# Impact of Stoichiometric Silicon Nitride Growth Conditions on Dispersion
Engineering of Broadband Microresonator Frequency Combs

Gregory Moille  Joint Quantum Institute, NIST/University of Maryland, College
Park, USA  Daron Westly  Microsystems and Nanotechnology Division, National
Institute of Standards and Technology, Gaithersburg, USA  Gregory Simelgor
Microsystems and Nanotechnology Division, National Institute of Standards and
Technology, Gaithersburg, USA  Kartik Srinivasan  Joint Quantum Institute,
NIST/University of Maryland, College Park, USA  Microsystems and
Nanotechnology Division, National Institute of Standards and Technology,
Gaithersburg, USA

###### Abstract

Microresonator frequency combs, or microcombs, have gained wide appeal for
their rich nonlinear physics and wide range of applications. Stoichiometric
silicon nitride films grown via low-pressure chemical vapor deposition
(LPCVD), in particular, are widely used in chip-integrated Kerr microcombs.
Critical to such devices is the ability to control the microresonator
dispersion, which has contributions from both material refractive index
dispersion and geometric confinement. Here, we show that modifications to the
LPCVD growth conditions, specifically the ratio of the gaseous precursors, has
a significant impact on material dispersion and hence the overall
microresonator dispersion. In contrast to the many efforts focused on
comparison between Si-rich films and stoichiometric (Si3N4) films, here we
entirely focus on films that are within the nominally stoichiometric growth
regime. We further show that microresonator geometric dispersion can be tuned
to compensate for changes in the material dispersion.

††journal: ol

The ability to realize frequency comb generation in integrated photonic
platforms through dissipative Kerr soliton (DKS) formation in $\chi^{(3)}$
microresonators has opened up numerous applications in timekeeping,
communications, and spectroscopy [1]. Recent demonstrations combining DKS
microcombs with chip-scale lasers [2], to the extent of realizing octave-
spanning bandwidth [3], highlight the potential for field deployment of such
systems. This is particularly true for silicon nitride (SiN), which not only
has been utilized in numerous demonstrations [4], but has been shown to fit
within foundry-like fabrication process flows suitable for mass production
[5]. Successful use of SiN within applications hinges on its large Kerr
nonlinearity, broadband optical transparency, and reproducible and
controllable dispersion, the latter is of which is particularly critical for
broadband applications such as the generation of octave-spanning combs [6, 7,
8]. The microcomb bandwidth and shape for a given material is entirely defined
through the resonator dispersion, which can be decomposed in two elements: the
material dispersion and the geometric dispersion (fig. 1), the former defined
during the material growth, and the latter relying on accurate dimensions of
the resonator compared to the theoretical design. SiN growth for microcombs is
typically done via low pressure chemical vapor deposition (LPCVD) using two
precursors gases, ammonia (NH3) and dichlorosilane (DCS or SiH2Cl2), where the
ratio between these two gases defines the composition of the deposited film.
Most Kerr comb works have focused on achieving stoichiometric films, i.e.,
Si3N4, though a substantial amount of work has been conducted in studying
silicon rich films [9, 10, 11, 12], in part due to the higher nonlinear
coefficient in this regime, with its refractive index dispersion also having
been thoroughly studied [10]. Yet, stoichiometric films are often preferred
due to linear losses that are typically lower and a wider optical transparency
window, the latter of particular importance considering the possibility of
nonlinear absorption. Interestingly, the gas ratio between NH3 and DCS at
which the stochiometric condition is reached remains unclear; often NH3:DCS
$>2:1$ is cited [13], though larger NH3:DCS ratios have also been considered
[14, 15]. Here, we study LPCVD-deposited SiN films within this nominally
stoichiometric regime with a NH3:DCS gas ratio $>2:1$. We find that the
refractive index dispersion continues to change as the NH3:DCS ratio is
increased, with 3:1, 5:1, 7:1, and 15:1 ratios considered. We consider the
impact of this varying material dispersion on the integrated dispersion of
microresonators fabricated in these LPCVD-deposited films, and show through
experimental generation of DKS microcomb states that differences in material
dispersion between the films can be somewhat compensated through geometric
dispersion, in particular, the thickness of the SiN film.

Figure 1: (a) LPCVD system, where SiO2 on Si wafers are loaded within a quartz
furnace tube which is heated to a temperature of
775$\mathrm{\SIUnitSymbolDegree}$C in our case. The two precursor gases, are
injected into the chamber at controllable flow levels, allowing change of the
gas ratio in the reaction and impacting the material refractive index and its
dispersion. (b) Microring resonator, which is coupled to a waveguide for
injection of pump light and collection of generated comb light. The three
geometric parameters, ring radius $RR$, thickness $H$, and ring width $RW$
determine the geometric dispersion contribution. (c) Both material and
geometric dispersion contribute to the total dispersion of the resonato,
allowing anomalous dispersion regime - free spectral range increasing with the
frequency (upper panel) - preferable for microcomb operation. In addition,
this dispersion determines the spectral position of any dispersive waves,
which appear at zero-crossings of the integrated dispersion $D_{\text{int}}$
(lower panel). Figure 2: (a) Ellipsometer fit coefficients for the different
SiN growth conditions (varying gas ratio), where the films are deposited on
silicon substrates and the ellipsometer data is fit to a standard Sellmeier
model. (b) Frequency-dependent refractive index obtained from the previous
fits for gas ratios of 3:1 (purple), 5:1 (cyan), 7:1 (teal) and 15:1 (red).
The transparent areas account for the uncertainty of about 1% in the
ellipsometer fitting coefficients. The Si-rich refractive index data (grey
dashed) has been extracted from ref. [15]. The octave spanning frequency
ranges for different microcombs are indicated on the top left, and correspond
to experimental data of refs[16, 17, 18]. (c) Average extracted intrinsic
quality factor of ring resonators with the same geometry but created under
different growth conditions. The standard variation is smaller than the
displayed data point Figure 3:  (a) Numerical simulations of the dispersion
parameter for a microring resonator with $RR=23$ µm and $RW=1060$ nm for a
thickness that varies from 410 nm (darker) to 485 nm (lighter) considering
different gas ratio, 3:1 (purple), 5:1 (cyan), 7:1 (teal) and 15:1 (red). (b)
Integrated dispersion for the same set of geometries and gas ratios (top
panel), with their corresponding LLE simulated spectra (bottom panel). (c)
Position of the DWs, obtained from the zero-crossings of the integrated
dispersion, as a function of thickness for the different gas ratios of
interest, considering a pump frequency at 281 THz. The insets highlight the
possibility to compensate for the DW shift due to the material dispersion by
adjusting the geometric dispersion.

Growth of SiN is conducted through LPCVD onto a 3 µm thick silicon dioxide
(SiO2) layer that has been grown via thermal oxidation of a 100 mm silicon
wafer. The precursor gases, as stated previously, are NH3 and DCS, where the
NH3/DCS ratio is adjusted with a fixed DCS flow of $8.3\times 10^{-7}$ m3.s-1
(50 sccm). We use a fixed temperature of 775 $\mathrm{\SIUnitSymbolDegree}$C
and pressure of 300 Pa for all the different grown films. In each growth run,
we also use a Si wafer without SiO2 as a reference, as this simpler layer
structure limits the number of free parameters in fitting of ellipsometer
measurements of the SiN thickness and refractive index. The ellipsometer
measurements are performed over a wavelength range of 600 nm to 1700 nm, and
are carried out for films grown with NH3:DCS ratios of 3:1, 5:1, 7:1 and 15:1
(fig. 2(a)). We fit the ellipsometer measurements to a Sellmeier model, which
has been found to be accurate for the SiN refractive index over a large
bandwidth [19]
$\varepsilon_{\mathrm{SiN}}(\lambda_{\mu})=n_{\mathrm{SiN}}^{2}(\lambda_{\mu})=1+A_{1}\frac{\lambda_{\mu}^{2}}{\lambda_{\mu}^{2}-B_{1}^{2}}-E_{1}\lambda_{\mu}^{2}$,
with $\lambda_{\mu}=10^{6}c/f$ being the wavelength in micrometers, and $c$
and $f$ being the speed of light and the frequency respectively. We note that
other models have been used to fit ellipsometer measurements of SiN films
taken over more extended infrared wavelengths [14]; however, as our concern in
this work is the behavior in the near-infrared and telecom bands, the
Sellmeier model is adequate.

A clear trend appears where the larger the NH3/DCS gas ratio, which
corresponds to a lower proportion of silicon in the chemical process, the
lower the refractive index, varying from around $n=1.998$ and $n=2.012$ to
$n=1.967$ and $n=1.982$ at 193 THz and 283 THz (i.e. 1555 nm and 1060 nm) for
3:1 and 15:1 gas ratios, respectively (fig. 2(b)). Although the material
refractive index variation is important for resonator design to accomplish the
desired dispersion for microcomb applications, it is also critical that the
resonator intrinsic quality factor remain high, that is, that the differences
in material growth do not result in added absorption. We fabricated microring
resonators with ring radius $RR=23$ µm, a SiN thickness of close to $H=440$
nm, and a ring width $RW=1060$ nm, in each of the SiN films, and performed
linear transmission measurements of the resulting cavity in the 280 THz band,
retrieving both the coupling and intrinsic quality factors [20]. The extracted
average intrinsic quality factors (fig. 2(c)), which are in the 106 range, do
not show a clear trend with precursor gas ratio. It is likely that the
observed differences are due to process variation from run-to-run, rather than
any specific attribute of the SiN films.

Figure 4:  DKS generated for the (a) 3:1 and (b) 7:1 gas ratios, exhibiting a
short wavelength DW whose frequency matches to within 2 THz (right inset), as
expected from the dispersion simulation presented in fig. 3(c) for each
thickness (left insets).

The cavity dispersion is composed of contributions from both geometric and
material dispersion, and we next consider how it is impacted by the variation
in SiN precursor gas ratio. We use finite element method (FEM) simulations to
compute the resonator dispersion characteristics for a variety of resonator
geometries with different refractive index dispersion profiles based on the
data from fig. 2. In this study, the microring with a $RR=23$ µm ring radius
and a $RW=1060$ nm ring width is simulated with the thickness varied from
$H=410$ nm to $H=485$ nm for the different gas ratios presented above. Because
the refractive index is changed through the precursor gas ratio, for a fixed
geometry - and hence a fixed geometric dispersion - the total ring dispersion
is changed (fig. 3(a)). For example, the ring exhibit complete normal
dispersion at a thickness below 410 nm for the 3:1 gas ratio and a thickness
below 415 nm for the 5:1 gas ratio. The impact of the material dispersion is
made more clear by studying the integrated dispersion, defined as the
discrepancy between the cavity resonances and a fixed frequency grid defined
by the free spectral range (FSR) around the pump, that is,
$D_{\mathrm{int}}(\mu)=\omega_{\mathrm{res}}-(\omega_{\mathrm{pmp}}+D_{\mathrm{1}}\mu)$,
with $\omega_{\mathrm{res,pmp}}$ being the angular frequency of resonance and
the pumped one respectively, $D_{\mathrm{1}}=2\pi\times\mathrm{FSR}$, and
$\mu$ is the mode number relative to the pumped mode (i.e
$\omega_{\mathrm{res}}(\mu=0)=\omega_{\mathrm{pmp}}$). Studying the integrated
dispersion (fig. 3(b)) allows for a more intuitive prediction of the supported
DKS behavior, as $D_{\mathrm{int}}=0$ correspond to the location of DKS-
induced dispersive waves (DWs), whose existence helps to broaden the comb
spectrum beyond the anomalous dispersion window [4], and which can be tailored
to aid in comb self-referencing [6]. From this standpoint, the impact of the
material dispersion is obvious as it significantly modifies the frequency
position of the DWs, exhibited through the integrated dispersion zero
crossings and the resulting predicted comb spectra based on Lugiato-Lefever
Eqaution (LLE) modeling [21]. For example, at the same thickness $H=455$ nm,
the high frequency DW exhibits a shift of more than 10 THz from 3:1 to 7:1 gas
ratio, ultimately decreasing the bandwidth from an octave spanning microcomb
with a span of 227 THz to a span of close to 100 THz. Interestingly, it is
possible to at least partially compensate for changes in the material
dispersion that arise from different growth conditions through appropriate
adjustment of the geometric dispersion. For instance, it is possible to
compensate for variations in the spectral position of one DW due to different
gas ratios (fig. 3(c)) by adjusting the ring thickness. Nearly the same high
frequency (short wavelength) DW can be obtained for any of the 3:1, 5:1, or
7:1 gas ratios, by adjusting the thickness between 445 nm and 453 nm. However,
it is important to note that the thickness alone cannot compensate for shifts
in the position of both DWs, and adjustments in the ring width would also be
needed to provide the two-dimensional parameter space needed for such
compensation.

