# Radiative Corrections to Semileptonic Beta Decays: Progress and Challenges

## Abstract

**:**

## 1. Introduction

## 2. EWRC in a Generic Semileptonic Beta Decay

#### 2.1. Basic Ingredients

#### 2.2. Weak Corrections

#### 2.2.1. First Diagram

#### 2.2.2. Second Diagram

#### 2.2.3. Third Diagram

#### 2.2.4. pQCD Corrections

#### 2.3. General Theory for Beta Decays

## 3. Sirlin’s Representation

- The first diagram is simply the wave function renormalization of the charged lepton. Elementary calculation gives:$${Z}_{\ell}=1-\frac{\alpha}{4\pi}\left[ln\frac{{M}_{W}^{2}}{{m}_{\ell}^{2}}+\frac{9}{2}-2ln\frac{{m}_{\ell}^{2}}{{M}_{\gamma}^{2}}\right]\phantom{\rule{3.33333pt}{0ex}},$$
- The second diagram represents the EMRC to the hadronic charged weak matrix element: ${F}^{\lambda}\to {F}^{\lambda}+\delta {F}^{\lambda}$. We will discuss it more later.
- The third diagram is the famous $\gamma W$-box correction. For future convenience, we split it into two pieces: $\delta {\mathfrak{M}}_{\gamma W}=\delta {\mathfrak{M}}_{\gamma W}^{a}+\delta {\mathfrak{M}}_{\gamma W}^{b}$ by applying the Dirac matrix identity (20) to the lepton structure:$$\begin{array}{ccc}\hfill \delta {\mathfrak{M}}_{\gamma W}^{a}& =& \eta \frac{{G}_{F}{e}^{2}}{\sqrt{2}}{L}_{\lambda}\int \frac{{d}^{4}{q}^{\prime}}{{\left(2\pi \right)}^{4}}\frac{{M}_{W}^{2}}{{M}_{W}^{2}-{q}^{\prime 2}}\frac{2{g}^{\nu \lambda}{p}_{\ell}^{\mu}-{g}^{\mu \lambda}{q}^{\prime \nu}-{g}^{\nu \lambda}{q}^{\prime \mu}+{g}^{\mu \nu}{q}^{\prime \lambda}}{[{({p}_{\ell}-{q}^{\prime})}^{2}-{m}_{\ell}^{2}][{q}^{\prime 2}-{M}_{\gamma}^{2}]}{T}_{\mu \nu}\hfill \\ \hfill \delta {\mathfrak{M}}_{\gamma W}^{b}& =& -i\frac{{G}_{F}{e}^{2}}{\sqrt{2}}{L}_{\lambda}\int \frac{{d}^{4}{q}^{\prime}}{{\left(2\pi \right)}^{4}}\frac{{M}_{W}^{2}}{{M}_{W}^{2}-{q}^{\prime 2}}\frac{{\u03f5}^{\mu \nu \alpha \lambda}{q}_{\alpha}^{\prime}}{[{({p}_{\ell}-{q}^{\prime})}^{2}-{m}_{\ell}^{2}]{q}^{\prime 2}}{T}_{\mu \nu}\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$In particular, using the Ward identities (8), we are able to isolate a part of $\delta {\mathfrak{M}}_{\gamma W}^{a}$ which is proportional to ${\mathfrak{M}}_{0}$ and is exactly integrable:$$\begin{array}{ccc}\delta {\mathfrak{M}}_{\gamma W}^{a}\hfill & =\hfill & \frac{\alpha}{2\pi}\left[ln\frac{{M}_{W}^{2}}{{m}_{\ell}^{2}}+1\right]{\mathfrak{M}}_{0}+\eta \frac{{G}_{F}{e}^{2}}{\sqrt{2}}{L}_{\lambda}\int \frac{{d}^{4}{q}^{\prime}}{{\left(2\pi \right)}^{4}}\frac{{M}_{W}^{2}}{{M}_{W}^{2}-{q}^{\prime 2}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}\hfill & \times \frac{{q}^{\prime \lambda}{T}_{\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mu}^{\mu}+2{p}_{\ell \mu}{T}^{\mu \lambda}-{(p-{p}^{\prime})}_{\mu}{T}^{\lambda \mu}+i{\Gamma}^{\lambda}}{[{({p}_{\ell}-{q}^{\prime})}^{2}-{m}_{\ell}^{2}][{q}^{\prime 2}-{M}_{\gamma}^{2}]}\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$As usual, terms of $\mathcal{O}\left({G}_{F}^{2}\right)$ are discarded.

