Radiative Corrections to Neutron and Nuclear Beta Decays Revisited

TL;DR

This paper revisits radiative corrections to neutron and nuclear beta decays using multiple computational methods, reducing previous discrepancies and refining implications for CKM unitarity and neutron lifetime measurements.

Contribution

It introduces an updated calculation of radiative corrections employing advanced QCD techniques and resonance models, decreasing the discrepancy with dispersion relation results.

Findings

Radiative correction estimate: 2.426(32)%

Discrepancy with dispersion relation reduced to 1.1σ

Stronger bounds on exotic neutron decays: <0.16%

Abstract

The universal radiative corrections common to neutron and super-allowed nuclear beta decays (also known as ``inner'' corrections) are revisited in light of a recent dispersion relation study that found $+2.467(22)\%$, i.e.~about $2.4\sigma$ larger than the previous evaluation. For comparison, we consider several alternative computational methods. All employ an updated perturbative QCD four-loop Bjorken sum rule (BjSR) defined QCD coupling supplemented with a nucleon form factor based Born amplitude to estimate axial-vector induced hadronic contributions. In addition, we now include hadronic contributions from low $Q^2$ loop effects based on duality considerations and vector meson resonance interpolators. Our primary result, $2.426(32)\%$ corresponds to an average of a Light Front Holomorphic QCD approach and a three resonance interpolator fit. It reduces the dispersion relation discrepancy to approximately $1.1\sigma$ and thereby provides a consistency check. Consequences of our new radiative correction estimate, along with that of the dispersion relation result, for CKM unitarity are discussed. The neutron lifetime-$g_A$ connection is updated and shown to suggest a shorter neutron lifetime $< 879$ s. We also find an improved bound on exotic, non-Standard Model, neutron decays or oscillations of the type conjectured as solutions to the neutron lifetime problem, $\text{BR}(n\to \text{exotics}) < 0.16 \%$.

Tables

Table 1. Table 1: Universal and neutron specific radiative corrections.

Line number	${\Delta }_R^V$	RC	Source
1	0.02467(22)	0.03992(22)	Seng:2018yzq  DR Result
2	0.02361(38)	0.03886(38)	Marciano:2005ec 2006 Result
3	0.02421(32)	0.03942(32)	AdS BjSR Approach, eqs. (33) and (34)
4	0.02431(32)	0.03952(32)	Three Resonance Interpolator, eq. (44)
5	0.02426(32)	0.03947(32)	Average of lines 3 and 4

Equations

$$|V_{\text{ud}}{|}^2+|V_{\text{us}}{|}^2+|V_{\text{ub}}{|}^2=1.$$

$$|V_{\text{ud}}|=0.97420(10{)}_{NP}(18{)}_{RC},$$

$${\Delta }_R^V=0.02361(38),$$

$$|V_{\text{us}}|=0.2243(9),$$

$$|V_{\text{ud}}{|}^2+|V_{\text{us}}{|}^2+|V_{\text{ub}}{|}^2=0.9994(4{)}_{\text{ud}}(4{)}_{\text{us}},$$

$$|V_{\text{ud}}|=0.97427(11)\text{ Unitarity condition from Kμ2/πμ2.}$$

$${\Delta }_R^V=0.02467(22),$$

$$|V_{\text{ud}}|=0.97370(14)$$

$$|V_{\text{ud}}{|}^2+|V_{\text{us}}{|}^2+|V_{\text{ub}}{|}^2$$

$${\Delta }_R^V=\frac{\alpha }{2\pi }\left\{3\ln \frac{m_Z}{m_p}+\ln \frac{m_Z}{m_A}+2C_{\text{Born}}+A_g\right\},$$

$$\frac{{\alpha }_{g_1}(Q^2)}{\pi }=\exp (-Q^2/Q_0^2)\text{ for }0<Q^2<Q_0^2,$$

$$Q_0^2=1.10(10){\text{ GeV}}^2,$$

$$g_A=1.2764(6)\text{ Perkeo III (2018) cite[cite]@@bibrefAuthors Phrase1YearPhrase2Markisch:2018ndu,Dubbers:2018kgh@@citephrase(@@citephrase)},$$

$$g_A^{\text{ave}}=1.2762(5)\text{ Post 2002 Experiments}.$$

$$\frac{1}{{\tau }_n}=\frac{G_{\mu }^2|V_{\text{ud}}{|}^2}{2{\pi }^3}{m}_e^5\left(1+3g_A^2\right)\left(1+\text{mbox}RC\right)f,$$

