponds to the QED fine structure constant.
Incorporating the interference between amplitudes outlined in Eq. ( 4), we arrive at a first-order approximation of the polarization asymmetry α NP as: where the ratios of M 2 Z ′ /M 2 ψ and Γ Z ′ /M Z ′ are written as r and y, respectively.It is crucial to emphasize that F 0 depends only on the ratios of the timelike baryonic form factors, which reduces certain uncertainties.We adopt F 0 = 0.67 from the 3 P 0 model, which aligns well with experimental measurements, as detailed in Ref. [36].
Due to the computation of Γ Z ′ requiring a comprehensive knowledge of the effective Lagrangian, which introduces additional unknown coefficients, and the observation that the dependence of α NP on Γ Z ′ can be safely neglected under the narrow width assumption, we have opted to set y = 0.01 in our subsequent evaluation.
We are now prepared to evaluate the discovery potential of the Z ′ boson, both within the existing BESIII experiment and in anticipation of future experiments at the STCF.The total number of events is provided as N event = N J/ψ × B ΛΣ 0 +c.c × ϵ, where N J/ψ represents the number of produced J/ψ particles, B ΛΣ 0 +c.c is given as 2.83 × 10 −5 and ϵ denotes the detector efficiency concerning the considered final states.For BESIII and STCF experiments, N J/ψ is estimated to be 10 10 and 3.4 × 10 12 , respectively [31,34].The detector efficiencies at the BESIII regarding to ΛΣ 0 and ΛΣ 0 are 17.6% and 21.7% [37], respectively.We take ϵ = 0.2 in the following for the sake of simplicity.
We have also adopted a theoretical value of α NP = 0.02, which is well within the reach by the BESIII experiment and can be easily surpassed by the STCF.Based on the anticipated events N event , along with the specified α NP , we conducted simulations of the angular distribution using the Monte Carlo method.
The simulation data are listed in Tab.I, where we have taken the endpoints of detector to be | cos θ| = 0.9.We plot both the simulation and fitting results in Fig. 2. -0.9 -0.5 0.0 0.5 0.9 We adopt the minimum χ 2 fitting method for the simulation data, where the fit function is given as , tot.err = sys.err 2 + 1/N sim event (6) where the systematic uncertainties in event numbers are assumed to be 5% at BESIII and 1% at STCF, and we take ndf=7 in both cases.The fitted values for (α NP , N fit ) are obtained by ∂χ 2 ∂α NP = 0, ∂χ 2 ∂N fit = 0 (7) and we obtain χ 2 /ndf = 1.18, 2.27 for BESIII and STCF, respectively.The uncertainties are given by the inverse of covariance matrix where a 1,2 represent α NP , N fit respectively, σ 1 is the standard deviation for α NP .
As shown in Fig. 2, the fitted results for BESIII and STCF are α NP = 0.030 ± 0.044 and α NP = 0.027 ± 0.009, where the error is mainly statistical.Take the goodness of fit χ 2 /ndf into consideration, we obtain the significances are 0.8σ and 2.2σ for BESIII and STCF respectively.Furthermore, with the anticipated significant improvements in both statistical and systematic precision at the forthcoming STCF, the prospects for detecting the Z ′ boson become much more promising, even for smaller values of α NP .
Importantly, it should be recognized that such an exploration bears significance even when no significant signal has been found.In such case, stringent constraints on the gauge coupling of Z ′ relative to its mass are established, taking into account the promising precision of α NP measurements at the BESIII and STCF.These constraints are depicted in Fig. 3.
In Fig. 3a, we consider the model-independent scenario, where the exclusion regions are clearly depicted above the solid lines.Our constraints on √ g V g A span the range of 10 −2 ∼ 10 −1 , which surpasses the existing bounds established by the CMS experiment [25].It is worth noting that the mass of the Z ′ boson exerts only a minimal influence on the exclusion curves, with the exception being the vicinity of M J/ψ .For where x can be any rational value.In Fig. 3b, we present the excluded parameter space for various values of x, assuming an upper limit of α NP at 0.02.Our approach is a valuable complement to other research efforts when studying Z ′ bosons with masses below 10 GeV.
In conclusion, we have explored the new possibility of discovering the Z ′ boson with a mass below 10 GeV, a range currently accessible at the BESIII.Our simulations indicate that these signals could be detected at the BESIII, provided that the systematic uncertainty is further reduced.There is also a potential for improved signal detection at the future STCF.If no clear signal emerges, we can still derive useful information by establishing general constraints on the couplings of the Z ′ boson to quarks, typically falling within the range of 10 −2 to 10 −1 , regardless of the Z ′ mass.Our approach offers a competitive and complementary method for hunting down the Z ′ boson with a mass below 10 GeV.Even in less favorable scenarios, it can still make valuable contributions to the constraints on the couplings of the Z ′ boson to quarks in the low mass range.
FIG. 3: Coupling-Mass curves for (a) the general case and (b) a specific Z ′ model.