To demonstrate such compensation of changes in material dispersion with
adjustments to the geometric dispersion, we study two microring resonators
with nominally identical in-plane dimensions, but different gas ratios were
used in the SiN film growth (3:1 and 7:1), along with a thickness difference
of about 10 nm (445 and 455 nm, respectively). We pump both resonators around
the same frequency of 281 THz (i.e. 1067 nm), corresponding to the same fist
order transverse electric (TE0) mode, with an in-waveguide power of
$P_{\mathrm{pmp}}=120$ mW. In order to reach the single soliton state, we
actively cool the resonator by counterclockwise pumping a cross-polarized mode
at a different frequency (307 THz or 977 nm), which has been demonstrated to
allow adiabatic access to DKS states [22, 8]. The obtained DKS spectrum for
each ring resonator exhibits almost the same high frequency DW (fig.
4(a)-(b)), to within 2 THz (i.e., a discrepancy of only 2 comb teeth), as
predicted from LLE simulations where only the thickness and the material
dispersion have been changed. In contrast, if the gas ratio was kept fixed
(e.g., at 7:1), a 10 nm thickness change, from 445 nm to 455 nm, is predicted
to result in a shift of the DW from approximately 335 THz to 365 THz
corresponding to a $\approx$30 FSRs shift of the short wavelength DW.

In conclusion, we have demonstrated that even when operating entirely within a
regime of LPCVD growth in which silicon nitride films are commonly considered
to be stoichiometric, changes to the gas ratio continue to impact chromatic
dispersion and its influence on microresonaor frequency comb generation.
Although the change in material dispersion is not as dramatic as when moving
to the silicon-rich regime, it is substantial enough to significantly impact
the properties of broadband microresonator frequency. Along with studying the
impact of precursor gas ratio on material dispersion and integrated dispersion
of microring resonators fabricated from these films, we show that changes in
geometric dispersion, for example resonator thickness, can be used to
compensate for changes in material dispersion. Going forward, we believe our
results point to the ability to use material growth conditions together with
geometric control in the dispersion engineering of microresonators for
broadband frequency comb applications.

Funding. Defense Advanced Research Projects Agency (DARPA-APHI); National
Institute of Standards and Technology (NIST-on-a-chip). This work was
performed in part at the Cornell NanoScale Facility, which is supported by the
National Science Foundation (Grant NNCI-2025233).

Disclosures. The authors declare no conflicts of interest.

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# Hybrid analysis of radiative corrections to neutron decay with current
algebra and effective field theory

Chien-Yeah Seng1,2 1Facility for Rare Isotope Beams, Michigan State
University, East Lansing, MI 48824, USA  2Department of Physics, University of
Washington, Seattle, WA 98195-1560, USA

(August 28, 2024)

###### Abstract

We introduce a useful framework for high-precision studies of the neutron beta
decay by merging the current algebra description and the fixed-order effective
field theory calculation of the electroweak radiative corrections to the
neutron axial form factor. We discuss the advantages of this hybrid method and
show that it only requires a minimal amount of lattice QCD inputs to achieve a
$10^{-4}$ theory accuracy for the Standard Model prediction of the neutron
lifetime and the axial-to-vector coupling ratio $\lambda$, both important to
the search for physics beyond the Standard Model.

##  I Introduction

Beta decays of strongly-interacting systems, e.g. pions, free neutron and
nuclei, provide a perfect avenue for high-precision tests of the Standard
Model (SM) at low energies and searches for physics beyond the Standard Model
(BSM). As an example, they allow high-precision extractions of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix element $V_{ud}$, which can be used for the
test of the CKM unitarity. Decays of spinful particles offer more experimental
observables which impose further constraints on BSM physics; for instance, the
axial-to-vector coupling ratio $\lambda\equiv g_{A}/g_{V}$ in free neutron
decay provides a sensitive probe of right-handed currents [1, 2]. Furthermore,
free neutron holds an advantage compared to nuclei for being free from
nuclear-structure uncertainties, and first-principles calculations with
lattice QCD are more reliable.

Similar to other experiments at the precision frontier, measurements of the
neutron decay lifetime $\tau_{n}$ and $\lambda$, both currently reaching
$10^{-4}$ precision [3], must be accompanied by equally-precise calculations
of SM theory inputs in order to identify small BSM effects from experimental
results. At $10^{-4}$, there are three SM corrections to free neutron decay
that require attention: (1) recoil corrections, (2) isospin-symmetry-breaking
(ISB) corrections, and (3) radiative corrections (RC), and in this paper we
focus on the last, which consists of virtual loop corrections and
bremsstrahlung corrections. Historically, it is divided into “outer” and
“inner” corrections [4]; the outer RC modifies the beta decay spectrum and
could be computed using elementary Quantum Electrodynamics (QED) assuming
point-charge interactions, while the inner RC modifies the vector and axial
coupling constants and involves non-perturbative Quantum Chromodynamics (QCD).
For many years, the study of RC in free neutron and nuclear beta decays has
been based on the current algebra formalism by Sirlin [5] (which we denote as
“Sirlin’s representation”, SR), which identified the form factor (FF)
corrections and the $\gamma W$-box diagram as the only sources of non-trivial
inner RC. Supplemented by recent improved analysis of the vector $\gamma
W$-box diagram [6, 7, 8, 9, 10, 11, 12], a substantial improvement in the
theory precision of the extraction of $V_{ud}$ was achieved.

Effective field theory (EFT) provides an alternative pathway to the problem;
by writing down the most general Lagrangian with the relevant degrees of
freedom (DOFs), one may compute the RC-corrected decay amplitude using
elementary Feynman rules, with the theory precision controlled by the built-in
power-counting scheme of the EFT. In the first attempt in Ref. [13], a
pionless EFT was adopted, with DOFs being nucleons, leptons and photon.
Natural scales in this formalism are $q\sim m_{n}-m_{p}\sim 1$ MeV and
$m_{N}\sim 1$ GeV, which means the natural expansion parameter of the EFT is
$q/m_{N}\sim 10^{-3}$. With that one could obtain the outer RC as well as the
“kinematic” recoil corrections. However, the lack of an intermediate scale
between $q$ and $m_{N}$ makes it unable to describe important physics such as
nucleon structure corrections induced by pion loops. To incorporate the
latter, one needs to start from an EFT that includes pions as dynamical DOFs,
namely the chiral perturbation theory (ChPT). This was performed in Ref.[14]
which observed, for the first time, a percent-level inner RC to the axial
coupling constant $g_{A}$ that comes from pion-loop contributions. While not
affecting the extraction of $V_{ud}$ from neutron beta decay, it plays an
important role when the experimentally-measured $g_{A}$ is compared to the
lattice-calculated, pure-hadronic $\mathring{g}_{A}$ [15] in order to
constrain new physics.

It should be stressed that this unexpectedly large RC was in fact incorporated
in SR; it is contained in the so-called “three-point function” contribution to
the FF correction, but just mistakenly regarded to be small due a
misinterpretation of the partially-conserved-axial-current (PCAC) relation. In
principle, SR provides an explicit expression of the three-point function in
terms of hadronic matrix elements which can be directly computed on lattice;
however, in practice this is difficult due to the complexity of such matrix
elements. To that end, it is beneficial to adopt a hybrid analysis which
combines SR and EFT description [16]. The key idea is to study not the full
RC, but only FF RC which is not easy to be computed on lattice, using EFT.
First, the current algebra formalism does not rely on a chiral expansion, so
the non-FF corrections in this formalism do not suffer from theory
uncertainties from such an expansion (although in principle the SU(2) chiral
expansion scales in powers of $m_{\pi}/\Lambda_{\chi}$, where
$\Lambda_{\chi}\sim 1$ GeV is the chiral symmetry breaking scale, and
converges rapidly). Second, SR provides a more convenient basis to connect to
lattice QCD as the lepton piece decouples from its master formula, so one may
concentrate on pure hadronic matrix elements without introducing dynamical
leptons. This strategy was proven successful in semileptonic kaon decay, where
a reduction of the theory uncertainty of long-distance RC by almost an order
of magnitude was achieved [17, 18, 19, 20, 21]. A similar analysis on the
neutron sector is highly desirable as it may provide further insights to the
general structure of the inner RC, and more importantly, what lattice inputs
are required to most efficiently pin down the unknown pieces.

The content of this work is arranged as follows. In Sec.II we introduce the
general structure of SR, identify the terms that require inputs from non-
perturbative QCD, and define the “outer” and “inner” corrections rigorously.
In Secs.III, IV, V we introduce the EFT formalism based on Heavy Baryon Chiral
Perturbation Theory (HBChPT), and use it to compute the RC to the axial FF,
which provides useful information on the inner correction to $g_{A}$. In
Sec.VI we compare the expressions of the hybrid analysis to the pure-EFT
calculation, which allows us to pin down a subset of low energy constants
(LECs) in the latter; we also discuss the future lattice QCD input needed to
determine the remaining LECs relevant for the neutron beta decay. We draw our
conclusions in Sec.VII.

##  II Radiative corrections in Sirlin’s representation

In this section we introduce the basic theory framework for the free neutron
decay and its RC. We start by writing down the tree-level decay amplitude:

$\mathfrak{M}_{0}=-\frac{G_{F}V_{ud}}{\sqrt{2}}L_{\lambda}F^{\lambda}\leavevmode\nobreak\
,$ (1)

with $G_{F}=1.1663787(6)\times 10^{-5}$ GeV-2 the Fermi constant obtained from
muon decay [22], and
$L_{\lambda}=\bar{u}_{e}\gamma_{\lambda}(1-\gamma_{5})v_{\nu}$ the leptonic
charged weak current. The single-nucleon matrix element of the hadronic
charged weak current $J_{W}^{\mu}=\bar{u}\gamma^{\mu}(1-\gamma_{5})d$ consists
of four FFs: vector, axial, weak magnetism and pseudoscalar (excluding second-
class currents). Upon neglecting recoil effects, the last two form factors can
be dropped while the vector and axial form factors can be approximated by
their values at zero momentum exchange. This gives:

$F^{\mu}\equiv\langle
p|J_{W}^{\mu}(0)|n\rangle\approx\bar{u}(p_{p})\gamma^{\mu}\left[\mathring{g}_{V}+\mathring{g}_{A}\gamma_{5}\right]u(p_{n})\equiv
F_{V}^{\mu}+F_{A}^{\mu}\leavevmode\nobreak\ ,$ (2)

where $\mathring{g}_{V}$ and $\mathring{g}_{A}$ are the pure-QCD vector and
axial coupling constants. In this paper we adopt the experimentalists’ sign
convention, $\mathring{g}_{A}<0$.