$$\text{RC}=\frac{\alpha }{2\pi }\left[\overline{g}(E_m)+3\ln \frac{m_Z}{m_p}+\ln \frac{m_Z}{m_A}+2C_{\text{Born}}+A_g\right],$$

$$\frac{\alpha }{2\pi }\overline{g}(E_m)=\mathrm{0.015035.}$$

$$\text{RC}=0.03186+1.017A_{NP}+1.08A_P,$$

$${\Delta }_R^V=0.01671+1.022A_{NP}+1.065A_P.$$

$${\int }_0^1\text{d}x\left[g_1^p(x,Q^2)-g_1^n(x,Q^2)\right]=\frac{g_A}{6}\left(1-\frac{{\alpha }_{g_1}(Q^2)}{\pi }\right),$$

$$\text{Box}(\gamma W{)}_{\text{VA}}=\frac{\alpha }{8\pi }\underset{0}{\overset{\infty }{\int }}d{Q}^2\frac{m_W^2}{Q^2+m_W^2}F(Q^2),$$

$$F(Q^2)\to \frac{1}{Q^2}\left(1-\frac{{\alpha }_{g_1}(Q^2)}{\pi }\right),$$

$$\frac{{\alpha }_{g_1}(Q^2)}{\pi }$$

$$+(41.44-7.607n_f+0.1775n_f^2)a_s^3$$

$$+(479.4-123.4n_f+7.697n_f^2-0.1037n_f^3)a_s^4,$$

$$\frac{{\alpha }_{g_1}}{\pi }=\exp (-Q^2/Q_0^2)\text{ for }{Q}^2<Q_0^2,$$

$$I_1$$

$$I_2$$

$$I_{ZW}=0.060\frac{\alpha }{\pi },$$

$$I_{\text{Born}}=0.85\frac{\alpha }{\pi }0<Q^2<Q_0^2.$$

$$A_{NP}=0.85\frac{\alpha }{\pi }=1.97\times 10^{-3},$$

$$A_P=2.025(21)\frac{\alpha }{\pi }=4.70(5)\times 10^{-3}.$$

Full text

††preprint: Alberta Thy 7-19

I Introduction

Precision tests of the Standard Model (SM) require accurate calculations of electroweak radiative corrections (RC) Kinoshita:1958ru ; Berman:1958ti ; berman62 ; Bjorken:1966jh ; Abers:1967zzb ; Abers:1968zz ; Sirlin:1974ni . For example, unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix leads to orthonormal relationships among row and column matrix elements and provides a means to search for indications of “New Physics” via departures from SM expectations. However, for those searches to be credible, strong interaction effects must be reliably evaluated.

Consider the precise CKM first row unitarity condition

[TABLE]

Employing the PDG 2018 average based on super-allowed $0^{+}\to 0^{+}$ nuclear beta decays Tanabashi:2018oca ; Hardy:2016vhg ,

[TABLE]

as extracted by Hardy and Towner Hardy:2016vhg , using a universal electroweak radiative correction Marciano:2005ec (also known as the “inner” correction),

[TABLE]

along with the $K_{\mu 2}/\pi_{\mu 2}$ and $K_{l3}$ weighted average Tanabashi:2018oca of $|V_{\text{us}}|=0.2253(7)$ and $|V_{\text{us}}|=0.2231(8)$ respectively,

[TABLE]

(where the uncertainty has been increased by a scale factor $S=1.8$ to account for $K_{\mu 2}$ and $K_{l3}$ inconsistencies) and negligible $|V_{\text{ub}}|^{2}\sim O(10^{-5})$ implies

[TABLE]

Alternatively, one may employ the updated Aoki:2019cca $K_{\mu 2}/\pi_{\mu 2}$ constraint $|V_{\text{us}}|/|V_{\text{ud}}|=0.2313(5)$ to derive the unitarity condition Marciano:2004uf ,

[TABLE]

Both eq. (2) and eq. (5) are in good accord with those Standard Model (SM) expectations. However, that confirmation has recently been questioned. A new analysis of the universal radiative corrections to neutron and super-allowed nuclear beta decays based on a dispersion relations (DR) study of hadronic effects by Seng, Gorchtein, Patel, and Ramsey-Musolf Seng:2018yzq finds a roughly 0.1% larger

[TABLE]

with reduced uncertainty. It leads to a smaller more precise Seng:2018yzq

[TABLE]

Both eq. (8) and eq. (9) exhibit an apparent roughly $3.2\sigma$ violation of unitarity. Taken literally, it could be interpreted as a strong hint of “new physics.” However, nuclear structure effects and other corrections to $V_{\text{ud}}$ and $V_{\text{us}}$ are still being investigated Seng:2018qru ; Gorchtein:2018fxl .

