Search for a heavy dark photon at future $e^+e^-$ colliders
Min He Xiao-Gang He Cheng-Kai Huang Gang Li
hep-ph


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    #1#2#1 #2-e^+e^-→μ^+μ^-γ m_μ^+μ^- 
                ^1T-D. Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong
                University, 800 Dongchuan Road, Shanghai 200240
                ^2Department of Physics, National Taiwan University, Taipei 106
                ^3National Center for Theoretical Sciences, Hsinchu 300
                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 ϵ. A non-zero ϵ 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 ϵ is small. Many efforts have been made to constrain the parameter ϵ 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^- →γ A' →γμ^+ μ^- 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 ϵ 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 GeV≲ m_A^'≲ 330 GeV, the 2σ exclusion limits on the mixing parameter are ϵ≲ 10^-3-10^-2. The CEPC with √(s)=240 GeV and FCC-ee with √(s)=160 GeV are more sensitive than the constraint from current LHCb measurement once the dark photon mass m_A^'≳ 50 GeV. For m_A^'≳ 220 GeV, the sensitivity at the FCC-ee with √(s)=350 GeV and 1.5 ab^-1 is better than that at the 13 TeV LHC with 300 fb^-1, while the sensitivity at the CEPC with √(s)=240 GeV and 5 ab^-1 can be even better than that at 13 TeV LHC with 3 ab^-1 for m_A^'≳ 180 GeV. 
                Search for a heavy dark photon
                
                at future e^+e^- colliders
                    Min He^1[Electronic address: hemind@sjtu.edu.cn], Xiao-Gang He^1, 2,3[Electronic address: hexg@phys.ntu.edu.tw], Cheng-Kai Huang^2[Electronic address: r01222045@ntu.edu.tw ], Gang Li^2[Electronic address: gangli@phys.ntu.edu.tw ]
                    December 30, 2023
                =========================================================================================================================================================================================================================================
                
                
                
                § 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 <cit.>. 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 <cit.>. 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 ϵ for a given dark photon mass m_A' <cit.>. There are strong constraints on ϵ 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 <cit.>. LHCb can provide stringent constraint on the kinetic mixing for dark photon mass larger than 10 GeV <cit.>. It has also been shown that the ATLAS and CMS may provide even better constraint <cit.> at dark photon mass around 40∼ 70 by studying the Drell-Yan process pp → X μ^+μ^-. There are several high energy e^+e^- colliders in the plan <cit.>. These colliders can provide constraints on the mixing parameter ϵ for a wide range of the dark photon mass. 
                
                In this work we extend our previous study <cit.>, using e^+e^- →γμ^+μ^-, to obtain constraints on ϵ for dark photon mass in the full range which can be covered by a future e^+ e^- collider. Compared with pp → X μ^+μ^-, final states in e^+e^- →γμ^+μ^- are easier to be studied. We find that a better constraint on ϵ 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 <cit.> 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^-→γ A^'→γμ^+μ^-. There are some other studies of heavy dark photon at future e^+e^- colliders <cit.>, 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 ϵ 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 and FCC-ee with √(s)=160 are more sensitive to ϵ than the constraint from current LHCb measurement once the dark photon mass m_A^'≳ 50. We also obtain the constraint on ϵ for 150≲ m_A^'≲ 350 from the direct searches in the Drell-Yan process pp→ Xμ^+μ^- using the 13 TeV LHC measurements with ℒ=36.1 <cit.> and project it to the measurements with ℒ=300 and 3. The corresponding constraints are ϵ≲ 8.3× 10^-3, ϵ≲ 4.8× 10^-3 and ϵ≲ 2.7× 10^-3 for 150≲ m_A^'≲ 300, and become weaker for m_A^'≳ 300. For m_A^'≳ 220, the sensitivity at the FCC-ee with √(s)=350 and 1.5 is better than that using pp→ X μ^+μ^- at the 13 TeV LHC with 300. We can also achieve a better sensitivity at the CEPC with √(s)=240 and 5 than that at 13 TeV LHC with 3 for m_A^'≳ 180. 
                