###  II.1 $\mathcal{O}(\alpha)$ radiative corrections

Figure 1: Non-trivial one-loop RCs to neutron beta decay. Left: Form factor
corrections, right: $\gamma W$-box diagram.

We now discuss the $\mathcal{O}(\alpha)$ SM electroweak virtual RC to the
decay amplitude based on SR. While details can be found in Ref. [23], here we
just summarize the most important results. One can show that, the only pair of
one-loop diagrams that are not analytically (exactly or perturbatively)
calculable are the electromagnetic RC to the weak FFs and the $\gamma W$-box
diagram, depicted by the first and second diagram in Fig.1 respectively. The
RC-corrected $n\rightarrow pe\bar{\nu}_{e}$ decay amplitude in SR reads:

$\displaystyle\mathfrak{M}$ $\displaystyle=$
$\displaystyle\sqrt{1+\Delta_{R}^{U}}\mathfrak{M}_{0}+\frac{\alpha}{4\pi}\left[3\ln\frac{m_{N}}{m_{e}}+2\ln\frac{m_{e}}{m_{\gamma}}-\frac{3}{4}\right]\mathfrak{M}_{0}+\left(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a}\right)_{\mathrm{int}}$ (3)
$\displaystyle-\frac{G_{F}V_{ud}}{\sqrt{2}}\delta
F_{3\text{pt}}^{\mu}L_{\mu}+\delta\mathfrak{M}_{\gamma
W}^{b}\leavevmode\nobreak\ ,$

and we explain the notations as follows. First, the quantity

$\Delta_{R}^{U}=\frac{\alpha}{2\pi}\left[3\ln\frac{m_{Z}}{m_{N}}+\ln\frac{m_{Z}}{m_{W}}+\tilde{a}_{g}\right]+\delta_{\text{HO}}^{\text{QED}}$
(4)

represents a universal (U), analytically-calculable virtual RC proportional to
the tree-level amplitude; $\tilde{a}_{g}\approx-0.083$ is a small perturbative
QCD correction, and $\delta_{\text{HO}}^{\text{QED}}=0.00109(10)$ represents a
leading higher-order QED effects [24] which we include for accuracy111A more
recent analysis suggested a slightly larger value of $\Delta_{R}^{U}$, which
was partially compensated by a redefinition of the Fermi function and led to a
+0.026% shift of the neutron decay rate [25]. The second squared bracket in
Eq.(3) collects the other part of the analytically-known RC which will later
be grouped into the “outer correction” following historical convention. Here
we introduce a fictitious photon mass $m_{\gamma}$ to regularize infrared
(IR)-divergences.

The remaining terms in $\mathfrak{M}$ are sensitive to physics at and below
the hadron scale. To explain their meaning we need to introduce a few more
prescriptions. First, using the on-mass-shell perturbation theory [26], one
splits the FF correction (i.e. the first diagram in Fig.1) into two terms:

$\delta F^{\mu}=\delta F_{\text{2pt}}^{\mu}+\delta
F_{\text{3pt}}^{\mu}\leavevmode\nobreak\ ,$ (5)

which we name as the “two-point function” and “three-point function”
respectively. The two-point function takes the following form for neutron:

$\delta
F_{\text{2pt}}^{\lambda}=\frac{e^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\partial}{\partial
k_{\lambda}}\left(\frac{m_{W}^{2}}{m_{W}^{2}-k^{2}}\frac{1}{k^{2}-m_{\gamma}^{2}}\right)T^{\mu}_{\>\>\mu}\leavevmode\nobreak\
,$ (6)

where

$T^{\mu\nu}\equiv\int d^{4}xe^{ik\cdot x}\langle
p|T[J_{\text{em}}^{\mu}(x)J_{W}^{\nu}(0)]|n\rangle$ (7)

is a “generalized Compton tensor” consists of the time-order product of the
electromagnetic and charged weak current. The definition of the “three-point
function” is more complicated and consists of two terms:

$\delta F_{\text{3pt}}^{\lambda}=-\lim_{\bar{q}\rightarrow
q}i\bar{q}_{\nu}\frac{\partial}{\partial\bar{q}_{\lambda}}\left[\bar{T}^{\nu}-B^{\nu}\right]+\lim_{\bar{q}\rightarrow
q}i\frac{\partial}{\partial\bar{q}_{\lambda}}\left[D-\bar{q}\cdot
B\right]\leavevmode\nobreak\ ,$ (8)

where $q\equiv p_{n}-p_{p}$ is the nucleon momentum exchange, and

$\displaystyle\bar{T}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{m_{W}^{2}}{m_{W}^{2}-k^{2}}\frac{1}{k^{2}-m_{\gamma}^{2}}\int
d^{4}xe^{i\bar{q}\cdot x}d^{4}ye^{ik\cdot y}\langle
p|T\\{J_{W}^{\mu}(x)J_{\text{em}}^{\nu}(y)J_{\nu}^{\text{em}}(0)\\}|n\rangle$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{ie^{2}}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{m_{W}^{2}}{m_{W}^{2}-k^{2}}\frac{1}{k^{2}-m_{\gamma}^{2}}\int
d^{4}xe^{i\bar{q}\cdot x}d^{4}ye^{ik\cdot y}\langle p|T\\{\partial\cdot
J_{W}(x)J_{\text{em}}^{\nu}(y)J_{\nu}^{\text{em}}(0)\\}|n\rangle$
$\displaystyle B^{\mu}$ $\displaystyle=$
$\displaystyle-\bar{u}_{p}\left[\frac{i\delta
m_{p}}{\not{p}_{n}-\bar{\not{q}}-m_{p}}\mathfrak{T}^{\mu}+\mathfrak{T}^{\mu}\frac{i\delta
m_{n}}{\not{p}_{p}+\bar{\not{q}}-m_{n}}\right]u_{n}\leavevmode\nobreak\ ,$ (9)

with $\delta m_{p,n}$ the nucleon mass shift due to electromagnetic
corrections, and $\mathfrak{T}^{\mu}(\bar{q})$ the single-nucleon weak vertex
function that consists of the vector, weak magnetic, axial and pseudoscalar
form factors:

$\mathfrak{T}^{\mu}(\bar{q})=F_{1}^{W}(\bar{q}^{2})\gamma^{\mu}-\frac{i}{2m_{N}}F_{2}^{W}(\bar{q}^{2})\sigma^{\mu\nu}\bar{q}_{\nu}+G_{A}(\bar{q}^{2})\gamma^{\mu}\gamma_{5}+\frac{G_{P}(\bar{q}^{2})}{2m_{N}}\gamma_{5}\bar{q}^{\mu}\leavevmode\nobreak\
.$ (10)

In particular, the pseudoscalar form factor is related to the axial form
factor as:

$\frac{G_{P}(\bar{q}^{2})}{2m_{N}}=\frac{2m_{N}G_{A}(\bar{q}^{2})}{\bar{q}^{2}-m_{\pi}^{2}}$
(11)

to ensure the conservation of the axial current in the chiral limit. The
$B$-terms in Eq.(8) remove the poles in $\bar{T}^{\nu}$ and $D$ at
$\bar{q}\rightarrow q$. Finally, we split the $\gamma W$-box diagram, i.e. the
second diagram in Fig.1, into two terms:

$\delta\mathfrak{M}_{\gamma W}=\delta\mathfrak{M}_{\gamma
W}^{a}+\delta\mathfrak{M}_{\gamma W}^{b}\leavevmode\nobreak\ ,$ (12)

the (b) term carries an $\epsilon$-tensor from the lepton structure while the
(a) term does not.

Now we can discuss the remaining terms in Eq.(3). First, a partial
cancellation occurs when combining $\delta F_{\text{2pt}}^{\mu}$ and
$\delta\mathfrak{M}_{\gamma W}^{a}$, resulting in an analytically-calculable
piece (which is already included in the first two terms in Eq.(3)) and a
“residual integral” that cannot be simply reduced to something proportional to
$\mathfrak{M}_{0}$:

$\displaystyle\left(\delta\mathfrak{M}_{2}+\delta\mathfrak{M}_{\gamma
W}^{a}\right)_{\mathrm{int}}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}V_{ud}e^{2}}{\sqrt{2}}L_{\lambda}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(p_{e}-k)^{2}-m_{e}^{2}}\frac{1}{k^{2}-m_{\gamma}^{2}}$
(13) $\displaystyle\times\left\\{\frac{2p_{e}\cdot
kk^{\lambda}}{k^{2}-m_{\gamma}^{2}}T^{\mu}_{\>\>\mu}+2p_{e\mu}T^{\mu\lambda}-(p_{n}-p_{p})_{\mu}T^{\lambda\mu}+i\Gamma^{\lambda}\right\\}\leavevmode\nobreak\
,$

where

$\Gamma^{\mu}\equiv\int d^{4}xe^{ik\cdot x}\langle
p|T[J_{\text{em}}^{\mu}(x)\partial\cdot J_{W}(0)]|n\rangle$ (14)

is similar to $T^{\mu\nu}$ except that the charged current is replaced by its
total derivative. They satisfy the following Ward identities:

$k_{\mu}T^{\mu\nu}=iF^{\nu}\leavevmode\nobreak\ ,\leavevmode\nobreak\
(k-q)_{\nu}T^{\mu\nu}=iF^{\mu}-i\Gamma^{\mu}\leavevmode\nobreak\ .$ (15)

The next non-trivial integral is the part of the $\gamma W$-box diagram with
an $\epsilon$-tensor:

$\displaystyle\delta\mathfrak{M}_{\gamma W}^{b}$ $\displaystyle=$
$\displaystyle-i\frac{G_{F}V_{ud}e^{2}}{\sqrt{2}}L_{\lambda}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{m_{W}^{2}}{m_{W}^{2}-k^{2}}\frac{\epsilon^{\mu\nu\alpha\lambda}k_{\alpha}}{[(p_{e}-k)^{2}-m_{e}^{2}]k^{2}}T_{\mu\nu}\leavevmode\nobreak\
.$ (16)

SR allows us to identify these two integrals, together with the three-point
function in Eq.(8), as the only non-trivial quantities in the
$\mathcal{O}(\alpha)$ virtual RC.

Figure 2: Feynman diagrams for the bremsstrahlung process.

Finally, in order to ensure the IR-finiteness of the total decay rate [27, 28,
29, 30, 31], it is necessary to include the bremsstrahlung process which emits
an extra photon, as depicted in Fig.2. The amplitude reads:

$\displaystyle\mathfrak{M}_{\mathrm{brems}}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}V_{ud}e}{\sqrt{2}}\frac{1}{2p_{e}\cdot
k}\left\\{-2p_{e}\cdot\varepsilon^{*}g^{\mu\nu}-k^{\mu}\varepsilon^{*\nu}+k^{\nu}\varepsilon^{*\mu}+i\epsilon^{\mu\nu\alpha\beta}k_{\alpha}\varepsilon_{\beta}^{*}\right\\}F_{\mu}L_{\nu}$
(17)
$\displaystyle+\frac{iG_{F}V_{ud}e}{\sqrt{2}}\varepsilon^{*\mu}L^{\nu}T_{\mu\nu}\leavevmode\nobreak\
,$

where $\varepsilon^{*\mu}$ is the polarization vector of the outgoing photon.

