Dark photon production via γγ→γA′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \gamma \rightarrow \gamma A'$$\end{document}

The dark photon is a new gauge boson which arises from an extra U′(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U'(1)$$\end{document} gauge symmetry. In this paper, a novel dark photon production mechanism based on MeV-scale γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}–γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} collider is considered: γγ→γA′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \gamma \rightarrow \gamma A'$$\end{document}. With the aid of PACKAGE-X, differential cross section of γγ→γA′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \gamma \rightarrow \gamma A'$$\end{document} is obtained, as a function of the kinetic mixing parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and dark photon mass mA′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{A'}$$\end{document}. Taking the light-by-light scattering as background, the constraints on the dark photon parameter space for different time intervals in a MeV-scale γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}–γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} collider are also given.

Dark matter candidates include: weakly interacting massive particles (WIMPs), asymmetric dark matter, axions, sterile neutrinos, dark photon, etc. WIMPs are expected to have been thermally produced in the early universe, they have large mass compared to standard particles, and they interact with cross sections no higher than the weak scale [15]. Models of asymmetric dark matter are based on the idea that dark matter may carry a matter-antimatter asymmetry [16]. The existence of axions was first postulated to solve the strong CP problem of quantum chromodynamics (QCD) [17]. Sterile neutrinos are neutrinos with right-handed chirality, and they interact only via gravity [18]. a e-mail: xiaorui_wong@pku.edu.cn b e-mail: huangys82@ihep.ac.cn (corresponding author) Dark photon is the gauge boson of an extra U (1) symmetry [19], and can interact with Standard Model photon via kinetic mixing [20][21][22]. Dark photon could be either massless or massive in different frames. The massive case has gained a lot of attention because it couples directly to the standard model currents and is more readily accessible in experiment [23]. In this work, massive dark photon is considered.
In this paper, a novel dark photon production mechanism based on γ -γ collider is considered: γ γ → γ A . Where γ means standard model photon and A stands for dark photon. γ -γ collider was firstly suggested in the early 1980s, with a concept of creating it by uniting linear electron accelerators with high peak and average power lasers [43]. There is still no γ -γ colliders in the world, but the technology of building a MeV-scale γ -γ collider is being mature. Furthermore, the background is respectively clean, which could serve better environment in searching dark photon. In this paper, differential cross section of the process γ γ → γ A is calculated and limits on dark photon parameter space in a γ -γ collider are also given. This paper is organized as follows: in Sect. 2, the dark photon model will be given, and amplitude of γ γ → γ A is calculated. In Sect. 3, we analyse our results and gave limits on dark photon in a MeV-scale γ -γ collider. Finally, summary is made in Sect. 4.

Amplitude of γ γ → γ A
If the Standard Model gauge group is extended by adding a new abelian U (1) symmetry: , a corresponding new gauge vector boson will occur, which is called dark photon and often labeled by A or γ .
With two U (1)s in the gauge group, there will be kinetic mixing, the mixing term can be written as [21]: The Lagrangian is: where ε is kinetic mixing parameter, m A is mass of dark photon, ε and m A are the only two free parameters. F μν is the field strength tensor of U (1) and Dark photon production is considered via γ γ → γ A , this reaction can be taken as the result of the production of a virtual electron-positron pair by two initial photons, followed by annihilation of the pair into the final state photon, and the dark photon.
Define k 1 and k 2 are momentums of incoming photons, k 3 is momentum of outgoing photon and k 4 is momentum of outgoing dark photon: hω is energy of an incoming photon, here we seth = 1. θ is the angle between directions of outgoing and incoming photons. We define Mandelstam variable: which satisfies: Differential cross section of this process is: Because k 1 +k 2 = k 3 +k 4 , it is easy to get The amplitude of γ γ → γ A has contributions from six diagrams. Three are shown in Fig. 1, the other three differ from these only in that the internal electron loop traverses in the opposite direction. Amplitude of the diagram with clockwise direction of electrons in the loop is the same as that with the anticlockwise direction loop, so the total scattering amplitude is: where μ (k 1 ), ρ (k 2 ) are polarisation vectors of incoming photons, * ν (k 3 ), * λ (k 4 ) are polarisation vectors of outgoing photon and dark photon, From quantum field theory, we can get: where m is electron mass, β is an introduced parameter to make the dimension correct, = 4 − n 2 .
μρνλ t and μρνλ u can be got similarly.
Amplitudes are calculated with PACKAGE-X [44]. Each diagram in Fig. 1 has divergent contribution, as shown in Eqs. (8)-(10) respectively: The metric tensor g μν = diag{1, −1, −1, −1}. It is obviously to tell total amplitude has a finite result. Under low energy limit, taking the n → 4, the full loop contribution reads as: Therefore the amplitude square is: where i 16π 2 comes from normalization in loop integration, and 1/4 comes from the average over initial spins.
Angular distribution of this reaction is shown in Fig. 2. Take limit ε → 1, m A → 0, Eq. (13) could go back to the case of photon-photon scattering in textbook [45].

Background analysis and constraints
In the center-of-mass system, kinetic analysis indicates final state photon and dark photon carry energies of ω − m 2 A 4ω and ω + m 2 A 4ω respectively. ω = 0.5 MeV, m A < 1 MeV. Therefore the signal we need to search is monophoton with energy less than 0.5 MeV plus the missing energy. As a simplification of the discussion, the detector efficiency is assumed to be 1, which means any final state photon that could travel into the detector can be detected. Taken into account of shape of the detector, final state photons that can get into the detector must satisfy | cos θ | < 0.8944 [46], where θ is the angle between directions of outgoing photon and incoming photon.
• Breit-Wheeler process as well as cases with final state radiations: γ γ → e + e − , γ γ → e + e − γ , γ γ → e + e − γ γ .  run-time respectively. Obviously we can see that longer time scale gives more stringent constraint on dark photon. Current limits on massive dark photon for m A < 1 MeV mainly come from cosmology, astrophysics, atomic experiments, XENON10, TEXONO, etc. COBE/FIRAS gives constraints on dark photon by measuring the distortions of the CMB spectrum, kinetic mixing parameter ε ≥ 10 −4 are excluded in mass window 10 −15 ∼ 10 −11 eV [47]. Constraints on solar hidden photon are given by CAST [48] and SHIPS [49]. There are also limits from solar lifetime(SUN-T, SUN-L), red giants (RG) and horizontal branches (HB) [50].
In atomic experiments, dark photons are constrained by the sensitive test of Coulomb's law on atomic length scales [51].

Summary
In this work, the feasibility of searching dark photon in γ -γ collider is considered. With PACKAGE-X, differential cross section of process γ γ → γ A is calculated. For the simplicity of discussion, any final state photon with | cos θ | < 0.8944 are assumed to be detected. Taken scattering of light-by-light as background, constraints are given with different time intervals of 6 months, 1 year, 3 years and 5 years, as shown in Fig. 3.
Compared with other constraints on dark photon lighter than 1 MeV, the upper limit of dark photon in a γ -γ collider is not as stringent as others. This is probably restricted to the capability of the machine, which could not offer higher luminosity under current level of technology. We can expect as energy and luminosity increase in a future γ -γ collider, it will be able to detect a wider dark photon mass window and give better limits.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: There are no associated data available.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .