Probing new physics scenarios of muon $g-2$ via $J/\psi$ decay at BESIII

The disagreement between the standard model prediction and the experimental measurement of muon anomalous magnetic moment can be alleviated by invoking an additional particle which is either a vector boson ($X_1$) or a scalar ($X_0$). This new particle, with the mass $m_X \lesssim 2 m_\mu$, can be searched for in the decay $J/\psi \to \mu^- \mu^+ X$, where $X$ is missing. Our numerical study shows that the search is quite feasible at the BESIII experiment in the parameter space allowed by muon $g-2$ measurements.


Introduction
The magnetic moment of muon (µ) is directly proportional to its intrinsic spin (S), µ = g µ ( e /2m µ )S, where g µ , e, m µ are the g-factor, elementary charge and mass of muon, respectively. The Dirac equation predicts g µ = 2 since muon is an elementary (i.e. structureless) spin-1 /2 fermion. However, radiative corrections from quantum loops are known to result in a tiny but non-zero deviation from this value. This deviation is quantified by the anomalous magnetic moment, a µ = g µ − 2 /2. The anomalous magnetic moment of muon has been very precisely measured by the E821 experiment at Brookhaven National Laboratory (BNL) [1]. The experimental measurement is found to be about 3.3σ larger than the Standard Model (SM) prediction [2], where the first error is from experiment and the second one is from theory prediction. This result, as well as the recent observation that inclusion of SM radiative corrections is not sufficient to resolve the anomaly in a µ [3], can be considered as possible hints of some underlying new physics. Further, the contribution of the leading order hadronic vacuum polarization to the muon a µ , extracted with high accuracy from the measurements of e + e − → hadrons [4], also cannot reduce the present discrepancy ∆a µ . In this paper, we shall probe two simple new physics scenarios, involving either a new vector boson (say X 1 ) or a new scalar (say X 0 ), which can contribute to muon anomalous magnetic moment and alleviate the existing discrepancy between theory prediction and experimental measurement. Our proposed search for the new particle is via the study of the decay J/ψ → µ − µ + X 0,1 , in the range m X 2m µ , with X 0,1 as the invisible or missing final state. This decay mode can be studied in the ongoing BESIII experiment or in any future experiment which can produce large number of on-shell J/ψ mesons at rest.
The experiment BESIII has clear advantages for the studies of production of such X particles in comparison with the experiments Belle II and BaBar where the continuum e + e − → µ + µ − X process can be considered [5]. First, at BESIII a very large number (∼ 10 11 ) of on-shell J/ψ particles (at rest) will be produced [6][7][8], and consequently the number of J/ψ → µ + µ − X events that can be produced is significantly higher than that at Belle II or BaBar. This is due to the fact that the continuum process e + e − → µ + µ − X at Belle II or BaBar does not take place via on-shell exchanged particle, making the number of such events considerably suppressed. Secondly, the center-of-mass energy √ s of the events at Belle II and BaBar is very high (≈ 10 GeV), making the cross section of the considered process suppressed, σ ∝ 1/s. In addition, since BESIII has the final state kinematics strongly constrained by the on-shell J/ψ (which has a very small decay width) without any initial soft photon radiation γ ISR , the background effects are much easier to analyze than at Belle II or BaBar.
We would like to point out that our approach to study the bosonic mediators X 0,1 differs from another recent proposal [9], which studied J/ψ decay at BESIII through the process of J/ψ → X 0,1 + γ → µ − µ + γ in the range m J/ψ m X 2m µ : (1) Unlike our paper, Ref. [9] does not consider the muon anomalous magnetic moment to probe and constrain the parameter space for X 0,1 . (2) In Ref. [9] the mediators X 0,1 can be scalar, pseudo-scalar, vector or axial-vector; while in our case, in order to explain the observed muon anomalous magnetic moment, we have constrained ourselves to the scalar and vector possibilities only, thus making the scenario much simpler. Probing such light scalar and vector particles have also been discussed in context of other decay modes in Refs. [10][11][12] in context of specific U(1) extensions of the SM.
