Searching for new light gauge bosons at $e^+e^-$ colliders

Neutral gauge bosons beyond the Standard Model are becoming interesting as possible mediators to explain several experimental anomalies. They have small masses, below one GeV, and are referred to as dark photons, $U$, $A'$ or $Z'$ bosons. Electron--positron collision experiments at the B-factories provide the most straightforward way to probe bosons of this kind. In the present article we study production of the bosons at $e^+e^-$ colliders operating at GeV center-of-mass energies. We have studied two channels: $e^+e^-\rightarrow \gamma Z'$ and $e^+e^-\rightarrow e^+e^-Z'$. Analytic expressions for the cross sections and various observables such as the energy spectra of the produced bosons and the final electrons from the $Z'$ decays are derived. We have also studied the transverse momentum distribution of the bosons and the spatial distribution of the $Z'\rightarrow e^+e^-$ decay vertices. It is shown that these distributions provide distinct signatures of the bosons in $e^+e^-\rightarrow\gamma Z'$. The reaction $e^+e^-\rightarrow e^+e^-Z'$ becomes important at small $Z'$ scattering angles where its contribution to the overall yield may be larger by orders of magnitude compared to $e^+e^-\rightarrow\gamma Z'$. The standard processes $e^+e^-\rightarrow\gamma\gamma$ and $e^+e^-\rightarrow e^+e^-\gamma$ that lead to the same signal are considered. We include numerical predictions for the production rates at the energy $\sqrt{s}=10.5$ GeV. The case with a light scalar boson is also discussed. The calculations are performed in detail and can be useful for additional studies.


I. INTRODUCTION
The notion of gauge bosons has become an integral part of particle physics long ago. A gauge boson is electrically neutral or charged particle with spin one responsible for transmission of forces in a theory. The well known representatives with precisely established properties are the photon, Z 0 and W ± .
Many extensions of the Standard Model accommodate new gauge bosons. After the electroweak SU(2) × U(1) Y model was proposed, there appeared numerous alternative theories with additional U(1) ′ symmetries leading to associated new neutral Z ′ bosons [1][2][3]. Production of heavy Z ′ s with masses in the TeV region have been studied [4,5] and searched for directly at the LHC in the ATLAS and CMS experiments which put stringent limits on their masses and couplings to the Standard Model particles [6][7][8][9]. These bosons have also been indirectly probed using high-precision electroweak data [10].
Apart from heavy Z ′ , models with much lighter gauge bosons of masses around one GeV or even a few tens of MeV are extensively discussed in articles and are popular today [11].
A search for a light boson in π 0 → Z ′ + γ by the NA48/2 experiment at CERN requires that Z ′ should couple to u and d quarks very weakly [49], which means that such a boson should be, as usually dubbed in the literature, "protophobic". On the other hand, if the Atomki anomaly is a manifestation of new physics, then the coupling of Z ′ to electrons is nonzero and the boson could be produced in a reversed process, for example in e + e − → γZ ′ [18,38]. Electron-positron collisions are the most straightforward reactions to probe Z ′ s. At the same time, one should keep in mind that the value of the coupling must be compatible with other measurements in which Z ′ may contribute as the electron magnetic dipole moment [50], beam dump experiments and νe scattering [51].
In this paper we focus our attention on the search for new light gauge bosons at e + e − colliders in the following reaction: This reaction contains subprocesses with the exchange of photons of small virtual mass leading to significant enhancement of the cross section, orders of magnitude larger than the cross section for the above mentioned channel e + e − → γZ ′ [18,38]. Therefore, (1) could be used to make the existing constrains on the parameters of Z ′ more stringent.
Assuming a general V − A interaction we present an analytic study of the production cross section, the energy distribution for the final Z ′ s as well their decay products. For completeness, we also calculate the same quantities in the case of a scalar theory considering the production of a spinless light boson φ The article is organized as follows. In Section II, we present the motivation of this work, carry out calculations of the cross section for vector boson production in full detail, derive the energy spectra of the bosons and electrons arising in the boson decay. In Section III, we analyze the case of scalar boson production. Section IV considers a possibility of additional boson decay channels, like decays into neutrino-antineutrino pairs. In Section V, we summarize our results and comment on a possibility of observation of the bosons at electron-positron colliders.

