Polarization effects in the search for dark vector boson at $e^+e^-$ colliders

We argue that the search for dark vector boson through $e^+e^-\to Z_d\gamma$ can determine the Lorentz structure of $Z_dl^+l^-$ couplings with the detection of leptonic decays $Z_d\to l^+l^-$. We assume a general framework that the dark vector boson interacts with ordinary fermions through vector and axial-vector couplings. As a consequence of Ward-Takahashi identity, $Z_d$ is transversely polarized in the limit $m_{Z_d}\ll \sqrt{s}$. On the other hand, the fraction of longitudinal $Z_d$ is non-negligible for $m_{Z_d}$ comparable to $\sqrt{s}$. Such polarization effects can be analyzed through angular distributions of final-state particles in $Z_d$ decays. Taking $l^{\pm}\equiv \mu^{\pm}$, we study the correlation between $Z_d$ angle relative to $e^-$ beam direction in $e^+e^-$ CM frame and $\mu^-$ angle relative to the boost direction of $Z_d$ in $Z_d$ rest frame. This correlation is shown to be useful for probing the Lorentz structure of $Z_dl^+l^-$ couplings. We discuss the measurement of such correlation in Belle II detector, taking into account the detector acceptance and energy resolution.


I. INTRODUCTION
Searching for dark matter (DM) is one of the major endeavors in the present day particle physics community. The efforts in direct and indirect detections as well as productions of DMs in LHC so far have not produced positive results. Recently there are growing interests to search for DM related phenomena with huge statistics and high precision measurements.
These phenomena involve the hidden sector [1][2][3][4][5][6], which is assumed to interact with Standard Model (SM) particles through certain messengers. A popular proposal for such a messenger particle is the so called dark photon, which mixes with the U (1) hypercharge field B µ in SM, where A µ is the dark photon field, and A µν ≡ ∂ µ A ν − ∂ ν A µ . The above mixing induces electromagnetic couplings, L em = ε γ eJ µ em A µ , between the dark photon and SM fermions, which generate rich phenomenology [7]. On the other hand, the neutral current couplings between the same set of particles are further suppressed by the factor m 2 A /m 2 Z for m A m Z with m A the dark photon mass. However, independent neutral current couplings can be generated through mass mixing between the messenger particle and the Z boson [8][9][10]. In this case, the messenger particle is often referred to as Z boson. The mass mixing term δm 2 Z µ Z µ can induce neutral current couplings L NC = (gε Z / cos θ w )J µ NC Z µ with ε Z ≡ δm 2 /m 2 Z . For a general scenario that both kinetic and mass mixings are present, the interactions between dark boson and SM fermions are given by with Z d the dark boson, which is the generalization of A and Z .
The search for the light vector boson with the reaction e + e − → Z d γ has been proposed before [11][12][13]. Particularly, there exist phenomenological studies on dark sectors under the environment of e + e − colliders [14][15][16][17][18][19][20][21][22]. Along this line, the experimental searches for Z d proceed through the detections of visible and invisible Z d decays. The visible mode requires a full reconstruction of Z d peak through measuring the energy and momentum of lepton or light hadron pairs from Z d decays [23][24][25][26][27][28], while the invisible mode looks for the peak of missing mass at m Z d given by M 2 mass = (P e − + P e + − P γ ) 2 [29]. We note that both phenomenological and experimental studies mentioned above consider only the dark photon scenario, i.e., Z d interacts with SM fermions only via electromagnetic current. On the other hand, since neutral-current coupling is also possible, it is of great importance to simultaneously detect Z d and measure the Lorentz structure of its coupling to SM fermions.
To determine the relative strengths of vector and axial-vector couplings, such as the ratio it is necessary to measure the angular distributions of final-state fermions from Z d decays 1 .
The dark vector boson Z d produced by e + e − → Z d γ is polarized. In fact, Z d must be in one of the transversely polarized states in the limit √ s m Z d . This is a direct consequence of Ward-Takahashi identity [30] to be elaborated in the next session. Furthermore, with the presence of both g f,A and g f,V , parity symmetry is broken. Hence there exists a forward-backward asymmetry for the production of each transversely polarized Z d state, while the production of longitudinal Z d is forward-backward symmetric. The magnitude of the above asymmetry is directly related to the degree of parity violation, characterized by the parameter ρ ≡ 4g f,A g f,V under the normalization g 2 f,V + g 2 f,A = 1. For a fixed ρ, the asymmetry reaches to the maximum for m Z d / √ s → 0. Besides the asymmetry in the production of transversely polarized Z d state, there is also forward-backward asymmetry for the angular distributions of final-state fermions from Z d decays, which is also controlled by the same parameter ρ. Hence the correlation between these two asymmetries can be exploited to probe ρ.
