1. Review: Probing Low Energy Neutrino Backgrounds with Neutrino Capture on Beta Decaying Nuclei Cocco A, Magnano G and Messina M 2007 J. Cosmol. Astropart. Phys. JCAP06(2007)015
Kim, Hanbeom

2. Introduction
(Anti)neutrino capture on beta decaying nuclei (NCB inteaction)
Ordinary beta decay
Minimum gap of 2 𝑚 𝜈
Able to distinguish beta decay and NCB interaction
𝑀(𝑁)
𝑀(𝑁′)
𝑄 𝛽
𝐸 𝑒 ± = 𝑄 𝛽 − 𝐸 𝜈
NCB
𝐸 𝑒 ± = 𝑄 𝛽 + 𝐸 𝜈
Ordinary
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3. Introduction Neutrino mass: eV range is still allowed
Oscillation experiment: a lower limit – the order of 0.05 eV
Direct measurements in 3H decay: < 2 eV
Data from Cosmic Microwave Background anisotropies and Large Scale Structure power spectrum: 0.3 – 2 eV
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4. Introduction The relic (anti)neutrino Number density
𝑛 𝜈 ~50 𝑐 𝑚 −3 per flavor
Very small mean kinetic energy
Nonrelativistic: 6.5 𝑇 𝜈 2 / 𝑚 𝜈 , relativistic: 𝑇 𝜈
𝑇 𝜈 = 𝑇 𝛾 ~1.7∙ 10 −4 eV
Chemical potential
𝜇 𝑇 𝜈 ≤0.1
Too small to experimentally detect degeneracy due to chemical potential
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5. Neutrino cross section on 𝛽 ± decaying nuclei
NCB and its corresponding beta decay are essentially the same phenomenon.
The same invariant squared amplitude
Use beta decay formalism to derive NCB cross section expression
Long wavelength limit approximation
𝜌 𝜈 𝑅≪1
Holds for 𝐸 𝜈 ≲10 MeV
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6. Neutrino cross section on 𝛽 ± decaying nuclei
NCB integrated rate
𝜆 𝜈 = 𝜎 𝑁𝐶𝐵 𝑣 𝜈 𝑓 𝑝 𝜈 𝑑 3 𝑝 𝜈 2𝜋 3
𝑓 𝑝 𝜈 = exp 𝑝 𝜈 𝑇 𝜈 −1 (the particular case of relic neutrinos)
Cross section
𝜎 𝑁𝐶𝐵 𝑣 𝜈 = 𝐺 𝛽 2 𝜋 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈
Behrens H and BüringW, 1982 Electron Radial Wave Functions and Nuclear Beta Decay Clarendon Oxford.
𝐹 𝐸 : Fermi function, exp 𝐸− 𝐸 𝐹 𝑘𝑇 +1 −1
Energy
𝐸 𝑒 = 𝐸 𝜈 + 𝑄 𝛽 + 𝑚 𝑒 = 𝐸 𝜈 + 𝑚 𝜈 + 𝑊 0
𝑊 0 : corresponding beta decay endpoint
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7. Neutrino cross section on 𝛽 ± decaying nuclei
𝜆 𝜈 = 𝐺 𝛽 2 𝜋 𝑊 0 +2 𝑚 𝜈 ∞ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 𝐸 𝜈 𝑝 𝜈 𝑓 𝑝 𝜈 𝑑 𝐸 𝑒
Nuclear shape factor
An angular momentum weighted average of nuclear state transition amplitudes
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 = 𝑘 𝑒 , 𝑘 𝜈 ,𝐾 𝜆 𝑘 𝑒 [ 𝑀 𝐾 2 𝑘 𝑒 , 𝑘 𝜈 + 𝑚 𝐾 2 𝑘 𝑒 , 𝑘 𝜈 − 2 𝜇 𝑘 𝑒 𝑚 𝑒 𝛾 𝑘 𝑒 𝑘 𝑒 𝐸 𝑒 𝑀 𝐾 2 ( 𝑘 𝑒 , 𝑘 𝜈 ) 𝑚 𝐾 2 ( 𝑘 𝑒 , 𝑘 𝜈 )]
𝑘: radial wave function (=𝑗+1/2)
K: nuclear transition multipolarity: ( 𝑘 𝑒 − 𝑘 𝜈 ≤𝐾≤ 𝑘 𝑒 + 𝑘 𝜈 )
