Neutrino and Antineutrino captures on $^{18}$O within QRPA models

Abstract

In this work we have evaluated the neutrino (NS) and antineutrino (AS) scattering cross sections on $^{18}$O and the inclusive muon capture rates at low energies within the Quasiparticle Random Phase Approximation (QRPA) and Projected QRPA (PQRPA) models. We present the first study of $^{18}$O$(\nu _e,e^-)^{18}$F and $^{18}$O$(\bar{\nu }_e,e^+)^{18}$N cross sections. These reactions are an important nuclear input for astrophysical calculations such as the CNO cycle. We have employed the weak formalism developed in Krmpotić et al. (Phys Rev C 71:044319, 2005) to analyze neutrino/antineutrino-nucleus scattering. Within this formalism, the nuclear residual interaction is described by $\delta$-force previously employed to evaluate single and double beta decays in QRPA models. The adopted parametrization leads to good results for the inclusive muon capture: we reproduce the available experimental data for the rate and the Gamow–Teller strength in $^{18}$F. We compared our results for the NS and AS cross sections on $^{18}$O with other theoretical evaluations on $^{16}$O, since there are no experimental data available for the processes on $^{18}$O. We noticed that the cross sections present a similar behavior as a function of the neutrino/antineutrino energy. For NS and AS we observed that the PQRPA procedure yields cross sections smaller than QRPA. We show that the Pauli blocking has an important role in the distribution of the partial contributions and the presence of two neutrons over the closed shell yields to higher NS cross sections than the AS cross sections. For NS and AS, the largest contribution comes from allowed and first-forbidden transitions, respectively.

Appendices

Appendix A: Relationship F and G NME

In this appendix, we give the relationship for G particle-particle (pp) and F particle-hole (ph) nuclear matrix elements with a general residual interaction V (following the conventions from the Unkelbach’s thesis [51]):

$$\begin{aligned} V=\frac{1}{4} \sum _{abcd}V_{abcd}^{as} \textbf{a}^{\dagger }_a \textbf{a}^{\dagger }_b \textbf{a}_c \textbf{a}_d, \end{aligned}$$

(44)

with the antisymmetrized matrix element:

$$\begin{aligned} V_{abcd}^{as}=_a\langle ab \vert V\vert cd \rangle _a = \langle ab \vert V\vert cd \rangle -\langle ab \vert V\vert dc \rangle , \end{aligned}$$

(45)

and $\vert cd \rangle _a = \frac{1}{\sqrt{2}}[\vert c \rangle \vert d \rangle -\vert d \rangle \vert c \rangle .$ Here, we use the coupling $\vert a \rangle \equiv \vert (\frac{1}{2}\ell _a)j_a \rangle$.

The particle-particle (PP) coupling is:

$$\begin{aligned} \vert ab,J \rangle&=\sum _{m_a m_b} \langle j_a m_a j_b m_b \vert JM \rangle \vert j_a m_a j_b m_b \rangle , \end{aligned}$$

(46)

and the particle-hole (ph) coupling is:

$$\begin{aligned} \vert ab^{-1},J \rangle&=\sum _{m_am_b}(-1)^{j_a-m_a} \langle j_a m_a j_b -m_b \vert JM \rangle \nonumber \\&\quad \vert j_a m_a(j_bm_b)^{-1} \rangle , \end{aligned}$$

(47)

where $\vert a^{-1} \rangle =\textbf{a}\vert 0 \rangle =\vert (j_am_a)^{-1} \rangle =(-1)^{j_a-m_a} \vert j_a-m_a \rangle$.

The antisymmetrized pp state is:

$$\begin{aligned} \vert ab,J \rangle _a =\frac{1}{\sqrt{1+\delta _{a,b}}}\sum _{m_am_b} \langle j_a m_a j_b m_b \vert JM \rangle \vert j_am_aj_bm_b \rangle , \end{aligned}$$

(48)

with the property: $\vert ab,J \rangle _a =(-1)^{j_a+j_b-J+1} \vert ba,J \rangle _a$.

