Neutron Polarization Observables in $d(\vec {\gamma },\vec {n})p$ at Low Energies of Interest to Astrophysics

A model-independent theoretical analysis of neutron polarization observables in $d(\vec {\gamma },\vec {n})p$ using circularly polarized photons at the range of energies of interest to Big Bang Nucleosynthesis is presented. An investigation of various spin dependent observables is carried out including the isoscalar multipole amplitudes $M{{1}_{s}}$ and $E{{2}_{s}}.$ It is suggested that the measurement of neutron polarization in the final state at near threshold energies will be very useful to assess the contribution of isoscalar amplitudes at range of energies of interest to Big Bang Nucleosynthesis.

1 INTRODUCTION

Deuteron being the simplest nuclei with a proton–neutron pair, is an example for a bound two-body system that has been studied for many decades now. Precise knowledge of the reactions $d + \gamma \leftrightarrow n + p$ play very important role in Nuclear Physics, to understand Nucleon–Nucleon interactions, in Solar Physics, pp-chain reactions [1, 2] and in Astrophysics in sharpening the predictions on Big Bang Nucleosynthesis [3] along with the inputs from other reactions involving ³He, ⁴He, ⁷Li, and ⁷Be [4]. It is necessary to accurately measure the cross sections in nuclear reactions involving these light nuclei for a clear understanding of Big Bang Nucleosynthesis. Among these light elements produced in Big Bang Nucleosynthesis, deuterium is a very good indicator of cosmological parameters as its abundance is highly sensitive to primordial barium density and hence is referred to as “Baryometer.” More recently, the baryon density of the universe was estimated using the reaction rate of deuterium burning in $D(p,\gamma )$³He [5]. The primordial nucleosynthesis is initiated by np-fusion reaction. The two reactions $d + \gamma \leftrightarrow n + p$ are related by time reversal and hence cross sections are related by the principle of detailed balance. Experimental measurements have been carried out on the radiative capture of neutrons by protons [6, 7]; however, the importance of spin polarization study was highlighted to identify the contributions of isoscalar $M1$ and $E2$ amplitudes in [8, 9]. On the other hand, several experimental measurements [10–15] have been carried out on $d + \gamma \to n + p$ at the Duke Free Electron Laser Laboratory using High Intensity γ-ray Source (HIGS).

Although it was known quite early that the thermal neutron capture by protons is dominated by the isovector magnetic dipole amplitude $M{{1}_{v}}$, Breit and Rustgi [16] were the first to propose a polarized target-beam experiment to look for an isoscalar $M{{1}_{s}}$ amplitude in view of the then existing 10% discrepancy between theory and experiment. The suggestion was more or less ignored in view of the surprising accuracy with which the 10% discrepancy was explained [17] as due to Meson exchange currents (MEC). However, the measured values for analyzing powers in $p(\vec {n},\gamma )d$ as well as for neutron polarization in photodisintegration of the deuteron were both found to differ [18, 19] from theoretical calculations which included MEC effects. Rustgi, Vyas, and Chopra [20] drew attention to the unambiguous disagreement between experiment and theory on $d(\gamma ,n)p$ at photon energy 2.75 MeV which widens when two body effects are taken into account. It may be mentioned that measurements of neutron polarization at energies 7 to 15 MeV [21], abruptly depart from theoretical predictions at around 10 to 12 MeV.