###  II.2 The three-point function

It is possible to infer some general properties of the three-point function
$\delta F_{\text{3pt}}^{\mu}$ just from its definition. We observe that, the
first term in Eq.(8) contains an explicit factor of the momentum exchange $q$,
while the second term depends on the total derivative of the charged current.
This means the first term vanishes in the forward limit ($q\rightarrow 0$) and
the second term vanishes in the current conservation limit ($\partial\cdot
J_{W}\rightarrow 0$). Due to the small neutron-proton mass splitting
$m_{n}-m_{p}\approx 1.3$ MeV, recoil effects in the neutron decay cannot be
larger than $(m_{n}-m_{p})/m_{\pi}\sim 10^{-2}$. So, a recoil effect on top of
a RC which is already suppressed by $\alpha/\pi\sim 10^{-3}$ gives a $10^{-5}$
corrections which is negligible given our precision goal. Therefore, it is
safe to drop the first term in $\delta F_{\text{3pt}}^{\mu}$.

The second term is more tricky, and one should discuss the conservation of the
vector and axial component of the charged current separately. The conservation
of the vector charged current is only broken by the quark mass difference,
i.e. $(\partial\cdot J_{W})_{V}\sim m_{u}-m_{d}$; so we again expect a
$10^{-5}$ effect which can be safely dropped. This would be the end of the
story if the decayed objects are spinless (e.g. pions or spinless nuclei)
because only the vector current contributes to $\delta F_{\text{3pt}}^{\mu}$,
which means we can drop the entire three-point function. But since nucleons
have spin, the axial charge current also enters. PCAC tells us that
$(\partial\cdot J_{W})_{A}\sim m_{\pi}^{2}P(x)$ where $P(x)$ is the
pseudoscalar density [32], and one might naïvely think the axial current
contribution is also suppressed since it is conserved in the chiral limit
(i.e. $m_{\pi}\rightarrow 0$). This is, however, incorrect because the actual
separation of scale reads $q\ll m_{\pi}\ll m_{N}$, so it is inappropriate to
take the chiral limit before the forward limit. Instead, one should drop terms
that scale as $q/m_{\pi}$ (which is why the pseudoscalar FF is not included in
Eq.(2)), and by doing so the axial current becomes explicitly non-conserved.
Therefore, there is a non-suppressed contribution to the three-point function
from the axial current, and in fact we will find out later that the percent-
level correction to $\mathring{g}_{A}$ discovered in Ref.[14] is contained
entirely in $\delta F_{\text{3pt}}^{\mu}$.

###  II.3 “Outer” and “inner” corrections

We may further investigate the properties of the virtual and real RC. The loop
integrals in Eqs.(13), (16) and the bremsstrahlung amplitudes depend only on
the tensors $T^{\mu\nu}$ and $\Gamma^{\mu}$, and it is useful to isolate the
“point-charge” (pc) contribution which depends only on the electromagnetic and
weak charges:

$T^{\mu\nu}=T^{\mu\nu}_{\text{pc}}+T^{\mu\nu}_{\text{non-
pc}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\Gamma^{\mu}=\Gamma^{\mu}_{\text{pc}}+\Gamma^{\mu}_{\text{non-
pc}}\leavevmode\nobreak\ ,$ (18)

where

$\displaystyle T_{\text{pc}}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{u}(p_{p})\gamma^{\mu}\frac{i(\not{p}_{p}+\not{k}+m_{N})}{(p_{p}+k)^{2}-m_{N}^{2}}\gamma^{\nu}(\mathring{g}_{V}+\mathring{g}_{A}\gamma_{5})u(p_{n})$
$\displaystyle\Gamma^{\mu}_{\text{pc}}$ $\displaystyle=$
$\displaystyle-2m_{N}\mathring{g}_{A}\bar{u}(p_{p})\gamma^{\mu}\frac{\not{p}_{p}+\not{k}+m_{N}}{(p_{p}+k)^{2}-m_{N}^{2}}\gamma_{5}u(p_{n})\leavevmode\nobreak\
.$ (19)

One may check that the pc-contribution satisfies the Ward identities (15) in
the isospin limit ($m_{p}=m_{n}=m_{N}$). Physically, the pc-term contributes
to the loop integral only at the IR scale: $k\sim q\sim p_{e}$; the
corresponding loop integrations are finite and distorts the beta decay
spectrum. Meanwhile the non-pc term probes physics at $k\sim m_{\pi}$ and
above; their only effect is to renormalize the bare coupling constants
$\mathring{g}_{V,A}$.

While the detailed calculation of the three-point function will be discussed
in later sections, here we may quote the main conclusion, namely: it splits
into an IR-divergent (labeled as “outer”) and IR-finite (labeled as “inner”)
piece:

$\displaystyle\delta F_{\text{3pt}}^{\mu}$ $\displaystyle=$
$\displaystyle(\delta F_{\text{3pt}}^{\mu})^{\text{outer}}+(\delta
F_{\text{3pt}}^{\mu})^{\text{inner}}$ (20) $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\left(\ln\frac{m_{\gamma}^{2}}{m_{N}^{2}}+2\right)F_{A}^{\mu}+\frac{1}{2}\Delta_{R,\text{3pt}}^{A}F_{A}^{\mu}\leavevmode\nobreak\
,$

the former corrects the beta decay spectrum while the latter introduces a
constant $\Delta_{R,\text{3pt}}^{A}$ that corrects $\mathring{g}_{A}$.
Collecting everything above, we can write the QED-corrected $n\rightarrow
pe\bar{\nu}_{e}$ amplitude as:

$\mathfrak{M}=\mathfrak{M}_{0}+\delta\mathfrak{M}=\mathfrak{M}_{0}+\delta\mathfrak{M}_{\text{inner}}+\delta\mathfrak{M}_{\text{outer}}\leavevmode\nobreak\
.$ (21)

The virtual “outer” correction arrives by combining the second square bracket
in Eq.(3), the pc-contribution to Eqs.(13), (16), and $(\delta
F_{\text{3pt}}^{\mu})^{\text{outer}}$. After neglecting recoil corrections,
the amplitude reads:

$\displaystyle\delta\mathfrak{M}_{\text{outer}}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}V_{ud}}{\sqrt{2}}L_{\mu}F_{\nu}\frac{\alpha}{4\pi}\left\\{g^{\mu\nu}\left[\frac{1}{2}\ln\frac{m_{N}^{2}}{m_{e}^{2}}+\ln\frac{m_{e}^{2}m_{N}^{2}}{m_{\gamma}^{4}}-\frac{11}{4}-4p_{e}\cdot
p_{p}C_{0}\right]\right.$ (22)
$\displaystyle\left.+\frac{2(\tanh^{-1}\beta-i\pi)(p_{p}^{\mu}p_{e}^{\nu}-p_{e}^{\mu}p_{p}^{\nu}-i\epsilon^{\mu\nu\alpha\beta}p_{p\alpha}p_{e\beta})}{p_{p}\cdot
p_{e}\beta}\right\\}\leavevmode\nobreak\ ,$

where $\beta=|\vec{p}_{e}|/E_{e}$ is the electron speed, and $C_{0}$ is the
standard Passarino-Veltman three-point loop function [33] (see also Eq.(C.7)
in Ref.[20]). Meanwhile, the “inner” RC simply shifts the vector and axial
coupling constants; it combines with the tree-level amplitude to give:

$\mathfrak{M}_{0}+\delta\mathfrak{M}_{\text{inner}}=-\frac{G_{F}V_{ud}}{\sqrt{2}}L_{\lambda}\bar{u}(p_{p})\gamma^{\lambda}\left[g_{V}+g_{A}\gamma_{5}\right]u(p_{n})\leavevmode\nobreak\
,$ (23)

which defines the “physical” $g_{V}$, $g_{A}$, the QED-corrected couplings
following the notations by Wilkinson and Macefield [4]:

$\displaystyle g_{V}^{2}$ $\displaystyle=$
$\displaystyle\mathring{g}_{V}^{2}\left\\{1+\Delta_{R}^{V}\right\\}=\mathring{g}_{V}^{2}\left\\{1+\Delta_{R}^{U}+2\Box_{\gamma
W}^{V}\right\\}$ $\displaystyle g_{A}^{2}$ $\displaystyle=$
$\displaystyle\mathring{g}_{A}^{2}\left\\{1+\Delta_{R}^{A}\right\\}=\mathring{g}_{A}^{2}\left\\{1+\Delta_{R}^{U}+2\Box_{\gamma
W}^{A}+2\Box^{A}_{\text{int}}+\Delta_{R,\text{3pt}}^{A}\right\\}\leavevmode\nobreak\
,$ (24)

where $\Delta_{\text{R}}^{U}$ is the “universal” inner RC in Eq.(4),
$\Box_{\gamma W}^{V,A}$ come from the non-pc contribution to
$\delta\mathfrak{M}_{\gamma W}^{b}$ and can be studied to high precision using
dispersion relation [6, 7, 8, 10] and lattice QCD [12],
$\Box^{A}_{\text{int}}$ comes from the contribution of
$\Gamma^{\mu}_{\text{non-pc}}$ to the “residual integral”, and
$\Delta_{R,\text{3pt}}^{A}$ comes from $\delta F_{\text{3pt}}^{\mu}$ which
contributes only to the axial FF. Notice that $\Delta_{R}^{V}$ is the famous
“nucleus-independent RC” that affects the determination of $V_{ud}$ from
neutron and nuclear decay processes [34]. From the equation above we also
obtain the following QED-corrected axial-to-vector ratio:

$\lambda^{2}=\frac{g_{A}^{2}}{g_{V}^{2}}=\frac{\mathring{g}_{A}^{2}}{\mathring{g}_{V}^{2}}\left\\{1+2(\Box_{\gamma
W}^{A}-\Box_{\gamma
W}^{V})+2\Box^{A}_{\text{int}}+\Delta_{R,\text{3pt}}^{A}\right\\}\leavevmode\nobreak\
.$ (25)

Previous works such as [35] missed the terms $\Box^{A}_{\text{int}}$ and
$\Delta_{R,\text{3pt}}^{A}$.