#### 3.1. On-Mass-Shell Perturbation Formula and Ward Identity

#### 3.2. Partial Cancellation between the Hadronic Vertex Correction and the $\gamma W$-Box Diagram

#### 3.3. Bremsstrahlung

#### 3.4. Large Electroweak Logarithms and the Higher-Order QED Effects

## 4. Effective Field Theory Representation

#### 4.1. Spontaneously-Broken Chiral Symmetry

#### 4.2. pNGBs and the Chiral Power Counting

- It is invariant under $\mathrm{SU}{\left(3\right)}_{\mathrm{L}}\times \mathrm{SU}{\left(3\right)}_{\mathrm{R}}$ in the limit of massless quarks;
- The chiral symmetry is spontaneously broken to SU(3)${}_{\mathrm{V}}$, and the pseudoscalar octets appear as the pNGBs;
- The chiral symmetry is explicitly broken by the insertion of the quark mass matrix ${M}_{q}$.

#### 4.3. External Sources

#### 4.4. Mesonic ChPT with External Sources

#### 4.5. Nucleon Sector

## 5. Pion Semileptonic Beta Decay

- It is spinless, so at tree level only the vector component of the charged weak current contributes;
- It is near-degenerate, that is, ${M}_{{\pi}^{+}}-{M}_{{\pi}^{0}}\ll {M}_{\pi}$, which simplifies the discussion a lot upon neglecting recoil corrections on top of the RCs;
- It does not suffer from nuclear structure uncertainties.

#### 5.1. Tree-Level Analysis

#### 5.2. EWRCs

#### 5.3. Early Numerical Estimation

#### 5.4. ChPT Treatment

#### 5.5. First-Principles Calculation

#### 5.5.1. Large-${Q}^{2}$ Contribution

#### 5.5.2. Small-${Q}^{2}$ Contribution

## 6. Beta Decay of I = J = 1/2 Particles

#### 6.1. Outer and Inner Corrections

#### 6.2. Dispersive Representation

- First-principles calculation with lattice QCD, in analogy to the calculation of ${\square}_{\gamma W}({\pi}^{+},$${\pi}^{0},{M}_{\pi})$ described in Section 5.5;
- Data-driven analysis that relates the hadronic matrix elements to experimental observables.

#### 6.3. Asymptotic Contribution

#### 6.4. Born Contribution

#### 6.5. Exact Isospin Relations

## 7. Free Neutron

#### 7.1. Earlier Attempts

- $0<{Q}^{2}<{Q}_{1}^{2}$ (long distances): Pure Born contribution, which is completely fixed by nucleon form factors, dominates.
- ${Q}_{1}^{2}<{Q}^{2}<{\left(1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}\right)}^{2}$ (intermediate distances): A VMD-inspired interpolating function is constructed:$$F\left({Q}^{2}\right)=\frac{{c}_{\rho}}{{Q}^{2}+{M}_{\rho}^{2}}+\frac{{c}_{A}}{{Q}^{2}+{M}_{A}^{2}}+\frac{{c}_{{\rho}^{\prime}}}{{Q}^{2}+{M}_{{\rho}^{\prime}}^{2}}\phantom{\rule{3.33333pt}{0ex}},$$
- ${Q}^{2}>{\left(1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}\right)}^{2}$ (short distances): $F\left({Q}^{2}\right)$ is given by the leading-twist OPE + pQCD correction.

- The result of the integral (156) at ${Q}^{2}>{\left(1.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GeV}\right)}^{2}$ is required to be the same using the VMD parameterization and the asymptotic expression;
- In the large-${Q}^{2}$ limit, the coefficient of the $1/{Q}^{4}$ term in Equation (157) is required to vanish by chiral symmetry;
- Equation (157) is required to vanish at ${Q}^{2}=0$ by ChPT;
- Finally, the connection scale ${Q}_{1}^{2}$ is chosen through the matching of ${F}_{\mathrm{Born}}$ and ${F}_{\mathrm{Interpolation}}$ at ${Q}^{2}={Q}_{1}^{2}$.