The Experimental set of polarization observables for photo and electro disintegration of deuterons, spin response of the deuterons and Gerasimov–Drell–Hearn (GDH) sum rule [22, 23] have been studied by Arenhövel et al. [24–27]. We may also recall that for the differential cross section model calculations have led to traditional forms, Rustgi [28] and Patrovi [29] parameterization. Equivalently the cross section may be expressed empirically in terms of associated Legendre Polynomials. There are theoretical calculations using Effective Field Theory wherein the $M{{1}_{s}}$ [30] and $E{{2}_{s}}$ [31–33] contribution has been taken into consideration. On the other hand several theoretical studies [34, 35] on radiative capture of neutrons by protons have been carried out. Attention was focused on photon polarization in n–p fusion reaction [8]. In this paper it was shown that the photon polarization which arises due to the interference of isovector $M1$ amplitude with isoscalar $M1$ and $E2$ amplitudes can be studied using polarized beam and target experiment. On the other hand, the role of isoscalar amplitudes was highlighted in the theoretical study on analyzing powers in $d + \gamma \to n + p$ [36] with unpolarized photons. Theoretical analysis of the photodisintegration of deuterons with aligned deuteron targets and linearly polarized photon beams was carried out [37] in which an analysis of the experimental data of [38] was also presented. It is pertinent to mention that several photonuclear reactions on polarized deuterons [39, 40] are being studied at higher energies using linearly polarized photons at the VEPP-3 storage rings. In addition, studies have also been carried out on deuteron targets pertaining to tensor analyzing powers in the process of pion photoproduction [41–43].

Since the advent of polarization measurements, there are new mysteries and polarization of emerging neutron $(P_{y}^{'})$ in reaction $d(\gamma ,\vec {n})p$ is a good example. There is also a mention about the unsolved puzzle of $P_{y}^{'}$ in the work of Gilman and Gross [44]. Working in the framework of pion less effective field theory with dibaryons (d-EFT) a recent study [45] for neutron polarization showed a significant discrepancy with experiment [46], which points “the necessity of further studies, both experimental and theoretical of the spin observables in the $\gamma d \to np$ reaction” [45]. This discrepancy is observed at low energies, energies close to those of interest to Big Bang nucleosynthesis, which hinders our understanding of processes in the early Universe.

Though the discrepancies have been reduced by using different potentials and by following different methods, like Standard Nuclear Physics Approach (SNPA) and Effective Field Theory (EFT), it is evident that in low energy deuteron photodisintegration, these potentials cannot be distinguished from each other by experimental data. We also would like to quote Young et. al. [47] “Because the measured induced deuteron polarization $P_{y}^{'}$ in $\gamma + d \to \vec {n} + p$ and theoretical calculation using the phenomenological potential models show a discrepancy [45, 48, 49], it is desirable to have a model independent calculation.” Therefore, we propose to study the polarization of emerging neutron of the reaction $\vec {\gamma } + d \to \vec {n} + p$ using a model independent irreducible tensor formalism with initially circularly polarized photons.

2 MODEL INDEPENDENT THEORETICAL FORMALISM

We choose the right-handed Cartesian coordinate system to represent momentum of polarized photon, k along z-axis and linear polarization along x-axis in c.m. frame. We also choose the neutron momentum $p$ to have polar coordinates $(p,\theta ,\phi )$ in c.m. frame. We make use of natural units where $\hbar = 1$ and $c = 1.$ The left and right circular states of photon polarization is defined as ${{\mu }_{\mu }} = - \mu {{\varepsilon }_{\mu }}$ following [50]. We represent the   state of linear polarization to be $\mu = \pm 1$ by $\frac{1}{{\sqrt 2 }}\left( {{{\mu }_{{ + 1}}} + {{\mu }_{{ - 1}}}} \right)$. Following [36, 51], we express the reaction matrix as

$$M(\mu ) = \sum\limits_{s = 0}^1 \sum\limits_{\lambda = |l - s|}^{l + s} \left( {{{\mathcal{S}}^{\lambda }}(s,1) \cdot {{\mathcal{F}}^{\lambda }}(s,\mu )} \right)$$

(1)

in terms of irreducible tensor operators, $\mathcal{S}_{\nu }^{\lambda }(s,1)$ of rank λ in hadron spin space connecting the initial spin 1 state of the deuteron with the final singlet and triplet states, $s = 0$, 1 of the $n - p$ system in the continuum. Using the multipole expansion for $u = \mu {{e}^{{ik \cdot r}}}$ [50] and expressing the continuum states of the $n - p$ system in terms of partial waves, the irreducible tensor amplitudes $\mathcal{F}_{\nu }^{\lambda }(s,\mu )$ of the rank λ is given by the expression