Although the use of DR for such an analysis represents a major advancement in the calculation of electroweak radiative corrections, it is important to reexamine the input leading to eq. (7) and compare with other computational approaches. In that way, one can better assess their consistency and individual reliabilities. Close examination may reveal issues with the RC or other inputs. For that reason, we update here an alternative study of the radiative corrections to neutron and super-allowed nuclear beta decays, estimate hadronic uncertainties and discuss various possible implications.

Before going into detail, let us briefly preview our study. We first recall the lowest order one loop universal radiative corrections to neutron and super-allowed nuclear beta decay rates in the framework of the SM. Leading log QED effects, beyond one loop order, controlled by the renormalization group are included. Overall, they increase the RC by about 0.1%. However, some care must be exercised in examining compound effects, particularly since the DR result to be compared with differs from the earlier calculations by a similar $\sim 0.1\%$. That difference could be offset by smaller changes in several other contributions to the decay rates.

Consider the weak vector amplitude stemming from tree and loop level effects. At very low momentum transfer, vector current induced effects are protected from strong interactions by vector current conservation (CVC). Hadronic effects, nevertheless, enter the vector amplitude via $\gamma W$ box diagrams (and to a lesser extent $ZW$ box diagrams), see Fig. 1, where the operator product expansion of quark axial and vector currents can produce a vector amplitude. In that way, short-distance QCD and long-distance hadronic structure dependence are induced by the non-conserved axial current.

Up until 2006 Marciano:2005ec , only the lowest order, $\mathcal{O}\left(\alpha_{s}\right)$, QCD perturbative correction to the box diagrams was considered Sirlin:1978sv ; Adler:1970hu ; Beg:1974vs . Non-perturbative long distance hadronic corrections, were estimated by evaluating a Born amplitude parameterized by inserting axial and vector nucleon dipole form factors in Fig. 1, an approach introduced in ref. Marciano:1986pd .

Those order $\alpha/\pi$ (with $\alpha\simeq 1/137.036$) vector and axial-vector induced corrections, universal to all beta decays, were estimated to be

[TABLE]

The $3\ln(m_{Z}/m_{p})$ short-distance vector current induced contribution is free of QCD corrections, while the remaining terms, due to axial-vector current, exhibit strong interaction effects. In eq. (10), $m_{A}\sim 1.2$ GeV is a hadronic short-distance cutoff as employed in Czarnecki:2004cw , $A_{g}\sim-0.34$ represents its perturbative QCD corrections and $C_{\text{Born}}\sim 0.86$ denotes the Born (elastic) amplitude contribution. All three terms depend on hadronic structure and/or perturbative QCD. Collectively, those axial-vector induced loop contributions increase the decay rate by about $2.9\alpha/\pi\sim 6.7\times 10^{-3}$. Although such contributions represent a relatively minor part of the full one loop universal radiative corrections, they carry most of the theoretical uncertainty.

A strategy for improving strong interaction effects emerged, when it was shown Marciano:2005ec that the perturbative QCD corrections to beta decays and the Bjorken sum rule (BjSR) Bjorken:1967px ; Bjorken:1969mm , the latter now known to four loop QCD order Larin:1991zw ; Larin:1991tj ; Baikov:2010je , are identical in the chiral + isospin symmetry limits modulo small singlet contributions that we do not consider Larin:2013yba ; Baikov:2015tea . Therefore, one can make use of theoretical and experimental BjSR results to define an effective physical QCD coupling Grunberg:1980ja that spans the perturbative and (as more recently argued) non-perturbative loop momentum domains and is continuous throughout the $Q^{2}$ transition region (see for example Ellis:1994py ; Ellis:1995jv ; Brodsky:1995tb ; Deur:2005cf ; Deur:2016tte ; Ayala:2017uzx ). An identical perturbative situation arises for the Gross-Llewellyn-Smith (GLS) non-singlet sum rule Gross:1969jf . In fact that sum rule is closer in structure to the $\gamma W$ box diagram and the leading twist term in its operator product expansion. However, accessing relevant low $Q^{2}$ data in that case is less straightforward.

