Dark photon production via $\gamma \gamma \rightarrow \gamma A'$

The dark photon is a new gauge boson which arises from an extra U'(1) gauge symmetry. In this paper, a novel dark photon production mechanism based on MeV-scale $\gamma$-$\gamma$ collider is considered: $\gamma \gamma \rightarrow \gamma A'$. With the aid of PACKAGE-X, differential cross section of $\gamma \gamma \rightarrow \gamma A'$ is obtained, as a function of the kinetic mixing parameter $\varepsilon$ and dark photon mass $m_{A'}$. Taking the light-by-light scattering as background, the constraints on the dark photon parameter space for different time intervals in a MeV-scale $\gamma$-$\gamma$ collider are also given.


I. INTRODUCTION
The dark matter problem has never lost its allure for nearly a century [1][2][3][4] . Back to 1922, Jacobus Kapteyn first suggested the existence of dark matter by stellar velocities 5 . In 1933, Fritz Zwicky postulated dark matter by the huge discrepancies between luminous mass and dynamical mass of the Coma Cluster 6 . There are many observational evidences for dark matter in cosmology: galaxy rotation curve [7][8][9] , gravitational lensing 10,11 , bullet cluster 12 , cosmic microwave background 13,14 and so on.
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 .
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.
The production mechanisms of dark photon mainly include: bremsstrahlung 24 , annihilation 25-27 , meson decay 28 and Drell-Yan 29 . Many related experiments [30][31][32][33][34][35][36][37][38][39][40][41][42] have been taken. But still, no robust signature of dark photon has come out. a) Electronic mail: xiaorui wong@pku.edu.cn b) Electronic mail: huangys82@ihep.ac.cn; Corresponding author In this paper, a novel dark photon production mechanism based on γ-γ collider is considered: γγ → γA . 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. Futhermore, the background is respectively clean, which could serve better environment in searching dark photon. In this paper, differential cross section of the process γγ → γA is calculated and limits on dark photon parameter space in a γ-γ colllider are also given.
This paper is organized as follows: in section II, the dark photon model will be given, and amplitude of γγ → γA is calculated. In section III, we analyse our results and gave limits on dark photon in a MeV-scale γ-γ collider. Finally, summary is made in section IV.

II. AMPLITUDE OF γγ → γA
If the Standard Model gauge group is extended by adding a new abelian U (1) symmetry: SU (3) × SU (2) × U (1) × U (1), 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

arXiv:2103.15079v2 [hep-ph] 10 May 2021
Dark photon production is considered via γγ → γA , 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: ω is energy of an incoming photon, here we set = 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: The amplitude of γγ → γA has contributions from six diagrams. Three are shown in Figure 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: 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 . Π µρνλ t and Π µρνλ u can be got similarly. Amplitudes are calculated with PACKAGE-X 44 . Each diagram in Figure 1 has divergent contribution, as shown in Eq.(8) to (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.

III. BACKGROUND ANALYSIS AND CONSTRAINTS
In the center-of-mass system, kinetic analysis indicates final state photon and dark photon carry energies of and ω + m 2 A 4ω respectively. ω = 0.5 MeV, m A < 1MeV. Therefore the signal we need to search is monophoton with energy less than 0.5 MeV plus the missing energy.
FIG. 2. Angular distribution of reaction γγ → γA (take ε = 1). θ is the angle between directions of outgoing and incoming photons, and dark photon is considered with mass range m A < 1MeV.
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.
• Compton scattering of a Compton photon and a beam electron: e − γ → e − γ.
The initial photon energy is set to be 0.5 MeV, which means under this energy scale, electron positron pair could only appear as virtual states, therefore the Breit-Wheeler processes are out of our consideration. On the other hand, beam electron has energy about 200 MeV, outgoing photons of this ultra-relativistic Compton scattering has very small θ, so most of Compton photons could not travel into the detector. Hence this process is not taken into consideration as well.
Therefore only scattering of light-by-light is considered as the background. 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 expriments, dark photons are constrained by the sensitive test of Coulomb's law on atomic length scales 51 .

IV. SUMMARY
In this work, the feasibility of searching dark photon in γ-γ collider is considered. With PACKAGE-X, differential cross section of process γγ → γA 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 six months, one year, three years and five years, as shown in Figure 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.

ACKNOWLEDGMENTS
I would like to thank C.Y.Gao, X.Guan, Y.D.Liu and H.Zhang for useful discussions. This work is supported in part by National Natural Science Foundation of China