$$\begin{gathered} \mathcal{F}_{\nu }^{\lambda }(s,\mu ) = \frac{1}{2}\sum\limits_{L = 0}^\infty \sum\limits_{l = 0}^\infty \sum\limits_{j = |l - s|}^{l + s} \sum\limits_{I = 0,1} {{(i)}^{{L - l}}}\left[ {1 - {{{( - 1)}}^{{l + s + I}}}} \right] \\ \times \;{{( - 1)}^{{j + L - l}}}[L][j{{]}^{2}}{{[s]}^{{ - 1}}}W(L1ls;j\lambda )F_{{ls;L}}^{{Ij}}\,f_{\nu }^{\lambda }(l,L,\mu ), \\ \end{gathered} $$

(2)

where $l,I$ denote the orbital angular momentum and isospin in the final state, j denotes the conserved total angular momentum, L denotes the total angular momentum of the photon and the shorthand notation $[L]$ stands for $\sqrt {2L + 1} .$ The partial wave multipole amplitudes $F_{{ls;L}}^{{Ij}}$ depend only on c.m. energy E, while the

$$f_{\nu }^{\lambda }(l,L,\mu ) = 4\pi \sqrt {2\pi } {{(i\mu )}^{{{{\pi }^{ + }}}}}C(l,L,\lambda ;{{m}_{l}}, - \mu ,\nu )$$

(3)

take care of the angular dependence and also the dependence on photon polarization. The projection operators

$${{\pi }^{ \pm }} = \frac{1}{2}\left[ {1 \pm {{{( - 1)}}^{{L - l}}}} \right]$$

(4)

assume either of the values 0, 1 such that, if ${{\pi }^{ + }} = 1$ implies ${{\pi }^{ - }} = 0$ and vice versa. The $F_{{ls;L}}^{{Ij}}$ denotes el-ectric ${{2}^{L}}$-pole amplitudes, if ${{\pi }^{ + }} = 1$ and magnetic ${{2}^{L}}$‑pole amplitudes, if ${{\pi }^{ - }} = 1.$

The differential cross section for $d(\vec {\gamma },n)p$ with circularly polarized photons is given, in c.m. frame, is given by

$$\frac{{d\sigma }}{{d\Omega }} = \frac{1}{6}\left( {M(\mu )\rho _{{\mu \mu '}}^{\gamma }{{M}^{\dag }}(\mu )} \right),$$

(5)

where ${{M}^{\dag }}(\mu )$ represents the complex conjugate of $M(\mu )$ and where ${{\rho }^{\gamma }}$ is given following [52], in the form

$${{\rho }^{\gamma }} = \frac{{{{s}_{0}}}}{2}[1 + {{\sigma }^{\gamma }}s] = \frac{{{{s}_{0}}}}{2}\left[ {\begin{array}{*{20}{c}} {1 + {{s}_{3}}}&{{{s}_{1}} - i{{s}_{2}}} \\ {{{s}_{1}} + i{{s}_{2}}}&{1 - {{s}_{3}}} \end{array}} \right],$$

(6)

where ${{\sigma }^{\gamma }}$ denote the Pauli matrices with rows and columns and the left and right circular polarizations. ${{s}_{0}},{{s}_{1}},{{s}_{2}},{{s}_{3}}$ may be referred to as the Stokes parameters, where ${{s}_{0}}$ denotes the total intensity.

In the region of interest to Big Bang Nucleosynthesis we restrict ourselves to lower order partial waves with $l = 0,1$ and $L = 1$. Thus we have two $M1$ amplitudes that is the isoscalar $M{{1}_{s}}$ amplitude leading to the final triplet state, isovector $M{{1}_{v}}$ amplitude leading to the final singlet state and three isovector $E1_{v}^{{j = 0,1,2}}$ amplitudes leading to the final $^{3}{{p}_{j}}$ state and an isoscalar $E{{1}_{s}}$ amplitude which is neglected. In addition to these we have considered the contribution from the isoscalar $E{{2}_{s}}$ amplitude with $L = 2$ because of its relevance to the study.

