Pseudo-neutrino versus recoil formalism for 4-body phase space and applications to nuclear decay

Neutron and nuclear beta decays provide an excellent avenue for precision tests of the Standard Model (SM) [1, 2]. Apart from the decay lifetime that determines the Cabibbo-Kobayashi-Maskawa (CKM) matrix $V_{ud}$, the study of the beta decay spectrum and various correlations constructed from spin and momentum vectors set stringent constraints on coupling strengths induced by physics beyond the Standard Model (BSM). The first comprehensive analysis of various terms in the beta decay differential rate in terms of the parameters in the most general Lee-Yang Lagrangian [3] was done by Jackson, Treiman and Wyld  [4, 5]. Taking neutron decay as an example, the well-known tree-level differential decay rate of a polarized neutron (with unpolarized final states) reads:

$$\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}\right)_{\text{tree}}\propto 1+a\vec{\beta}\cdot\hat{p}_{\nu}+b\frac{m_{e}}{E_{e}}+\hat{s}_{n}\cdot\left[A\vec{\beta}+B\hat{p}_{\nu}+D\vec{\beta}\times\hat{p}_{\nu}\right]\leavevmode\nobreak\ ,$$

(1)

where $\hat{p}_{\nu}$ is the unit neutrino momentum vector, $\vec{\beta}=\vec{p}_{e}/E_{e}=\beta\hat{p}_{e}$ is the electron velocity vector, and $\hat{s}_{n}$ is the neutron unit polarization vector. The SM provides well-defined predictions for the correlation coefficients $a$, $b$, $A$ and $B$ and $D$; experimental confirmation of any deviation from such predictions will be an unambiguous signal of BSM physics.

As the present experimental precision of beta decay observables have reached $10^{-4}$, one needs to carefully account for the higher-order SM corrections to the tree-level expression; the most prominent ones are recoil effects and the radiative corrections (RC). The latter consists of the Fermi function [6] and “outer corrections” that are functions of $E_{e}$ and the angles, and “inner corrections” that only serve to renormalize weak coupling constants; the first two are more important for the determination of correlation coefficients.

Pioneering works by Sirlin [7], Shann [8] and Garcia-Maya [9] established the $\mathcal{O}(\alpha)$ outer corrections to the polarized neutron differential rate, which takes a very elegant form:

$$\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}\right)_{\text{outer}}\propto\frac{\alpha}{2\pi}\left\{\delta_{1}+a\vec{\beta}\cdot\hat{p}_{\nu}(\delta_{1}+\delta_{2})+\hat{s}_{n}\cdot\left[A\vec{\beta}(\delta_{1}+\delta_{2})+B\hat{p}_{\nu}\delta_{1}\right]\right\}\leavevmode\nobreak\ ,$$

(2)

where $\delta_{1}$, $\delta_{2}$ are known functions of $E_{e}$. This equation serves as an important theory input for the study of the beta spectrum and the beta asymmetry parameter $A$. On the other hand, for observables such as the neutrino-electron correlation coefficient $a$ and the neutrino asymmetry parameter $B$, it was noticed long ago [10] that the representation above, which I call the neutrino ($\nu$)-formalism, contains a serious drawback as it depends on the neutrino momentum $\vec{p}_{\nu}$ which is not directly measured. In a three-body decay $\phi_{i}\rightarrow\phi_{f}e\nu$, it could be deduced from the momentum conservation $\vec{p}_{\nu}=-\vec{p}_{f}-\vec{p}_{e}$, where both $\vec{p}_{f}$ and $\vec{p}_{e}$ are measurable quantities. However, this relation breaks down in the bremsstrahlung process where an extra real photon is emitted. in fact, since in most experiments the neutrino and the extra photon(s) are not directly detected, there is no way to determine the neutrino solid angle $\Omega_{\nu}$ using the momenta of $\phi_{f}$ and $e$. Putting it in other words, using the $\nu$-formalism for the differential rate together with the 3-body momentum conservation relations could lead to a theoretical error of the order $\alpha/\pi$ for the $\hat{p}_{\nu}$-dependent observables due to the mistreatment of the 4-body kinematics.

