Search for a heavy dark photon at future $e^+e^-$ colliders

A coupling of a dark photon $A'$ from a $U(1)_{A'}$ with the standard model (SM) particles can be generated through kinetic mixing represented by a parameter $\epsilon$. A non-zero $\epsilon$ also induces a mixing between $A'$ and $Z$ if dark photon mass $m_{A'}$ is not zero. This mixing can be large when $m_{A'}$ is close to $m_Z$ even if the parameter $\epsilon$ is small. Many efforts have been made to constrain the parameter $\epsilon$ for a low dark photon mass $m_{A'}$ compared with the $Z$ boson mass $m_Z$. We study the search for dark photon in $e^+e^- \to \gamma A' \to \gamma \mu^+ \mu^-$ for a dark photon mass $m_{A'}$ as large as kinematically allowed at future $e^+e^-$ colliders. For large $m_{A'}$, care should be taken to properly treat possible large mixing between $A'$ and $Z$. We obtain sensitivities to the parameter $\epsilon$ for a wide range of dark photon mass at planed $e^+\;e^-$ colliders, such as Circular Electron Positron Collider (CEPC), International Linear Collider (ILC) and Future Circular Collider (FCC-ee). For the dark photon mass $20~\text{GeV}\lesssim m_{A^{\prime}}\lesssim 330~\text{GeV}$, the $2\sigma$ exclusion limits on the mixing parameter are $\epsilon\lesssim 10^{-3}-10^{-2}$. The CEPC with $\sqrt{s}=240~\text{GeV}$ and FCC-ee with $\sqrt{s}=160~\text{GeV}$ are more sensitive than the constraint from current LHCb measurement once the dark photon mass $m_{A^{\prime}}\gtrsim 50~\text{GeV}$. For $m_{A^{\prime}}\gtrsim 220~\text{GeV}$, the sensitivity at the FCC-ee with $\sqrt{s}=350~\text{GeV}$ and $1.5~\text{ab}^{-1}$ is better than that at the 13~TeV LHC with $300~\text{fb}^{-1}$, while the sensitivity at the CEPC with $\sqrt{s}=240~\text{GeV}$ and $5~\text{ab}^{-1}$ can be even better than that at 13~TeV LHC with $3~\text{ab}^{-1}$ for $m_{A^{\prime}}\gtrsim 180~\text{GeV}$.


I. INTRODUCTION
If there is an additional U (1) A symmetry beyond the standard model (SM) gauge symmetry SU (3) × SU (2) L × U (1) Y , a non-zero coupling to the gauge particle A of U (1) A can be generated due to kinetic mixing between the gauge field of U (1) A and the SM hypercharge field at the renormalizable level [1,2]. This gauge symmetry U (1) A is referred as a dark gauge symmetry since the SM particles have zero charge of this U (1) A group and naively invisible. The kinetic-mixing induced A coupling to the SM particles is proportional to the electromagnetic coupling, therefore A is usually referred as dark photon. This is a portal between a possible dark sector and the SM sector. The existence of U (1) A has many interesting effects in particle physics, astrophysics and cosmology [1][2][3][4]. Great efforts have been made to search for a dark photon through various processes and stringent limits have been obtained for the kinetic mixing parameter for a given dark photon mass m A [5][6][7]. There are strong constraints on for a low dark photon mass m A (less than 10 GeV or so) from various low energy facilities and rare decays of known particles. There are fewer studies of constraints on dark photon with a larger mass. Experiments, such as LHCb, ATLAS, CMS and SHiP at CERN, may provide some important information [6,7]. LHCb can provide stringent constraint on the kinetic mixing for dark photon mass larger than 10 GeV [7,8]. It has also been shown that the ATLAS and CMS may provide even better constraint [7,9] at dark photon mass around 40 GeV ∼ 70 GeV by studying the Drell-Yan process pp → Xµ + µ − . There are several high energy e + e − colliders in the plan [10]. These colliders can provide constraints on the mixing parameter for a wide range of the dark photon mass.