The density matrix, ρ characterizing the neutron polarization in the final state is then defined in terms of its elements,

$${{\rho }_{{{{m}_{n}}m_{n}^{'}}}} = \sum\limits_{K,q} \frac{1}{2}{{( - 1)}^{q}}[K]C\left( {\frac{1}{2}K{\kern 1pt} \frac{1}{2};m_{n}^{'}{\kern 1pt} - q{\kern 1pt} {{m}_{n}}} \right)\mathcal{P}_{q}^{K},$$

(7)

where

$$\mathcal{P}_{q}^{K} = \frac{1}{{3\sqrt 2 }}\sum\limits_{s,s',\lambda ,\lambda '} {{( - 1)}^{{s + s' + 1}}}{{[s]}^{2}}{{[s{\kern 1pt} ']}^{2}}[\lambda ][\lambda {\kern 1pt} ']W(s{\kern 1pt} '\lambda {\kern 1pt} 's\lambda ;1K),$$

$$W\left( {s{\kern 1pt} \frac{1}{2}{\kern 1pt} s{\kern 1pt} '{\kern 1pt} \frac{1}{2};\frac{1}{2}{\kern 1pt} K} \right)\left( {{{\mathcal{F}}^{\lambda }}(s,\mu ) \otimes {{\mathcal{F}}^{{\dag \lambda '}}}(s{\kern 1pt} ',\mu )} \right)_{q}^{K}.$$

(8)

The neutron polarization $\mathcal{P}$ is thus obtained on comparing $\rho$ with the standard form

$$\rho = \frac{1}{2}\left[ {1 + \sigma \mathcal{P}} \right].$$

(9)

The neutron polarization is obtained in terms of bilinear forms involving the irreducible tensor operators.

3 RESULTS AND DISCUSSION

3.1 Neutron Polarization, $P_{y}^{u}$ with Unpolarized Photons

If we have initially unpolarized photons and unpolarized deuterons, $P_{y}^{u}$ is given by

$$\mathcal{P}_{x}^{u} = - A\sin \theta \sin \phi - B\sin \theta \cos \theta \sin \phi ,$$

(10)

$$\mathcal{P}_{y}^{u} = A\sin \theta \cos \phi + B\sin \theta \cos \theta \cos \phi ,$$

(11)

$$\mathcal{P}_{z}^{u} = 0,$$

(12)

where A involves the imaginary part of interference between isovector $M{{1}_{v}}$ and isoscalar $M{{1}_{s}}$ with $E{{1}_{v}}$ amplitudes and B involves the terms with the imaginary parts of interference between electric multipole amplitudes. Explicitly,

$$A = 4\sqrt {\frac{2}{3}} {{\pi }^{2}}{\text{Im}}[E2_{s}^{ * }(4E1_{v}^{{j = 0}} - 3E1_{v}^{{j = 1}} - E1_{v}^{{j = 2}})]$$

$$ - \;4\sqrt {\frac{2}{3}} {{\pi }^{2}}{\text{Im}}[M1_{s}^{ * }(4E1_{v}^{{j = 0}} + 3E1_{v}^{{j = 1}} + 5E1_{v}^{{j = 2}})]$$

(13)

$$ - \;16{{\pi }^{2}}{\text{Im}}[E1_{v}^{{j = 1}}M1_{v}^{ * }],$$

$$B = - 8{{\pi }^{2}}{\text{Im}}[E1_{v}^{{j = 2*}}(2E1_{v}^{{j = 0}} + 3E1_{v}^{{j = 1}})].$$

(14)

3.2 Neutron Polarization with Circularly Polarized Photons

The values of ${{\mathcal{P}}_{x}},{{\mathcal{P}}_{y}},{{\mathcal{P}}_{z}}$ using left circularly polarized photons and unpolarized deuterons are given by the expressions

$$\mathcal{P}_{x}^{{(l)}} = a{{\pi }^{2}}\sin \theta \cos \phi + b{{\pi }^{2}}\sin \theta \cos \theta \cos \phi ,$$

(15)

$$\mathcal{P}_{y}^{{(l)}} = a{{\pi }^{2}}\sin \theta \sin \phi + b{{\pi }^{2}}\sin \theta \cos \theta \sin \phi ,$$