Several alternative approaches were proposed to overcome the drawback in Eq.(2). An important example is the “recoil formalism”, namely to fix the momenta $\vec{p}_{e}$ and $\vec{p}_{f}$ (instead of $\vec{p}_{e}$ and $\vec{p}_{\nu}$) in the calculation; in this way one may choose the variables as $E_{e},E_{f}$ or $E_{e},\cos\theta_{ef}\equiv\hat{p}_{e}\cdot\hat{p}_{f}$. This is analogous to the treatment of RC in semileptonic kaon decays [11, 12, 13, 14]. The recoil formalism was used to study decays of unpolarized [15, 16] and polarized [17, 18, 19] baryons. The complicated bremsstrahlung phase space integrals were performed either semi-analytically [16, 18] or numerically [15, 20, 21, 22]. Recently, the need to apply the correct recoil formalism (instead of the $\nu$-formalism) to the decay of free neutrons was reiterated [23], and subsequent re-analysis was performed to the aCORN [24] and aSPECT [25] data.

Another approach is the pseudo-neutrino ($\nu^{\prime}$)-formalism. It defines a “pseudo-neutrino” momentum $\vec{p}_{\nu}^{\prime}\equiv-\vec{p}_{e}-\vec{p}_{f}$ which is a direct experimental observable; this definition holds with or without extra photons. Subsequently, one expresses the differential rate in terms of $E_{e},\Omega_{e},\Omega_{\nu}^{\prime}$ instead of $E_{e},\Omega_{e},\Omega_{\nu}$. This approach possesses several advantageous comparing to the recoil formalism:

• •

  It is the most natural generalization to the traditional $\nu$-formalism and has the closest resemblance to the latter, which makes cross-checking easier. It is also the most natural formalism to describe decays of polarized nuclei.

• •

  It is much more natural when combined with non-recoil approximation (unlike the recoil formalism where the variation of $E_{f}$ cannot be discussed without involving recoil effects). It is sometimes beneficial to study both the kinematical dynamical recoil corrections simultaneously within a unified framework, e.g. using methods like the heavy baryon expansion [26], to avoid possible double-counting. So it is desirable to split recoil effects completely from RC, and the $\nu^{\prime}$-formalism is best suited for this purpose.

• •

  The analytic expressions of the phase space integrals in the $\nu^{\prime}$-formalism are usually much more elegant, and only a minimal amount of numerical integration is required.

Existing literature that discussed the $\nu^{\prime}$-formalism includes Refs.[27, 17, 18, 16], with applications to decays of unpolarized and polarized neutron and hyperons. Despite their comprehensiveness, I failed to identify a complete treatment of the outer RC to the differential rate of a polarized parent nucleus in the non-recoil limit (appropriate for neutron and nuclear decays) with the $\nu^{\prime}$-formalism which, in principle, should display a similar degree of theory elegance as Eq.(2). Also, analytic expressions of many important intermediate quantities (e.g. expansion coefficients and results of phase-space integrals) are not explicitly shown, which makes their results not so straightforwardly applicable in numerical analysis; furthermore, one also finds that some numerical results in these earlier papers are not completely accurate. In the meantime, while many recent computer packages are capable to produce reliable numerical results [23], they are practically a black box to outsiders.