The most sensitive search for Z d through the visible mode e + e − → Z d γ → e + e − γ, µ + µ − γ is performed by BaBar [24]. Using 514 fb −1 of data, the upper limits on the mixing parameter ε is 10 −4 − 10 −3 for m Z d between 0.02 GeV and 10.2 GeV. Comparable sensitivity to ε is expected at Belle II [31][32][33][34] with 500 fb −1 of integrated luminosity. Belle II is an electronpositron collider experiment running at the SuperKEKB accelerator. It is a next-generation B-factory experiment aiming to record a dataset of 50 ab −1 . In this article we focus on the prospect of detecting Z d and measuring the parity violation parameter in its interaction with SM fermions with e + e − → Z d γ followed by Z d → µ + µ − decay at Belle II. Backgrounds to this process are QED process e + e − → µ + µ − γ 2 and the resonant production process e + e − → γX [X = J/ψ, ψ(2S), Υ(1S), Υ(2S)] followed by X → µ + µ − . We will not consider the decay mode Z d → e + e − in this article since backgrounds to this mode are more complicated, including e + e − → e + e − (γ) and e + e − → γγ(γ). Since we are mainly interested in probing 1 Here we choose the normalization g 2 f,V + g 2 f,A = 1. 2 Here we neglect the Z boson exchange diagrams since their entire contributions to the total cross section is less than 1% from our numerical studies. the parity violation effect by Z d ff coupling, the study on Z d → µ + µ − is sufficient to make our point. We note that there are recent interests in the signals for 17 MeV protophobic vector boson [35][36][37] motivated from anomalies in 8 Be and 4 He nuclear transitions [38,39].
Searching for vector boson in e + e − colliders for this particular parameter range has been proposed [40,41]. Although we shall not focus on such a specific scenario, we do notice that the protophobic vector boson interacts with the electron through both vector and axial-vector couplings. However parity violation effects resulting from the presence of both couplings were not considered in the above analyses. For an earlier candidate of light neutral gauge boson [42], its parity violating effects to low energy neutral current processes were studied quite some time ago [43]. Among those low energy processes, the search for atomic parity violations [44,45] still attracts high attentions in recent years. For phenomenological discussions on this issue under the current dark boson scenario, see Refs. [9,46]. This article is organized as follows. In Section II, we present the polarized differential show that the production of longitudinal Z d is suppressed due to Ward-Takahashi identity, i.e., Z d is transversely polarized in such a limit. In Section III, we discuss the method for probing the parity violation parameter ρ in e + e − colliders. We present angular distributions of leptons arising from polarized Z d decays. Combining with angular distributions of Z d in production process, we construct the double angular distribution for the signal process e + e − → γZ d → γl + l − . It will be shown that this double angular distribution depends on ρ 2 rather than ρ. We bin the signal events according to the sign of J ≡ cos θ × cos θ d where θ is the angle of Z d with respect to the e − direction in e + e − CM frame while θ d is the helicity angle of lepton arising from Z d decay. The asymmetry A PN ≡ (S(J > 0) − S(J < 0))/(S(J > 0) + S(J < 0)) with S the number of signal events will be shown to be directly proportional to ρ 2 , and will be important for the numerical studies in the next section. In Section IV, event numbers of e + e − → γµ + µ − from signal and background are calculated with specific integrated luminosity in Belle II detector, taking into account the detector acceptance and energy resolutions. We also calculate the asymmetry parameter A PN which depends on the detector acceptance. It will be shown that the simultaneous fitting to J > 0 and J < 0 event bins should improve the significance of dark boson detection from simply counting the total event excess. The degree of improvement is closely related to A PN . In addition, A PN also dictates how well one can distinguish between the dark photon scenario (ρ = 0) and those scenarios with non-vanishing ρ. We summarize and conclude in Section V.
A. Ward-Takahashi identity and the polarization of Z d Let us write the amplitude for e − (p 1 ) Here M µ contains the photon polarization vector.