Our paper is organized as follows. In Sec. 2 we elaborate on the two new physics scenarios under our consideration and clearly lay down the search strategy using the decay J/ψ → µ − µ + X 0,1 . This is followed in Sec. 3 by a numerical study of the two scenarios as well as that of the competing SM background processes, in context of the BESIII experiment. Finally we conclude in Sec. 4 emphasizing the various salient features of our study.
2 Theoretical motivation and experimental search strategy

Simplest muonic interactions
For an effective solution to the problem of muon anomalous magnetic moment without affecting any other existing studies, it would be ideal if the new interactions that get introduced only involve muons. In this context it is well known that if there exists either a scalar X 0 or a vector X 1 that interacts only with the muons, we can write down the following interaction Lagrangians: These interactions give contributions to the muon anomalous magnetic moment. The leading order Figure 1. Contribution from X 0 or X 1 to muon anomalous magnetic moment.
contributions, from the loops as shown in Fig. 1, are given by [13], where m X is used to denote the mass of both X 0 and X 1 and these results are applicable for m X 2m µ . The region of parameter space in g 0,1 -m X planes allowed by the current discrepancy in anomalous magnetic moment (at 2σ level, adding the errors in Eq. (1.1) by quadrature) is shown in Fig. 2. It must be noted that the condition m X < 2m µ is imposed to kinematically forbid the only possible tree-level decay X 0,1 → µ − µ + . Other decay modes, such as X 0,1 → e − e + , ν ν for = e, µ, τ are not allowed at the tree level, but in principle these are possible via loop processes which are suppressed if not forbidden kinematically. ∆a µ (2σ allowed) Figure 2. The parameters g 0 , g 1 and m X 2m µ , as allowed by ∆a µ at 2σ level. Obtained by adding the errors in Eq. (1.1) in quadrature and by using Eq. (2.2). It is important to note here that X 0,1 are assumed to exclusively interact with muons alone as per Eq. (2.1).
It is important to note that we have considered in Eq. (2.1) only parity even scenarios here, i.e. no pseudo-scalar or axial-vector possibilities are being considered. This is so because of the fact that the contribution of pseudo-scalar and axial-vector particles to the anomalous magnetic moment of muon has opposite sign which makes the discrepancy between theory prediction and experimental measurement much larger. Therefore, we shall limit ourselves to scalar and vector cases only. It is also interesting to note that the scalar and vector scenarios may have their origin in a more complete model. There are many UV complete models which can accommodate Eqs. (2.1), but they have additional features which might constrain the model severely. In the next subsection we shall consider one of the simplest probable models elucidating the main ideas behind such an approach.
2.2 Towards a possibly complete model: Here we are concerned with a specific extension of the SM gauge group, namely by an additional symmetry group U(1) L µ −L τ which conserves the difference between the muon and tau lepton numbers, while keeping the overall model anomaly free [14][15][16]. The new gauge symmetry, under which only the second and third generations of leptons are charged, gives rise to an additional massive vector gauge boson, X 1 , which naturally couples to second and third generations of leptons alone at the tree level. The mass of the X 1 boson, m X , can be generated via either spontaneous symmetry breaking or Stueckelberg mechanism [17,18]. The underlying Lagrangian including the kinetic term, mass term and gauge interaction term, for the gauge boson X 1 is therefore given by, is the field strength tensor, and J α µ−τ is the µ − τ current given by, where P L ≡ 1 2 1 − γ 5 is the left projection operator. The first term in J α µ−τ in Eq. (2.4) is same as the term in Eq. (2.1b). However, in the U(1) L µ −L τ extension of the Standard Model, there appear additional terms which would contribute to the anomalous magnetic moment of tau via a loop diagram similar to the one in Fig. 1. However, currently the anomalous magnetic moment of tau is not well measured to constrain these new physics scenarios [2]. Therefore, we shall refrain from using anomalous magnetic moment of tau in this paper. Nevertheless, in the U(1) L µ −L τ extension, the X 1 vector boson is not necessarily stable even for m X < 2m µ as X 1 → ν µ ν µ , ν τ ν τ are allowed at the tree level. The decay X 1 → e − e + is still forbidden at the tree level and can only happen (if kinematically allowed) via quantum loop and it would therefore be suppressed. Thus, predominantly the X 1 boson would decay invisibly and its direct signature at experiments would be missing 4-momentum. Further, X 1 could also couple to dark matter constituting yet another invisible decay mode.
