Neutron Beta Decay and Correlation Coefficients

Chapter I: Introduction

1.3 Neutron Beta Decay

1.3.2 Neutron Beta Decay and Correlation Coefficients

Taking the fully generalized Hamiltonian of the weak interaction [LY56], we can express the fully generalized differential decay rate of the (polarized) neutron as a function of the emitted electron’s energy, momentum and spin, the neutrino’s momentum, and the spin of the decay proton (see [JTW57a; JTW57b] for a complete description). The full description of the differential decay rate simplifies greatly when we note that, in UCNA, we have polarized neutrons (initial state), and we integrate over all other kinematic and polarization parameters except the decay electron’s momentum (notably, we are insensitive to the decay proton and neutrino).

Under these conditions, we obtain a final simplified differential decay rate given by 𝑑Γ

𝑑𝐸_𝑒𝑑Ω𝑒

= 1 2

𝐹(±𝑍 , 𝐸_𝑒) (2𝜋)⁴

𝑝_𝑒𝐸_𝑒(𝐸

0−𝐸 𝑒)²𝜉

"

1+𝑏 𝑚_𝑒

𝐸_𝑒 +𝐴

h ®𝐽i 𝐽

· 𝑝®_𝑒 𝐸_𝑒

#

(1.12) where 𝐸_𝑒 is the total decay eletron energy, 𝑚_𝑒 is the rest mass of the electron, 𝑝_𝑒 is the momentum of the electron, 𝐽®here represents the spin state of the initial decay neutron, and the quantity on the left hand side is the full differential decay

rate of the free (polarized) neutron. In addition, 𝐹(±𝑍 , 𝐸_𝑒) is the Fermi function which is a shape correction to the decay electron spectrum that arises from Coulomb interactions [Wil82].

The individual constants,𝑏and𝐴in this case, are the correlation coefficients that, at this stage, must be experimentally determined and provide insights into the underly- ing physics in the neutron𝛽-decay. We note there are several other decay correlation

coefficients in the fully general decay rate such as𝑎, 𝑐, 𝐵, 𝐷 , 𝐽 , 𝐼 , 𝐾⁰, 𝑀 , 𝑁 , 𝑄 , 𝑅, 𝑆, 𝑇 , 𝑈 , 𝑊 , 𝑉, and more, which are not shown in equation 1.12.

1.3.2.1 The Asymmetry Term, 𝐴

In the decay rate in equation 1.12, the𝐴coefficient is called the “asymmetry” term and it represents the fractional decay rate of electrons whose momentum is aligned vs anti-aligned with the neutron polarization (initial spin direction). The asymmetry analysis is the topic of [Men14; Bro18] and indeed the work in the later chapters of this dissertation build upon the analysis procedures, studies, and insights derived in those works.

In equation 1.12, the asymmetry term 𝐴 does not account for energy dependent corrections and is traditionally distinguished from the experimentally measured asymmetry by redefining it as 𝐴

0. The experimentally measured asymmetry which is energy dependent is redefined as 𝐴. The connection between the two is given by

𝐴(𝐸_𝑒)= 𝑃_𝑛𝐴

0𝛽 < cos𝜃 > (1.13) where𝑃_𝑛is the neutron polarization, 𝛽 = ^𝑣

𝑐 where𝑣is the electron velocity, and 𝑐 is the speed of light. < cos𝜃 >≈ ¹

2 in the UCNA apparatus.

The asymmetry can be expressed in terms of the coupling constants of the vector and axial-vector weak interaction components

𝐴0=−2

𝜆(𝜆+1)

1+3𝜆2 (1.14)

where𝜆is defined in equation 1.11 and for the neutron this result is derived by taking 𝑀_𝐹 = 1 (Fermi matrix element) and 𝑀_𝐺𝑇 =

√

3 (Gamow-Teller matrix element).

These simple coefficients arise from the simple structure of the neutron and point towards one of the advantages of using neutrons for studies: the lack of complicated nuclear𝛽-decay theory corrections.

Given the relation in equation 1.14, high-precision measurements of𝐴then become a powerful probe of the value of𝜆and hence the relative coupling strengths of the

vector vs axial-vector components of the weak interaction. In fact,𝐴measurements currently provide the most precise measurements of𝜆with𝜆= _𝑔^𝑔𝐴

𝑉

=−1.2783(22) [Bro+18]. These measurements can be combined with the neutron lifetime, 𝜏_𝑛, which is also sensitive to𝜆in order to test the electroweak Standard Model [Gon+21].

1.3.2.2 The Fierz Interference Term,𝑏

The𝑏coefficient in equation 1.12 is called the Fierz interference term and manifests itself in the decay rate as an energy shift in the electron energy spectrum. The description of𝑏and analysis associated with extracting a value for𝑏from the various UCNA datasets comprise the work in Chapter 4. The topic of Fierz interference in the UCNA experiment is covered in [Hic13] and the work in this dissertation builds upon it and presents an alternative extraction methodology.

