APPENDIX I

APPENDIX I: NEUTRINO INITIATED PROCESS AND ATTENUATION DETAILS

For the \(\{ \} = 0\) subset of processes, define a dimensional transitional time, \({T_{fi}}\), (seconds) in the following manner: \[\begin{array}{c}{\tau _{HALF}}^{ – 1} \equiv T_{fi}^{ – 1}{F_S}{\rm{,}}\\{F_S} = {\int_1^{{y_m}} {ydy\left( {{y^2} – 1} \right)} ^{1/2}}{\left( {{y_m} – y} \right)^2},\end{array}\tag{I.1}\] where \({F_S}\) is the dimensionless density of final states function, \({\tau _{HALF}}\) is the summer half-life of the positron decay for the inverse NIP (seconds), \({y_m} = \Delta E/{m_e}{c^2}\), and \(\Delta E = \left[ {m{{(A.Z + 1)}_j} – m{{(A,Z)}_0}} \right]{c^2}\). Coulomb final state effects have been neglected in \({F_S}\). The reaction cross section in winter can be derived as follows: \[\begin{array}{l}v\left( {{E_\nu }} \right){\sigma _{fi}}\left( {{E_\nu }} \right) \equiv {\xi _{RN}}{V_W}T_{fi}^{ – 1}{F_0}\left( {{E_\nu }} \right){\rm{,}}\\{F_0}\left( {{E_\nu }} \right) = (x – {y_m}){[{(x – {y_m})^2} – 1]^{1/2}}{\rm{,}}\\v\left( {{E_\nu }} \right) = c{\left[ {1 – {{\left( {m_{\nu e}^W{c^2}/{E_\nu }} \right)}^2}} \right]^{1/2}}{\rm{,}}\\{\xi _{RN}} = {\left( {g_W^Wm_W^S/g_W^Sm_W^W} \right)^4},\end{array}\tag{I.2}\] where \({E_\nu }\) is the total energy of the massive neutrino, v is the massive neutrino velocity, \({V_W} = 2{\pi ^2}{\left( {\hbar /{m_{e_{}^{}}}c} \right)^3}\) is a characteristic volume, \({F_0}\) is the dimensionless density of states function, \(x = {E_\nu }/{m_e}{c^2}\), \({y_m}\) is the same as in Eq. (I.1). In summer, \({\xi _{RN}} = 1\), but in winter it will need to be extremely large.
For initial target states that are stable, there is experimental data for some of the lighter nuclei that allows the determination of \({T_{fi}}\). Consider the important NIP, \({\nu _e} + {\rm{C12}} \to {\rm{N12}} + {e^ – }\), where \(\Delta E = 16.827{\rm{ Mev}}\), \({\tau _{HALF}} = 0.01163{\rm{ s}}\), and then \[{T_{fi}} = 14940{\rm{ s}}\] for use in Eq. (I.2). For proton-rich states unstable in summer, the experimental data is sparse, so estimates are necessary.
In the case where {} corresponds to the final state H1 or He4, the reaction rate is given by Eq. (I.2) with \({F_0}\) replaced by \({F_1}\) defined as follows: \[{F_1} = \int_0^y {dx(1 + x){{(y – x)}^{1/2}}{{[{{(1 + x)}^2} – 1]}^{1/2}}} {\rm{,}}\tag{I.3}\] where \(y = ({E_\nu } – \Delta E)/{m_e}{c^2}\), and \(\Delta E = [m(A – a,Z – z + 1) + m(a,z) + {m_e} – m(A,Z)]{c^2}\). Until more accurate calculations are made, these approximations can be used. The density of states function is known given \({E_\nu }\), leaving \({T_{fi}}\) as the only unknown for each NIP. Note that the \({T_{fi}}\) for the \({F_1}\) group are not directly comparable to the \({T_{fi}}\) for the \({F_0}\) group since they represent different processes.
There is an approximation for all the NIPS that is useful—intermediate states. The NIP can be separated into two sequential steps: \[\begin{array}{l}{\nu _e} + {[A,Z]_0} \to {[A,Z + 1]_j} + {e^ – } \to \\{[A – a,Z – z + 1]_0} + \{ a,z\} + {e^ – }{\rm{.}}\end{array}\tag{I.4}\] Here, it is assumed that the intermediate state is created via Eq. (I.2) and then, if the intermediate state is unstable against proton emission, one or more protons are emitted (possibly in a cascade) until a nuclide stable against proton emissions results. Only the reaction cross section of the first step is needed for attenuation since any subsequent strong decays occur instantly on the cosmic time scale. There is no experimental data for positron decays of proton emitters in summer, since the extremely rapid strong decay dominates, i.e., \({T_{fi}}\) in Eq. (I.1) is unknown. In many cases the nuclear spin and parity are known for target and intermediate state, so one can estimate \({T_{fi}}\) based on whether the decay would be favored, allowed, or forbidden.
An important NIP is the attenuation of He4 (ground state = 0+ spin/parity). In that case, the intermediate state is Li4 (ground state = 2- spin/parity) and a transition would be first forbidden and not favored. Li4 has an excited state at 0.320 Mev (1- spin/parity), and that transition would be first forbidden and favored—100 times smaller \({T_{fi}}\) than the ground to ground transition. First forbidden \[{T_{fi}}\] is typically \({10^4}\)times the favored value. Thus, a reasonable estimate for the excited intermediate transition is \({T_{fi}} = 5 \times {10^7}\) seconds and \(\Delta E = 22.71\) Mev to be used in Eq. (I.1). The resulting cross section is smaller than for all other target nuclides, and that is why He4 is the only significant neutron-bearing nuclide left when the neutrinos are exhausted.
