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Here, we discuss different DMPs, such as pseudoscalar, vector, and scalar particles, that may be generated in Compton processes.
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If pseudoscalar particles couple with electrons, the corresponding effective Lagrangian can be expressed as [22]
L=gae∂μϕa2meˉψeγμγ5ψe=−igaeˉψeγ5ψeϕa,
(1) where the dimensionless
gae is the pseudoscalar-electron coupling constant. Here, we consider a pseudoscalar particle that is emitted in a Compton-like process, i.e.,γ+e−→e−+a . The corresponding Feynman diagram is shown in Fig. 2. The differential cross section can be expressed asdσdΩ=164π21EkEp|vk−vp||→k′|2k′0p′0×|1|→k′|k′0+|→k′|−(→p+→k)⋅^k′p′0|(12∑spin|M2|),
(2) The definitions of
→k ,→k′ ,→p ,vk ,vp ,Ek ,Ep , and M are provided in the Appendix.spin is the spin of the incoming electron.Subsequently, the total cross section in the laboratory coordinator can be obtained by the integral
σlab=∫dσ=∏f(∫d3→pf(2π)312Ef)12Ek2Ep|vk−vp|×(12∑spin|M2|)(2π)4δ4(p+k−p′−k′),
(3) where subscript index f is the final quantum state of the dark matter candidate (pseudoscalar particle) and electron.
Eq. (2) indicates that the differential cross section
dσlabdΩ(ma,θ0,θ,ϕ) is a function of the DMP massma , colliding angle between the electron and the laserθ0 , and scattering angles,θ andϕ , whereas the total cross sectionσlab(ma,θ0) depends only onma andθ0 .By using the parameters of SLEGS,
Ee=3.5 GeV and a photon wavelength of 10.64μ m, the cross sectionσlab(ma,θ0) is shown in Fig. 3, which shows the results of four colliding angles:θ0=90∘ ,120∘ ,150∘ , and180∘ . As the figure shows, atθ0=180∘ , i.e., head-to-head collision, the cross section approaches its maximum. Moreover, the cross section decreases as the mass of the pseudoscalar particle increases. The differential cross sectiondσlabdΩ(ma,θ0,θ,ϕ) is provided in the Appendix.Figure 3. (color online) Cross sections of the pseudoscalar particle through the reaction
e+γ→e+a . The cross sectionσlab(ma,θ0) is a function of the pseudoscalar particle massma , and colliding angleθ0 between the pseudoscalar particle and electron. The different color lines representθ0=90∘ (blue dash dot dot),120∘ (green dash dot),150∘ (red dash), and180∘ (black solid). The cross sectionσ is in barns. -
For the coupling of vector field particles, such as dark photons, to an electron, the corresponding effective Lagrangian can be expressed as [7]
L=gA′eeˉψeγμψeA′μ,
(4) where the dimensionless
gA′e is the mixing parameter between the dark photon and the Standard Model photon, and e is the electron charge. Similar to the Compton process, the dark-photon-electron interaction process can be expressed asγ+e−→e−+A′ . The corresponding Feynman diagram is shown in Fig. 2.The differential and total cross sections of the interaction are described in detail in the Appendix. Similar to the pseudoscalar particle scenario, with an electron energy of
Ee=3.5 GeV and the laser polarization direction perpendicular to the reaction plane, the cross sectionσlab(ma,θ0) for interactionγ+e−→e−+A′ is shown in Fig. 4, which shows theσlab(ma,θ0) values at colliding angles ofθ0=90∘ ,120∘ ,150∘ , and180∘ . Again, the head-to-head collision atθ0=180∘ has the maximum cross section compared with other collision angles. The cross section for the dark photon decreases as its mass increases. The dependence of the differential cross section on scattering angles with the dark photon is discussed in the Appendix.Figure 4. (color online) Cross sections of dark photons through the reaction
e+γ→e+ϕ . The cross sectionσlab(ma,θ0) is a function of the dark photon massmA′ , and colliding angleθ0 between dark photon and electron. The different color lines representθ0=90∘ (blue dash dot dot),120∘ (green dash dot),150∘ (red dash), and180∘ (black solid). The cross sectionσ is in barns. -
For a scalar DMP,
γ+e−→e−+ϕ , the corresponding effective Lagrangian can be expressed as [23]L=gϕeˉψeψeϕ,
(5) where the dimensionless
gϕe is the scalar-field-particle-electron coupling constant. The corresponding Feynman diagram is shown in Fig. 5.Figure 5. (color online) Scalar DMP generating cross sections through the reaction
e+γ→e+ϕ . The cross sectionσlab(ma,θ0) is a function of scalar DMP massmϕ , and colliding angleθ0 between the scalar DMP and electron. The different color lines represent180∘ (black solid),150∘ (red dash),120∘ (green dash dot), andθ0=90∘ (blue dash dot dot). The cross sectionσ is in barns.Using an electron energy of
Ee=3.5 GeV, the cross sectionσlab(ma,θ0) for interactionγ+e−→e−+ϕ can be calculated, and shown in Fig. 5, for the scattering anglesθ0=90∘ ,120∘ ,150∘ , and180∘ . For the same mass of a scalar DMP, the interaction cross section increases with increasing scattering angle and reaches its maximum atθ0=180∘ . For detection at other given angles, the expected cross section decreases as the scalar DMP mass increases.Further information on the interaction's differential and total cross sections are described in the Appendix, from which we observe that the cross sections have different dependences on the scattering angles. In the calculations, the polarization of the laser beam is parallel to the y-axis. The cross section of pseudoscalar particles is insensitive to the polarization of the incoming photon (Fig. A1). However, the cross sections of dark photons and scalar particles are highly dependent on the polarization of the laser beam (Fig. B1 & C1). The dark photons are primarily emitted parallel to the x-direction, which is similar to the Thomson scattering. In contrast, the scalar particles are emitted primarily parallel to the y-direction (Fig. C1). This is because the radiation of scalar particles is parallel to the laser’s polarization. The angular distributions can be used to distinguish different types of dark matter candidates since the distributions highly depend on particle types.
