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
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 2018-12-08 KIMS
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 2018-12-08 KIMS
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: 3.15 𝑇 𝜈 𝑇 𝜈 = 4 11 1 3 𝑇 𝛾 ~1.7∙ 10 −4 eV Chemical potential 𝜇 𝑇 𝜈 ≤0.1 Too small to experimentally detect degeneracy due to chemical potential 2018-12-08 KIMS
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 2018-12-08 KIMS
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 −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 2018-12-08 KIMS
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 2018-12-08 KIMS
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 1 2 = 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 1 2 𝐶 𝛽 2018-12-08 KIMS
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. 2018-12-08 KIMS
Superallowed transitions Large superposition between initial and final nuclear states → The lowest known 𝑓 𝑡 1 2 value 0+ → 0+ transition 𝐶 𝐸 𝑒 , 𝑝 𝜈 = 𝑉 𝐹 000 0 2 =<𝐅 > 2 =(𝑇− 𝑇 3 )(𝑇+ 𝑇 3 +1) Jπ → Jπ, J≠0 transition 𝐶 𝐸 𝑒 , 𝑝 𝜈 = 𝑉 𝐹 000 0 2 + 𝐴 𝐹 101 0 2 =<𝐅 > 2 + 𝑔 𝐴 𝑔 𝑉 2 <𝐆𝐓> 𝑇, 𝑇 3 : isospin quantum numbers 𝑔: the axial (vector) coupling constant 2018-12-08 KIMS
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 2018-12-08 KIMS
Specific case of tritium 3H→ 3He 𝑄 𝛽 =18.591 1 keV, 𝑡 1 2 =12.32 4 years <𝐅 > 2 =0.9987, <𝐆𝐓> = 3 ⋅(0.964±0.016) 𝐺 𝐹 =1.16637 1 × 10 −5 GeV −2 𝑔 𝐴 =1.2695 29 𝑉 𝑢𝑑 =0.97377 27 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 2018-12-08 KIMS
Allowed transitions 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 = 𝑉 𝐹 000 0 2 + 𝐴 𝐹 101 0 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 2018-12-08 KIMS
K-th forbidden transitions 𝐶 𝐸 𝑒 , 𝑝 𝜈 𝛽 = 𝐴 𝐹 𝐿𝐿−11 0 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 2018-12-08 KIMS
Estimating: 𝑄 3 /𝐴 vs. 𝑄 2018-12-08 KIMS
Estimating: 𝑓/𝑄 3 vs. 𝑄 2018-12-08 KIMS
Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝐸 𝜈 ( 𝛽 − ) 2018-12-08 KIMS
Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝐸 𝜈 ( 𝛽 + ) 2018-12-08 KIMS
Estimating: 𝜎 𝑁𝐶𝐵 𝑣 𝜈 vs. 𝑄 2018-12-08 KIMS
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 1 2 ln2 = 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 2018-12-08 KIMS
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 2018-12-08 KIMS
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 2018-12-08 KIMS
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. 2018-12-08 KIMS