In this work we extend our previous study [11], using e + e − → γµ + µ − , to obtain constraints on for dark photon mass in the full range which can be covered by a future e + e − collider. Compared with pp → Xµ + µ − , final states in e + e − → γµ + µ − are easier to be studied. We find that a better constraint on as a function of m A may be possible at some of the planned e + e − colliders. The same e + e − → γµ + µ − process had been used by BaBar [12] to set stringent constraints on the relevant parameters, but the reach of the dark photon mass is limited to be lower than 10 GeV or so. We will study the possibility to search for a heavier dark photon at future e + e − colliders, CEPC, ILC and FCC-ee, through the process e + e − → γA → γµ + µ − . There are some other studies of heavy dark photon at future e + e − colliders [13,14], which will be discussed later.
In our study, we perform a detailed detector simulation with more moderate selection cuts based on a realistic muon momentum resolution. We find that the 2σ exclusion limits on for the dark photon from 20 GeV to 330 GeV can reach 10 −3 − 10 −2 at future e + e − colliders. The CEPC with √ s = 240 GeV and FCC-ee with √ s = 160 GeV are more sensitive to than the constraint from current LHCb measurement once the dark photon mass m A 50 GeV. We also obtain the constraint on for 150 GeV m A 350 GeV from the direct searches in the Drell-Yan process pp → Xµ + µ − using the 13 TeV LHC measurements with L = 36.1 fb −1 [15] and project it to the measurements with L = 300 fb −1 and 3 ab −1 . For m A 220 GeV, the sensitivity at the FCC-ee with √ s = 350 GeV and 1.5 ab −1 is better than that using pp → Xµ + µ − at the 13 TeV LHC with 300 fb −1 . We can also achieve a better sensitivity at the CEPC with √ s = 240 GeV and 5 ab −1 than that at 13 TeV LHC with 3 ab −1 for m A 180 GeV.
The paper is arranged as the following. In section II, we discuss the interactions between the dark photon with the SM sector, where the complete couplings of dark photon with arbitrary mass to the SM particles are studied. In section III, we discuss the production and decays of dark photon at e + e − colliders. In section IV, a detailed collider simulation with dark photon mass ranging from 20 GeV to that as kinematically allowed at future e + e − colliders are performed. In section V, we summarize our results.

II. COUPLINGS OF DARK PHOTON TO THE SM PARTICLES
We now study the kinetic mixing effects on the interactions of A and Z with other SM particles. For A with mass smaller than the Z boson mass, the mixing effects have been studied in details. Since we will allow the A mass from small to be larger than Z bosom mass, care should be taken in particular when m A is very close to m Z where the mixing can be large.
A dark photon field A 0 from an extra U (1) A gauge group can indirectly interact through a gauge kinetic mixing term F 0,µν B µν 0 with the SM sector. Here F 0,µν = ∂ µ A 0ν − ∂ ν A 0µ and B 0,µν = ∂ µ B 0ν − ∂ ν B 0µ , and A 0 and B 0 are the U (1) A and U (1) Y gauge fields, respectively. It is interesting to note that this term should naturally exist since there is no symmetry to prevent it to appear in the relevant Lagrangian even one requires renormalizability. With gauge kinetic mixing, the renormalizable terms involving these two U (1) gauge fields are given by [1] The U (1) Y gauge field B 0 is a linear combination of the photon A 0 and the Z 0 boson The interaction ofÃ,Z andÃ with SM currents is given by where J µ em , J µ Z are the SM electromagnetic and Z boson interaction currents, respectively. J µ D is the dark current in the dark sector. We will work with models where J µ D does not involve SM particles and assume that the width of dark photon decaying into the dark sector is zero. Therefore J µ D does not play a role in e + e − → γµ + µ − and will be ignored in our later discussions.
After the electroweak symmetry breaking, the Z 0 boson obtains a non-zero mass m Z .