The purpose of this paper is to provide a transparent, elegant and self-contained theoretical description of the $\nu^{\prime}$-formalism in the non-recoil limit. As a simple application, I study the outer RC to the differential rate of the $\beta^{\pm}$ decay of a polarized spin-1/2 nucleus, but the final result is equally applicable to spin-0 systems. Therefore, this analysis covers a wide range of interesting beta decay processes including semileptonic pion decay, free neutron decay, superallowed $0^{+}\rightarrow 0^{+}$ decays [28], spin-half nuclear mirror transitions [29] (e.g. ¹⁹Ne$\rightarrow$¹⁹F), etc. I provide the full set of analytic expressions of all intermediate functions needed in the analysis, so that the quoted numerical results are fully transparent and can be easily reproduced by any interested reader, without the need of complicated numerical packages. For completeness, I also provide a similar set of analytic formula for unpolarized nuclei in the recoil formalism.

I start by briefly reviewing the total rate formula of a 3-body decay $\phi_{i}(p_{i})\rightarrow\phi_{f}(p_{f})+e(p_{e})+\nu(p_{\nu})$¹¹1The term “electron” throughout this paper may refer to an electron or a positron depending on the decay mode, and similar for the neutrino.:

$$\Gamma_{3}=\int d\Pi_{3}|\mathcal{M}|_{\text{3-body}}^{2}$$

(3)

with the 3-body phase space:

$$\int d\Pi_{3}=\frac{1}{2m_{i}}\int\frac{d^{3}p_{f}}{(2\pi)^{3}2E_{f}}\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f}-p_{e}-p_{\nu})\leavevmode\nobreak\ .$$

(4)

One is interested in beta decay processes where $m_{i},m_{f}\gg m_{i}-m_{f}$. The evaluation of this phase space in the $\nu$-formalism consists of first integrating out $\vec{p}_{f}$ using the spatial delta function. The remaining temporal delta function reads:

$$\delta(m_{i}-E_{f}-E_{e}-E_{\nu})\approx\delta(E_{m}-E_{e}-E_{\nu})$$

(5)

which only set bounds on the energy variables but not the angles; here I have taken the non-recoil approximation, $E_{f}\approx m_{f}$ and $E_{m}\equiv(m_{i}^{2}-m_{f}^{2}+m_{e}^{2})/(2m_{i})\approx m_{i}-m_{f}$ is the electron end-point energy. Of course, in practice one needs to account for recoil corrections to the tree-level 3-body rate for precision, but that is not the focus of this paper. This delta function is then used to integrate out the neutrino energy, which leaves us with:

$$\int d\Pi_{3}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int d\Omega_{e}d\Omega_{\nu}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|E_{\nu}\leavevmode\nobreak\ ,$$

(6)

where $E_{\nu}=E_{m}-E_{e}$; the upper limit of $E_{e}$ is set by the temporal delta function: $E_{m}-E_{e}=E_{\nu}\geq 0$. Eq.(6) is the underlying reason why one utilized $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}$ as free variables in the traditional analysis of beta decays.

To bypass the conceptual deficit in the $\nu$-formalism that $\Omega_{\nu}$ is not a practical observable, one may turn to the $\nu^{\prime}$-formalism by defining the pseudo-neutrino momentum:

$$p_{\nu}^{\prime}\equiv p_{i}-p_{f}-p_{e}=(E_{\nu}^{\prime},\vec{p}_{\nu}^{\prime})$$

(7)

which is an experimental observable, because both $e$ and $\phi_{f}$ can be directly detected. In the non-recoil limit, $E_{\nu}^{\prime}=E_{m}-E_{e}$ is only a function of electron energy. In this formalism, I change the integration over $\vec{p}_{f}$ to that over $\vec{p}_{\nu}^{\prime}$:

$$\int d^{3}p_{f}d^{3}p_{e}=\int d^{3}p_{\nu}^{\prime}d^{3}p_{e}\leavevmode\nobreak\ ,$$

(8)

and rewrite Eq.(4) in the non-recoil limit as:

$$\int d\Pi_{3}\approx\frac{1}{(2\pi)^{6}8m_{i}m_{f}}\int d^{3}p_{\nu}^{\prime}\frac{d^{3}p_{e}}{E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu})\leavevmode\nobreak\ ,$$