Figure A1. (color online) Differential cross sections of pseudoscalar particles through the reaction
e−+γ→e−+a . The mass is selected to be0 . The origin represents the z-direction. The exit angleθ of the DMP is represented by√x2+y2 . The unit isbarn/(g2ae⋅sr2) . -
The scattering cross section can be obtained directly from an S-matrix.
dσdΩ=164π21EkEp|vk−vp||→k′|2k′0p′0×|1|→k′|k′0+|→k′|−(→p+→k)⋅^k′p′0|(12∑|M2|),
(A1) dσ=12Ek2Ep|vk−vp|(∏fd3pf(2π)312Ef)×(12∑spin|M2|)(2π)4δ4(p+k−p′−k′),
where we replace
|M|2 with12∑spin|M|2 because the electron's spin is not controllable. It is the average of all possible electron spins. The z-axis is selected to be the moving direction of the incoming electron. The photon and electron are located in the x-z plane. The four momenta of all particles in the Cartesian coordinates arep= (p0,0,0,|→p|) ,k=(k0,|→k|sinθp,0,|→k|cosθp) , andk′=(k′0,|→k′) |sinθcosϕ,|→k′)|sinθsinϕ|,→k′)|cosθ) . The four momenta of the emitted particles,p′ andk′ , can be determined by solving the energy momentum conservation and energy momentum relation:p2=p′2=m2e andk′2=m2A .For different types of particles (pseudoscalar, scalar, and vector),
12∑spin|M|2 can be expressed as follows. -
There are two possible Feynman diagrams for the interaction
e−γ→e−ψa , Fig. 2. For a pseudoscalar particle,12∑spin|M|2 is12∑spin|M|2=−g2aee2(I((p+k)2−m2e)2+II((p+k)2−m2e)((p−k′)2−m2e)+III((p−k′)2−m2e)2),
The nominators are
I=4(p′⋅k)(k⋅p)+8(p′⋅k)(p⋅ϵ)(p⋅ϵ)−8(p′⋅ϵ)(k⋅p)(p⋅ϵ)+8(p′⋅p)(p⋅ϵ)(p⋅ϵ)−8m2e(p⋅ϵ)(p⋅ϵ),
II2=4(p⋅k)(p′⋅k)+4(p′⋅ϵ)(p′⋅k)(p⋅ϵ)−4(p′⋅ϵ)(p′⋅ϵ)(p⋅k)+4(p⋅ϵ)(p′⋅k)(p⋅ϵ)−4(p⋅ϵ)(p′⋅ϵ)(p⋅k)+8(p⋅ϵ)(p′⋅ϵ)(p′⋅p)−8m2e(p⋅ϵ)(p′⋅ϵ),
III=4(p⋅k)(p′⋅k)−8(p′⋅ϵ)(p′⋅ϵ)(k⋅p)+8(p′⋅ϵ)(p′⋅k)(p⋅ϵ)+8(p′⋅p)(p′⋅ϵ)(p′⋅ϵ)−8m2e(p′⋅ϵ)(p′⋅ϵ),
where
ϵ is the polarization direction of the photon. Here, it points in the y-direction. The differential cross section at different collision angles (θ=90∘,120∘,150∘ , and180∘ ) are shown in Fig. A1. As expected, because the energy of the electron is significantly larger than that of the photon, in the laboratory frame, the expected scalar DMPs are highly concentrated in the forward angle. This property will benefit experimental detection. -
Similar to the pseudoscalar scenario, the
12∑spin|M|2 of the vector particle (dark photon) can be expressed asiM=ˉu(p′)(−igA′eϵ∗ν(k′)γν)i(⧸p+⧸k+me)(p+k)2−m2e×(−ieϵμ(k)γμ)u(p)+ˉu(p′)(−ieϵμ(k)γμ)i(⧸p−⧸k′+me)(p−k′)2−m2e×(−igA′eϵ∗ν(k′)γν)u(p),
where