(16)

$$\mathcal{P}_{z}^{{(l)}} = \alpha {{\sin }^{2}}\theta + \beta \cos \theta + c.$$

(17)

It is interesting to note that the neutron polarization with right circularly polarized photons and unpolarized deuterons is equal and negative to left circularly polarized photons and unpolarized deuterons. Ex-plicitly,

$$a = 2\sqrt {\frac{2}{3}} {\text{Re}}[M1_{s}^{ * }(3E1_{v}^{{j = 1}} + E1_{v}^{{j = 2}})]$$

$$ - \;2\sqrt {\frac{2}{3}} {\text{Re}}[E2_{s}^{ * }(E1_{v}^{{j = 1}} - E1_{v}^{{j = 2}})]$$

(18)

$$ + \;\frac{8}{3}{\text{Re}}[M1_{v}^{ * }(2E1_{v}^{{j = 0}} + E1_{v}^{{j = 2}})],$$

$$\begin{gathered} b = - \frac{8}{3}{\text{Re}}[E1_{v}^{{j = 1 * }}(E1_{v}^{{j = 0}} - E1_{v}^{{j = 2}})] \\ + \;2({\text{|}}E1_{v}^{1}{{{\text{|}}}^{2}} - \;{\text{|}}E1_{v}^{2}{{{\text{|}}}^{2}}), \\ \end{gathered} $$

(19)

$$\begin{gathered} \alpha = \frac{2}{3}{\text{Re}}[E1_{v}^{{j = 1 * }}(2E1_{v}^{{j = 0}} - 3E1_{v}^{{j = 2}})] \\ - \;({\text{|}}E1_{v}^{{j = 1}}{{{\text{|}}}^{2}} - \;{\text{|}}E1_{v}^{{j = 2}}{{{\text{|}}}^{2}}), \\ \end{gathered} $$

(20)

$$\beta = \frac{8}{3}{\text{Re}}[M1_{v}^{ * }(E1_{v}^{{j = 0}} - E1_{v}^{{j = 2}})]$$

$$ + \;2\sqrt {\frac{2}{3}} {\text{Re}}[E2_{s}^{ * }(E1_{v}^{{j = 1}} - E1_{v}^{{j = 2}})]$$

(21)

$$ + \;2\sqrt {\frac{2}{3}} {\text{Re}}[M1_{s}^{ * }(E1_{v}^{{j = 1}} - E1_{v}^{{j = 2}})],$$

$$c = - 4\sqrt {\frac{2}{3}} {\text{Re}}(E{{2}_{s}}M1_{s}^{ * }) - 4\sqrt {\frac{2}{3}} {\text{Re}}(E{{2}_{s}}M1_{v}^{ * })$$

$$ - \;4\sqrt {\frac{2}{3}} {\text{Re}}(M{{1}_{s}}M1_{v}^{ * }) - \frac{2}{3}{\text{|}}E{{2}_{s}}{{{\text{|}}}^{2}} + \;{\text{|}}E1_{v}^{{j = 1}}{{{\text{|}}}^{2}}$$

(22)

$$ - \;{\text{|}}E1_{v}^{{j = 2}}{{{\text{|}}}^{2}} - \frac{2}{3}{\text{|}}M{{1}_{s}}{{{\text{|}}}^{2}} + 2{\text{Re}}[E1_{v}^{{j = 2}}E1_{v}^{{j = 1 * }}].$$

Experimental measurements can be carried out on neutron polarization for different photon energies. At each energy, for different azimuthal angles ϕ and polar angles θ the polarization observables can be studied. At the range of energies of interest for Big Bang Nucleosynthesis, by appropriately choosing $\theta = 90^\circ$ and $\phi = 90^\circ$, we have

$$\mathcal{P}_{y}^{u} = 0,$$

(23)

$$\mathcal{P}_{y}^{{(l)}} = a = {{a}_{0}}M1_{s}^{ * } + {{a}_{1}}E2_{s}^{ * } + {{a}_{2}}M1_{v}^{ * },$$