To evaluate the bremsstrahlung amplitude, we notice that it depends only on
physics at the scale $k\sim q\sim p_{e}$ since the photon must be on-shell,
thus it is appropriate to replace $T^{\mu\nu}\rightarrow
T_{\text{pc}}^{\mu\nu}$. Adding everything above, we are now able to write
down the differential rate of the inclusive decay process $n\rightarrow
pe\bar{\nu}_{e}(\gamma)$ for a polarized neutron as:

$\displaystyle\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}$ $\displaystyle=$
$\displaystyle\frac{(G_{F}V_{ud})^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}(E_{m}-E_{e})^{2}F(E_{e})g_{V}^{2}(1+3\lambda^{2})\left(1+\frac{\alpha}{2\pi}g(E_{e})\right)$
$\displaystyle\times\left\\{1+\left(1+\frac{\alpha}{2\pi}\delta^{(2)}(E_{e})\right)a\vec{\beta}\cdot\hat{p}_{\nu}+\hat{e}_{s}\cdot\left[\left(1+\frac{\alpha}{2\pi}\delta^{(2)}(E_{e})\right)A\vec{\beta}+B\hat{p}_{\nu}\right]\right\\}\leavevmode\nobreak\
,$

where $\beta=\vec{p}_{e}/E_{e}$, $\hat{e}_{s}$ is the neutron unit spin
vector, $E_{m}=(m_{n}^{2}-m_{p}^{2}+m_{e}^{2})/(2m_{n})\approx m_{n}-m_{p}$ is
the electron end-point energy, $F(E_{e})\approx 1+\alpha\pi/\beta$ is the
Fermi function [36] (see also more recent discussions in [25, 37, 38]), and

$a=\frac{1-\lambda^{2}}{1+3\lambda^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\
A=-\frac{2\lambda(\lambda+1)}{1+3\lambda^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\
B=\frac{2\lambda(\lambda-1)}{1+3\lambda^{2}}\leavevmode\nobreak\ $ (27)

are the correlation coefficients that can be experimentally measured. The
functions $g(E_{e})$ and $\delta^{(2)}(E_{e})$ describe the full outer
corrections (after excluding the Fermi function), and are given by [39, 40]:

$\displaystyle g(E_{e})$ $\displaystyle=$ $\displaystyle
3\ln\frac{m_{N}}{m_{e}}-\frac{3}{4}+4\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)\left(\ln\frac{2(E_{m}-E_{e})}{m_{e}}+\frac{E_{m}-E_{e}}{3E_{e}}-\frac{3}{2}\right)$
$\displaystyle-\frac{4}{\beta}\mathrm{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)+\frac{1}{\beta}\tanh^{-1}\beta\left(2+2\beta^{2}+\frac{(E_{m}-E_{e})^{2}}{6E_{e}^{2}}-4\tanh^{-1}\beta\right)$
$\displaystyle\delta^{(2)}(E_{e})$ $\displaystyle=$ $\displaystyle
2\left(\frac{1-\beta^{2}}{\beta}\right)\tanh^{-1}\beta+\frac{4(E_{m}-E_{e})(1-\beta^{2})}{3\beta^{2}E_{e}}\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)$
(28)
$\displaystyle+\frac{(E_{m}-E_{e})^{2}}{6\beta^{2}E_{e}^{2}}\left(\frac{1-\beta^{2}}{\beta}\tanh^{-1}\beta-1\right)\leavevmode\nobreak\
.$

In particular, $g(E_{e})$ is the well-known Sirlin’s function [39]. It is
worth pointing out that, despite its mathematical rigorosity,
Eq.(LABEL:eq:differential) is actually problematic when applied to the
analysis of actual experimental outputs, especially the
$\vec{p}_{\nu}$-dependent correlation functions. This is because most
experiments do not detect the outgoing neutrino, so it is the following
“pseudo-neutrino” momentum:

$\vec{p}_{\nu}^{\prime}\equiv-\vec{p}_{p}-\vec{p}_{e}\leavevmode\nobreak\ ,$
(29)

instead of $\vec{p}_{\nu}$, which is an actual experimental observable. The
equality $\vec{p}_{\nu}^{\prime}=\vec{p}_{\nu}$ breaks down in the presence of
the bremsstrahlung photon, so the naïve application of
Eq.(LABEL:eq:differential) would lead to an $\mathcal{O}(\alpha/\pi)$ error in
the $\vec{p}_{e}$-dependent correlations. A possible solution is to use
$\vec{p}_{\nu}^{\prime}$ in replacement of $\vec{p}_{\nu}$ as a free variable
in the differential rate; by doing so one modifies Eq.(LABEL:eq:differential)
as:

$\displaystyle\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{(G_{F}V_{ud})^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}(E_{m}-E_{e})^{2}F(E_{e})g_{V}^{2}(1+3\lambda^{2})\Bigg{\\{}1+a\beta
c^{\prime}+\frac{\delta_{\text{tot}}^{0}(E_{e},c^{\prime})}{1+3\lambda^{2}}$
(30)
$\displaystyle+\hat{e}_{s}\cdot\left[\left(A+\frac{\delta_{\text{tot}}^{\beta}(E_{e},c^{\prime})}{1+3\lambda^{2}}\right)\vec{\beta}+\left(B+\frac{\delta_{\text{tot}}^{\nu}(E_{e},c^{\prime})}{1+3\lambda^{2}}\right)\hat{p}_{\nu}^{\prime}\right]\Bigg{\\}}\leavevmode\nobreak\
,$

where $c^{\prime}\equiv\hat{p}_{e}\cdot\hat{p}_{\nu}^{\prime}$, and the
analytic expressions of $\delta_{\text{tot}}^{0,\beta,\nu}$ can be found in
Ref.[41]. Powerful numerical packages that return equivalent results are also
available [42].

##  III Inner correction from the residual integral

Given that the quantities $\Box_{\gamma W}^{V,A}$ are now well-understood, we
need only to determine $\Box^{A}_{\text{int}}$ and $\Delta^{A}_{R,\text{3pt}}$
to fully pin down the inner RCs, and we start with $\Box^{A}_{\text{int}}$
which originates from the “residual integral”. Among the four terms in the
curly bracket of Eq.(13), the first three terms have an explicit suppression
of $p_{e}$ or $p_{n}-p_{p}$ in the numerator, therefore their integrals can
only depend on physics at the scale $k\sim p_{e}$, which means the replacement
$T^{\mu\nu}\rightarrow T^{\mu\nu}_{\text{pc}}$ is sufficient. This is,
however, not true for the fourth term: it depends not only on physics at
$k\sim p_{e}$ (which is evaluated by setting
$\Gamma^{\lambda}\rightarrow\Gamma^{\lambda}_{\text{pc}}$), but also up to
$k\sim m_{\pi}$ where the pseudoscalar form factor $G_{P}$ cannot be
neglected; it does not probe the scale $k\gg m_{\pi}$ where chiral symmetry is
restored and $\Gamma^{\lambda}\rightarrow 0$. Therefore, to compute the full
contribution of the fourth term in Eq.(13) we need to include all the nucleon
electromagnetic and weak form factors (here we take $p_{p}=p_{n}=p$ for
simplicity):

$\displaystyle\Gamma^{\mu}$ $\displaystyle=$
$\displaystyle-2m_{N}G_{A}\frac{m_{\pi}^{2}}{m_{\pi}^{2}-k^{2}}\bar{u}_{p}\left\\{\frac{[F_{1p}\gamma^{\mu}-\frac{i}{2m_{N}}\sigma^{\mu\nu}k_{\nu}F_{2p}](\not{p}+\not{k}+m_{N})\gamma_{5}}{(p+k)^{2}-m_{N}^{2}}\right.$
(31)
$\displaystyle\left.+\frac{\gamma_{5}(\not{p}-\not{k}+m_{N})[F_{1n}\gamma^{\mu}-\frac{i}{2m_{N}}\sigma^{\mu\nu}k_{\nu}F_{2n}]}{(p-k)^{2}-m_{N}^{2}}\right\\}u_{n}\leavevmode\nobreak\
,$

and $\Gamma^{\mu}_{\text{pc}}$ in Eq.(19) simply corresponds to taking
$k\rightarrow 0$ everywhere except the nucleon propagators in the expression
above.

Combining Eqs.(19) and (31), we obtain the analytic expression of
$\Gamma^{\nu}_{\text{non-pc}}=\Gamma^{\mu}-\Gamma^{\mu}_{\text{pc}}$ which is
then plugged into Eq.(13) (and take $p_{e}\rightarrow 0$) to get its inner RC
to $g_{A}$. The result takes the following integral form:

$\displaystyle\Box^{A}_{\text{int}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{2\pi}\int_{0}^{\infty}\frac{dQ^{2}}{Q^{2}}\left\\{\frac{Q^{2}}{4m_{N}^{2}}\left(\frac{2m_{N}^{2}}{Q^{2}}+1-\sqrt{1+\frac{4m_{N}^{2}}{Q^{2}}}\right)\times\right.$
(32)
$\displaystyle\left[\left(F_{1p}(Q^{2})-F_{1n}(Q^{2})\right)\hat{G}_{A}(Q^{2})\frac{m_{\pi}^{2}}{m_{\pi}^{2}+Q^{2}}-1\right]-\frac{1}{3}\left(F_{2p}(Q^{2})-F_{2n}(Q^{2})\right)\times$
$\displaystyle\left.\hat{G}_{A}(Q^{2})\frac{m_{\pi}^{2}}{m_{\pi}^{2}+Q^{2}}\left[\frac{(Q^{2})^{2}}{16m_{N}^{4}}-\frac{3Q^{2}}{8m_{N}^{2}}+\left(\frac{Q^{2}}{2m_{N}^{2}}-\frac{(Q^{2})^{2}}{16m_{N}^{4}}\right)\sqrt{1+\frac{4m_{N}^{2}}{Q^{2}}}\right]\right\\}\leavevmode\nobreak\
,$

where $\hat{G}_{A}(Q^{2})\equiv G_{A}(Q^{2})/\mathring{g}_{A}$. The integral
can be evaluated analytically by taking the $Q^{2}\rightarrow 0$ limit to all
the form factors: $F_{1p}(0)=\hat{G}_{A}(0)=1$, $F_{1n}(0)=0$,
$F_{2p}(0)=\kappa_{p}=\mu_{p}-1$, $F_{2n}(0)=\kappa_{n}=\mu_{n}$. The result
is:

$\displaystyle\Box^{A}_{\text{int}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\left\\{\Lambda-\frac{2m_{N}^{2}-m_{\pi}^{2}}{2m_{N}^{2}}\ln\frac{m_{N}^{2}}{m_{\pi}^{2}}+1\right.$
(33)
$\displaystyle\left.-\frac{1-\mu_{p}+\mu_{n}}{24m_{N}^{4}}\left[(16m_{N}^{4}+2m_{N}^{2}m_{\pi}^{2})\Lambda+(6m_{N}^{2}m_{\pi}^{2}+m_{\pi}^{4})\ln\frac{m_{N}^{2}}{m_{\pi}^{2}}+2m_{N}^{2}m_{\pi}^{2}\right]\right\\}$
$\displaystyle\approx$ $\displaystyle 2.47\times 10^{-3}\leavevmode\nobreak\
,$

where
$\Lambda\equiv(m_{\pi}/m_{N}^{2})\sqrt{m_{\pi}^{2}-4m_{N}^{2}}\ln[(\sqrt{m_{\pi}^{2}-4m_{N}^{2}}+m_{\pi})/(2m_{N})]$.
We also checked that, retaining the full nucleon form factor-dependence [43,
44, 45, 46, 47, 48, 49] only shifts its value slightly to $2.44\times 10^{-3}$
with an uncertainty of the order $10^{-6}$ from the form factors; this proves
that the $\Gamma^{\mu}$ contribution is indeed insensitive to physics at $k\gg
m_{\pi}$. For the later comparison with EFT, it is also useful to obtain its
expansion in powers of $m_{\pi}/m_{N}$:

$\Box^{A}_{\text{int}}=\frac{\alpha}{2\pi}\left\\{\frac{1}{2}\left(\ln\frac{m_{N}^{2}}{m_{\pi}^{2}}-1\right)+\frac{\pi
m_{\pi}}{6m_{N}}(1+2\mu_{p}-2\mu_{n})+\mathcal{O}\left(\frac{m_{\pi}^{2}}{m_{N}^{2}}\right)\right\\}\leavevmode\nobreak\
.$ (34)

##  IV Introducing the effective field theory framework

We now deal with $\Delta_{R,\text{3pt}}^{A}$ that involves a three-current
product and is more difficult to be treated with the method above. We resort
to the hybrid analysis introduced in Ref.[16], namely to compute not the full
amplitude, but only the FF correction $\delta F^{\mu}$ in an EFT framework,
and perform manually the splitting of two- and three-point function. This
provides an EFT prediction of $\Delta_{R,\text{3pt}}^{A}$ which can be used
alongside with other, more accurately-predicted inner RC. For that purpose, we
need to introduce the EFT Lagrangian from which the Feynman rules of the
charged weak current operator and other QCD+QED interactions are derived. In
this work we stick to the HBChPT framework adopted in Ref.[14], with some
minor changes of notations.