(9)

and integrate out $\vec{p}_{\nu}$, instead of $\vec{p}_{f}$, using the spatial delta function. The remaining temporal delta function reads $\delta(E_{\nu}^{\prime}-|\vec{p}_{\nu}^{\prime}|)$, which is then used to integrate out $|\vec{p}_{\nu}^{\prime}|$. This yields:

$$\int d\Pi_{3}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int d\Omega_{e}d\Omega_{\nu}^{\prime}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|E_{\nu}^{\prime}\leavevmode\nobreak\ .$$

(10)

This is identical to Eq.(6) upon replacing $E_{\nu}^{\prime}\rightarrow E_{\nu}$ and $\Omega_{\nu}^{\prime}\rightarrow\Omega_{\nu}$, which do not change the physics since $p_{\nu}^{\prime}=p_{\nu}$ in 3-body decay. The real distinction between the two formalisms only shows up when extra photons are emitted.

To facilitate the discussion later, I also provide the representation of the 3-body phase space in the recoil formalism, i.e. using $E_{e}$ and $E_{f}$ as variables. It is most convenient to adopt the notations used in semileptonic decays of kaons [11, 12, 13, 14], where the differential decay rate is expressed in terms of the following dimensionless Lorentz scalars:

$$x\equiv\frac{p_{\nu}^{\prime 2}}{m_{i}^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ y\equiv\frac{2p_{i}\cdot p_{e}}{m_{i}^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ z\equiv\frac{2p_{i}\cdot p_{f}}{m_{i}^{2}}\leavevmode\nobreak\ ,$$

(11)

with $y=2E_{e}/m_{i}$ and $z=2E_{f}/m_{i}$ energy parameters in the parent’s rest frame. In 3-body decay, $p_{\nu}^{\prime 2}=p_{\nu}^{2}=0$ assuming massless neutrino, and I obtain the exact relation:

$$\int d\Pi_{3}=\frac{m_{i}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz\leavevmode\nobreak\ ,$$

(12)

where the integration region $\mathcal{D}_{3}$ can be represented in two equivalent ways:

	$\displaystyle a(y)-b(y)<z<a(y)+b(y)\leavevmode\nobreak\ ,$		$\displaystyle 2\sqrt{r_{e}}<y<1+r_{e}-r_{f}$
	$\displaystyle a(y)=\frac{(2-y)(1+r_{f}+r_{e}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\ ,$		$\displaystyle b(y)=\frac{\sqrt{y^{2}-4r_{e}}(1+r_{e}-r_{f}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\ ,$		(13)

or

	$\displaystyle c(z)-d(z)<y<c(z)+d(z)\leavevmode\nobreak\ ,$		$\displaystyle 2\sqrt{r_{f}}<z<1+r_{f}-r_{e}$
	$\displaystyle c(z)=\frac{(2-z)(1+r_{e}+r_{f}-z)}{2(1+r_{f}-z)}\leavevmode\nobreak\ ,$		$\displaystyle d(z)=\frac{\sqrt{z^{2}-4r_{f}}(1+r_{f}-r_{e}-z)}{2(1+r_{f}-z)}\leavevmode\nobreak\ ,$		(14)

with $r_{f}\equiv m_{f}^{2}/m_{i}^{2}$, $r_{e}\equiv m_{e}^{2}/m_{i}^{2}$.

The contribution from the bremsstrahlung process $\phi_{i}(p_{i})\rightarrow\phi_{f}(p_{f})+e(p_{e})+\nu(p_{\nu})+\gamma(k)$ to the total decay rate is:

$$\Gamma_{4}=\int d\Pi_{4}|\mathcal{M}|^{2}_{\text{4-body}}\leavevmode\nobreak\ ,$$

(15)

where the 4-body phase space reads:

$$\int d\Pi_{4}=\frac{1}{2m_{i}}\int\frac{d^{3}p_{f}}{(2\pi)^{3}2E_{f}}\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}\frac{d^{3}k}{(2\pi)^{3}2E_{k}}(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f}-p_{e}-p_{\nu}-k)\leavevmode\nobreak\ .$$