                The paper is arranged as the following. In section <ref>, 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 <ref>, we discuss the production and decays of dark photon at e^+e^- colliders.  In section <ref>, 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 <ref>, we summarize our results.
                
                
                
                § 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_0^μν 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 <cit.>
                    L_ = -1 4 B_0^μνB_0,μν -1 2σ F'_0, μνB_0^μν - 1 4 F'_0,μνF_0^'μν .
                
                The U(1)_Y gauge field B_0 is a linear combination of the photon A_0 and the Z_0 boson fields, B_0 = c_W A_0 - s_W Z_0 with c_W = cosθ_W and s_W = sinθ_W, where θ_W is the weak interaction Weinberg angle.  
                
                To make the above Lagrangian in the canonical form, that is, there is no crossing term, one needs to redefine the fields. Letting the redefined fields to be Ã, Z̃ and Ã', we have <cit.>
                    ( [  A_0;  Z_0; A'_0 ] ) =  ( [              1              0 -c_W σ√(1-σ^2);              0              1  s_W σ√(1-σ^2);              0              0      1√(1-σ^2) ] )
                      ([  Ã; Z̃; Ã' ] ) .
                
                 
                The interaction of Ã, Z̃ and Ã' with SM currents is given by
                
                    J^μ_em (Ã_μ - c_W σ√(1-σ^2)Ã'_μ) + J^μ_Z (Z̃_μ + s_Wσ√(1-σ^2)Ã'_μ) +  1√(1-σ^2)J^μ_DÃ'_μ ,
                
                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 non-trivial U(1)_A' quantum number s_A' to break the symmetry, A'_0 boson
                will receive  a mass m_A' = g_A's_A'v_s/√(2) from (D_μ S)^† (D^μ S) term in the Lagrangian. Here g_A' is the U(1)_A' gauge coupling constant and v_s/√(2) is the vacuum expectation value ⟨ S ⟩ of S field. 
                In the Z̃ and Ã' basis, they mix with each other with the mixing matrix  given by
                
                    ( [                                 m^2_Z                    σ s_W√(1-σ^2)m^2_Z;                    σ s_W√(1-σ^2)m^2_Z 1 1-σ^2m^2_A' + s^2_W σ^2 1-σ^2 m^2_Z ] ).
                
                The above mass matrix can be diagonalized by an unitary transformation
                
                    ( 
                    [   Ã;  Z̃; Ã^' ])
                    = (
                    [                     1                     0                     0;                     0 σ s_W m_Z^2/𝐌√(1-σ^2)           m_Z^2-λ_1/𝐌;                     0           λ_1-m_Z^2/𝐌 σ s_W m_Z^2/𝐌√(1-σ^2) ])(
                    [   A;   Z; A^' ]),
                
                where the normalization factor 𝐌 is 
                
                    𝐌   = √((λ_1-m_Z^2)^2+σ^2 s_W^2/1-σ^2 m_Z^4) ,
                
                and λ_1,2 are the eigenvalues
                
                    λ_1,2   = 1/2(m_Z^2+1/1-σ^2 m_A^'^2
                    + σ^2 s_W^2/1-σ^2 m_Z^2 ±Δ),  λ_1≥λ_2,
                    Δ   ≡√((m_Z^2 - 11-σ^2 m_A^'^2
                    - σ^2 s_W^2/1-σ^2 m_Z^2)^2 + 4 σ^2 s_W^2/1-σ^2 m_Z^4).
                