We start by introducing external source terms to the two-flavor massless QCD
Lagrangian $\mathcal{L}_{\text{QCD}}^{0}$:

$\mathcal{L}=\mathcal{L}_{\text{QCD}}^{0}+\bar{q}_{R}\gamma_{\mu}(r^{\mu}+v_{s}^{\mu}/3)q_{R}+q_{L}\gamma_{\mu}(l^{\mu}+v_{s}^{\mu}/3)q_{L}-\bar{q}_{R}(s+ip)q_{L}-\bar{q}_{L}(s-ip)q_{R}\leavevmode\nobreak\
,$ (35)

where $r^{\mu}$, $l^{\mu}$, $v_{s}^{\mu}$, $s$ and $p$ are external Hermitian
source fields. In particular, $r^{\mu}$, $l^{\mu}$ are isotriplet (i.e.
traceless in the flavor space) while $v_{s}^{\mu}$ is isosinglet. To compute
$\delta F^{\mu}$ we need to include dynamical photons but not leptons. This
amounts to splitting the external fields as:

$\displaystyle r^{\mu}$ $\displaystyle=$ $\displaystyle-
eA_{\mu}\left\\{Q_{\text{EM}}^{R}-\frac{1}{2}\langle
Q_{\text{EM}}^{R}\rangle\right\\}+\bar{r}^{\mu}$ $\displaystyle l^{\mu}$
$\displaystyle=$ $\displaystyle-
eA_{\mu}\left\\{Q_{\text{EM}}^{L}-\frac{1}{2}\langle
Q_{\text{EM}}^{L}\rangle\right\\}+\bar{l}^{\mu}$ $\displaystyle v_{s}^{\mu}$
$\displaystyle=$ $\displaystyle-\frac{3e}{4}A_{\mu}\langle
Q_{\text{EM}}^{R}+Q_{\text{EM}}^{L}\rangle+\bar{v}_{s}^{\mu}$ $\displaystyle
s+ip$ $\displaystyle=$ $\displaystyle
M_{q}+\bar{s}+i\bar{p}\leavevmode\nobreak\ ,$ (36)

where $Q_{\text{EM}}^{R}=Q_{\text{EM}}^{L}=\text{diag}(2/3,-1/3)$ is the quark
charge matrix, $M_{q}=\text{diag}(m_{u},m_{d})$ is the quark mass matrix,
$A_{\mu}$ is the photon field, and $\langle...\rangle$ denotes the trace over
flavor space. Throughout this work we will assume strong isospin symmetry,
i.e. $m_{u}=m_{d}\equiv m_{q}$. The physical Lagrangian is recovered when the
“classical fields” $\bar{r}^{\mu}$, $\bar{l}^{\mu}$, $\bar{v}_{s}^{\mu}$,
$\bar{s}$ and $\bar{p}$ are set to zero. Particular interest is on the field
$\bar{l}^{\mu}$ attached to the left-handed current: by spelling out its
isotriplet components,

$\bar{l}_{\mu}=2\tau_{+}\bar{l}^{+}_{\mu}+2\tau_{-}\bar{l}^{-}_{\mu}+\tau_{3}\bar{l}^{0}_{\mu}\leavevmode\nobreak\
,$ (37)

the charged current operator $J_{W}^{\mu}$ can be obtained as:

$J_{W}^{\mu}=\left(\frac{\partial}{\partial\bar{l}^{+}_{\mu}}\mathcal{L}\right)_{\bar{l}_{\mu}=0}\leavevmode\nobreak\
.$ (38)

The equation above allows us to derive the Feynman rules of the $J_{W}^{\mu}$
vertices in an EFT.

Now we proceed to introduce the chiral Lagrangian; standard notations in
HBChPT can be found in Ref.[50] and will not be repeated here. We assign the
chiral power counting $e\sim p$ which allows us to retain finite terms in the
full chiral Lagrangian given our precision goal. In the pion sector we only
need the Lagrangian to the chiral order $n=2$:

$\mathcal{L}_{\pi}=\mathcal{L}_{\pi}^{(2)}+...=\mathcal{L}_{\pi}^{p^{2}}+\mathcal{L}_{\pi}^{e^{2}}+...\leavevmode\nobreak\
,$ (39)

where

$\mathcal{L}_{\pi}^{p^{2}}=\frac{F_{0}^{2}}{4}\langle
u_{\mu}u^{\mu}+\chi_{+}\rangle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\mathcal{L}_{\pi}^{e^{2}}=e^{2}Z_{\pi}F_{0}^{4}\langle
Q_{\text{EM}}^{R}UQ_{\text{EM}}^{L}U^{\dagger}\rangle\leavevmode\nobreak\ ,$
(40)

with

$\displaystyle u_{\mu}$ $\displaystyle=$ $\displaystyle
i\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}\right]$
$\displaystyle\chi_{+}$ $\displaystyle=$ $\displaystyle u^{\dagger}\chi
u^{\dagger}+u\chi^{\dagger}u$ $\displaystyle\chi$ $\displaystyle=$
$\displaystyle B_{0}(s+ip)\leavevmode\nobreak\ .$ (41)

In particular, the $\mathcal{O}(e^{2})$ term describe short-distance QED
effects where the photons are integrated out [51]. The LEC $Z_{\pi}\approx
0.81$ is obtained from $\pi^{+}-\pi^{0}$ mass splitting.

In the nucleon sector, we need the chiral Lagrangian up to $n=3$:

$\mathcal{L}_{\pi N}=\mathcal{L}_{\pi N}^{(1)}+\mathcal{L}_{\pi
N}^{(2)}+\mathcal{L}_{\pi N}^{(3)}+...$ (42)

The $n=1$ Lagrangian is well-known:

$\mathcal{L}_{\pi
N}^{(1)}=\bar{N}_{v}iv\cdot\mathcal{D}N_{v}-\mathring{g}_{A}\bar{N}_{v}S\cdot
uN_{v}\leavevmode\nobreak\ ,$ (43)

where the chiral covariant derivative on the nucleon field reads:

$\mathcal{D}_{\mu}N_{v}=\left(\partial_{\mu}+\frac{1}{2}\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u+u(\partial_{\mu}-il_{\mu})u^{\dagger}\right]-iv_{\mu}^{s}\right)N_{v}\leavevmode\nobreak\
.$ (44)

At $n=2$ there are two terms:

$\mathcal{L}_{\pi N}^{(2)}=\mathcal{L}_{\pi N}^{p^{2}}+\mathcal{L}_{\pi
N}^{e^{2}}\leavevmode\nobreak\ .$ (45)

The $\mathcal{O}(e^{2})$ term gives short-distance QED corrections to nucleon
masses, which effect is canceled out in the full amplitude. So we display only
the $\mathcal{O}(p^{2})$ term:

$\displaystyle\mathcal{L}_{\pi N}^{p^{2}}$ $\displaystyle=$
$\displaystyle\bar{N}_{v}\left[\frac{1}{2m_{N}}\left((v\cdot\mathcal{D})^{2}-\mathcal{D}^{2}\right)+\frac{i\mathring{g}_{A}}{2m_{N}}\\{S\cdot\mathcal{D},u\cdot
v\\}+c_{1}\langle\chi_{+}\rangle\right.$ (46)
$\displaystyle+\left(c_{2}+\frac{\mathring{g}_{A}}{8m_{N}}\right)(u\cdot
v)^{2}+c_{3}u\cdot
u+\left(c_{4}+\frac{1}{4m_{N}}\right)[S^{\mu},S^{\nu}]u_{\mu}u_{\nu}+c_{5}\tilde{\chi}_{+}$
$\displaystyle\left.-\frac{i}{4m_{N}}[S^{\mu},S^{\nu}]\left((1+\mathring{\kappa}_{V})f^{+}_{\mu\nu}+2(1+\mathring{\kappa}_{S})v_{\mu\nu}^{s}\right)\right]N_{v}\leavevmode\nobreak\
,$

where

$\displaystyle\tilde{\chi}_{+}$ $\displaystyle=$
$\displaystyle\chi_{+}-\frac{1}{2}\langle\chi_{+}\rangle$ $\displaystyle
f_{\mu\nu}^{+}$ $\displaystyle=$ $\displaystyle
u\left(\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}]\right)u^{\dagger}+u^{\dagger}\left(\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}]\right)u$
$\displaystyle v_{\mu\nu}^{s}$ $\displaystyle=$
$\displaystyle\partial_{\mu}v_{\nu}^{s}-\partial_{\nu}v_{\mu}^{s}\leavevmode\nobreak\
,$ (47)

$\mathring{\kappa}_{S}=\mathring{\kappa}_{p}+\mathring{\kappa}_{n}$,
$\mathring{\kappa}_{V}=\mathring{\kappa}_{p}-\mathring{\kappa}_{n}$ are the
isoscalar and isovector nucleon magnetic moments in the chiral limit (the
physical magnetic moments are $\kappa_{p}\approx 1.793$,
$\kappa_{n}\approx-1.913$), and the LECs $c_{i}$ can be extracted from pion-
nucleon scattering [52, 53]. Notice that the magnetic moment terms here appear
to be different from that in Ref.[14] because we use the quark charge matrix,
instead of the nucleon charge matrix, as the spurion field. Finally, at $n=3$
there are also two terms:

$\mathcal{L}_{\pi N}^{(3)}=\mathcal{L}_{\pi N}^{p^{3}}+\mathcal{L}_{\pi
N}^{e^{2}p}\leavevmode\nobreak\ ,$ (48)

but we only need the part in $\mathcal{L}_{\pi N}^{e^{2}p}$ that contributes
to $F_{A}^{\mu}$ at tree level:

$\mathcal{L}_{\pi
N}^{e^{2}p}=-2e^{2}\left(g_{1}+g_{2}+\frac{g_{11}}{2}\right)\bar{N}_{v}S^{\mu}\cdot\left(\tau_{+}\bar{l}_{\mu}^{+}+\tau_{-}\bar{l}_{\mu}^{-}\right)N_{v}+...$
(49)

The LECs $\\{g_{i}\\}$ are defined in Ref.[54], except that we rescale them as
$F_{0}^{2}(g_{i})_{\text{there}}=(g_{i})_{\text{here}}$ for convenience. The
above are the full chiral Lagrangian which our following EFT analysis is based
on.

##  V Radiative corrections to the axial form factor

Figure 3: LO loop diagrams, both self-energy and one-particle-irreducible
(1PI), for $\delta F_{A}^{\mu}$. Solid line: nucleon. Dashed line: pion.
Internal wavy line: photon. External wavy line: connection to
$\bar{l}^{+}_{\mu}$. Dot: vertices from $\mathcal{L}_{\pi}^{p^{2}}$ and
$\mathcal{L}_{\pi N}^{p}$. Diamond: vertex from $\mathcal{L}_{\pi}^{e^{2}}$.

Now we are in the position to apply the EFT framework. As discussed before,
the vector FF correction is fully under control, so we may concentrate on the
RC to the axial FF, $\delta F_{A}^{\mu}$. There are two steps in our analysis:
We first compute the full $\delta F_{A}^{\mu}$ in EFT, and then separate it
into two- and three-point function; with this we can infer the value of
$\Delta_{R,\text{3pt}}^{A}$.