However, M µ k µ 1 = 0 in the limit m e → 0 as implied by Ward-Takahashi identity [30]. Therefore the amplitude for a longitudinal polarized Z d is of The square of e − (p 1 ) + e + (p 2 ) → Z d (k 1 ) + γ(k 2 ) amplitude for a given Z d polarization can be expressed as follows: whereM is the amplitude with the polarizations of initial fermions and final-state photon summed, α is the fine-structure constant, m Z d and µ are dark boson mass and polarization vector, respectively, s, t, and u are Mandelstam variables. It is clear that those terms proportional to g f,V · g f,A vanish by summing the Z d polarization, In the center of momentum (CM) frame of colliding electrons and positrons, the momenta of initial and final-state particles are given by where ω is the photon energy, Using energy and momentum conservation, we have Let us denote the amplitude for each polarization as M + , M − , and M for right-handed, left-handed, and longitudinal polarized dark boson final state, respectively. We have where the normalization g 2 f,V + g 2 f,A = 1 has been taken and ρ = 4g f,V g f,A . The absolute value of ρ essentially describes the degree of parity violation. It is shown that |M| 2 is suppressed by m 2 Z d /s compared to |M| 2 ± . In Fig. 1, we present the fraction of matrix element square as a function of cos θ for each helicity state of Z d . We take the V − A case with g f,V = −g f,A = 1/ √ 2 for illustration. The upper, middle, and lower panels correspond to m Z d / √ s = 0.1, 0.3, and 0.8, respectively. For the first two cases, one can see that the longitudinal fraction is no more than 10%. In addition, |M| 2 − dominates the forward direction (0 ≤ cos θ ≤ 1) while |M| 2 + dominates the backward direction. For the third case, the longitudinal fraction is non-negligible and the fractions for helicity +1 and −1 states are almost identical due to the suppression of ρ dependent terms, i.e., the forward-backward asymmetry approaches to zero in the limit s → m 2 Z d . The polarized differential cross section is readily calculated with with i = +, −, and . To check our result, we take ε = 1 and sum over contributions from all polarizations. In the limit s m 2 e , m 2 Z d , our result approaches to the differential cross section of e + e − → γγ. The results for polarized differential cross section are shown in Fig. 2.
s is taken to be 0.1, 0.3, and 0.8 on upper, middle, and lower panels, respectively. In this calculation we take ε = 7 × 10 −4 for m Z d / √ s = 0.1, 0.3, and 0.8, respectively, for illustrations. This ε value is reachable by Belle II with 500 fb −1 luminosity for m Z d around 1 GeV [31]. Similar to the case of amplitude square, the longitudinal polarized contribution is suppressed for small m Z d / √ s. We present in Fig. 3 with i = +, −, and , y = 1 − 4m 2 l /m 2 Z d , and θ d the angle between l − direction in the Z d rest frame and the Z d direction in e + e − CM frame. Thus we obtain Given g 2 l,V + g 2 l,A = 1 and ρ = 4g l,V g l,A , we have g 2 l,V = (1 + 1 − ρ 2 /4)/2 for |g l,V | ≥ |g l,A |, while g 2 l,V = (1 − 1 − ρ 2 /4)/2 for |g l,V | ≤ |g l,A |. The double distribution of final-state leptons is given by with κ = cos θ, ξ = cos θ d , Γ l + l − the unpolarized Z d → l + l − decay width, and σ T the total Z d production cross section. We first observe that the double distribution d 2 P/dκdξ only depends on ρ 0 and ρ 2 . Secondly, the ρ 2 dependent term in the double distribution is given by ρ 2 β(1 − m 2 Z d /s) 2 J/(1 − κ 2 ) with J = κξ and β ≡ p l /E l the lepton velocity. The sign of this contribution is determined by the sign of J. This contribution vanishes at the dark boson production threshold, s = m 2 Z d , or at the threshold for Z d decaying into the lepton pair, i.e., β = 0. It also vanishes if either κ or ξ is integrated to the full range.  To analyze the double distribution, we integrate κ from −1 to 0, i.e., we consider leptonic decays of Z d produced in the backward direction. The forward-backward asymmetry of l − in its helicity angle is presented in Fig. 4. In three panels we present the angular distribution of l − for |ρ| = 0, 1, and 2, respectively. The case |ρ| = 0 implies either g l,A = 0 or g l,V = 0 with the former corresponding to the dark-photon scenario, while |ρ| = 2 corresponds to either V − A or V + A cases. On the upper panel, we separate results into |g l,V | ≥ |g l,A | and |g l,V | ≤ |g l,A |. These two cases coincide in the middle and lower panels with β = 1, i.e., m l → 0. The asymmetries in the upper and lower panels are small either due to a small β Significant asymmetry is seen in the middle panel with large |ρ|. In general, we may define the following asymmetry parameter where the subscript PN indicates that A PN describes the difference in signal event rate as κξ reverses its sign. In limits of β → 1 and m Z d √ s, we have where κ m is the maximum of κ. The minimum of κ is assumed to be −κ m . It is found that A PN is not very sensitive to κ m . A PN = 0.64 × (ρ 2 /4) for κ m = 0.95, and 0.55 × (ρ 2 /4) for κ m = 0.80. In practice one cannot measure A PN directly due to contamination from QED background events. However this asymmetry will show its effect statistically as we shall see in the next section.