Depending on how the U (1) A symmetry is broken, A 0 boson can receive a non-zero mass which may cause a mixing with Z 0 . If one introduces a SM singlet S with a nontrivial U (1) A quantum number s A to break the symmetry, A 0 boson will receive a mass gauge coupling constant and v s / √ 2 is the vacuum expectation value S of S field. In theZ andÃ basis, they mix with each other with the mixing matrix given by The above mass matrix can be diagonalized by an unitary transformation where the normalization factor M is and λ 1,2 are the eigenvalues For the case m A < m Z , the masses of dark photon A and Z boson are λ 2 and λ 1 , respectively; while for m A > m Z , they correspond to λ 1 and λ 2 . The interaction of physical dark photon with SM sector currents will be modified further compared with Eq. (3). In the rest of this paper, we will work within this simple model for dark photon mass generation and study the consequences.
The final transformation between the basis (A 0 , Z 0 , A 0 ) T and the mass eigenstate where the transformation matrix After the mass diagonalization, Eq. (3) will be modified with the couplings to J µ em and J µ Z currents to be given by where V 11 = 1, ≡ V 13 and τ ≡ V 23 .
For m A < m Z , we have while for m A > m Z , and the transformation matrix V ± can be expressed in an uniform form as follows: which gives It is apparent that both and τ depend on the mixing parameter σ linearly. In this case one can express τ in terms of as We find that τ is very small if m A m Z , thus it is usually neglected for light dark photon searches. For m A being close to m Z , it becomes significantly large and should be taken into account.
The approximation in Eq. (14) is no longer valid once |m A − m Z | s W σm Z , in this case we should first take the limit of m A → m Z and then expand in series of σ. To the first order in σ, we obtain that and the coupling constants are From the above we see that the mixing between A µ 0 and Z µ 0 are nearly maximal. There is a discontinuous behavior of τ for m A below and above m Z .
Physically, at m A = m Z and σ approaching zero, the mixing parameter should be zero. At that point, one should take the average of the mixing effect of m A below and above m Z . Then the correct limit for m A = m Z and σ = 0 can be obtained.
For the value σ ∼ 10 −3 − 10 −2 that we are interested in, one obtains that s W σm Z ∼ 0.043 GeV − 0.43 GeV. Our detector simulation to be discussed later shows that a mass window cut ∆m µ + µ − < 0.5 GeV ∼ 1.5 GeV is appropriate for the dark photon searches in e + e − → γA → γµ + µ − at future e + e − colliders. Thus we can work with the assumption It is important to emphasize that m A and m Z are not the physical masses but the mass parameters of A 0 and Z 0 . From Eq. (7), the masses for Z and A can be expressed as (m phys.
respectively. For σ ∼ 10 −3 − 10 −2 , the relative mass shift is at most 0.3%. In later discussions, we will use m Z and m A as the physical Z boson mass and dark photon mass, respectively.

III. PRODUCTION AND DECAY OF DARK PHOTON
We now study the production and decays of A . Since we will consider m A to be much larger than m Z , therefore A can decay into fermion pairs, and also W + W − and Zh at the tree level. The couplings to fermion pairs, W + W − and Zh are given by where f = e, µ, τ, ν e,µ,τ , u, d, s, c, b, Q f is the charge of the fermion, g is the coupling for fermions being isospin 1/2 and −1/2, respectively.
The higher-order correction of Γ(A → ff ) for m A 12 GeV is small, for example the QCD correction to 3-loop level is 1.5% for m A = 60 GeV [19], and not considered in this study. In this section, we will study the sensitivities of searching for dark photon in the process e + e − → γA , A → µ + µ − 2 at the future e + e − colliders with several different energies √ s = 160 GeV, 240 GeV and 350 GeV. The projected/updated integrated luminosities at the CEPC [20], FCC-ee [21,22] and ILC [23][24][25] are summarized in Table I.