II. LIGHT VECTOR GAUGE BOSONS
A. The total cross section Consider the production of a light vector gauge boson Z ′ in reaction (1). We assume the general form of the electron-boson interaction: where ε denotes the coupling strength of Z ′ to the vector current, e is the elementary electric charge. The leading Feynman diagrams are shown in Fig. 1.
We calculate the cross section in the Weizsäcker-Williams equivalent photon approximation (EPA). According to EPA we factorize (1) into two subprocesses. The first one is emission of a photon by the electron (positron), e ∓ → e ∓ + γ, the second one is absorption of the emitted photon by the positron (electron) with production of a Z ′ boson: Within EPA the sought-for total cross section is represented as where s is the center-of-mass energy (cms), f γ/e (η, s) is the equivalent photon distribution of the electron (positron), η is the fraction of the electron (positron) energy carried by the photon. The factor 2 arises because the distributions for electrons and positrons coincide.
Throughout this paper we adopt [4] Here α is the fine structure constant.
Besides the unknown coupling ε which varies over a wide range of values, depending on a model, EPA is quite good for order of magnitude estimations. In addition we emphasize that the interference between the two upper and two lower diagrams in Fig. 1 are negligibly small in the limit s ≫ m 2 Z (which is the condition for the cases we study). The point is that the Z ′ boson will be predominantly emitted in the direction of the electron or positron so that the processes become distinguishable. Thus the cross section will be determined by the square of the sum of the two upper diagrams plus that of the lower ones. This is another justification for using EPA which considerably simplifies calculations.
To find the cross section for the subprocess (4), there are two lowest order Feynman diagrams contributing to it shown in Fig. 2. Squaring these diagrams, averaging and summing over the spin states yield where t and u are the Mandelstam variables. After standard algebra one can obtain in the Note that full consideration would also require taking into account diagrams of the same order depicted in σ(s) may be orders of magnitude higher. This is clearly illustrated by the ratio of the cross sections in Fig. 4 at cms energies typical for experiments like BaBar [52] where the bosons can be searched. In Fig. 4, we set g V = 1, g A = 0 and m Z = 17 MeV as hinted by the recent observations of the Atomki Collaboration [32]. The main reason is that even though (1) is higher order in α compared to e + e − → γZ ′ , the suppression is compensated by the soft photon exchange in the t-channel. The same property produces the dependence on the Z ′ mass shown in figure 5, where one observes the significant dominance of reaction (1) as well.
Thus, reaction (1) may be a promising channel for the production of new light gauge bosons that couple to electrons.

B. Energy spectrum of the produced bosons
We now describe some observables which may be useful for analyzing experimental data in order to probe Z ′ bosons in e + e − collisions. The first one is the energy distribution of the bosons produced in (1).
The cross section differential in the Z ′ boson energy E can be written as where dσ dE corresponds to the subprocess γe ± → e ± Z ′ . In order to find dσ dE we start from the following formula: where |M| 2 (s, t, u) is given by (7). By definition Here p L is the longitudinal momentum of Z ′ and we have used that E e = √ s/2. On the other hand note that E γ = ηE e = η √ s/2. There is also the condition Adding (11) to (12) and using (13) yield Substituting (14) into (12) we find the relation between t and E t = Since we obtain One can see that (17) coincides with equation (53) of [4] at q(x, P 2 ) = δ(x − 1). The integration limits in (9) are the solutions of the following equations: In the limit s ≫ m 2 Z ≫ m 2 e one can obtain that Using (17) in (9) we find which actually determines the energy spectrum of Z ′ bosons. The Z ′ energy varies in the range Note that the boson mass is kept in the denominator of (20) so that the cross section behaves correctly at E → E max . A plot of the energy distribution of Z ′ bosons given by (20) for the cms energy of the BaBar experiment and m Z = 17 MeV is shown in Fig. 6.