MEASURING THE PARITY VIOLATION PARAMETER ρ IN BELLE II
In this section, we discuss the search for dark vector boson and the possible measurement of parity violation parameter ρ in Belle II. We shall begin by considering the detector acceptance of Belle II and compare our sensitivity estimation for the dark photon search through e + e − → A γ with A → µ + µ − with the BaBar result at 514 fb −1 and the projected sensitivity of Belle II at 500 fb −1 . Next we consider Belle II at the full integrated luminosity 50 ab −1 and extend our discussions to the dark boson scenario with a non-vanishing ρ.

A. Sensitivity estimation for the dark photon search at Belle II
To illustrate our points in previous sections, we take the dark photon mass as 0.5 GeV and 2 GeV, respectively as benchmark values. These two mass values satisfy m Z d / √ s 1 so that the dark photons are produced in transversely polarized states. Hence for the generalization to dark boson scenario in the next subsection, we shall see that the asymmetry parameter A PN will be significant.
Let us begin by taking m Z d = 0.5 GeV with ρ = 0. In this case, the branching ratio for [14,47]. The Belle II calorimeter angular coverage is 12.
which detects final-state photon in the rapidity range −1.51 ≤ η lab γ ≤ 2.22. Since the boost velocity from the laboratory frame to CM frame is β CM = (E e − − E e + )/(E e − + E e + ) = 3/11, the photon rapidity in the CM frame is given by η CM Furthermore the angular coverage of K L -Muon detector [48] is 25 • ≤ θ lab µ ± ≤ 150 • . This leads to the muon rapidity range −1.60 ≤ η CM µ ± ≤ 1.23 in the CM frame. Since the signal mass resolution is between 1.5 MeV and 8 MeV in BaBar analysis [24], we take it to be 5 MeV for our sensitivity estimation. We note that the rapidity cuts preserve S(κ · ξ > 0) = S(κ · ξ < 0) for ρ = 0. Using CalcHEP [49], we find that the signal e + e − → γZ d with Z d → µ + µ − has the cross section 1.84 · 10 3 · ε 2 pb, and the cross section for QED background process e + e − → γµ + µ − with the same acceptance cut is 7.76 · 10 −2 pb. We note that the above parametrization for signal cross section is valid only for ε < 0.3 such that the ε-dependent Z d width is less than 10% of the signal mass resolution. With 500 fb −1 of integrated luminosity, the Belle II 90% C.L. sensitivity to ε is estimated by the following χ 2 function where n is the observed event number while w is the expected event number. With n = S + B = (1.84 · 10 3 · ε 2 + 7.76 · 10 −2 ) pb · 500 fb −1 , w = B = 7.76 · 10 −2 pb · 500 fb −1 , and χ 2 = (1.645) 2 for 90% C.L. sensitivity 3 , we obtain ε = 5.9 · 10 −4 , which is consistent with the sensitivity ε = 5.6 · 10 −4 given in Belle II physics book for the visible modes Z d → e + e − , µ + µ − [31]. The latter is also comparable to the constraint from BaBar search via visible modes at 514 fb −1 [24].

B. Probing the parity violation parameter ρ in Belle II
To probe ρ, we modify the χ 2 function in Eq. (12) as where n a (w a ) and n b (w b ) are observed (expected) event numbers in κ · ξ > 0 and κ · ξ < 0 bins, respectively. By considering separate event bins, the dark boson detection significance is expected to be improved. With n a,b = S a,b + B a,b and w a,b = B a,b , we can show that with the assumption S a,b B a,b and the identity B a = B b . Hence the detection significance · σ by considering separate event bins.