###  V.1 Full $\delta F_{A}^{\mu}$

The lowest-order $F_{A}^{\mu}$ is derived from $\mathcal{L}_{\pi N}^{p}$ and
thus scales as $\mathcal{O}(p)$; we denote the terms in the FF correction
$\delta F_{A}^{\mu}$ that scale as $\mathcal{O}(e^{2}p)$ and
$\mathcal{O}(e^{2}p^{2})$ as the “leading-order” (LO) and “next-to-leading-
order” (NLO) corrections, respectively. One-loop diagrams at LO are given in
Fig.3; the loop integrals contain ultraviolet (UV)-divergences which are
canceled by the $\mathcal{O}(e^{2}p)$ LECs in Eq.(49). The result reads:

$\displaystyle(\delta F_{A}^{\mu})_{\text{LO}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left[\frac{Z_{\pi}}{2}(1+3\mathring{g}_{A}^{2})\left(\ln\frac{\mu^{2}}{m_{\pi}^{2}}-1\right)-\mathring{g}_{A}^{2}Z_{\pi}-\frac{1}{2}\left(\ln\frac{m_{\gamma}^{2}}{m_{\pi}^{2}}+1\right)\right.$
(50)
$\displaystyle\left.-\frac{8\pi^{2}}{\mathring{g}_{A}}\left(g_{1}^{r}+g_{2}^{r}+\frac{g_{11}^{r}}{2}\right)\right]F_{A}^{\mu}\leavevmode\nobreak\
,$

where $\mu$ is the renormalization scale in the dimensional-regularization
prescription. The renormalized LECs $g_{i}^{r}$ are related to the bare LECs
$g_{i}$ as:

$g_{i}=\eta_{i}\frac{\mu^{d-4}}{(4\pi)^{2}}\left(\frac{1}{d-4}-\frac{1}{2}\left(-\gamma+\ln
4\pi+1\right)\right)+g_{i}^{r}(\mu)\leavevmode\nobreak\ .$ (51)

The values of the coefficients $\eta_{i}$ can be found in Ref.[54]. As we
advocate in Sec.II.3, one can separate the expression above into “outer” and
“inner” piece:

$(\delta F_{A}^{\mu})_{\text{LO}}=(\delta F_{A}^{\mu})^{\text{outer}}+(\delta
F_{A}^{\mu})_{\text{LO}}^{\text{inner}}\leavevmode\nobreak\ ,$ (52)

where

$\displaystyle(\delta F_{A}^{\mu})^{\text{outer}}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{4\pi}\left(\ln\frac{m_{\gamma}^{2}}{m_{N}^{2}}+2\right)F_{A}^{\mu}$
$\displaystyle(\delta F_{A}^{\mu})_{\text{LO}}^{\text{inner}}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left[\frac{Z_{\pi}}{2}(1+3\mathring{g}_{A}^{2})\left(\ln\frac{\mu^{2}}{m_{\pi}^{2}}-1\right)-\mathring{g}_{A}^{2}Z_{\pi}-\frac{1}{2}\left(\ln\frac{m_{N}^{2}}{m_{\pi}^{2}}-1\right)\right.$
(53)
$\displaystyle\left.-\frac{8\pi^{2}}{\mathring{g}_{A}}\left(g_{1}^{r}+g_{2}^{r}+\frac{g_{11}^{r}}{2}\right)\right]F_{A}^{\mu}\leavevmode\nobreak\
.$

Figure 4: NLO diagrams for $\delta F_{A}^{\mu}$. Circled dot: Vertices from
$\mathcal{L}_{\pi N}^{p^{2}}$.

Following Ref.[14], we also compute the NLO corrections. The Feynman diagrams
are given in Fig.4 and the loop integrals are UV-finite. The resulting
corrections are purely inner:

$\displaystyle(\delta F_{A}^{\mu})_{\text{NLO}}$ $\displaystyle=$
$\displaystyle(\delta F_{A}^{\mu})_{\text{NLO}}^{\text{inner}}$
$\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}\left[4\pi
m_{\pi}Z_{\pi}\left(c_{4}-c_{3}+\frac{3}{8m_{N}}+\frac{9}{16m_{N}}\mathring{g}_{A}^{2}\right)-\frac{\pi
m_{\pi}}{6m_{N}}(1+2\mathring{\mu}_{p}-2\mathring{\mu}_{n})\right]F_{A}^{\mu}\leavevmode\nobreak\
.$

###  V.2 Two- and three-point function

Now we proceed to split the full $\delta F_{A}^{\mu}$ into two- and three-
point function. One way to do it is to compute directly $\delta
F_{A,\text{3pt}}^{\mu}$ from Feynman diagrams as described in Ref.[16]; for
that we need the Feynman rules of $\partial\cdot J_{W}$, on which equation of
motion must be applied. In this paper we demonstrate an alternative approach,
namely to compute $\delta F_{A,\text{2pt}}^{\mu}$ in EFT. We start from its
definition in Eq.(6), and realize that the EFT only probes physics at $k\ll
m_{W}$, so we take $m_{W}^{2}/(m_{W}^{2}-k^{2})\rightarrow 1$ and adopt the
dimensional regularization prescription for UV-divergences. This leads to the
following EFT representation:

$\delta
F_{A,\text{2pt}}^{\lambda}=-e^{2}\mu^{4-d}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{k^{\lambda}}{(k^{2}-m_{\gamma}^{2})^{2}}(T^{\mu}_{\>\>\mu})^{A}\leavevmode\nobreak\
,$ (55)

where the superscript $A$ on $T^{\mu}_{\>\>\mu}$ denotes that only the axial
component of the charged current is included.

Figure 5: LO and NLO diagrams for $(T^{\mu\nu})^{A}$ with non-zero
contribution to $\delta F_{A,\text{2pt}}^{\mu}$.

According to its definition, it is obvious that the two-point function cannot
depend on short-distance electromagnetic effect, i.e. those derived from
$\mathcal{L}_{\pi}^{e^{2}}$ and $\mathcal{L}_{\pi N}^{e^{2}}$. Therefore, our
task is to construct $(T^{\mu}_{\>\>\mu})^{A}$ at LO and NLO using the pure-
QCD chiral Lagrangian. Non-vanishing contributions come from the Feynman
diagrams in Fig.5, which give:

$\displaystyle(T^{\mu}_{\>\>\mu})_{a}^{A}$ $\displaystyle=$
$\displaystyle\frac{-2i\mathring{g}_{A}}{k^{2}-m_{\pi}^{2}}\left(1-\frac{\delta
m_{N}}{v\cdot k}\right)\bar{u}_{p,v}S\cdot ku_{n,v}$
$\displaystyle(T^{\mu}_{\>\>\mu})_{b}^{A}$ $\displaystyle=$
$\displaystyle\frac{2i\mathring{g}_{A}}{k^{2}-m_{\pi}^{2}}\bar{u}_{p,v}S\cdot
ku_{n,v}$ $\displaystyle(T^{\mu}_{\>\>\mu})_{c}^{A}$ $\displaystyle=$
$\displaystyle\frac{-i\mathring{g}_{A}}{k^{2}-m_{\pi}^{2}}\left(\frac{(v\cdot
k)^{2}-k^{2}}{m_{N}v\cdot k}+\frac{2\delta m_{N}}{v\cdot
k}\right)\bar{u}_{p,v}S\cdot ku_{n,v}$
$\displaystyle(T^{\mu}_{\>\>\mu})_{d}^{A}$ $\displaystyle=$
$\displaystyle\frac{i\mathring{g}_{A}}{k^{2}-m_{\pi}^{2}}\frac{(v\cdot
k)^{2}-k^{2}}{m_{N}v\cdot k}\bar{u}_{p,v}S\cdot ku_{n,v}\leavevmode\nobreak\
.$ (56)

An interesting point to notice is that diagram (a), despite being LO, gives
also an NLO contribution proportional to $\delta m_{N}$, namely the quark-mass
correction to the nucleon mass from $\mathcal{L}_{\pi N}^{p^{2}}$, due to the
on-shell condition of the external nucleon momenta. Here we assume the strong
isospin symmetry: $\delta m_{p}=\delta m_{n}=\delta m_{N}$.

We observe that the four terms in Eq.(56) add up to zero, which implies that
the full axial FF RC at both LO and NLO belongs to the three-point function.
This is a major conclusion of this paper, which can also be verified
independently by computing $\delta F_{A,\text{3pt}}^{\mu}$ using Feynman
rules. After removing the outer corrections as in Eq.(20), we obtain the
desired EFT prediction of the inner correction to $g_{A}$ due to the three-
point function:

$\Delta_{R,\text{3pt}}^{A}=(\Delta_{R,\text{3pt}}^{A})_{\text{LO}}+(\Delta_{R,\text{3pt}}^{A})_{\text{NLO}}+...\leavevmode\nobreak\
,$ (57)

where

$\displaystyle(\Delta_{R,\text{3pt}}^{A})_{\text{LO}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left[Z_{\pi}(1+3\mathring{g}_{A}^{2})\left(\ln\frac{\mu^{2}}{m_{\pi}^{2}}-1\right)-2\mathring{g}_{A}^{2}Z_{\pi}-\ln\frac{m_{N}^{2}}{m_{\pi}^{2}}+1\right.$
$\displaystyle\left.-\frac{16\pi^{2}}{\mathring{g}_{A}}\left(g_{1}^{r}+g_{2}^{r}+\frac{g_{11}^{r}}{2}\right)\right]$
$\displaystyle(\Delta_{R,\text{3pt}}^{A})_{\text{NLO}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2\pi}\left[8\pi
m_{\pi}Z_{\pi}\left(c_{4}-c_{3}+\frac{3}{8m_{N}}+\frac{9}{16m_{N}}\mathring{g}_{A}^{2}\right)-\frac{\pi
m_{\pi}}{3m_{N}}(1+2\mathring{\mu}_{p}-2\mathring{\mu}_{n})\right]\leavevmode\nobreak\
,$

which completes the missing piece in the SR + dispersion relation/lattice
analysis.

It is instructive to compare our result with Ref.[14]. We find that, adding
the expanded expression of $2\Box^{A}_{\text{int}}$ in Eq.(34) to
$\Delta_{R,\text{3pt}}^{A}$ reproduces exactly Eqs.(6), (7) in Ref.[14] (apart
from differences in normalization) except the LECs $X_{i}$, $\tilde{X}_{i}$
and $g_{9}$ which correspond to the short-distance electroweak RC (i.e.
$\Delta_{R}^{U}$ and $\Box_{\gamma W}^{V,A}$) as we will show later. Notice
that, in this work we define the full $g_{V,A}$ in terms of the QED-corrected
$n\rightarrow pe\bar{\nu}_{e}$ amplitude (21), while in Ref.[14] $g_{V,A}$ are
defined as coupling constants in the pionless EFT Lagrangian
$\mathcal{L}_{\not{\pi}}$, which will then be used to compute the decay
amplitude coming from tree-level and photon-loop diagrams. As a consequence,
$g_{V,A}$ in Ref.[14] are scale-dependent but those here are not. Fortunately,
what we really interested in is the ratio $\lambda=g_{A}/g_{V}$ which turns
out to be the same in the two representations, because the photon-loop
contribution to the inner RC in the pionless EFT is the same for $g_{V}$ and
$g_{A}$, and thus cancels out in the ratio.