                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. (<ref>). 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 (A,Z,A^')^T can be expressed as
                
                    ( 
                    [   A_0;   Z_0; A_0^' ])
                       =V (
                    [   A;   Z; A^' ]) ,  
                    V= (
                    [ V_11 V_12 V_13; V_21 V_22 V_23; V_31 V_32 V_33 ]),
                
                where the transformation matrix V≡ V_-(V_+) for m_A^'<m_Z (m_A^'>m_Z) are given by
                
                    V_- = (
                    [                                         1             -c_W σ (λ_1 -m_Z^2)/𝐌√(1-σ^2)              -σ^2 s_W c_W m_Z^2/𝐌 (1-σ^2);                                         0                        s_W σλ_1/𝐌√(1-σ^2) 1/𝐌( m_Z^2 - λ_1 + σ^2 s_W^2 m_Z^2/1-σ^2);                                         0                     λ_1 - m_Z^2/𝐌√(1-σ^2)                     σ s_W m_Z^2/𝐌 (1-σ^2) ]) ,  
                    V_+ = V_- (
                    [  1  0  0;  0  0  1;  0 -1  0 ]).
                
                After the mass diagonalization, Eq. (<ref>) will be modified with the couplings to J^μ_em and J^μ_Z currents to be given by 
                
                    J^μ_em ( V_11 A_μ + V_12 Z_μ + ϵ A'_μ) + J^μ_Z ( V_22 Z_μ + τ A'_μ)  ,
                
                where V_11=1, ϵ≡ V_13 and τ≡ V_23.
                
                For m_A^' < m_Z, we have
                
                    V_12   =-c_W σ (λ_1 -m_Z^2)/𝐌√(1-σ^2) ,     V_22   =s_W σλ_1/𝐌√(1-σ^2) ,
                    ϵ   =-σ^2 s_W c_W m_Z^2/𝐌 (1-σ^2) ,     τ   =1/𝐌( m_Z^2 - λ_1 + σ^2 s_W^2 m_Z^2/1-σ^2),
                
                while for m_A^' > m_Z,
                
                    V_12   =σ^2 s_W c_W m_Z^2𝐌 (1-σ^2) ,    V_22   =-1𝐌( m_Z^2 - λ_1 + σ^2 s_W^2 m_Z^21-σ^2) ,
                    ϵ   = -σ c_W (λ_1 - m_Z^2)/𝐌√(1-σ^2) ,
                       τ   = σ s_W λ_1/𝐌√(1-σ^2).
                
                
                We find that if |m_Z - m_A^'| ≫ s_W m_Z σ,
                
                    Δ    = |m_Z^2 - m_A^'^2| + (m_A^'^4 + s_W^2 m_Z^4 - c_W^2 m_Z^2 m_A^'^2) σ^2/|m_Z^2 - m_A^'^2|+𝒪(σ^3),
                
                and the transformation matrix V_± can be expressed in an uniform form as follows:
                
                    V_+=V_-   = (
                    [                            1                            0                       -c_W σ;                            0                            1 s_W σ m_A^'^2m_A^'^2 - m_Z^2;                            0  -s_W σ m_Z^2m_A^'^2 - m_Z^2                            1 ]) +𝒪(σ^2),
                
                which gives
                
                    V_12=0,    V_22=1,   ϵ   =-c_Wσ,  τ=s_Wσ m_A^'^2 m_A^'^2-m_Z^2.
                
                It is apparent that both ϵ and τ depend on the mixing parameter σ linearly. In this case one can express τ in terms of ϵ as 
                
                    τ=-s_W m_A^'^2ϵc_W (m_A^'^2-m_Z^2).
                
                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. (<ref>) is no longer valid once |m_A^'- m_Z| ≪ s_Wσ m_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
                
                    V_- = (
                    [                                    1                          -c_W σ/√(2)                          -c_W σ/√(2);                                    0  1/√(2) + (3 s_W^2 - 1) σ/4 √(2) s_W -1/√(2) + (3 s_W^2 - 1) σ/4 √(2) s_W;                                    0    1/√(2) + (s_W^2 + 1) σ/4 √(2) s_W    1/√(2) - (s_W^2 + 1) σ/4 √(2) s_W ])+𝒪(σ^2) ,  
                    V_+ = V_- (
                    [  1  0  0;  0  0  1;  0 -1  0 ])
                
                and the coupling constants are
                
                    ϵ   =-c_Wσ√(2),  τ=-1√(2)+(3s_W^2-1)σ4√(2)s_W,  for   m_A^'<m_Z,
                    ϵ   =-c_Wσ√(2),  τ=1√(2)+(3s_W^2-1)σ4√(2)s_W,     for   m_A^'>m_Z.
                