At √ s = 160 GeV, the integrated luminosity can be 10 ab −1 at the FCC-ee with about 1-year running [22], which is larger than that at the ILC. At √ s = 240 ∼ 250 GeV, the integrated luminosity can be 5 ab −1 at the CEPC and FCC-ee with 10-year [20] and 3-year [22] running, respectively; while events with only 1.5 ab −1 will accumulate at the ILC. At √ s = 350 GeV, the integrated luminosity is projected to be 1.5 ab −1 with 4 ∼ 5- year running at the FCC-ee [22] and larger than that at the ILC. Thus in our study, we concentrate on the CEPC at √ s = 240 GeV and the FCC-ee with √ s = 160 GeV and 350 GeV. The most dominant SM background, for e + e − → γA → γµ + µ − , is e + e − → γ(Z, γ) followed by virtual Z and γ decaying into µ + µ − . The analytic expressions of total cross sections and differential cross sections for the signal and background processes have been obtained in Ref. [11]. In this work, we carry out numerical analyses with more realistic event selections at various future e + e − colliders. For event generation, we use MG5 aMC v2 4 3 [26] 3 . The following basic cuts are imposed at the parton-level: where for the CEPC and -∆p T p T = 0.1% ⊕ p T 10 5 GeV for |η| < 1.0 and 10 times larger for 1.0 < |η| < 2.4; for the FCC-ee.
After passing these pre-selection criteria, the invariant mass distributions of µ + µ − are displayed in Fig. 3  in our collider study.
The energy of dark photon can be expressed as Based on the kinematical distributions, we further impose the selection cuts: where the ∆m µ + µ − cut is explicitly shown in Table II. The missing transverse momentum (E miss T ) cut is used to remove the SM backgrounds τ + τ − γ and W + W − γ, which have larger On the other hand, in Ref. [13] the following selection cuts were imposed: We find that after our mass window cut ∆m µ + µ − < 0.5 ∼ 1.5 GeV is imposed, the above cut on the photon energy spectrum is not effective. In Ref. [14], the total and differential cross sections of e + e − → γA → γµ + µ − were expressed as those of e + e − → A → µ + µ − convoluted with the probability function of the emitted photon [33]. For signal extraction, an estimated mass resolution ∆m µ + µ − = m 2 A /(10 5 GeV) was adopted [14], which was based on the specification of the muon momentum resolution ∆(1/p T ) = 2 × 10 −5 GeV −1 .
This estimation, in our opinion, is too optimistic especially for low p T muons. 5 There is also SM background from e + e − → hγ, h → µ + µ − . We have checked that it is small and can be neglected.  The signal significance is evaluated using where the benchmark value of the significance (S/ √ B) 0 is evaluated with 2 = 10 −4 and L = 1 ab −1 . hard to pass the selection p γ T ≥ 10 GeV. As discussed in section I, the LHC Drell-Yan process can provide constraints for dark photon mass above 10 GeV. For the dark photon mass 10 GeV < m A < 80 GeV, the sensitivities to has been explored [34] by recasting the CMS 7 TeV measurements [35] and The LO cross section is then multiplied with a next-to-leading-order (NLO) QCD Kfactor K NLO 1.2 [39]. The experimental 95% confidence level (C.L.) upper limit on the cross section times the branching ratio of pp → A → µ + µ − with 36.1 fb −1 , denoted as [σ(A )Br(µ + µ − )] 95%C.L. , is extracted from Ref. [15] directly and independent of . The 95% C.L. upper limit on with the integrated luminosity L is thus Besides the direct searches in the Drell-Yan process, the mixing between Z 0 and A 0 leads to shifts in the mass of Z boson and its couplings to the SM fermions, which are confronted with the electroweak precision tests (EWPTs) [19,40].