Employing the known BjSR or equivalent GLS non-singlet sum rule four loop QCD corrections as input allows a precise evaluation of the perturbative QCD corrections to the $\gamma W$ box diagrams for loop momentum above the demarcation scale $Q_{0}^{2}$ (see below eq. (12)), $Q_{0}^{2}<Q^{2}<\infty$, with little uncertainty. Below $Q^{2}=Q_{0}^{2}$, a non-perturbative evaluation of hadronic loop effects is required. For that purpose, we depend primarily on a nucleon-based form factor Born amplitude contribution. In addition, one of our new approaches, employs a somewhat speculative analytic extension of the BjSR coupling based on Light Front Holographic QCD (LFHQCD) Brodsky:2003px ; Brodsky:2014yha (see also Zou:2018eam for a pedagogical introduction). Our use of that non-perturbative interpolator represents a novel application and fundamental test of that approach. It introduces a nonperturbative $\alpha_{g_{1}}(Q^{2})$ given by

[TABLE]

where $g_{1}$ designates its dependence on the polarized structure function $g_{1}(x,Q^{2})$ from which it is derived (see eq. (20)). The transition scale we use,

[TABLE]

is fixed by matching non-perturbative and perturbative couplings Herren:2017osy ; Chetyrkin:2000yt using $\alpha_{s}(m_{Z})=0.1181(10)$ and the four-loop QCD code in Chetyrkin:2000yt ; Herren:2017osy . The matching is quite smooth and leads to additional $Q^{2}<Q_{0}^{2}$ nonperturbative loop effects that were neglected in 2006 Marciano:2005ec  under the assumption that they were included via the Born amplitude. However, as demonstrated by the DR study Seng:2018yzq  such effects are distinct and should be separately included. Fortunately, they are relatively small. Nevertheless, they are a source of some uncertainty and estimates of their magnitude represent the main difference between distinct calculations. In that regard, the DR uses the GLS non-singlet sum rule data at low $Q^{2}$ for guidance while our method follows ideas developed from BjSR studies Marciano:2005ec . Both have the same perturbative QCD corrections modulo singlet contributions (although the DR approach applies only three of the known four loop effects Seng:2018yzq ) and include similar estimates of the Born amplitude; but differ in the low $Q^{2}$ evaluation of other hadronic effects. In addition to the AdS based LFHQCD approach, we also evaluate hadronic effects using a three resonance interpolator function fixed by boundary conditions. Consistency of the two approaches reinforces their individual credibility. The results are subsequently averaged to give our current best estimate of the radiative corrections.

After presenting our updated evaluation of the RC to neutron decay, we take this opportunity to discuss its implications for our recent analysis of the neutron lifetime-$g_{A}$ connection Czarnecki:2018okw  in light of the new very precise Perkeo III Markisch:2018ndu ; Dubbers:2018kgh  experimental result

[TABLE]

which increases the average of post 2002 experiments to

[TABLE]

That value, taken together with the average trap neutron lifetime, $\tau_{n}^{\text{trap}}=879.4(6)$ s is used to (conservatively) improve our previous bound on exotic neutron decays from $<0.27\%$ to $<0.16\%$. It actually suggests, as we later discuss, that one should probably anticipate a future reduction in the neutron lifetime to the range 878-879 s or a decrease in the value of $g_{A}$.

II Radiative Corrections to Neutron Decay

We begin by reviewing the electroweak radiative corrections for neutron decay and then isolate a subset that is also universal to super-allowed Fermi decays called $\Delta^{V}_{R}$. The inclusive neutron decay rate or inverse lifetime $\tau_{n}^{-1}$ in the SM is predicted to be