(16)

Let us again evaluate the phase space first in the traditional $\nu$-formalism. I integrate out $\vec{p}_{f}$ using the spatial delta function, which leaves the temporal part as:

$$\delta(m_{i}-E_{f}-E_{e}-E_{\nu}-E_{k})\approx\delta(E_{m}-E_{e}-E_{\nu}-E_{k})\leavevmode\nobreak\ .$$

(17)

Here the non-recoil approximation can be safely taken, because in neutron/nuclear beta decays a recoil correction on top of a RC gives a $\sim(\alpha/\pi)(m_{i}-m_{f})/m_{i}\lesssim 10^{-5}$ correction to the total decay rate, which exceeds the experimental precision and can be neglected. The remaining delta function is then used to integrate out $E_{\nu}$, which gives:

$$\int d\Pi_{4}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int d\Omega_{e}d\Omega_{\nu}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|\int_{E_{k}<E_{m}-E_{e}}\frac{d^{3}k}{(2\pi)^{3}2E_{k}}E_{\nu}\leavevmode\nobreak\ ,$$

(18)

where $E_{\nu}=E_{m}-E_{e}-E_{k}$. The unobserved photon momentum $\vec{k}$ is then integrated out, leaving $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}$ as variables in the differential decay rate formula.

As I stressed in the introduction, since in Eq.(18) the unobserved photon momentum is integrated out, further applying the 3-body momentum conservation $p_{\nu}=p_{i}-p_{f}-p_{e}$ to the 4-body phase space in the $\nu$-formalism leads to an error. This issue is non-existent in the recoil formalism, since the variables $\{x,y,z\}$ are experimentally measurable. In this formalism, one can derive the following exact formula of the 4-body phase space [30]:

	$\displaystyle\int d\Pi_{4}$	$\displaystyle=$	$\displaystyle\frac{m_{i}^{3}}{512\pi^{4}}\left\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}(y,z)}dx+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\right\}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$		(19)
			$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\leavevmode\nobreak\ ,$

where

$$\alpha_{\pm}(y,z)\equiv 1-y-z+r_{f}+r_{e}+\frac{yz}{2}\pm\frac{1}{2}\sqrt{y^{2}-4r_{e}}\sqrt{z^{2}-4r_{f}}\leavevmode\nobreak\ ,$$

(20)

and the region $\mathcal{D}_{4-3}$ can also be expressed in two equivalent ways:

$$2\sqrt{r_{f}}<z<a(y)-b(y)\leavevmode\nobreak\ ,\leavevmode\nobreak\ 2\sqrt{r_{e}}<y<1-\sqrt{r_{f}}+\frac{r_{e}}{1-\sqrt{r_{f}}}$$

(21)

or

$$2\sqrt{r_{e}}<y<c(z)-d(z)\leavevmode\nobreak\ ,\leavevmode\nobreak\ 2\sqrt{r_{f}}<z<1-\sqrt{r_{e}}+\frac{r_{f}}{1-\sqrt{r_{e}}}\leavevmode\nobreak\ .$$

(22)

This formula is extremely useful, because it turns out the $p_{\nu}$- and $k$-integration with respect to the squared amplitude is always analytically doable [13], so one is left with at most one single numerical integration over $x$, if the differential rate is expressed as $d\Gamma/dydz$. The recoil formalism has also been applied to baryon decays [15, 16, 17, 18, 19]; in particular, the non-overlapping regions $\mathcal{D}_{3}$ and $\mathcal{D}_{4-3}$ in Eq.(19) are just what known as the “in” and “out” region respectively in Ref.[23]. However, since $\{y,z\}$ are all energy and not angular variables, the connection to the traditional representation (1), (2) is not straightforward.