𝑀 𝐾 2 , 𝑚 𝐾 2 : nuclear form factor function
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8. Neutrino cross section on 𝛽 ± decaying nuclei
𝜆 𝜈 = 𝐺 𝛽 2 𝜋 𝑊 0 +2 𝑚 𝜈 ∞ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 𝐸 𝜈 𝑝 𝜈 𝑓 𝑝 𝜈 𝑑 𝐸 𝑒
𝜆 𝛽 = 𝐺 𝛽 2 2 𝜋 3 𝑚 𝑒 ∞ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 𝐸 𝜈 𝑝 𝜈 𝑓 𝑝 𝜈 𝑑 𝐸 𝑒
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 =𝐶 𝐸 𝑒 ,− 𝑝 𝜈 𝛽
Mean shape factor
𝐶 𝛽 ≡ 1 𝑓 𝑚 𝑒 ∞ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 𝐸 𝜈 𝑝 𝜈 𝑓 𝑝 𝜈 𝑑 𝐸 𝑒
𝑓 𝑡 = 2 𝜋 3 ln 2 𝐺 𝛽 2 𝐶 𝛽 , 𝑓≡ 𝑚 𝑒 ∞ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐸 𝜈 𝑝 𝜈 𝑓 𝑝 𝜈 𝑑 𝐸 𝑒
𝜎 𝑁𝐶𝐵 𝑣 𝜈 = 2 𝜋 3 ln 2 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 𝑓 𝑡 𝐶 𝛽
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9. Neutrino cross section on 𝛽 ± decaying nuclei
𝐴= 𝑓 𝐶 𝛽 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 = 𝑚 𝑒 𝑊 0 𝑝′ 𝑒 𝐸 𝑒 ′ 𝐹 𝑍, 𝐸 𝑒 ′ 𝐶 𝐸 𝑒 ′ , 𝑝 𝜈 ′ 𝛽 𝐸 𝜈 ′ 𝑝 𝜈 ′ 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 𝐸 𝜈 𝑝 𝜈 𝑑 𝐸 𝑒 ′
𝜎 𝑁𝐶𝐵 𝑣 𝜈 = 2 𝜋 2 ln 2 𝐴⋅ 𝑡 1 2
In some relevant cases, the evaluation of A is particularly simple.
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10. Superallowed transitions
Large superposition between initial and final nuclear states
→ The lowest known 𝑓 𝑡 value
0+ → 0+ transition
𝐶 𝐸 𝑒 , 𝑝 𝜈 = ​ 𝑉 𝐹 ​ 2 =<𝐅 > 2 =(𝑇− 𝑇 3 )(𝑇+ 𝑇 3 +1)
Jπ → Jπ, J≠0 transition
𝐶 𝐸 𝑒 , 𝑝 𝜈 = ​ 𝑉 𝐹 ​ ​ 𝐴 𝐹 ​ 2 =<𝐅 > 𝑔 𝐴 𝑔 𝑉 <𝐆𝐓>
𝑇, 𝑇 3 : isospin quantum numbers
𝑔: the axial (vector) coupling constant
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11. Superallowed transitions
𝐴= 𝑓 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒
𝜎 𝑁𝐶𝐵 𝑣 𝜈 = 2 𝜋 2 ln 2 𝑝 𝑒 𝐸 𝑒 𝐹 𝑍, 𝐸 𝑒 𝑓⋅ 𝑡 1 2
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12. Specific case of tritium
3H→ 3He
𝑄 𝛽 = keV, 𝑡 = years
<𝐅 > 2 =0.9987, <𝐆𝐓> = 3 ⋅(0.964±0.016)
𝐺 𝐹 = × 10 −5 GeV −2
𝑔 𝐴 =
𝑉 𝑢𝑑 =
Assuming a total 1.6% systematic uncertainty on the Gamow-Teller matrix element evaluation
𝜎 𝑁𝐶𝐵 ( 3 𝐻) 𝑣 𝜈 𝑐 = 7.7±0.2 × 10 −45 cm 2
Only experimental uncertainties on 𝑄 𝛽 & 𝑡 1 2
𝜎 𝑁𝐶𝐵 ( 3 𝐻) 𝑣 𝜈 𝑐 = 7.84±0.03 × 10 −45 cm 2
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13. Allowed transitions
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 = ​ 𝑉 𝐹 ​ ​ 𝐴 𝐹 ​ 2 +𝑂 𝑝 𝑒 𝑅 𝑂(𝛼𝑍)
If only the leading terms are taken into account:
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 =𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 =constant
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 /𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 ≅1