Using the chart of nuclides, it is possible to obtain the attenuation chain for the ground state of any nuclide. One follows the path of the first half of Eq. (I.4) until a proton emitter is encountered, then drop down a path of constant neutron number, (A-Z), to a proton non-emitter. No entry on the chart of nuclides counts as a proton emitter since that would be a state of negative binding energy. These chains are independent of the cross sections. By absorbing one neutrino, a specific target nuclide can be converted into only one specific resulting nuclide, but more than one target nuclide can contribute to a specific resulting nuclide. Two important chains will illustrate this process: O16-O15-O14-O13-C10-C9-He4 (summarized by \(6\nu (e) + O16 \to He4 + 12{p^ + } + 6{e^ – }\)) and C12-N12- C10-C9-He4. C10 can be made from both O13 and N12.
The intermediate state approximation has unknown accuracy, but it is a reasonable guide to the relative sizes of some of the cross sections—important for the overall attenuation pattern. There are many steps along the chains for which there is insufficient chart-data to use this approximation—one must use generic cross sections instead.
The physical time evolution during winter of the set, \({Y_j}(\tau ) = N(j,\tau )/{N_{RN}}\), is as follows: \[\begin{array}{c}d{Y_j}/d\tau = – {Y_j}/{T_j} + \sum\limits_n {{B_{jn}}} {Y_n}{\rm{/}}{T_n}{\rm{,}}\\{{\rm{T}}_j}{(\tau )^{ – 1}} = \int_0^\infty {d{E_\nu }{\rm{ }}{n_\nu }({E_\nu },\tau )} v\left( {{E_\nu }} \right){\sigma _j}\left( {{E_\nu }} \right){\rm{.}}\end{array}\tag{I.5}\] The combining matrix, \({B_{jn}}\), has either one or two nonzero entries (\( = 1\)) determined from the attenuation chains for each nuclide. The time constant, \({T_j}{\rm{(}}\tau {\rm{)}}\), is measured in seconds and depends on physical time. It can be simplified using Eq. (7.6) as follows: \[{T_j}{(\tau )^{ – 1}} = \Gamma {T_{fi}}{(j)^{ – 1}}\sum\limits_{k = 1}^K {{{\rm P}_k}(\tau )\,} F\left( {j,{E_{\nu k}}} \right),\tag{I.6}\] where the parameters defined after Eq. (I.2) are used to obtain \(\Gamma = {\xi _{RN}}{V_w}{N_{RN}}\), a dimensionless scaling factor for time duration. \(F\left( {j,{E_{\nu k}}} \right)\) is either \({F_0}\left( {{E_{\nu k}}} \right)\) as in Eq. (I.2), or \({F_1}\left( {{E_{\nu k}}} \right)\) as in Eq. (I.3) depending on the final state of the attenuation of nuclide j. The evolution of the neutrino spectrum is as follows: \[d{{\rm P}_k}/d\tau = – {{\rm P}_k}\Gamma \sum\limits_j {{Y_j}} \left( \tau \right){T_{fi}}{(j)^{ – 1}}F\left( {j,{E_{\nu k}}} \right),\tag{I.7}\] where \(F = {F_0}{\rm{ or }}{F_1}\) as in Eq. (I.6).
The coupled set, Eq. (I.5) and Eq. (I.7), can be solved numerically subject to the following boundary conditions: (1) \({P_k}\) and \({Y_j}\) are continuous across (red or blue) spring or fall transitions. (2) The set, \({E_{\nu k}}\), are changed when \({m_{\nu e}}\) changes. (3) \({Y_j}\left( 0 \right)\) is determined by the abundance of stable nuclides at the end of red or blue fall. (4) The relic neutrino density ratio, \[RNDR = {N_\nu }\left( {{\rm{winter – end}}} \right)/{N_\nu }\left( {{\rm{winter – start}}} \right)\], constrains the duration of winter. The proton and neutron densities are of interest: \[\begin{array}{l}{N_p} = {N_{RN}}\sum\limits_j {{Z_j}{Y_j}} ,\\{N_n} = {N_{RN}}\sum\limits_j {\left( {{A_j} – {Z_j}} \right){Y_j}} {\rm{.}}\end{array}\tag{I.8}\] Although it is not obvious, Eq. (I.5) and Eq. (I.7) satisfy Eq. (7.2), so there are two constraints when antineutrinos are neglected: \[\begin{array}{c}1 = \sum\limits_j {{A_j}{Y_j}} {\rm{,}}\\{\vartheta _{RN}} = \sum\limits_j {\left( {{A_j} – {Z_j}} \right){Y_j}} – \sum\limits_k {{P_k}{\rm{.}}} \end{array}\tag{I.9}\] I have chosen \({\vartheta _{RN}} = 0.125\) and \(RNDR = {10^{ – 4}}\) for this model. A numerical solution of Eq. (I.5) and Eq. (I.7) has been carried out for the model and parameters described above.
The results are summarized in four figures that appear in section VII(C.4), but the main observation is simple enough—almost all of winter is spent attenuating He4 and He3. All other nuclides could be neglected to a good approximation. The poorly known universal constant, \({N_{RN}},\) has no real impact on the pattern of evolution. This model requires \({\xi _{RN}}{N_{RN}} = 8.873 \times {10^{21}}\), so if \({N_{RN}} = 0.25\), then \({\xi _{RN}} = 3.549 \times {10^{22}}\). The parameter \({\xi _{RN}}\) represents a constraint on two of the new universal constants (\(m_W^W\)and \(g_W^W\)).