                
                From the above we see that the mixing between A^'μ_0 and Z_0^μ 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_Wσ m_Z∼ 0.043-0.43. 
                Our detector simulation to be discussed later shows that a mass window cut  Δ m_μ^+μ^-<0.5∼ 1.5 is appropriate for the dark photon searches in e^+e^-→γ A^'→γμ^+μ^- at future e^+e^- colliders. Thus we can work with the assumption that |m_Z - m_A^'| ≫ s_W m_Z σ in section <ref>.
                
                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. (<ref>), the masses for Z and A^' can be expressed as
                
                    (m_Z^phys.)^2    = m_Z^2 + m_Z^4 s_W^2 σ^2/m_Z^2 - m_A'^2 + 𝒪(σ^3),
                    
                    (m_A^'^phys.)^2    = m_A'^2 + (c_W^2 m_Z^2 - m_A'^2) m_A'^2 σ^2/m_Z^2 - m_A'^2 + 𝒪(σ^3),
                
                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.
                
                
                
                
                § 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
                
                    ℒ_fermions =  (ϵ e  Q_ff̅γ^μ f+  τg/2c_Wf̅γ^μ (g_V^f-g_A^fγ_5) f  )A_μ^',
                       ℒ_gauge = -ie(ϵ+τθ_W) [-∂_μA_ν^'(W_μ^+W_ν^–W_ν^+W_μ^-)
                                    +A_ν^'(-W_μ^+∂_νW_μ^-+W_μ^-∂_νW_μ^+
                    +W_μ^+∂_μW_ν^–W_μ^-∂_μW_ν^+)],
                       ℒ_higgs = τ g m_Zc_Wh A_μ^'Z^μ,
                
                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 constant of SU(2)_L, g_V^f=T_3^f-2s_W^2 Q_f and g_A^f=T_3^f with T_3= 1/2, -1/2 for fermions being isospin 1/2 and -1/2, respectively.
                
                The cross section of e^+e^-→ A^'γ is given by <cit.>
                    σ_A^'γ=-e^2(m_A^'^4+s^2)(1-lnsm_e^2)4π s^2(s-m_A^'^2){ e^2ϵ^2 +g^2τ^2[(g_V^e)^2+(g_A^e)^2]4c_W^2 -e g g^e_V ϵτc_W},
                
                where g_V^e=-1/2+2s_W^2, g_A^e=-1/2, and τ depends on σ and m_A^'. 
                
                In Fig. <ref> we show the cross sections of e^+e^-→ A^'γ at √(s)=160, 240 and 350  for σ=10^-2. To keep the finiteness of the cross sections at m_A^'=m_Z, we use the complete forms of ϵ and τ. However, the forms in Eq. (<ref>) are very good approximations as we have discussed. For m_A^' being around m_Z the cross sections exhibit peaks since the coupling τ is proportional to m_A^'^2/(m_A^'^2-m_Z^2). Besides, the cross sections grow with m_A^' for m_A^' being above m_Z due to σ_A^'γ∼(m_A^'^4+s^2)/[s^2(s-m_A^'^2)]. However, if m_A^' is very close to √(s), that is the center-of-mass (c.m.) energy of e^+e^- collider, the photon energy is small. In order to detect a isolated photon, a minimal energy of the photon should be demanded. In our case, the photon energy E_γ>10 is required. 
                