To overcome the aforementioned shortcomings, I re-evaluate the 4-body phase space in the $\nu^{\prime}$-formalism, keeping close analogy to the 3-body case as well as the kaon representation, in order to retain the advantages in both methods. Notice that the definition of the pseudo-neutrino momentum $p_{\nu}^{\prime}$ in Eq.(7) remains unchanged in 4-body, and the relation $E_{\nu}^{\prime}=E_{m}-E_{e}$ still holds. I first write:

$$\int d^{3}p_{f}\frac{d^{3}p_{e}}{E_{e}}=\int d^{3}p_{\nu}^{\prime}\frac{d^{3}p_{e}}{E_{e}}=\int d\Omega_{e}d\Omega_{\nu}^{\prime}\int dE_{e}|\vec{p}_{e}|\int d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}\leavevmode\nobreak\ ,$$

(23)

and deduce the boundaries of each variable. First, the boundaries of $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}^{\prime}$ must be exactly the same as in the 3-body case; an easy way to understand this is that it must be so to ensure the exact cancellation of the infrared (IR)-divergence between the 3-body (one-loop) and 4-body (tree-level) contributions to the differential decay rate $d\Gamma/dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}$ [31, 32, 33, 34, 35]. The boundaries of $|\vec{p}_{\nu}^{\prime}|$ are determined by the delta function:

$$p_{\nu}^{\prime 2}=E_{\nu}^{\prime 2}-|\vec{p}_{\nu}^{\prime}|^{2}=(p_{\nu}+k)^{2}=(E_{\nu}+E_{k})^{2}-|\vec{p}_{\nu}+\vec{k}|^{2}\geq 0\leavevmode\nobreak\ ,$$

(24)

so one has $|\vec{p}_{\nu}^{\prime}|\leq E_{\nu}^{\prime}$, with the equal sign occurring when $\vec{p}_{\nu}\parallel\vec{k}$. Meanwhile, for any given $\vec{p}_{e}$, a $\vec{p}_{\nu}^{\prime}=\vec{0}$ configuration can always be achieved by setting $\vec{p}_{f}=-\vec{p}_{e}$ (which does not affect the energy $E_{f}\approx m_{f}$ in the non-recoil limit), with the remaining energy of the system carried away by a $\nu\gamma$-pair with equal and opposite momenta. So, the lower and upper limits of the $|\vec{p}_{\nu}^{\prime}|$-integration are 0 and $E_{\nu}^{\prime}$ respectively. Thus, I obtain the $\nu^{\prime}$-representation of the 4-body phase space in the non-recoil limit:

	$\displaystyle\int d\Pi_{4}$	$\displaystyle\approx$	$\displaystyle\frac{1}{512\pi^{6}m_{i}m_{f}}\int d\Omega_{e}d\Omega_{\nu}^{\prime}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|\int_{0}^{E_{\nu}^{\prime}}d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$		(25)
			$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\leavevmode\nobreak\ .$

Eq.(25) is the first central result of this work. It possesses the following advantages:

1. 1.

   The differential rate is expressible in terms of the experimentally-measurable energy and angular variables $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}^{\prime}$, which solves the problem in the traditional $\nu$-formalism.

2. 2.

   The mathematical structure of Eq.(25) is much more elegant than the recoil formalism, Eq.(19). For instance, here one does not need to separate the so-called “in” and “out” region as two different terms, and the boundaries are also much simpler.

3. 3.

   The full $p_{\nu}$- and $k$-integrations are retained as in the kaon formalism. This allows one to directly make use of the known analytic formula of such integrations in the latter.

Finally, one can easily check the correctness of Eq.(25) by comparing it to Eq.(19). For instance, after removing the $p_{\nu}$- and $k$-measures and the delta function, I evaluate the two expressions numerically by substituting the neutron decay parameters, and find that their difference is at the level 0.1% which is the typical size of a recoil correction $\sim(m_{n}-m_{p})/m_{n}$. This shows that Eq.(25) is indeed the correct 4-body phase space formula in the non-recoil limit.