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14. K-th forbidden transitions
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 = ​ 𝐴 𝐹 𝐿𝐿− ​ 2 × 𝑛=1 𝐿 𝐵 𝐿 𝑛 𝜆 𝑛 𝑝 𝑒 𝑅 2 𝑛−1 𝑝 𝜈 𝑅 2 𝐿−𝑛
K: degree of forbidness, L=K+1
𝐵 𝐿 𝑛 :numerical coefficient, 𝜆 𝑛 : numerical function
If only the leading terms are taken into account:
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 =𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 =constant
𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 /𝐶 𝐸 𝑒 , 𝑝 𝜈 𝜈 ≅1
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15. Estimating: 𝑄 3 /𝐴 vs. 𝑄
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16. Estimating: 𝑓/𝑄 3 vs. 𝑄
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17. Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝐸 𝜈 ( 𝛽 − )
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18. Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝐸 𝜈 ( 𝛽 + )
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19. Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝑄
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20. NCB vs 𝛽 decay for relic neutrinos
In spite of no threshold, the ratio is very small.
𝜆 𝜈 𝜆 𝛽 = lim 𝑝 𝜈 →0 𝜎 𝑁𝐶𝐵 𝑣 𝜈 𝑛 𝜈 𝑡 ln⁡2 = lim 𝑝 𝜈 →0 2 𝜋 2 𝐴 𝑛 𝜈
Relic neutrinos have a very small mean momentum of order 𝑇 𝜈 .
The case of 3H
𝜆 𝜈 =0.66⋅ 10 −23 𝜆 𝛽
Too small!
The little mass of neutrino & the experimental energy solution → hard to distinguish NCB from standard beta events
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21. NCB vs 𝛽 decay for relic neutrinos
Optimistic scenario
An energy resolution Δ in the future
eV range neutrino mass
For the last beta decay electron energy bin 𝑊 0 −Δ< 𝐸 𝑒 < 𝑊 0
𝜆 𝜈 𝜆 𝛽 (Δ) ~2.2⋅ 10 −10 for Δ=0.2 eV, 𝑚 𝜈 =0.5 eV
Total event rate 𝜆 𝜈 𝑁 𝐴 𝑀[𝑔] 𝐴 =2.85⋅ 10 −2 𝜎 𝑁𝐶𝐵 𝑣 𝜈 𝑐 10 −45 cm 2 y −1 mol −1
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22. NCB vs 𝛽 decay for relic neutrinos
Gravitational clustering enlarges the massive neutrino density.
10~20 for 0.6 eV
3~4 for 0.3 eV
Nearly homogeneous for mass < 0.1 eV
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23. Conclusion 𝜎 𝑁𝐶𝐵 𝑣 𝜈 can be as large as 10 −42 ~ 10 −43 cm 2 𝑐
High event rate: 10 events/year with 100 g of 3H
Can be larger for 𝑚 𝜈 =0.3~0.7 eV and gravitational clustering: 20~150 events/year
A reasonable rejection of the background due to standard 𝛽 decay
Necessary to reach a sensitivity better than the value of 𝜈 masses
Ex) 𝑚 𝜈 =0.5 eV, Δ=0.1~0.2 eV
If smaller, the mass will be evaluated very hard.
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