1. Review: Prospects of detection of relic antineutrinos by resonant absorption in electron capturing nuclei. J D Vergados & Yu N Novikov, J. Phys. G: Nucl. Part. Phys. 41 (2014)
Kim, Hanbeom

2. Introduction The relic neutrino The cosmic neutrino background
Analogous to the cosmic background radiation
Neutrino decoupling about 1 second after the Big Bang,
1010 K
Very low average energy
𝐸 𝜈 ≈ eV (corresponding to T = 1.95 K)
(during the calculation, 𝐸 𝜈 ≈ eV )
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3. Introduction Antineutrino absorption Ordinary electron capture
Exothermic
Week interaction
𝜈 𝑒 + 𝑒 − + 𝑝 + →𝑛
𝜈 𝑒 +(𝐴,𝑍, 𝑍 𝑒 )→ (𝐴,𝑍−1, 𝑍 𝑒 −1) ∗
Ordinary electron capture
𝑝 + + 𝑒 − →𝑛+ 𝜈 𝑒
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4. The Formalism
Cross section for a neutrino of given velocity υ 𝜎 𝐸 𝜈 =2𝜋 1 𝑣 𝑀𝐸 𝐸 𝑥 nuc 2 𝜙 𝑒 2 𝐺 𝐹 2 2 𝛿 𝐸 𝜈 + 𝑚 𝜈 +Δ− 𝐸 𝑥 −𝑏 𝐸 𝜈 +𝑏≥Δ+ 𝑚 𝜈 𝑚 𝜈 ≈1 eV 𝜙𝑒: electron wavefunction Δ: mass difference of the two neutral atoms 𝑏: electron binding energy 𝐸𝑥: final state energy 𝐺𝐹: Fermi’s constant
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5. The Formalism
𝑁 𝜈 𝐸 𝜈 = Φ 𝜈 𝜎 𝐸 𝜈 = 𝑛 𝜈 ( 𝑟 0 )2𝜋 𝑀𝐸 𝐸 𝑥 nuc 2 𝜙 𝑒 2 𝐺 𝐹 2 2 𝛿 𝐸 𝜈 + 𝑚 𝜈 +Δ− 𝐸 𝑥 −𝑏 𝑁= 𝑁 𝜈 𝐸 𝜈 𝑓 𝐸 𝜈 𝑑 𝐸 𝜈 =𝜉 𝑛 𝜈 2𝜋 𝑀𝐸 𝐸 𝑥 nuc 2 𝜙 𝑒 2 𝐺 𝐹 2 2 𝑓( 𝐸 𝑥 +𝑏− 𝑚 𝜈 −Δ) 𝜉= 𝑛 𝜈 ( 𝑟 0 )/ 𝑛 𝜈 𝑓 𝐸 = 1 𝑘 𝑇 0 𝑒 − 𝐸 𝑘 𝑇 0 , 𝑘 𝑇 0 =1.5× 10 −4 eV 𝑛 𝜈 ( 𝑟 0 ): the density of neutrinos in our vicinity 𝑛 𝜈 : the relic neutrino density 𝑓(𝐸): neutrino energy distribution (gravitationally bound)
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6. The Formalism Neutrino capture
𝑁= 𝑁 𝜈 𝐸 𝜈 𝑓 𝐸 𝜈 𝑑 𝐸 𝜈 =2𝜋𝜉 𝑀𝐸 𝐸 𝑥 nuc 𝐺 𝐹 𝜙 𝑒 𝑘 𝑇 0 𝑒 − 𝐸 𝑥 +𝑏− 𝑚 𝜈 −Δ 𝑘 𝑇 𝑛 𝜈
Electron capture with the final state energy 𝐸 𝑥 ′
𝑁 𝑒−capture = 1 2𝜋 𝑀𝐸 𝐸 𝑥 ′ nuc 2 𝜙 𝑒 𝐺 𝐹 (Δ− 𝐸 𝑥 ′ −𝑏) 2
𝜆 𝑎 𝜆 𝑐 = 𝑁 𝑁 𝑒−capture = 2𝜋 𝑀𝐸 𝐸 𝑥 nuc 2 𝑀𝐸 𝐸 𝑥 ′ nuc 2 𝜉 𝑛 𝜈 (Δ− 𝐸 𝑥 ′ −𝑏) 2 𝑘 𝑇 0 𝑒 − 𝐸 𝑥 +𝑏− 𝑚 𝜈 −Δ 𝑘 𝑇 0
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7. The Formalism Let 𝜖= 𝐸 𝑥 +𝑏− 𝑚 𝜈 −Δ
𝑁 𝑁 𝑒−capture = 2𝜋 𝑀𝐸 𝐸 𝑥 nuc 2 𝑀𝐸 𝐸 𝑥 ′ nuc 2 𝜉 𝑛 𝜈 (Δ− 𝐸 𝑥 ′ −𝑏) 2 𝑘 𝑇 0 𝑒 − 𝜖 𝑘 𝑇 0