#### 7.2. ${\square}_{\gamma \phantom{\rule{3.33333pt}{0ex}}W}^{V}$: DR Analysis

- The “non-asymptotic” pieces (Born, low-energy continuum, resonances): They are clearly different for different ${F}_{3}$, and need to be calculated case-by-case;
- The “asymptotic” pieces (${F}_{3,\mathrm{pQCD}}$ at large ${Q}^{2}$ and ${F}_{3,\mathbb{R}}$ at large ${W}^{2}$): They are largely universal for different ${F}_{3}$ (up to multiplicative factors), so we can either calculate them explicitly (${F}_{3,\mathrm{pQCD}}$), or infer one from the other.

#### 7.3. ${\square}_{\gamma \phantom{\rule{3.33333pt}{0ex}}W}^{A}$: DR Analysis

- The Born contribution is just given by Equation (148). Isospin symmetry requires ${G}_{V}\left({Q}^{2}\right)={f}_{1}^{V}\left({Q}^{2}\right)$ and ${G}_{M}\left({Q}^{2}\right)={f}_{2}^{V}\left({Q}^{2}\right)$, so the integral is completely fixed by the nucleon electromagnetic form factors;
- For the inelastic contribution from ${g}_{1}^{\gamma}$, what we need is the following quantity:$${\overline{\Gamma}}_{1}^{p-n}\left({Q}^{2}\right)\equiv {\int}_{0}^{{x}_{\pi}}dx\frac{4(5+4r)}{9{(1+r)}^{2}}\left\{{g}_{1,p}^{\gamma}(x,{Q}^{2})-{g}_{1,n}^{\gamma}(x,{Q}^{2})\right\}$$(${x}_{\pi}={Q}^{2}/[{({m}_{N}+{M}_{\pi})}^{2}-{m}_{N}^{2}+{Q}^{2}]$ is the pion production threshold) as a function of ${Q}^{2}$. At ${Q}^{2}>2$ GeV${}^{2}$, one resorts again to the pQCD-corrected polarized Bjorken sum rule, and include additionally a small higher-twist (HT) correction which is required to match the theory prediction with the experimental data [236,237,238]. At ${Q}^{2}<2$ GeV${}^{2}$, data are taken from the EG1b experiment at JLab [228,229] that provided the first three moments of ${g}_{1,N}^{\gamma}$ from 0.05 GeV${}^{2}$ to 3.5 GeV${}^{2}$, which allow for a precise reconstruction of ${\overline{\Gamma}}_{1}^{p-n}\left({Q}^{2}\right)$ at any value of ${Q}^{2}$ within the range. Figure 13 shows the combination of the experimental and theory prediction of ${\overline{\Gamma}}_{1}^{p-n}\left({Q}^{2}\right)$. One then performs the ${Q}^{2}$-integral to obtain the full ${g}_{1}^{\gamma}$ contribution.
- Finally, the ${g}_{2}^{\gamma}$ contribution is split into two pieces, namely the twist-two and twist-three (and higher) contributions. The former is related to ${g}_{1}^{\gamma}$ through the Wandzura–Wilczek relation [239], while the latter is related to the so-called nucleon color polarizability [240,241] calculated within the baryon chiral effective theory [242].

#### 7.4. Future Prospects with Lattice QCD

## 8. Superallowed Beta Decays

- At tree level only the vector charged weak current is involved, whose matrix element is exactly known assuming isospin symmetry;
- Experimental data of 23 superallowed transitions had been accumulated over five decades [164,255,256,257,258,259,260,261], with 15 among them whose $ft$-value precision is 0.23% or better; the large sample size leads to a huge gain in statistics [261]. In fact, this makes it the only avenue where the experimental uncertainty in $|{V}_{ud}|$ is smaller than the theory uncertainty.