Considering only the U(1) L µ −L τ new physics and depending on whether the mass m X is small or large, there exist other diverse observations which can also probe or constrain the allowed region in the g 1 -m X plane, such as (1) precision measurements at Z pole [5], (2) neutrino-nucleus scattering involved in neutrino trident production [19,20], (3) rare kaon decays in beam-dump experiments [21], (4) tests of lepton universality such as R(K), R(K * ) [22][23][24] etc., as well as (5) big-bang nucleosynthesis with constraint on deviation from effective number of light neutrinos (∆N eff ) [25,26].
CCFR ν-trident production (95% C.L. exclusion) ∆a µ (2σ allowed) Figure 3. Allowed region in the g 1 -m X plane from anomalous magnetic moment of muon. Here we have combined the experimental and theoretical uncertainties in quadrature.
In the mass range of our interest, i.e. m X < 2m µ , the constraints from neutrino trident production as measured by the CCFR experiment [19] as well as the constraint from big-bang nucleosynthesis [25] are relevant and are shown in Fig. 3. It is very clear from Fig. 3 that the exclusion region from neutrino trident production experiment strongly constrains a portion of the parameter space allowed by muon anomalous magnetic moment in the higher mass ranges. In order to probe the allowed parameter space in a more thorough manner, our chosen process must not only have high yield, but should have a distinct experimental signature in the parameter region of interest. In the next subsection, we analyze a decay mode which satisfies these criteria.

New search strategy
Since we are concerned with probing the scalar X 0 and vector boson X 1 (which can be called "muon-philic") as they satisfy the Eq. (2.1), it is only natural to think of a process that involves muons in the final state to search for X 0,1 . Moreover, as we have discussed above, X 0,1 with mass m X < 2m µ would be fully invisible as it is electrically neutral and stable (if it decays, then it decays to neutrino-antineutrino pair and possibly to dark matter particles, which are also invisible). Thus, the process we consider must have missing 4-momentum in the final state, and it should be possible, in principle and practice, to measure the missing 4-momentum as precisely as experimentally possible. An excellent process that satisfies all these criteria is the decay J/ψ → µ − µ + X 0,1 , where J/ψ needs to be produced at rest so that the initial 4-momentum is fully known and fixed. Because the final state has two muons which are well reconstructed in modern detectors, this would imply that the missing 4-momentum can be precisely inferred in such a case.
It is important to note that (1) extremely large sample of on-shell J/ψ can be produced in e + e − colliders such as BESIII, which provides statistically significant number of signal events, (2) the extremely narrow width of J/ψ ensures that events with the missing initial soft photon radi- ation γ ISR from the colliding electron-positron beams can be safely ignored (unlike the continuum process of e − e + → µ − µ + X 0,1 , where the initial state radiation of soft photons would be a major background) and (3) the missing final state soft photon radiation from the muons (as shown in Fig.  4(c)), which constitutes the dominant background for our decay J/ψ → µ − µ + X 0,1 , can also be dealt with very precisely due to the fact that J/ψ → µ − µ + is very well studied and the missing mass in J/ψ → µ − µ + γ soft events always peaks at the photon pole, i.e. at missing mass equal to zero. The continuum process e − e + → µ − µ + X 0,1 , which can be studied at experiments such as Belle II has a much larger set of background processes and the strategy to be dealt with such a study can be found in Ref. [5]. The quark-level Feynman diagrams for the signal and background processes are shown in Fig. 4. It should be noted that, except the final state soft radiation (shown in Fig. 4(c)), the other backgrounds (shown in Fig. 4(d,e,f)) are extremely suppressed as they involve two or more weak vertices. Quantitatively, the soft photon background dominates over the weak background by roughly eight orders of magnitude. Therefore, we shall not dwell upon any of the weak background processes, shown in Fig. 4, in our numerical studies. Now, considering the signal events alone in the context of the BESIII experiment where it is estimated that 10 11 number of J/ψ would be produced, we find that roughly 300-2000 events of J/ψ → µ − µ + X 1 or about 30-300 events of J/ψ → µ − µ + X 0 can be expected (see Fig. 5), corresponding to the region of parameter space allowed by the muon anomalous magnetic moment. Due to this large number of events expected, if we observe fewer or no events, then this would rule out the simplistic scalar and vector explanation of muon anomalous magnetic moment which we considered here. In order to fully understand the feasibility of the decay J/ψ → µ − µ + X 0,1 , we made a comparative study of the dominant SM background and the new physics events in the following Section.