The pp matrix element is defined as:

$$\begin{aligned} G(abcd;J)&=\sum _{m_am_bm_cm_d} \langle j_a m_a j_b m_b \vert JM \rangle \nonumber \\&\qquad \langle j_c m_c j_d m_d \vert JM \rangle V_{abcd}, \end{aligned}$$

(49)

and the ph matrix element as:

$$\begin{aligned} F(abcd;J)= & {} \sum _{m_am_bm_cm_d} (-1)^{j_a-m_a} (-1)^{j_c-m_c}\nonumber \\{} & {} \qquad \langle j_a m_a j_b -m_b \vert JM \rangle \nonumber \\{} & {} \quad \times \langle j_c m_c j_d -m_d \vert JM \rangle V_{abcd}^{PH}, \end{aligned}$$

(50)

where

$$\begin{aligned} V_{abcd}^{ph} =\langle ab^{-1} \vert V\vert cd^{-1} \rangle = \langle ad \vert V\vert bc \rangle =V_{adbc}^{pp}. \end{aligned}$$

(51)

Thus

$$\begin{aligned} F(abcd;J)= & {} \sum _{m_am_bm_cm_d} (-1)^{j_a-m_a} (-1)^{j_c-m_c}\nonumber \\{} & {} \quad \langle j_am_aj_b-m_b \vert JM \rangle \nonumber \\{} & {} \quad \times \langle j_cm_cj_d-m_d \vert JM \rangle V_{adbc}. \end{aligned}$$

(52)

The pp and the ph matrix elements are related as:

$$\begin{aligned} F(abcd;J)= & {} -\sum _{J'}\hat{J}^{'2} \left\{ \begin{array}{llll} {j_a}&{}{j_d}&{}{J'}\\ {j_c}&{}{j_b}&{}{J} \end{array}\right\} G(dabc;J'). \end{aligned}$$

(53)

Appendix B: NME for $\delta$-force

The nuclear matrix elements for the $\delta$-residual interaction are taken from Ref. [52, 53]. They were used in several works describing single and double beta decay as well semileptonic interactions as muon capture decay and neutrino capture [3, 4, 24, 45,46,47]. Here, we give a brief summary of main results. The residual interaction $\delta -$force (in units of MeV.fm$^3$)reads

$$\begin{aligned}&V=-4\pi (v_t P_t+v_s P_s )\delta (r), \end{aligned}$$

(B11)

where $v_t$ and $v_s$ are the singlet and triplet couplings constants in the pp and ph channels, respectively. The argument $r=\vert \textbf{r}_1 -\textbf{r}_2 \vert$, and $P_t$ and $P_t$ are the spin and isospin-projection operators

$$\begin{aligned} P_t= \Pi _1 \cdot \Lambda _0&,&P_s= \Pi _0 \cdot \Lambda _1, \end{aligned}$$

(B12)

with

$$\begin{aligned} \Pi _0= & {} \frac{1}{2}(1-P_{\sigma })=\frac{1}{4}(1-\varvec{\sigma }\cdot \varvec{\sigma }); \nonumber \\ \Pi _1= & {} \frac{1}{2}(1+P_{\sigma })=\frac{1}{4}(3+\varvec{\sigma }\cdot \varvec{\sigma }), \nonumber \\ \Lambda _0= & {} \frac{1}{2}(1-P_{\tau })=\frac{1}{4}(1-\varvec{\tau }\cdot \varvec{\tau }); \nonumber \\ \Lambda _1= & {} \frac{1}{2}(1+P_{\tau })=\frac{1}{4}(3+\varvec{\tau }\cdot \varvec{\tau }), \end{aligned}$$

(B13)

are the spin-projection $\Pi _S$ and isospin-projection operators $\Lambda _T$.

From pg. 642 of Ref. [52] the matrix element for the $\delta$-force between non-identical particles are

$$\begin{aligned}{} & {} 4\pi \langle abJ \vert \delta (\vert \textbf{r}_1 -\textbf{r}_2 \vert ) \vert cdJ \rangle _{dir} \nonumber \\{} & {} \quad =\frac{1}{2}R\hat{j}_a \hat{j}_b\hat{j}_c\hat{j}_d(-1)^{j_a+j_b+j_c+j_d} \nonumber \\{} & {} \quad \times \left[ \left( \begin{array}{lll} {j_a}&{} {j_b}&{} {J}\\ {\frac{1}{2}}&{}{\frac{1}{2}}&{}{-1}\end{array}\right) \left( \begin{array}{lll} {j_c}&{}{j_d}&{}{J}\\ {\frac{1}{2}}&{}{\frac{1}{2}}&{}{-1}\end{array}\right) \right. \nonumber \\{} & {} \quad \left. -(-1)^{\ell _b+\ell _d+j_b+j_d} \left( \negthinspace \begin{array}{ccc} j_a&{}j_b&{}J\\ \frac{1}{2}&{}-\frac{1}{2}&{}0\end{array}\right) \left( \negthinspace \begin{array}{ccc} j_c&{}j_d&{}J\\ \frac{1}{2}&{}-\frac{1}{2}&{}0\end{array}\right) \right] , \nonumber \\ \end{aligned}$$