                
                
                Assuming that the dark photon A^' decays exclusively to the SM particles, the partial widths of A^' are <cit.>[There exist tri-boson decays A^'→ W^+W^-γ and A^'→ W^+W^-Z, which originate from the SM quartic gauge couplings before the kinetic mixing. However, the tri-boson decay widths of the dark photon are smaller than 𝒪(1%) so we neglect them in this paper.]
                    Γ(A^'→ ff̅)   =g^2m_A^'12π c_W^2 N_c^f {ϵ^2 Q_f^2 c_W^2s_W^2 +ϵτ Q_f c_W s_W g_V^f +1/4τ^2 [(g_V^f)^2+(g_A^f)^2]},
                    Γ(A^'→ Zh)   =g^2τ^2m_A^'192π c_W^2λ^1/2(1,x_Z,x_h){λ(1,x_Z,x_h)+12x_Z},
                    Γ(A^'→ W^+W^-)   =g^2 s_W^2(ϵ+τθ_W)^2m_A^'192πx_W^-2(1-4x_W)^3/2(1+20x_W+12x_W^2),
                
                where N_c^f=3 for quarks and 1 for leptons, x_W,Z,h=(m_W,Z,h/m_A^')^2 and λ(x,y,z)=x^2+y^2+z^2-2xy-2yz-2zx. The tree-level decay width into fermion pair in the first line of Eq. (<ref>) is a good approximation above the bb̅ threshold Υ(nS), i.e., m_A^'≳ 12. The higher-order correction of Γ(A^'→ ff̅) for m_A^'≳ 12 is small, for example the QCD correction to 3-loop level is 1.5% for m_A^'=60 <cit.>, and not considered in this study.
                
                
                
                Figure <ref> displays the widths of A^' and the braching ratio of A^'→μ^+μ^- in the left and right panels, respectively. Similar to the case of cross section, the complete forms of ϵ and τ are employed, so the branching ratio depends on σ slightly. From the left panel, all the partial widths of A^' into fermions exhibit peaks around m_Z due to τ∼ m_A^'^2/(m_A^'^2-m_Z^2) and depend on m_A^' linearly for m_A^'≳ 120 where the contributions from the second and third terms in the first line of Eq. (<ref>) are negligible. The decays into W^+W^- and Zh are possible once they are kinematically allowed. The widths Γ(A^'→ Zh) and Γ(A^'→ W^+W^-) are a few percent of the total width due to the suppression of phase space. Besides, the coupling of dark photon to W^+W^- is proportional to ϵ+τθ_W=c_Wσ m_Z^2/(m_A^'^2-m_Z^2), which becomes smaller for larger m_A^'. From the right panel, there is a dip around m_Z due to the contribution from the coupling τ. The branching ratio becomes flat for m_A^'≳ 150 where τ is small and the dependence on m_A^' in Γ(A^'→μ^+μ^-) and the total width cancels.
                
                
                
                
                § SENSITIVITIES AT FUTURE E^+E^-EE COLLIDERS
                
                
                In this section, we will study the sensitivities of searching for dark photon in the process e^+e^-→γ A^', A^'→μ^+μ^- [The 2→ 3 process with off-shell dark photon and the interferene between the signal and background processes are checked to be small, which are thus neglected.] at the future e^+e^- colliders with several different energies √(s)=160, 240 and 350. The projected/updated integrated luminosities at the CEPC <cit.>, FCC-ee <cit.> and ILC <cit.> are summarized in Table <ref>. At √(s)=160, the integrated luminosity can be 10 at the FCC-ee with about 1-year running <cit.>, which is larger than that at the ILC. At √(s)=240∼ 250, the integrated luminosity can be 5 at the CEPC and FCC-ee with 10-year <cit.> and 3-year <cit.> running, respectively; while events with only 1.5 will accumulate at the ILC. At √(s)=350, the integrated luminosity is projected to be 1.5 with 4∼ 5-year running at the FCC-ee <cit.> and larger than that at the ILC. Thus in our study, we concentrate on the CEPC at √(s)=240 and the FCC-ee with √(s)=160 and 350.
                