##  VI Fixing the low energy constants

An immediate application of the analysis above is to compare the predicted
observables with those in the pure EFT representation. Since the two must be
equivalent at $\mathcal{O}(\alpha)$, the matching allows us to pin down
specific combinations of unknown LECs in the chiral Lagrangian. This idea was
previously applied to semileptonic kaon decays, which successfully fixed
several important LECs at $\mathcal{O}(e^{2}p)$ with $\sim 10$% precision [17,
18].

The first observable to compare is the differential decay rate, i.e.
Eq.(LABEL:eq:differential) in this work and Eq.(26) in the Appendix of
Ref.[14]. Notice that, to compute the decay rate in an EFT one needs not only
photons, but also leptons as dynamical DOFs. This introduces three types of
(renormalized) LECs: $\\{g_{i}^{r}\\}$ from the pure hadron sector,
$X_{6}^{r}$ from the pure lepton sector, and $\\{\tilde{X}_{i}^{r}\\}$ from
the lepton-hadron sector; they enter the differential decay rate through the
linear combination:

$\hat{C}_{V}(\mu)\equiv
8\pi^{2}\left[-\frac{X_{6}^{r}}{2}+2\left(\tilde{X}_{1}^{r}-\tilde{X}_{2}^{r}\right)+g_{9}^{r}\right]\leavevmode\nobreak\
.$ (59)

The comparison yields the following EFT representation of the famous “nucleus-
independent RC”:

$\Delta_{R}^{V}=\frac{\alpha}{2\pi}\left[2\hat{C}_{V}(\mu)+\frac{5}{4}+3\ln\frac{\mu}{m_{N}}\right]\leavevmode\nobreak\
.$ (60)

Substituting the expression of $\Delta_{R}^{V}$ in SR gives:

$\displaystyle
8\pi^{2}\left[-\frac{X_{6}^{r}}{2}+2(\tilde{X}_{1}^{r}-\tilde{X}_{2}^{r})+g_{9}^{r}\right]$
$\displaystyle=$
$\displaystyle\frac{3}{2}\ln\frac{m_{Z}}{\mu}+\frac{1}{2}\ln\frac{m_{Z}}{m_{W}}-\frac{5}{8}+\frac{1}{2}\tilde{a}_{g}+\frac{2\pi}{\alpha}\Box_{\gamma
W}^{V}\leavevmode\nobreak\ .$ (61)

Notice that we do not include $\delta_{\text{HO}}^{\text{QED}}$ in the
matching because it was not a part of the fixed-order HBChPT calculation. For
the vector box diagram $\Box_{\gamma W}^{V}$, we can either take
$3.85(9)\times 10^{-3}$ obtained from an average over several non-lattice
determinations [55], or $3.65(8)\times 10^{-3}$ from a direct lattice
calculation [12], the two exhibit a $1.7\sigma$ deviation. Here we decide to
average them: $\Box_{\gamma W}^{V}=3.74(10)\times 10^{-3}$, where the
uncertainty is enlarged by a scale factor $S=1.7$ following the usual practice
by the Particle Data Group [3]. We then obtain the numerical value of the
following LEC combination at $\mu=m_{N}$:

$8\pi^{2}\left[-\frac{X_{6}^{r}}{2}+2(\tilde{X}_{1}^{r}-\tilde{X}_{2}^{r})+g_{9}^{r}\right](\mu=m_{N})=9.48(13)_{\text{HO}}(9)_{\text{box}}\leavevmode\nobreak\
,$ (62)

where the uncertainties come from terms of higher chiral orders (estimated by
multiplying the central value by $m_{\pi}^{2}/\Lambda_{\chi}^{2}$, with
$\Lambda_{\chi}=4\pi F_{0}$ the chiral symmetry breaking scale) and
$\Box_{\gamma W}^{V}$ respectively.

The second observable to compare is the axial-to-vector ratio $\lambda$, i.e.
Eq.(25) of this work and Eq.(12) in Ref.[14]. Notice that $g_{A}$ in EFT
depends on a separate linear combination of LECs:

$\hat{C}_{A}(\mu)\equiv
8\pi^{2}\left[-\frac{X_{6}^{r}}{2}-\frac{1}{\mathring{g}_{A}}\left[\tilde{X}_{3}^{r}+\left(g_{1}^{r}+g_{2}^{r}+\frac{g_{11}^{r}}{2}\right)\right]\right]\leavevmode\nobreak\
.$ (63)

The comparison yields the following matching of LECs:

$\displaystyle
8\pi^{2}\left[-\frac{X_{6}^{r}}{2}-\frac{1}{\mathring{g}_{A}}\tilde{X}_{3}^{r}\right]$
$\displaystyle=$
$\displaystyle\frac{3}{2}\ln\frac{m_{Z}}{\mu}+\frac{1}{2}\ln\frac{m_{Z}}{m_{W}}-\frac{5}{8}+\frac{1}{2}\tilde{a}_{g}+\frac{2\pi}{\alpha}\Box_{\gamma
W}^{A}\leavevmode\nobreak\ ,$ (64)

where we have made use of the expanded form of $\Box^{A}_{\text{int}}$ in
Eq.(34). The dispersion relation [35] and lattice [12] determination of the
axial box $\Box_{\gamma W}^{A}$ return $\Box_{\gamma W}^{A}$: $3.96(6)\times
10^{-3}$ and $3.72(14)\times 10^{-3}$ respectively, showing a $1.6\sigma$
discrepancy. Averaging the two yields $\Box_{\gamma W}^{A}=3.92(9)\times
10^{-3}$ where the uncertainty is enlarged by a scale factor $S=1.6$. It
returns:

$8\pi^{2}\left[-\frac{X_{6}^{r}}{2}-\frac{1}{\mathring{g}_{A}}\tilde{X}_{3}^{r}\right](\mu=m_{N})=9.64(13)_{\text{HO}}(8)_{\text{box}}\leavevmode\nobreak\
.$ (65)

Up to this point, the only undetermined combination of LECs relevant to
neutron beta decay is $g_{1}^{r}+g_{2}^{r}+g_{11}^{r}/2$ that appears in the
three-point function. In principle, since $\delta F_{\text{3pt}}^{\mu}$ is
rigorously defined in terms of single-nucleon matrix elements in Eq.(9), one
should be able to compute it with lattice QCD; but in practice, what we call
“three-point function” here is in fact a “five-point function” in lattice QCD
(two external states + three current insertions), which is extremely difficult
to handle. An alternative approach is to compute the full axial current matrix
element $\langle p|(J_{W}^{\mu})_{A}|n\rangle$ in the presence of
electromagnetic interactions, which gives us directly $\delta F_{A}^{\mu}$.
This is possible using, e.g. lattice QCD + QED with massive photons [56],
which is also referred to as QED${}_{\text{M}}$. We have shown in Sec.V.2 that
the relation $\delta F_{A}^{\mu}=\delta F_{A,\text{3pt}}^{\mu}$ holds at least
up to NLO; so, a lattice calculation of $\delta F_{A}^{\mu}$ provides a first-
principles determination of $\Delta_{R,\text{3pt}}^{A}$ after removing the IR-
divergent outer corrections:

$\langle
p|(J_{W}^{\mu})_{A}|n\rangle_{\text{QCD}+\text{QED}_{\text{M}}}=\left[1-\frac{\alpha}{4\pi}\left(\ln\frac{m_{\gamma}^{2}}{m_{N}^{2}}+2\right)+\frac{1}{2}\Delta_{R,\text{3pt}}^{A}\right]\langle
p|(J_{W}^{\mu})_{A}|n\rangle_{\text{QCD}}\leavevmode\nobreak\ .$ (66)

In fact, (25) and (66) are the only equations needed as far as the RC to
$\lambda$ is concerned. Substituting the lattice-deduced value of
$\Delta_{R,\text{3pt}}^{A}$ to Eq.(LABEL:eq:DeltaRA) yields the numerical
value of the combination $g_{1}^{r}+g_{2}^{r}+g_{11}^{r}/2$ which can be used
to analyze other processes.

We conclude this section by stressing that SR provides a useful basis for
lattice studies of RC in beta decays because lepton fields are decoupled from
its master formula, thus bypassing the difficulty to introduce leptonic DOFs
on lattice. To give a recent example, the RC in semileptonic kaon decays
($K_{\ell 3}$) are determined to $10^{-4}$ precision under this framework [19,
20, 21] where the only required lattice inputs are the $K\pi$ and $\pi\pi$ box
diagrams that are relatively straightforward to compute on lattice [57, 18].
In contrast, a full-fledged lattice calculation of the $K_{\ell 3}$ RC
(virtual and real) with dynamical photons and leptons has a projected time
scale of $\sim$ 10 years to reach a $10^{-3}$ precision [58]. We expect the
same advantage to also hold in the nucleon sector.

##  VII Summary

Recent years have witnessed tremendous improvements in the experimental
determination of the neutron lifetime $\tau_{n}$ and the axial-to-vector ratio
$\lambda$, both reaching a precision level of $10^{-4}$. To make full use of
these results for precision tests of SM, a control of the SM RC uncertainty at
$10^{-4}$ is necessary. In early days, the RC to the spin-independent part of
the neutron beta decay received more attention as it affects the extraction of
$V_{ud}$. Recently, due to breakthroughs in first-principles calculation of
$\mathring{g}_{A}$ that approaches sub-percent level precision, the
understanding of the spin-dependent part of the RC becomes equally important.
Recent EFT analysis unveiled a percent-level correction to $g_{A}$ which was
not covered by the previous dispersive analysis of $\gamma W$-box diagrams. It
plays a crucial role when the lattice-calculated $\mathring{g}_{A}$ is
compared to experimentally-measured $\lambda$ for the search of new physics.

In principle, the full RC to inclusive neutron decay can be studied on lattice
by including photonic and leptonic DOFs, which is extremely challenging in
practice. It is easier to include only photonic DOFs, and such a setup is
ideal to compute the QED-corrected single-nucleon FFs, in which the
aforementioned large RC resides. Correspondingly, one needs an EFT description
for just the RC to the axial FF instead of the full RC, which is attempted in
this paper. Furthermore, we merge this EFT calculation with the well-developed
current algebra formalism of RC, which is extremely powerful in controlling
all the remaining one-loop RC when combine with dispersion relation analysis.
By comparing the expressions between our hybrid analysis and the full EFT
calculation, we are able to determine numerically a subset of LECs that
correspond to the vector and axial $\gamma W$-box diagrams. Furthermore, we
show that a future lattice calculation of $\delta F_{A}^{\mu}$ will provide
the last missing piece $\Delta_{R,\text{3pt}}^{A}$ in the hybrid method,
namely the inner correction to $g_{A}$ from the three-point function.

Our analysis provides an elegant framework for non-perturbative studies of the
neutron decay RC, where only a minimal amount of lattice QCD inputs are
required. It will be useful for current and future tests of SM and the search
of BSM physics at the precision frontier.

###### Acknowledgements.

We thank Vincenzo Cirigliano, Wouter Dekens, Emanuele Mereghetti and Oleksandr
Tomalak for many useful discussions, especially for pointing out the missing
term $\Box^{A}_{\text{int}}$ in the first arXiv version. The work of C.-Y.S.
is supported in part by the U.S. Department of Energy (DOE), Office of
Science, Office of Nuclear Physics, under the FRIB Theory Alliance award DE-
SC0013617, and by the DOE grant DE-FG02-97ER41014. We acknowledge support from
the DOE Topical Collaboration “Nuclear Theory for New Physics”, award No. DE-
SC0023663.

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