The exclusion limits at future e + e − colliders and those from direct searches in the Drell-Yan process pp → A → + − at the LHC, LHCb prompt searches [8] and the EWPTs are 10 GeV < m A < 80 GeV (blue region) are taken from Ref. [19] (in fact, QCD K-facotrs should also be included for recasting the results in the Drell-Yan process at the LHC, which will lead to stricter constraints), which is also rescaled to that with L = 300 fb −1 (blue dashed region). The 90% C.L. constraint from the LHCb prompt search is taken from Ref. [8]. The 95% C.L. constraints for m A 150 GeV at the 13 TeV LHC with L = 36.1 fb −1 (blue region), 300 fb −1 (blue dashed region) and 3 ab −1 (blue dot-dashed region) are also shown.
photon mass below 70 GeV, the LHCb prompt searches have imposed severe constraints on , which is 2 × 10 −3 − 6 × 10 −3 . The FCC-ee (160 GeV) and CEPC (240 GeV) can have better sensitivity with m A 50 GeV. There does not exist constraints on from the LHC direct searches in the Drell-Yan process for 80 GeV m A 150 GeV, while the constraint from the EWPTs is weak for m A 100 GeV. The exclusion limit can be improved significantly at future e + e − colliders, which can even reach 2 × 10 −3 at the FCCee (160 GeV) with L = 10 ab −1 . For larger dark photon mass 150 GeV m A 300 GeV, the current constraint from the LHC direct searches is 8.3 × 10 −3 , which is projected to be 4.8 × 10 −3 and 2.7 × 10 −3 with the integrated luminosity L = 300 fb −1 and 3 ab −1 , respectively. The sensitivity at the FCC-ee (350 GeV) with 1.5 ab −1 is better than that at the 13 TeV LHC with 300 fb −1 for m A 220 GeV. While the sensitivity at the CEPC (240 GeV) with 5 ab −1 can be even better than that at 13 TeV LHC with 3 ab −1 for m A 180 GeV. For the dark photon mass larger than about 300 GeV, the sensitivity to may be further improved at a e + e − collider with larger c.m. energy.

V. SUMMARY AND CONCLUSIONS
In this work we study dark photon search using e + e − → γA → γµ + µ − for a dark photon mass m A as large as kinematically allowed at future e + e − colliders. For small dark photon mass, the mixing is small for a small mixing parameter σ. For large m A , care should be taken to properly treat possible large mixing between A and Z. We show that stringent constraints on the parameter for a wide range of dark photon mass can be obtained at planed e + e − colliders, such as CEPC, ILC and FCC-ee.
As compared to previous studies with the estimated mass resolution ∆m µ + µ − = m 2 A /(10 5 GeV) [14] at √ s = 90 GeV and 250 GeV or the mass window cut ∆m µ + µ − < 5 GeV [13] at √ s = 250 GeV and 500 GeV, our study with a detailed detector simulation and a realistic muon momentum resolution shows that a mass window cut ∆m µ + µ − < 0.5 GeV ∼ 1.5 GeV is appropriate for the dark photon searches in e + e − → γA → γµ + µ − at future e + e − colliders with √ s = 160 GeV, 240 (250) GeV and 350 GeV. Consequently, our results are optimized and more realistic. Epecifically, we find that the 2σ exclusion limits on for the dark photon from 20 GeV to 330 GeV are 10 −3 − 10 −2 at future e + e − colliders. The CEPC (240 GeV) and FCC-ee (160 GeV) are more sensitive than the constraint from current LHCb measurement once the dark photon mass m A 50 GeV.
We also obtain the constraint on for 150 GeV m A 350 GeV from the direct searches in the Drell-Yan process pp → Xµ + µ − using the 13 TeV LHC measurements with L = 36.1 fb −1 [15] and project it to the measurements with L = 300 fb −1 and 3 ab −1 .
The corresponding constraints are 8.3 × 10 −3 , 4.8 × 10 −3 and 2.7 × 10 −3 for 150 GeV m A 300 GeV, and become weaker for m A 300 GeV. For m A 220 GeV, the sensitivity at the FCC-ee (350 GeV) with 1.5 ab −1 is better than that at the 13 TeV LHC with 300 fb −1 , while the sensitivity at the CEPC (240 GeV) with 5 ab −1 can be even better than that at 13 TeV LHC with 3 ab −1 for m A 180 GeV. Besides, we have compared the sensitivities at √ s = 160 GeV, 240 GeV and 350 GeV with the same integrated luminosity and find that the sensitivity at the 160 GeV e + e − collider is better than the other two due to its largest cross section for on-shell dark photon production.