                
                
                The most dominant SM background, for e^+e^-→γ A' →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. <cit.>. In this work, we carry out numerical analyses with more realistic event selections at various future e^+e^- colliders. For event generation, we use  <cit.> [Initial State Radiation (ISR) and beamstrahlung effect are not considered in this study.]. The following basic cuts are imposed at the parton-level:
                
                    |η_μ^±,γ|<3,   E_γ>2,  Δ R_ij>0.2,  Δ m_μ^+μ^-< 10 GeV,
                
                where Δ R_ij=√((η_i-η_j)^2+(ϕ_i-ϕ_j)^2) with i,j=μ^±,γ and Δ m_μ^+μ^-≡ | m_μ^+μ^--m_A^'|. The parton-level events are then interfaced with  <cit.> for parton shower and hadronization. The detector effects are simulated with  <cit.> and the built-in  and  for the CEPC and FCC-ee, respectively. The detector parametrization for muon momentum resolution and electromagnetic calorimeter (ECAL) energy resolution are
                
                  -Δ p_Tp_T=0.1%⊕p_T10^5 for |η|<1.0 and 10 times larger for 1.0<|η|<3.0;
                
                  -Δ EE=0.20√(E/GeV)⊕ 0.5%. for |η|<3.0.
                
                for the CEPC and 
                
                  -Δ p_Tp_T=0.1%⊕p_T10^5 for |η|<1.0 and 10 times larger for 1.0<|η|<2.4;
                
                  -Δ EE=0.15√(E/GeV)⊕ 1%. for |η|<3.0.
                
                for the FCC-ee.
                
                In order to identify objects in the final state, we impose the following pre-selection criteria: 
                
                  *  A pair of opposite-sign muons are selected with E_μ^±>2 <cit.>, |η_μ^±|<2.5 <cit.> for the CEPC with a better muon identification than the ILD-like detector performance: p_Tμ^±>10 <cit.>, |η_μ^±|<2.5 <cit.>, which is used for the simulation at the FCC-ee;
                
                  *  Exactly one photon with p_T^γ≥ 10 <cit.> and |η_γ|<2.5 <cit.> is selected;
                
                  * Δ R_ij>0.4 for i,j=μ^±,γ.
                
                
                After passing these pre-selection criteria, the invariant mass distributions of μ^+μ^- are displayed in Fig. <ref>, where we have chosen three benchmark values of m_A^', i.e.,  m_A^'=30, 70, 300, with ϵ/c_W=10^-2 and 10^-3 at the FCC-ee with √(s)=350 (the results are similar at the CEPC with √(s)=240 and the FCC-ee with √(s)=160). The invariant mass distributions of μ^+μ^- are affected by the muon momentum resolution and the total width of dark photon, which is proportional to ϵ^2. Figure <ref> shows that for ϵ/c_W∼ 10^-3-10^-2 the impact of the dark photon total width on the invariant mass distributions is negligible. Thus it is flexible to choose ϵ/c_W=10^-2 [Actually in the very small region of m_A^'≃ m_Z, the total width of dark photon becomes significantly large, thus the cut efficiency of the signal depends on the value of ϵ.] in our collider study. 
                
                The energy of dark photon can be expressed as
                
                    E_A^'=√(s)2(1+m_A^'^2s),
                
                where √(s) is the c.m. energy of the collider. Thus the invariant mass distribution becomes broader with larger m_A^' or the increase of √(s) from 160 to 350. 
                
                
                
                Based on the kinematical distributions, we further impose the selection cuts:
                
                    Δ m_μ^+μ^- < 0.5∼ 1.5 GeV,   E_T^miss< 5,
                
                where the Δ m_μ^+μ^- cut is explicitly shown in Table <ref>. The missing transverse momentum (E_T^miss) cut is used to remove the SM backgrounds τ^+τ^-γ and W^+W^-γ, which have larger E_T^miss [There is also SM background from e^+e^-→ hγ, h→μ^+μ^-. We have checked that it is small and can be neglected.].
                
                On the other hand, in Ref. <cit.> the following selection cuts were imposed:
                
                    Δ m_ℓ^+ℓ^-≡ |m_ℓ^+ℓ^--m_A^'| < 5 GeV,   |E_γ-s-m_A^'^2/2√(s)|<2.5.
                