As an important application of the $\nu^{\prime}$-formalism, I study the so-called “outer RC” in a $J_{i}=J_{f}=1/2$ decay of a polarized parent nucleus (with unmeasured final spins) at the $\mathcal{O}(\alpha/\pi)$ level. This includes well-known one-loop corrections and the one-photon bremsstrahlung process. The former does not distinguish between the $\nu$- and $\nu^{\prime}$-method, so I focus on the latter.

I start by writing down the hadronic and leptonic charged weak current operator²²2In this work I adopt the experimentalists’ convention, $g_{A}<0$.:

$$J_{\text{had}}^{\mu}=\left\{\begin{array}[]{ccc}\bar{n}\gamma^{\mu}(g_{V}+g_{A}\gamma_{5})p&,&\beta^{+}\\ \bar{p}\gamma^{\mu}(g_{V}+g_{A}\gamma_{5})n&,&\beta^{-}\end{array}\right.\leavevmode\nobreak\ ,\leavevmode\nobreak\ J_{\text{lep}}^{\mu}=\left\{\begin{array}[]{ccc}\bar{\nu}\gamma^{\mu}(1-\gamma_{5})e&,&\beta^{+}\\ \bar{e}\gamma^{\mu}(1-\gamma_{5})\nu&,&\beta^{-}\end{array}\right.\leavevmode\nobreak\ .$$

(26)

The Fermi (F) and Gamow-Teller (GT) matrix element of $J_{\text{had}}^{\mu}$ are defined through:

$$\langle f|\int d^{3}xJ_{\text{had}}^{0}(\vec{x})|i\rangle=g_{V}M_{\text{F}}\mathbbm{1}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \langle f|\int d^{3}x\vec{J}_{\text{had}}(\vec{x})|i\rangle=\frac{g_{A}}{\sqrt{3}}M_{\text{GT}}\vec{\sigma}\leavevmode\nobreak\ .$$

(27)

With this one can define a “generalized” axial-to-vector ratio:

$$\lambda\equiv\frac{M_{\text{GT}}}{\sqrt{3}M_{\text{F}}}\frac{g_{A}}{g_{V}}\leavevmode\nobreak\ ,$$

(28)

which reduces to $g_{A}/g_{V}$ for neutron decay ($M_{\text{F}}=1$ and $M_{\text{GT}}=\sqrt{3}$). Taking the non-recoil limit at the amplitude level, which corresponds to setting $p_{f}\approx p_{i}\equiv p=(m,\vec{0})$ with $m=m_{i}$ for simplicity, I can write down the tree-level squared amplitude as:

$$|\mathcal{M}|_{\text{tree}}^{2}\approx 16G_{V}^{2}m^{2}E_{e}E_{\nu}^{\prime}\left\{1+3\lambda^{2}+(1-\lambda^{2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\vec{\beta}-2\lambda(\lambda\zeta+1)\hat{p}_{\nu}^{\prime}\right]\right\}\leavevmode\nobreak\ ,$$

(29)

where $\zeta=+1(-1)$ for $\beta^{+}(\beta^{-})$-decay, $G_{V}\equiv G_{F}V_{ud}g_{V}M_{F}$ is the effective weak vector coupling, and $\hat{s}$ is the unit vector of the parent’s polarization. Plugging it into Eq.(10) gives the tree-level differential rate in the non-recoil limit (taking $m^{2}\approx m_{i}m_{f}$):

	$\displaystyle\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{\text{tree}}$	$\displaystyle\approx$	$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime 2}\Bigl{\{}1+3\lambda^{2}+(1-\lambda^{2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}$		(30)
			$\displaystyle+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\vec{\beta}-2\lambda(\lambda\zeta+1)\hat{p}_{\nu}^{\prime}\right]\Bigr{\}}\leavevmode\nobreak\ .$