The Fierz interference term,𝑏, with its multiplicative factor,𝜉, can be expressed as [JTW57a]

𝑏 𝜉 =±2𝛾Re

|𝑀_𝐹|²𝜆_𝐽0𝐽(𝐶_𝑆𝐶^∗

𝑉 −𝐶⁰

𝑆𝐶^0∗

𝑉) + |𝑀_𝐺𝑇|²(𝐶_𝑇𝐶^∗

𝑉 −𝐶⁰

𝑇𝐶^0∗

𝐴)

(1.15) where the + (-) sign indicates𝛽⁻(𝛽⁺) decay,𝛾 =

√

1−𝛼2𝑍2,𝛼is the fine structure constant,𝑍 is the atomic number, and𝜆_𝐽0𝐽 is given by

𝜆_𝐽0𝐽 =

















1 𝐽 → 𝐽⁰=𝐽−1

𝐽+11 𝐽 → 𝐽⁰=𝐽

−𝐽

𝐽+1 𝐽 → 𝐽⁰=𝐽+1

















(1.16)

where 𝐽 , 𝐽⁰ are the angular momenta of the original and final nuclei respectively, and𝜉is given by

𝜉 = |𝑀_𝐹|²(|𝐶_𝑆|²+ |𝐶_𝑉|²+ |𝐶⁰

𝑆|²+ |𝐶⁰

𝑉|²) + |𝑀_𝐺𝑇|²(|𝐶_𝑇|²+ |𝐶_𝐴|²+ |𝐶⁰

𝑇|²+ |𝐶⁰

𝐴|²) (1.17) where |𝑀_𝐹|² and |𝑀_𝐺𝑇|² are the conventional Fermi and Gamow-Teller nuclear matrix elements, the subscripts 𝑆, 𝑇 , 𝑉 , 𝐴refer to scalar, tensor, vector, and axial- vector (see table 1.2), the𝐶_𝑖, 𝐶⁰

𝑖 denote coupling constants (see [LY56] for further details on the couplings, [JTW57a] for details on𝑏).

Assuming a V-A nature of the weak interaction, we can see from equation 1.15 that the 𝑏 term is identically 0. Hence, searches for Fierz interference represent probes of beyond Standard Model physics and, in particular for neutron 𝛽-decay,

would represent probes of scalar and tensor couplings in the weak interaction.

Measurements of Fierz interference can be related to beyond Standard Model scalar and tensor couplings by using effective field theories as described most recently in [GNS19] and the references therein describe other physically motivated theories for non-zero Fierz interference in neutron decay.

For the neutron, the Fierz interference term simplifies to become 𝑏= 𝑏_𝐹 +3𝜆²𝑏_𝐺𝑇

1+3𝜆2 (1.18)

where 𝑏 here specifically refers to the Fierz interference for the neutron, 𝑏_𝐹 rep- resents the Fermi component of the Fierz interference which is sensitive to scalar interactions, and 𝑏_𝐺𝑇 represents the Gamow-Teller component of the Fierz inter- ference which is sensitive to tensor interactions. This simplification arises due to the simple nature of the neutron and the lack of complicated nuclear structure to consider.

1.3.2.3 The Neutron Lifetime,𝜏

As described previously, the free neutron will decay into a proton via equation 1.4 due to the favorable energy differences in initial and final state. The lifetime of this decay is also partially covered in Chapter 5 when we explore the potential for a neutron to decay via an unknown dark matter decay channel. Thus, in this introduction, we provide a short overview of the neutron mean lifetime.

The neutron lifetime measurement is the topic of [Fri22] and additional details are provided there. The neutron lifetime can be roughly calculated using the Feynmann diagram in figure 1.2 and the vertices given in equations 1.8 and 1.10.

In the low-momentum limit (𝑄 𝑚_𝑊), the propagator term of the decay can be approximated as

𝑖𝑔𝜇 𝜈

𝑚²

𝑊

which yields the following matrix element for the decay:

𝑀 =𝑖𝜓¯_𝑛𝐾_𝜇𝜓_𝑝 𝑖𝑔_{𝜇 𝜈}

𝑚²

𝑊

¯ 𝜓_𝑒𝐽^𝜈𝜓_𝜈

𝑒

= 𝑔²

𝑊𝑉_{𝑢 𝑑} 8𝑚²

𝑊

¯

𝜓_𝑛𝛾_𝜇(1−𝜆𝛾⁵)𝜓_𝑝𝜓¯_𝑒𝛾^𝜇(1−𝛾⁵)𝜓_𝜈

𝑒

(1.19)

where 𝜓_{𝑛, 𝑝,𝑒,𝜈}

𝑒 are the spin wave functions of the neutron, proton, electron and electron anti-neutrino, and we have inserted equation 1.8 for 𝐽^𝜈 and equation 1.10 for𝐾_𝜇. We can use the matrix element of the decay to directly compute the decay

rate (and hence the lifetime) via Fermi’s Golden Rule 1

𝜏_𝑖→𝑓

= 2𝜋

ℏ |𝑀|²𝜌(𝐸_𝑓) (1.20)

where𝑖represents the initial state, 𝑓 represents the final state, and𝜌 is a density of states. Using equation 1.19, this yields

𝜏= 64𝜋³ℏ𝑚⁴

𝑊

𝑚⁵_𝑒𝑐2𝑔⁴_𝑤|𝑉_{𝑢 𝑑}|²(1+3𝜆2)𝑓 (1.21) where 𝑓 is a statistical factor to account for the integral over the energy phase space of the decay.

This simple derivation illustrates the theoretical calculation of the neutron lifetime.

In practice, this computation is difficult due to non-trivial higher-order corrections to the neutron 𝛽-decay. However, modern calculations can achieve a theoretical prediction on 𝜏_𝑛 within the uncertainty of current 𝜏_𝑛 measurements. We note that calculations are limited by theory uncertainties whereas experimental measure- ments are reaching new precision benchmarks from improvements in measurement techniques.

In Chapter 5, we further examine the neutron lifetime and, in particular, analyze the UCNA dataset under the paradigm of an exotic, beyond Standard Model decay mode involving a dark matter decay channel. This would be a supplementary (ideally present at the 1% branching ratio) decay channel to the conventional neutron𝛽-decay described in equation 1.4.