                We find that after our mass window cut Δ m_μ^+μ^- < 0.5∼ 1.5 GeV is imposed, the above cut on the photon energy spectrum is not effective. In Ref. <cit.>, the total and differential cross sections of e^+e^-→γ A^'→γμ^+μ^- were expressed as those of e^+e^-→ A^'→μ^+μ^- convoluted with the probability function of the emitted photon <cit.>. For signal extraction, an estimated mass resolution Δ m_μ^+μ^- = m_A^'^2/(10^5) was adopted <cit.>, which was based on the specification of the muon momentum resolution Δ(1/p_T)=2× 10^-5^-1. This estimation, in our opinion, is too optimistic especially for low p_T muons.
                
                
                
                The signal significance is evaluated using
                
                    S/√(B)   =(S/√(B))_0 ϵ^210^-4√(ℒ1),
                
                where the benchmark value of the significance (S/√(B))_0 is evaluated with ϵ^2=10^-4 and ℒ=1.
                
                Figure <ref> shows the sensitivities to ϵ with S/√(B)=2 and S/√(B)=5 at the FCC-ee (160 GeV), CEPC (240 GeV) and FCC-ee (350 GeV) with the same integrated luminosity ℒ=1. We find that  FCC-ee (160 GeV) has the best sensitivity for 20<m_A^'<140 as compared to the CEPC (240 GeV) and FCC-ee (350 GeV) and the sensitivities increase for larger m_A^' if m_A^'≳ 120. This is mainly because that the cross section of e^+e^-→ A^'γ at √(s)=160 is largest and the cross sections at √(s)=160, 240 and 350 increase for m_A^'≳ 120 as shown in Fig. <ref>. Note that the sensitivities are only shown with m_A^'≲√(s)-20 beyond which the signal events are hard to pass the selection p_T^γ≥ 10. 
                
                
                As discussed in section <ref>, the LHC Drell-Yan process can provide constraints for dark photon mass above 10 GeV. For the dark photon mass 10<m_A^'<80, the sensitivities to ϵ has been explored <cit.> by recasting the CMS 7 TeV measurements <cit.> and making projects for the 8 TeV LHC with the integrated luminosity ℒ=20 fb^-1 and at the High-Luminosity LHC (HL-LHC) with √(s)=14 TeV and ℒ=3. The limits on ϵ for m_A^'≳ 180 were derived <cit.> using the ATLAS 8 TeV measurements with ℒ=20 fb^-1 <cit.> (the published version is Ref. <cit.>). The results were then projected to the HL-LHC in Ref. <cit.>. In this work, we recast the 13 TeV measurents at high mass region (150∼ 350) with ℒ=36.1 <cit.> and project it to the sensitivities at 300 and 3. Specifically, we generate the leading-order (LO) process pp→ A^'→μ^+μ^- in  with ϵ/c_W=10^-2 and obtain the cross section of pp→ A^'→μ^+μ^-, denoted as σ_LO(A^')Br(μ^+μ^-), for m_A^' in the range 150∼ 350. The LO cross section is then multiplied with a next-to-leading-order (NLO) QCD K-factor K_NLO≃ 1.2 <cit.>. 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, denoted as [σ(A^')Br(μ^+μ^-)]^95%C.L., is extracted from Ref. <cit.> directly and independent of ϵ. The 95% C.L. upper limit on ϵ with the integrated luminosity ℒ is thus
                
                    ϵ^95%C.L.= ( [σ(A^')Br(μ^+μ^-)]^95%C.L.K_NLOσ_LO(A^')Br(μ^+μ^-)/(10^-4c_W^2)√(36.1ℒ))^1/2.
                 
                
                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) <cit.>.
                