ϵν is the polarization direction of the photon. Here, it points in the y-direction. The nominators areI=8(p′⋅k)(p⋅k)−16(p⋅k)(p′⋅ϵ)(p⋅ϵ)+16(p′⋅k)(p⋅ϵ)(p⋅ϵ)+16(p′⋅p)(p⋅ϵ)(p⋅ϵ)−32m2e(p⋅ϵ)(p⋅ϵ),
II2=−8(p′⋅ϵ)(k⋅p)(p⋅ϵ)+8(p′⋅k)(p⋅ϵ)(p⋅ϵ)+8(p′⋅k)(p⋅ϵ)(p′⋅ϵ)−8(p′⋅ϵ)(k⋅p)(p′⋅ϵ)+16(p′⋅p)(p′⋅ϵ)(p⋅ϵ)−32m2e(p′⋅ϵ)(p⋅ϵ),
III=8(p′⋅k)(p⋅k)−16(k⋅p)(p′⋅ϵ)(p′⋅ϵ)+16(p′⋅k)(p⋅ϵ)(p′⋅ϵ)+16(p′⋅p)(p′⋅ϵ)(p′⋅ϵ)−32m2e(p′⋅ϵ).
The differential cross section at different collision angles (
θ=90∘,120∘,150∘ , and180∘ ) are shown in Fig. B1. As expected, because the energy of the electron is significantly larger than that of the photon, in the laboratory frame, the expected scalar DMPs are highly concentrated in the forward angle. This property will benefit experimental detection. -
Similar to the pseudoscalar scenario, the
12∑spin|M|2 of the scalar particles can be expressed asiM=ˉu(p′)(−igϕe)i(⧸p+⧸k+me)(p+k)2−m2e×(−ieϵμ(k)γμ)u(p)+ˉu(p′)(−ieϵμ(k)γμ)×i(⧸p−⧸k′+me)(p−k′)2−m2e(−iga)u(p),
Here, it points in the y-direction. The nominators are
I=4(p′⋅k)(p⋅k)+8(p⋅ϵ)(p′⋅k)(ϵ⋅p)−8(p⋅ϵ)(p′⋅ϵ)(p⋅k)+8(p⋅ϵ)(p⋅ϵ)(p′⋅p)+8m2e(p⋅ϵ)(p⋅ϵ),
II2=4(p′⋅k)(k⋅p)−4(p⋅ϵ)(p′⋅ϵ)(k⋅p)+4(p⋅ϵ)(p′⋅k)(p⋅ϵ)−4(p′⋅ϵ)(p′⋅ϵ)(p⋅k)+8(p′⋅p)(p′⋅ϵ)(p⋅ϵ)+8m2e(p′⋅ϵ)(p⋅ϵ),+4(p′⋅ϵ)(p′⋅k)(p⋅ϵ)
III=4(p⋅k)(p′⋅k)−8(p′⋅ϵ)(p′⋅ϵ)(k⋅p)+8(p′⋅ϵ)(p′⋅k)(p⋅ϵ)+8(p′⋅ϵ)(p′⋅ϵ)(p′⋅p)+8m2e(p′⋅ϵ)(p′⋅ϵ),
where
ϵ is the polarization direction of the photon. The differential cross section at different collision angles (θ=90∘,120∘,150∘ , and180∘ ) are shown in Fig. C1. As expected, because the energy of the electron is significantly larger than that of the photon, in the laboratory frame, the expected scalar particles are highly concentrated in the forward angle. This property will benefit experimental detection.
Searching for dark matter particles using Compton scattering
- Received Date: 2021-02-13
- Available Online: 2021-09-15
Abstract: The dark matter puzzle is one of the most important fundamental physics questions in the 21st century. There is no doubt that solving the puzzle will be a new milestone for human beings in achieving a deeper understanding of nature. Herein, we propose the use of the Shanghai laser electron gamma source (SLEGS) to search for dark matter candidate particles, including dark pseudoscalar particles, dark scalar particles, and dark photons. Our simulations indicate that, with some upgrading, electron facilities such as SLEGS could be competitive platforms in the search for light dark matter particles with a mass below tens of keV.