#### 8.1. The Nuclear Structure Correction

#### 8.1.1. Earlier Treatments

#### 8.1.2. Recent Developments

#### 8.2. The Isospin-Breaking Correction

## 9. Kaon Semileptonic Decays

- ${\Gamma}_{{K}_{\u01423}}$ is the ${K}_{\ell 3}$ decay width. The kaon lifetimes, branching ratios and differential decay widths are measured by BNL E865 [304], KTeV [305,306,307,308,309], KLOE [310,311,312,313,314,315,316,317,318]. KLOE-2 [319], NA48 [320,321,322,323] and ISTRA+ [324,325] over the past two decades.
- ${f}_{+}^{{K}^{0}{\pi}^{-}}\left(0\right)$ is the ${K}^{0}{\pi}^{-}$ charged weak form factor in the (unphysical) $t\to 0$ limit, with the general definition of the charged weak form factors in spinless systems given in Equation (95). High-precision lattice calculations of this quantity were performed over the past decade by the FNAL/MILC [326,327,328] and the ETM [329] collaborations and showed perfect mutual consistencies and a steady improvement in precision. The most recent Flavor Lattice Averaging Group (FLAG) online review (updated from the 2019 version [330]) quoted the following average: Refs. [328,329]$$|{f}_{+}^{{K}^{0}{\pi}^{-}}\left(0\right)|=0.9698\left(17\right)\phantom{\rule{3.33333pt}{0ex}}.\phantom{\rule{1.em}{0ex}}{n}_{f}=2+1+1.$$However, a recent calculation from the PACS collaboration based on a single lattice spacing returned a somewhat smaller value [331].
- ${C}_{K}$ is a simple isospin factor defined through the value of ${f}_{+}^{K\pi}\left(0\right)$ in the SU(3) limit:$$|{C}_{K}|=|{f}_{+}^{K\pi}{\left(0\right)|}_{{m}_{u}={m}_{d}={m}_{s}}=\left\{\begin{array}{cc}1\phantom{\rule{3.33333pt}{0ex}},& {K}^{0}\\ 1/\sqrt{2}\phantom{\rule{3.33333pt}{0ex}},& {K}^{+}\end{array}\right.\phantom{\rule{3.33333pt}{0ex}}.$$
- ${I}_{K\ell}^{\left(0\right)}$ is the tree-level phase space factor defined as: (Notice that a number of important review papers, for example, Refs. [102,332], contain a typo in their formula for ${I}_{K\ell}^{\left(0\right)}$.)$${I}_{K\ell}^{\left(0\right)}={\int}_{{m}_{\ell}^{2}}^{{({M}_{K}-{M}_{\pi})}^{2}}\frac{dt}{{M}_{K}^{8}}{\overline{\lambda}}^{3/2}\left(1+\frac{{m}_{\ell}^{2}}{2t}\right){\left(1-\frac{{m}_{\ell}^{2}}{t}\right)}^{2}\left[{\overline{f}}_{+}^{2}\left(t\right)+\frac{3{m}_{\ell}^{2}{\Delta}_{K\pi}^{2}}{\left(2t+{m}_{\ell}^{2}\right)\overline{\lambda}}{\overline{f}}_{0}^{2}\left(t\right)\right]\phantom{\rule{3.33333pt}{0ex}},$$$${f}_{0}^{K\pi}\left(t\right)\equiv {f}_{+}^{K\pi}\left(t\right)+\frac{t}{{M}_{K}^{2}-{M}_{\pi}^{2}}{f}_{-}^{K\pi}\left(t\right)$$
- Finally, ${\delta}_{\mathrm{SU}}{\left(2\right)}^{K\pi}$ is defined as the isospin breaking correction to the ${f}_{+}^{K\pi}\left(0\right)$ form factor at $t=0$: (The appearance of the factor ${C}_{{K}^{0}}/{C}_{K}$ is simply due to our choice of normalization of the form factors in Equation (95).)$${\delta}_{\mathrm{SU}}{\left(2\right)}^{K\pi}\equiv {\left(\frac{{C}_{{K}^{0}}}{{C}_{K}}\frac{{f}_{+}^{K\pi}\left(0\right)}{{f}_{+}^{{K}^{0}{\pi}^{-}}\left(0\right)}\right)}^{2}-1\phantom{\rule{3.33333pt}{0ex}},$$

#### 9.1. Kinematics

#### 9.2. ChPT Treatment of the EWRC

- The mesonic loop diagrams depicted in Figure 17. They correct the form factors ${f}_{\pm}^{K\pi}\left(t\right)$ that enter the phase space factor ${I}_{K\ell}^{\left(0\right)}$.