Numerical study of feasibility of probing the new physics scenarios
As we have mentioned in the previous Section, the final state radiation of soft photon from either of the muon lines is the dominant as well as the only relevant background in our case. We have devised a non-traditional approach to study the dominant background process J/ψ → µ − µ + γ soft .
Since the soft photon is not detected, the observed events are essentially J/ψ → µ − µ + events as the soft-photon has low energy (BESIII cannot detect soft photons with energy < 20 MeV [27]) and hence it also has a small magnitude of 3-momentum, essentially keeping the two final muons backto-back, within the accuracy of the experimental resolution of the muon tracks (provided J/ψ is produced at rest, which is true for the BESIII experiment). Therefore, if we take the muon momentum resolution of the experiment into account for the muon pair in J/ψ → µ − µ + , we can essentially get all possible soft-photon events as required for our numerical study. At BESIII the error in the momentum resolution is about 1% of the momentum being measured [27], i.e. the experimental uncertainty in the measurement σ p ∼ 0.01p where p is the central value of the magnitude of the 4-momentum. For simplicity, we have specifically assumed σ p ∼ 15 MeV for our numerical simulation, which would presumably provide bigger momentum uncertainty for most events than what is expected experimentally. We utilize a multitude of observables and relevant cuts, as discussed below, to distinguish the signal and background events so as to facilitate the discovery of new physics in our decay mode J/ψ → µ − µ + X 0,1 . For our numerical study we have considered 10 11 number of J/ψ that would be produced at rest at the BESIII experiment. In the numerical simulation of signal events we have also considered the central value of g 0,1 for the corresponding value of m X as allowed by the muon anomalous magnetic moment, see Fig. 5. Below we first illustrate our methodology for the vector boson case and finally discuss the scalar case and compare.

Square of the missing mass
If we denote the 4-momenta of J/ψ, µ − , µ + and the missing component (which can be the new particles X 0,1 or soft photon γ soft ) by p J , p − , p + and p miss , then the missing mass m miss is given by, Theoretically m 2 miss distribution for signal events will be characterized by very sharp peaks due to the tiny decay width of X 0,1 , e.g. in the U(1) L µ −L τ model the X 1 → ν µ ν µ , ν τ ν τ decays provide Γ X 1 = g 2 1 m X /(12π), and the background events should be crowded at m 2 miss = 0. However errors in measurements of momenta will smear the distribution of events for both signal and background.
The results from our simulation for both signal and background events are shown in Fig. 6. We have shown m 2 miss distribution for three different signal cases corresponding to m X = 50 MeV, 100 MeV and 200 MeV. It is easy to observe a corresponding shift in the position of the signal peaks when we go from smaller mass to the larger mass. The background is large in the neighborhood of m 2 miss = 0, as expected. It can be seen from Fig. 6 that the background events dominate up to around m 2 miss ∼ (120 MeV) 2 . This especially kills signal events with m 2 miss < (120 MeV) 2 and makes it difficult to accurately identify m X , the mass of X 1 . In order to identify m X , we need to use the missing energy information and apply energy cut as discussed below.