(24)

where

$${{a}_{0}} = 2\sqrt {\frac{2}{3}} \left( {3E1_{v}^{{j = 1}} + E1_{v}^{{j = 2}}} \right);$$

(25)

$${{a}_{1}} = - 2\sqrt {\frac{2}{3}} \left( {E1_{v}^{{j = 1}} - E1_{v}^{{j = 2}}} \right);$$

(26)

$${{a}_{2}} = \frac{8}{3}\left( {2E1_{v}^{{j = 0}} + E1_{v}^{{j = 2}}} \right).$$

(27)

From the first experimental observations of the splittings of the $E1$ p-wave amplitudes [38], the approximate values of the squares of the $3E1_{v}^{j}$ amplitudes can be read. ${{a}_{0}},{{a}_{1}}$, and ${{a}_{2}}$ have different linear combinations of the three amplitudes, rather than their modules squares. The combinations involve their phase values also. By assuming all of them to be real, the values of ${{a}_{0}}$, ${{a}_{1}}$, and ${{a}_{2}}$ at 14 and 16 MeV are given in Table 1. A detailed analysis has been provided in [37] and we have determined the values of ${{a}_{0}}$, ${{a}_{1}}$, and ${{a}_{2}}$ following the same method.

Table 1. Values of ${{a}_{0}},{{a}_{1}}$, and ${{a}_{2}}$

It is interesting to note from the above Table that the numerical value of the coefficient of isoscalar $M{{1}_{s}}$ amplitude, ${{a}_{0}}$ is approximately of the same order as the numerical value of ${{a}_{2}}$.

A complete analysis of the beam analyzing powers was carried out [53] on $d(\vec {\gamma },n)p$ using two states of linear polarization of the photon and two states of circular polarization of the photon. It was shown that in the differential cross section, $\cos \theta$ dependent term survives even if all the higher order multipoles are disregarded and this term arises due to interference of isoscalar $M{{1}_{s}}$ amplitude with isovector $E{{1}_{v}}$ amplitude. It was shown in the Table 1 of [53] that $M{{1}_{s}} \ne 0$ and that $E1_{v}^{j}$ for $j = 0,1,2$ cannot all be equal. This formalism may prove useful in assessing the contributions from these amplitudes in the final state as well.

In the theoretical study using model independent formalism [8] it was observed that the contribution of isoscalar $M1$ and $E2$ amplitudes will be highlighted if the polarization of neutrons and protons are orthogonal or opposite to each other. It was also suggested to go for a polarized target beam experiment, to verify the same. It is pertinent to mention that there are a few measurements on $d + \gamma \leftrightarrow n + p$ to study the reaction near threshold region. On the other hand, in the published paper on theoretical calculations on $d + \gamma \to n + p$ using effective field theory [47], there is also a mention about a proposal from Duke University to measure the neutron polarization in the final state (see P. N. Seo, Private communications (2016)—a reference mentioned in [47]).

In this contribution, we have studied the neutron polarization using model independent formalism for $d(\vec {\gamma },\vec {n})p$ reaction with unpolarized photons and with two circular polarization states of the photon. The component of neutron polarization along k is zero with initially unpolarized photons. It is worth pointing out that the neutron polarization using circularly polarized photons highlights the real part of the interference between the amplitudes, whereas neutron polarization studied with unpolarized photons contain the information about the imaginary part of the interference between the amplitudes. Thus, the neutron spin is sensitive to the initial polarization of the photon. It is worth mentioning that the neutron polarizations along the x and z directions vanish with the parity-conserving interactions but they can be non-vanishing with the parity-violating interactions. The experimental observation [38] at 14 and 16 MeV, that all the 3 $E1_{v}^{{j = 0,1,2}}$ amplitudes are not equal is quiet encouraging enough. Since the possible role of $M{{1}_{s}}$ and $E{{2}_{s}}$ amplitudes has been discussed by several authors using different formalism in the past, we feel it is necessary to carry out measurements on neutron polarization in addition to differential cross section. We hope that the experimental measurements on neutron polarization with circularly polarized photons will clarify the role of the isoscalar multipole amplitudes at near threshold energies.