C. Energy spectrum of electrons from Z ′ decays
The produced Z ′ bosons may decay back into electron-positron pairs and another convenient measurable quantity is the energy spectrum of these electrons. We compute it as where E e is the electron energy, Γ denotes the decay width of the Z ′ boson with energy E, dσ dE is given by (20). The latter equation leads to [4] dσ with integration limits defined as The lower limit E min is a consequence of the condition cos θ ≤ 1 with θ being the angle between the three-momenta of Z ′ and the electron from the decay. In contrast to E min , where the electron mass can be safely neglected (as we have done), in E max the mass should be kept to ensure the regular behavior of the electron spectrum in the upper edge.
The integration in (23) yields the distribution of the electrons arising from the decay We plot the electron spectrum (25) in Fig. 7. It is shown that for an integrated luminosity

III. LIGHT SCALAR BOSONS
We can extend the analysis and study the production of neutral scalar boson φ in e + e − collisions. We introduce the following interaction: where g is a Yukawa coupling.
As in the previous section, we consider first the subprocess e ± γ → e ± φ. The corresponding Feynman diagrams are the same as in Figs. 1 and 2 when one replaces Z ′ by φ. Then the amplitude squared is The part −2g 2 e 4 in (27) gives a term in the total cross section decreasing as 1/s, so that the calculations become similar to the case of the vector boson. For example, one can anticipate that the cross section for e + e − → e + e − φ has the following form analogous to equation (8): Proceeding exactly as before, we find the distribution of the boson energy: as well as the energy spectrum of electrons coming from the boson decay φ → e + e − : These spectra are plotted in figures 6 and 7 for g = 10 −5 , m φ = 17 MeV and √ s = 10.5 GeV. Figure 8 demonstrates dependence of the spectra on the boson mass at a fixed energy E e = 2 GeV. One can observe an order of magnitude coincidence with the light vector boson production rates as well as similar behavior of the spectra.

IV. SWITCHING ON ADDITIONAL DECAY CHANNELS
So far we considered a model with only one decay mode, namely Z ′ → e + e − . In principle, the boson may couple to other leptons, for example neutrinos. In this case there is a possibility that reaction (1) will be followed by the decay In a e + e − collision experiment this channel can be observed as a missing energy. It is easy to generalize our results to include this possibility. Since the production of Z ′ and its subsequent decay Z ′ → νν are independent processes, the missing energy distribution will where Br(Z ′ → νν) is the branching ratio of the decay Z ′ → νν and dσ dE is given by equation (20). Therefore, the missing energy spectrum will have exactly the same shape as shown in Fig. 6, but normalized by the branching fraction. In this paper, we have described two processes for electron-positron collision experiments to study the production of scalar and vector bosons with masses in the range MeV to GeV.
We have analyzed the reaction e + e − → e + e − Z ′ in detail. We have considered a model with a single Z ′ that directly decays into a e + e − pair. The analysis indicates that the discussed reaction may be the most favorable channel for the production of light gauge bosons at e + e − colliders, because the corresponding cross section is larger by orders of magnitude compared to similar reactions proposed in articles. Since all the production rates are much higher, the experimental results can put stringent bounds on the parameters of Z ′ or observe it. For example, the rates are discussed for the BaBar experiment. For an integrated luminosity of ∼ 500 fb −1 the production of ∼ 10 3 Z ′ bosons is predicted with a mass of 17 MeV and a coupling ε = 10 −5 . Even for a smaller coupling 10 −6 there are still ∼ 10 events produced.
The dominant background to the considered reactions is the production of QED pairs, e + e − → e + e − e + e − . This background can be reduced by selecting e + e − pairs whose vertices are clearly separated from the collision point of the incoming e + and e − beams [53,54]. For the bounds on ε and m Z presently available, the lifetime of the Z ′ in the laboratory frame is sizable. In fact for ε ≤ 10 −3 , m Z = 17 MeV at the BaBar energy of √ s = 10.5 GeV the separation between the Z ′ and the vertex can reach ∼ 10 cm, which exceeds the spatial separation of the beams at the interaction point; this is illustrated in Fig. 9. The energy spectra for the produced boson and for the electrons or positrons, derived in this article, correlated with the e + e − vertex should be helpful for designing the interaction region in a way that maximizes the sensitivity to Z ′ events.
We have also analyzed the possibility of the boson decay into neutrino-antineutrino pairs.
In this case, the boson will manifest itself as missing with a characteristic spectrum.