We note that the χ 2 function in Eq. (13) can also determine the significance level of rejecting ρ = 0 hypothesis. One can fit (n a , n b ) by the dark photon scenario (ρ = 0) with the minimal χ 2 , denoted as χ 2 dp,min , given by w a = w b = (n a + n b )/2. It is easy to show that and the p value corresponding to χ 2 dp,min can be calculated accordingly. We note that the asymmetry parameter A PN as defined by Eq. (10) is actually independent of the integrated luminosity. It can be determined by the scattering cross section with appropriate kinematic cuts imposed. For the general dark boson scenario, σ(κ · ξ > 0) generally differs from σ(κ · ξ < 0). Let us take the special case ε γ = ε Z in Eq. (2). Recasting the coupling of Z d to leptons into the standard form eεl(g l,V γ µ +g l,A γ µ γ 5 )lZ µ d , we have ε = 1.18ε γ , g l,V = −0.87, and g l,A = −0.5, which leads to ρ = 1.74. In the extreme asymmetry case, such as V − A interaction between Z d and SM fermions, we have g l,A = −g l,V = 1/ √ 2, which leads to ρ = −2. This scenario occurs when ε γ = ε Z tan θ W such that ε = 0.83ε Z .
In A PN to Eqs. (14) and (15) to calculate the improved detection significance and χ 2 dp,min . The results of our calculations are summarized in Table I. For |ρ| = 1.74, fitting S a and S b simultaneously increases the detection significance to 5.4σ and 5.5σ for m Z d = 0.5 GeV and 2 GeV, respectively. For |ρ| = 2, the improved detection significance is 5.8σ for both benchmark masses. Finally, the p value associated with χ 2 dp,min determines the statistical significance of rejecting the dark photon scenario. Taking the significance level α = 0.05, one can see from Table I that all benchmark cases can meet such a criterion for rejecting the dark photon scenario.

V. SUMMARY AND CONCLUSION
In this article we have pointed out that the dark boson produced by e + e − → Z d γ is transversely polarized in the limit m Z d √ s. This is a direct consequence of Ward-Takahashi identity. We also demonstrated this property by explicit calculations. The suppressed production of longitudinally-polarized dark boson state is shown in Fig. 1 for the V − A limit, i.e., g l,V = −g l,A = 1/ √ 2. For m Z d √ s, the negative-helicity dark boson dominates the forward region (cos θ > 0) while the positive-helicity one dominates the backward region (cos θ < 0). As m Z d approaches to √ s, the production of longitudinally-polarized dark boson becomes noticeable. Furthermore, the angular distributions of negative-and positive-helicity dark bosons become indistinguishable.
Since we aim for determining the parity violation parameter ρ, we analyze µ − (µ + ) angular distributions from polarized Z d decays. The double distribution of final state muons d 2 P/dκdξ (κ = cos θ, ξ = cos θ d ), defined in Eq. (9), was shown to be sensitive to ρ. Explicitly we found that d 2 P/dκdξ = Q 0 + Q 2 ρ 2 with Q 0 an even function of both κ and ξ and Q 2 an odd function of these variables. This implies that the signal event number in the kinematic range κ · ξ > 0 differs from that with κ · ξ < 0, which motivates our definition of asymmetry parameter A PN proportional to ρ 2 . Besides depending on ρ 2 , A PN also depends on the range for the parameter κ·ξ, which is related to the detector acceptance. We calculate numbers of signature and background events for two benchmark masses m Z d = 0.5 GeV and 2 GeV in Belle II detector. The resulting 90% C.L. sensitivity to ε at 500 fb −1 integrated luminosity is found to be consistent with that in Belle II physics book for the dark photon scenario.
In the general scenario with non-vanishing ρ, we have seen that the detection significance of dark bosons increases by separately considering events with different signs of κ · ξ rather than just counting the overall event excess. The increased χ 2 value is proportional to A 2 PN , as seen from Eq. (14). We also calculate the minimal χ 2 , referred to as χ 2 dp,min , by simultaneously fitting the event numbers in both κ · ξ > 0 and κ · ξ < 0 event bins with dark photon signal plus background events. We found that χ 2 dp,min = S 2 A 2 PN /B. The p value associated with χ 2 dp,min can be calculated and it determines how well one can reject the dark photon scenario, indicating the possibility of parity-violating couplings between the dark boson and SM fermions.
In conclusion, we have shown that the detection of dark boson decays into muon pairs in e + e − colliders can probe the parity-violating couplings between the dark boson and SM fermions. Assuming a 5σ event excess in the search for e + e − → γZ d with Z d → µ + µ − at Belle II, we have seen that the simultaneous fitting to event numbers in positive and negative κ · ξ bins should improve the detection significance to 5.4σ and 5.8σ for input true models with |ρ| = 1.74 and |ρ| = 2.0, respectively. We have also seen that the dark photon scenario can be rejected at the significance level α = 0.05 for both |ρ| values and benchmark masses.