Choose 𝐸 𝑥 ′≠ 𝐸 𝑥 since 𝐸 𝑥 >∆−𝑏+ 𝑚 𝜈 , 𝐸 𝑥 <∆−𝑏
Given a very fine setting: 𝜖≈𝑘 𝑇 0 ≈ 10 −3 eV,Δ− 𝐸 𝑥 ′ −𝑏=100 keV and known parameters: 𝑛 𝜈 ≈56 cm −3 & assuming nuclear matrix elements of the same order
𝑁 𝑁 𝑒−capture =0.4× 10 −11 𝜉
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8. The Formalism The Uncertainty Principle
Suppose that there is a resonance in the final nucleus at an energy 𝜖 above the value Δ− 𝑏+ 𝑚 𝜈 with a width Γ= 𝜖 1+𝛿 , 𝛿≪1
𝛿 𝐸 𝜈 + 𝑚 𝜈 +Δ− 𝐸 𝑥 −𝑏 → 2 𝜋 Γ ( 𝐸 𝑥 −(Δ−𝑏+𝜖+ 𝑚 𝜈 )) 2 + ( Γ 2 ) 2
Integrate from 𝐸 𝑥 =Δ−𝑏+ 𝑚 𝜈 to 𝐸 𝑥 =Δ−𝑏+ 𝑚 𝜈 +Γ
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9. The Formalism Antineutrino capture
𝑁=2𝜋𝜉 𝑀𝐸 𝐸 𝑥 nuc 𝐺 𝐹 𝜙 𝑒 𝑘 𝑇 0 𝑒 − 𝐸 𝑥 +𝑏− 𝑚 𝜈 −Δ 𝑘 𝑇 𝑛 𝜈 →2𝜋𝜉 𝑀𝐸 𝐸 𝑥 nuc 𝐺 𝐹 𝜙 𝑒 𝑛 𝜈 1 𝑘 𝑇 0 𝐾 𝛽,𝛿
𝐾 𝛽,𝛿 =− 1 𝜋 𝑖 𝑒 − 1 2 𝑖 𝛿+1 𝛽−𝛽 − 𝐸 1 1− 𝑖 2 𝛿+1 𝛽 + 𝑒 𝑖 𝛿+1 𝛽 𝐸 1 1− 𝑖 2 𝛿+1 𝛽 − 𝐸 𝑖 𝛿+ 1+2𝑖 𝛽 + 𝐸 𝛿+1 −𝑖− 2 𝛿+1 𝛽 + 𝑒 𝑖 𝛿+1 𝛽 𝐸 1 1− 𝑖 2 𝛿+1 𝛽 − 𝐸 𝑖 𝛿+ 1+2𝑖 𝛽 − 𝐸 1 1− 𝑖 2 𝛿+1 𝛽 + 𝑒 𝑖 𝛿+1 𝛽 𝐸 1 1− 𝑖 2 𝛿+1 𝛽 − 𝐸 𝑖 𝛿+ 1+2𝑖 𝛽 + 𝐸 𝛿+1 −𝑖− 2 𝛿+1 𝛽 𝐸 ㅜ = 1 ∞ 𝑒 −𝑧𝑡 𝑡 𝑛 𝑑𝑡 ,𝛽= 𝜖 𝑘 𝑇 0
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10. The Formalism Electron capture
𝑁 𝑒−capture = 1 2𝜋 𝑀𝐸 𝐸 𝑥 ′ nuc 2 𝜙 𝑒 𝐺 𝐹 (Δ− 𝐸 𝑥 ′ −𝑏) 2 → 1 2𝜋 𝑀𝐸 𝐸 𝑥 ′ nuc 2 𝜙 𝑒 𝐺 𝐹 𝜖 0 2 Λ( 𝜖 𝜖 0 ,𝛿)
Λ 𝜖 𝜖 0 ,𝛿 = 4 𝜋 𝜖 𝜖 𝛿 𝛿+1 + tan − 𝛿+1 − tan − ≈ 4 𝜋 𝜖 𝜖 𝛿(1− 𝛿)
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11. The Formalism
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12. The Formalism
𝑁 𝑁 𝑒−capture = (2𝜋) 3 𝜉 𝑛 𝜈 𝜖 𝑘 𝑇 0 𝐾 𝛽,𝛿 Λ 𝜖 𝜖 0 ,𝛿
Average energy available for de- excitation after ordinary 𝑒−capture= Δ−𝑏+ 𝜖 , 𝜖 =𝜖 4 𝜋 3 5 𝛿(1− 𝛿)
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13. Some Results
𝑘 𝑇 0 ≈ 10 −3 eV, 𝑛 𝜈 ≈56 cm −3 → 𝑛 𝜈 𝜖 𝑘 𝑇 0 =1.6× 10 −13
ε/keV δ
0.02
0.04
0.06
0.08
0.10
0.4
N/N(e-capture)/10-17ξ
10.9
5.68
3.92
3.04
2.56
0.1
6.96
3.60
2.52
1.96
1.64
0.05
5.56
2.88
2.00
1.56
1.32
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14. Some Results
(7.7× 10 −22 , 5.8× 10 −23 , 1.4× 10 −23 ) obtained for Q=2.3, 2.5, 2.8 keV respectively for the target 163Ho
L. Lusignoli and M. Vignati, Phys. Lett. B 697, 11 (2011), arXiv:1012/0760 (hep-ph)