Missing energy
The missing energy E miss is defined as Events / (50 MeV) 2 Figure 6. Distribution of missing mass square for both signal events J/ψ → µ − µ + X 1 and background events J/ψ → µ − µ + γ soft . (Here the background is composed of not only J/ψ → µ − µ + γ soft but also experimentally smeared J/ψ → µ − µ + .) The central value of g 1 that solves ∆a µ for the corresponding value of m X (central value of Fig. 2) is used for the demonstration. Shifts in g 1 will only scale the distribution accordingly. The bin size of 50 MeV for missing mass does not imply that mass of X 1 can not be probed below 50 MeV.  Figure 7. Distribution of missing energy for the signal events J/ψ → µ − µ + X 1 and the background events J/ψ → µ − µ + γ soft . Here the bin size for missing energy is 50 MeV. All the background events are confined to the region E miss < 140 MeV. Thus for signal events with m X 50 MeV the missing energy peaks cannot be observed after imposing the E miss < 140 MeV cut.
where M J/ψ is the mass of J/ψ, E ± denote the energies of µ ± . The E miss distribution of a simulated background as well as few benchmark signal cases (corresponding to m X = 50 MeV, 100 MeV and  Table 1. Canonical branching fractions and the number of signal events before and after applying the the missing energy cut, E miss = 140 MeV. The central value of g 1 that solves ∆a µ for the corresponding value of m X (central value of Fig. 2) is used for computing the number of signal events.
200 MeV) are shown in Fig. 7. The figure shows that the energy cut around 140 MeV will completely eliminate the background events (as the missing soft-photon would have energy < 20 MeV at BESIII), while leaving most of signal events.
In Table 1 we list the canonical branching ratio, Br ≡ Br/g 2 1 , and the number of signal events before and after applying the missing energy cut by which we throw away all those events with E miss < 140 MeV. Only the number of signal events with lighter m X (around 50 MeV) is significantly reduced. After applying the missing energy cut the m 2 miss distributions gets modified as is shown in Fig. 8. The figure shows clearly that no background events survive after this cut is imposed. Now the new gauge boson mass can be extracted from the resultant distribution, shown in Fig. 8. Please note that although the demonstrations were made with the specific values of g 1 , which solve the muon anomalous magnetic moment for corresponding m X , the strategy is generic and changes in g 1 will end up with nothing more than overall scaling of the number of events (in Fig. 6, 7, 8, 9 and Table 1), while leaving the overall shapes of distributions unchanged.

Mass shift of muon pair from J/ψ
Alternatively one can utilize another observable to probe X 1 , namely the deviation in the measurement of invariant mass of observed muon pair from mass of J/ψ, If this value significantly deviates from zero for a distribution of events, those events would qualify as signal events. In the case of background event J/ψ → µ − µ + γ soft , ∆(M J/ψ ) indicates difference between the actual and observed mass of J/ψ. So it will be peaked at zero with some smearing due to error in momentum measurement. Taking the energy and momentum resolution at BESIII experiment, the standard deviation of ∆(M J/ψ ) distribution comes around 11 MeV and so there will be some events up to around 70 MeV. This is shown in the Fig. 9.  It is easy to show that where E X is the energy of X 1 and it is the same as E miss , the latter was analyzed in Sec. 3.1.2 and Fig. 7 for a few benchmark signal scenarios. The minimum value of measured ∆(M J/ψ ) is equal to m X when we substitute the minimum value of E X = m X in Eq. (3.4). Thus the mass of X 1 can, in principle, be inferred from Fig. 9 by reading the minimum value of ∆ M J/ψ for the corresponding distribution. However, this information is also subject to the smearing effect from momentum resolution. Nevertheless, looking at the ∆(M J/ψ ) distribution would complement our search for new physics using previously discussed observables. An important feature of the ∆(M J/ψ ) distribution is that it runs up to much larger values for a signal compared to the background, even after smearing is taken into account. Especially, if the mass of X 1 is larger than 70 MeV, the signal can be easily distinguished from the background. Comparing the canonical branching ratios of the scalar and vector cases, we find that in the scalar case they are about 10 times smaller than in the vector case, see Table 2. Nevertheless, applying the same techniques as discussed above for the vector case, we can also probe the scalar new physics possibility. It is important to note that the missing energy distribution of a scalar is different from that of a vector case. In Fig. 10 we plotted the canonical differential decay rates dΓ 0,1 /dE miss (where Γ 0,1 ≡ Γ/g 2 0,1 is the canonical decay width) for the scalar and vector cases, respectively. It is clear that scalar new physics prefers higher missing energy whereas vector new physics prefers lower missing energy. Thus, one can easily distinguish them in an experiment once either of them gets detected.