(B14)

and

$$\begin{aligned}{} & {} 4\pi \langle abJ \vert \delta (\vert \textbf{r}_1 -\textbf{r}_2 \vert ) \varvec{\sigma }\cdot \varvec{\sigma }\vert cdJ \rangle _{dir}\nonumber \\{} & {} \quad =\frac{1}{2}R\hat{j}_a \hat{j}_b\hat{j}_c\hat{j}_d(-1)^{j_a+j_b+j_c+j_d} \nonumber \\{} & {} \qquad \times \left[ \left( \negthinspace \begin{array}{ccc} j_a&{}j_b&{}J\\ \frac{1}{2}&{}\frac{1}{2}&{}-1\end{array}\right) \left( \negthinspace \begin{array}{ccc} j_c&{}j_d&{}J\\ \frac{1}{2}&{}\frac{1}{2}&{}-1\end{array}\right) \right. \nonumber \\{} & {} \qquad \left. +\left( 1+2(-1)^{\ell _a+\ell _b+J}\right) (-1)^{\ell _b+\ell _d+j_b+j_d}\right. \nonumber \\{} & {} \left. \qquad \left( \negthinspace \begin{array}{ccc} j_a&{}j_b&{}J\\ \frac{1}{2}&{}-\frac{1}{2}&{}0\end{array}\right) \left( \negthinspace \begin{array}{ccc} j_c&{}j_d&{}J\\ \frac{1}{2}&{}-\frac{1}{2}&{}0\end{array}\right) \right] , \nonumber \\ \end{aligned}$$

(B15)

where

$$\begin{aligned} R\equiv R(abcd)=\int R_aR_bR_cR_d r^2dr. \end{aligned}$$

(B16)

Introducing the shorthand notation:

$$\begin{aligned} g(abJ)= & {} \sqrt{4\pi }\hat{J}^{-1}(-1)^{j_a+j_b+\ell _a+1} \langle a\Vert Y_J\Vert b \rangle \nonumber \\= & {} (-1)^{j_a-\frac{1}{2}+\ell _a}\hat{j}_a\hat{j}_b \left( \negthinspace \begin{array}{ccc} j_a&{}j_b&{}J\\ \frac{1}{2}&{}-\frac{1}{2}&{}0\end{array}\right) , \nonumber \\ h(abJ)= & {} \sqrt{4\pi }\hat{J}^{-1}(-1)^{\ell _a} \langle a\Vert [\varvec{\sigma }\times Y_L]_{J=L}\Vert b \rangle \nonumber \\= & {} (-1)^{j_a+j_b}\hat{j}_a\hat{j}_b \left( \negthinspace \begin{array}{ccc} j_a&{}j_b&{}J\\ \frac{1}{2}&{}\frac{1}{2}&{}-1\end{array}\right) . \nonumber \end{aligned}$$

Finally, the matrices F and G for $\delta$-force residual interaction read

$$\begin{aligned}{} & {} G(pnp'n';J) \nonumber \\{} & {} \quad =-v_s\frac{1}{4}R \left( 1+(-1)^{\ell _p+\ell _n+J}\right) g(pnJ)g(p'n'J) \nonumber \\{} & {} \qquad -v_t\frac{1}{4}R [ 2h(pnJ)h(p'n'J)\nonumber \\{} & {} \qquad + \left( 1-(-1)^{\ell _p+\ell _n+J}\right) g(pnJ)g(p'n'J) \nonumber \\{} & {} \quad \times g(pnJ)g(p'n'J)], \end{aligned}$$

(B17)

$$\begin{aligned}{} & {} F(pnp'n';J)\nonumber \\{} & {} \quad =\frac{R}{4}(-1)^{\ell _p+\ell _{p'}} \left\{ [v_s+v_t] h(pnJ)h(p'n'J)\right. \nonumber \\{} & {} \quad + \left. \left[ -v_s(-1)^{\ell _p+\ell _n+J} +v_t\left( 2+(-1)^{\ell _p+\ell _n+J}\right) \right] \right. \nonumber \\{} & {} \quad \left. \times g(pnJ)g(p'n'J)\right\} . \end{aligned}$$

(B18)

According to the hypothesis of Halbleib and Sorensen on the particle-hole force [65], the parameters $v_s$ and $v_t$ of the previous equations are different. For pp we have the t and s parameters defined in Eq. (23) related with $v_t$ and $v_t$ of Eq. (B17). While for the ph channel, we have the ph-coupling constants, $v_s^\textrm{ph}$ and $v_t^\textrm{ph}$ for Eq. (B18).