6.6× 10 −24 for tritium
A. Cocco, G. Magnamo, and M. Messina, JCAP 0706, 015 (2007), ; J. Phys. Conf. Ser (2008) 08214, arXiv:hep/ph/
However……
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15. Specific Example – 157TB 157Tb (71 y)→ 157Gd (g.s.) & 157Gd (54 keV)
𝑄 𝐸𝐶 =60.1 −62.9 keV (Δ), 𝑏 𝐾 =50.24 keV, 𝑏 𝐿1 =8.38 keV , 𝑏 𝐿2 =7.93 keV , 𝑏 𝐾 = 7.24 keV
Assume 𝑏 𝐿 =8 keV
Only L-capture to the excited state is allowed
The ratios of K- & L- capture = 7.36
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16. Specific Example – 157TB Branching Ratio
𝑁 𝑁 𝑒−capture = (2𝜋) 3 𝜉 𝑛 𝜈 𝜖 𝑘 𝑇 0 𝐾 𝛽,𝛿 Λ 𝜖 𝜖 0 ,𝛿
Adopt the view that the branching ratio for L-capture toe the 54 keV state < that dictated by the phase-space vector by a factor of 10
Ex) 𝜖=50 eV, 𝛿=0.02, 𝑚 𝜈 =1 eV→ 𝜖 =6 eV → 𝑇 𝜈 = 𝜖 − 𝑚 𝜈 =5 eV
(1/10)(5/104)2=2.5⨯10-8
(104 eV: K-capture energy to the g.s. of 157Gd
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17. Specific Example – 157TB
Branching Ratio (𝜉= 10 6 )
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18. Discussion
If exists a resonance around (Δ−𝑏+𝜖+ 𝑚 𝜈 ) with a width 𝜖 1+𝛿 , there can be a relatively large rate for 𝜈 absorption.
However, EC capture just below (Δ−𝑏+ 𝑚 𝜈 ) cannot be suppressed completely for 𝛿≠ 0.
If the final state is populated by EC capture, the average energy available for de- excitation is Δ−𝑏− 𝜖 ( 𝜖 =𝜖 4 𝜋 3 5 𝛿 1− 𝛿 ), smaller than (Δ−𝑏+𝜖+ 𝑚 𝜈 )
If 𝛿≤ 10 −10 & 𝜉 is large enough, 𝜆 𝑎 𝜆 𝑐 can be larger than 1.
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19. Conclusion
The possibility of observing relic neutrino with neutrino absorption in a nucleus strongly depends on the properties of the target nuclide.
Trituim, 187Re in beta-decay sector and 163Ho – considered as possible candidates
But the rate is too small
If some resonance conditions are met, a considerable enhancement of the associated rates can be obtained.
Mass difference = b
𝜖=relic neutrino total energy
157Tb→157Gd meets the conditions, while the nuclide above don’t.
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