Conclusion
We investigated the possible J/ψ → µ + µ − X events at the BESIII experiment, where X is a vector (or scalar) "muon-philic" particle which could explain the present discrepancy ∆a µ ≡ a exp µ − a SM µ of the anomalous magnetic moment of µ lepton, cf. Eq. (1.1). It turns out that, if the coupling of X to muon is g 0,1 ∼ 4 × 10 −4 -10 −3 and its mass is m X < 2m µ , this particle can explain the mentioned discrepancy ∆a µ .
The advantage of BESIII, in comparison with other experiments (Belle II and BaBar) where the continuum process e + e − → µ + µ − X is considered, is that at BESIII there will be produced a very large number (∼ 10 11 ) of on-shell J/ψ particles (at rest) without any initial soft photon radiation. Thus, the number of J/ψ → µ + µ − X events can be significantly higher than at Belle II or BaBar, because the latter experiments do not have on-shell intermediaries for the continuum process e + e − → µ + µ − X. Further, the final state kinematics is more constrained at BESIII because of the mentioned on-shellness and the very small decay width of J/ψ, making the background effects smaller and easier to analyze than at Belle II or BaBar. In addition, the center-of-mass energy at Belle II and BaBar is very high, which makes the cross sections of e + e − → µ + µ − X suppressed (σ ∝ 1/s).
We showed that the number of events J/ψ → µ + µ − X that take place at BESIII in the mentioned range of parameters (g 0,1 , m X ) is ∼ 10 3 when X is vector, and ∼ 10 2 when X is scalar. The main background to these events at BESIII is the final state radiation J/ψ → µ + µ − γ. This is in contrast with Belle II [5] (and BaBar) background to e + e − → µ + µ − X, where initial state radiation and e + e − → τ + τ − → µ + ν µντ µ −ν µ ν τ are additional background sources. We showed that the distribution m 2 miss , Eq. (3.1), is a priori not a good quantity to identify the signal events J/ψ → µ + µ − X, because of strong background (J/ψ → µ + µ − γ) contributions to this quantity. On the other hand, the distribution dN J/ψ→µµX /dE miss , where E miss = M J/ψ − E + − E − [cf. Eq. (3.2)], is a good quantity to identify the signal events when m X > 50 MeV (and m X < 2m µ ) once the cut E miss < 140 MeV is applied which eliminates the background; for m X < 50 MeV, the signal rate gets significantly diminished by the cut and the maximum is swamped by the background. The mentioned cut E miss < 140 MeV also eliminates completely the background to the quantity m 2 miss . Further, we showed that the quantity ∆(M J/ψ ), defined in Eq. (3.3), is a good complementary quantity to identify the signal events if m X > 70 MeV. If X is a scalar, the number of events in the mentioned parameter range of (g 0,1 , m X ) is by about a factor of 10 lower, as mentioned earlier. However, the form of the distribution dN J/ψ→µµX /dE miss is in this case shifted to higher values of E miss , allowing the scalar case to be easily distinguished from the vector case.
In summary, in this paper we demonstrated that it is possible to not only probe both the scalar and vector new physics cases contributing to anomalous magnetic moment of muon by searching for the signal J/ψ → µ − µ + + "missing," but also to distinguish between the SM background and the new physics possibilities by using missing mass, missing energy and mass shift of muon pair from J/ψ. Our numerical analysis clearly shows that BESIII could be the best place, at present, to implement this study experimentally. The amazing aspect of this probe is the possibility to either discover new physics or to completely rule out the simplest explanations for the longstanding muon anomalous magnetic moment discrepancy.