                
                
                The exclusion limits at future e^+e^- colliders and those from direct searches in the Drell-Yan process pp→ A^'→ℓ^+ℓ^- at the LHC, LHCb prompt searches <cit.> and the EWPTs are illustrated in Fig. <ref> for the dark photon mass 20<m_A^'≲ 340. For the dark photon mass below 70, the LHCb prompt searches have imposed severe constraints on ϵ, which is ϵ≲ 2× 10^-3-6× 10^-3. The  FCC-ee (160 GeV) and CEPC (240 GeV) can have better sensitivity with m_A^'≳ 50. There does not exist constraints on ϵ from the LHC direct searches in the Drell-Yan process for 80≲ m_A^'≲ 150, while the constraint from the EWPTs is weak for m_A^'≳ 100. The exclusion limit can be improved significantly at future e^+e^- colliders, which can even reach 2× 10^-3 at the FCC-ee (160 GeV) with ℒ=10. For larger dark photon mass 150≲ m_A^'≲ 300, the current constraint from the LHC direct searches is ϵ≲ 8.3× 10^-3, which is projected to be ϵ≲ 4.8× 10^-3 and ϵ≲ 2.7× 10^-3 with the integrated luminosity ℒ=300 and 3, respectively. The sensitivity at the FCC-ee (350 GeV) with 1.5 is better than that at the 13 TeV LHC with 300 for m_A^'≳ 220. While the sensitivity at the CEPC (240 GeV) with 5 can be even better than that at 13 TeV LHC with 3 for m_A^'≳ 180. For the dark photon mass larger than about 300 GeV, the sensitivity to ϵ may be further improved at a e^+e^- collider with larger c.m. energy.
                
                
                
                
                § SUMMARY AND CONCLUSIONS
                
                
                In this work we study dark photon search using e^+e^- →γ A' →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 ϵ 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 Δ m_μ^+μ^- = m_A^'^2/(10^5) <cit.> at √(s)=90 and 250 or the mass window cut Δ m_μ^+μ^-<5 <cit.> at √(s)=250 and 500, our study with a detailed detector simulation and a realistic muon momentum resolution shows that a mass window cut Δ m_μ^+μ^-<0.5∼ 1.5 is appropriate for the dark photon searches in e^+e^-→γ A^'→γμ^+μ^- at future e^+e^- colliders with √(s)=160, 240 (250) and 350. Consequently, our results are optimized and more realistic. Epecifically, we find that the 2σ exclusion limits on ϵ for the dark photon from 20 GeV to 330 GeV  are ϵ≲ 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. We also obtain the constraint on ϵ for 150≲ m_A^'≲ 350 from the direct searches in the Drell-Yan process pp→ Xμ^+μ^- using the 13 TeV LHC measurements with ℒ=36.1 <cit.> and project it to the measurements with ℒ=300 and 3. The corresponding constraints are ϵ≲ 8.3× 10^-3, ϵ≲ 4.8× 10^-3 and ϵ≲ 2.7× 10^-3 for 150≲ m_A^'≲ 300, and become weaker for m_A^'≳ 300. For m_A^'≳ 220, the sensitivity at the FCC-ee (350 GeV) with 1.5 is better than that at the 13 TeV LHC with 300, while the sensitivity at the CEPC (240 GeV) with 5 can be even better than that at 13 TeV LHC with 3 for m_A^'≳ 180. Besides, we have compared the sensitivities at √(s)=160, 240 and 350 with the same integrated luminosity and find that the sensitivity at the 160e^+e^- collider is better than the other two due to its largest cross section for on-shell dark photon production.
                
                
                We would like to thank Qing-Hong Cao, Gang Li (IHEP), Tanmoy Modak, Manqi Ruan and Hao Zhang for valuable discussions. This work was supported in part by the MOST (Grant No. MOST104-2112-M-002-015-MY3 and 106-2112-M-002-003-MY3 ), and in part by Key Laboratory for Particle Physics,
                Astrophysics and Cosmology, Ministry of Education, and Shanghai Key Laboratory for Particle
                Physics and Cosmology (Grant No. 15DZ2272100), and in part by the NSFC (Grant Nos. 11575111 and 11735010).
                
                
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