[TABLE]

where $G_{\mu}=1.1663787(6)\times 10^{-5}\text{ GeV}^{-2}$ is the Fermi constant obtained from the muon lifetime and $g_{A}$ is the axial-current coupling obtained from the neutron decay asymmetry, $A_{0}=2g_{A}(1-g_{A})/(1+3g_{A}^{2})$. $f=1.6887(1)$ is a phase space factor that includes the Fermi function, a relatively large roughly $+3\%$ final state enhancement due to Coulomb interactions. RC stands for Radiative Corrections which have been taken, up until recently, to be $+0.03886(38)$ based on a study Marciano:2005ec  in 2006. The more recent DR approach Seng:2018yzq  found $+0.03992(22)$, a significant increase outside of the error budgets. In the case of neutron decay, RC are computed explicitly for the vector current amplitude and $g_{A}$ is defined via eq. (15) so that $g_{A}^{2}$, $g_{V}^{2}$ and $f$ have the same and factorable RC Czarnecki:2004cw . That $g_{A}$ as defined via eq. (15) is measured in neutron decay asymmetry studies, after correcting for residual recoil, weak magnetism and small $\mathcal{O}\left(\alpha\right)$ corrections as discussed by Wilkinson Wilkinson:1982hu and Shann Shann:1971fz . Corrections to the asymmetry reduce its magnitude by about 1% and correspondingly decrease $g_{A}$ by about 0.25% Markisch:2018ndu ; Dubbers:2018kgh .

The purpose of this paper is to update and improve the 2006 RC calculation approach Marciano:2005ec , compare it to the recent DR result Seng:2018yzq , and try to understand any remaining difference. It is predicated in part by the DR finding that non-perturbative low $Q^{2}$ effects not covered by the Born contribution are present and should be included along with a post 2006 four loop calculation of perturbative QCD corrections to the non-singlet Bj Baikov:2010je and GLS sum rules.

The factorized components of the lowest order RC to neutron decay are given by

[TABLE]

where $\overline{g}(E_{m})$ represents long distance loop corrections as well as bremsstrahlung effects averaged over the neutron decay beta decay electron spectrum, and $E_{m}=1.292581$ MeV is the end point electron energy specific to neutron decay. We find its updated value is slightly shifted to

[TABLE]

which reduces RC by $2\times 10^{-5}$. That contribution to the neutron decay RC in eq. (17) is specific to the neutron spectrum and is not maintained for other beta decays, although the function $g(E)$ used to derive it is universal to all beta decays (see however Gorchtein:2018fxl ). It, along with the $3\ln(m_{Z}/m_{p})$ term in eq. (10), are independent of strong interaction effects Sirlin:1967zza . The rest of that RC expression represents axial current induced effects that are dependent on strong interactions. They provide the main focus for this paper.

III Axial Current Loop Contributions to RC

The complete radiative corrections to neutron decay in eq. (10), including axial-current induced and QED leading log summation effects can be written to a good approximation as Czarnecki:2004cw

[TABLE]

where the $+0.03186$ corresponds to the pure vector current induced part of the RC including higher order effects. $A_{\scriptscriptstyle NP}$ and $A_{\scriptscriptstyle P}$ represent lowest order $\alpha$ long distance non-perturbative (NP) and short distance perturbative (P) contributions to RC from axial current effects in $\gamma W$ and $ZW$ box diagrams (see Fig. 1). Coefficients of the $\mathcal{O}\left(\alpha\right)$ contributions in eq. (18) follow from QED leading log enhancements and interference with other parts of the vector current induced RC. The short-distance parts in our approach correspond to loop momentum $Q^{2}>Q_{0}^{2}$ (see eq. (12)) while long distance parts correspond to $Q^{2}<Q_{0}^{2}$.

For the universal $\Delta^{V}_{R}$ used by Hardy and Towner in their analysis of super-allowed beta decays Hardy:2014qxa ; Hardy:2016vhg , one finds a corresponding approximate relationship

[TABLE]

The terms in eq. (18) and (19) were derived using the leading log QED summation described in Appendix 1 of ref. Czarnecki:2004cw .

We subsequently employ eqs. (18) and (19) to present updated radiative corrections for the neutron and super-allowed beta decays. Note, when applied to the DR $\mathcal{O}\left(\alpha\right)$ corrections, eqs. (18) and (19) give somewhat larger effects than those reported in ref. Seng:2018yzq . However, for the most part, whenever we refer to DR results, values cited correspond to the original literature Seng:2018yzq .

The short-distance axial current $\gamma W$ box diagram is the primary source of $A_{\scriptscriptstyle P}$. It is well described using an effective QCD coupling $\alpha_{g_{1}}(Q^{2})$ defined for $Q^{2}>Q_{0}^{2}$ via the isovector BjSR,

[TABLE]

where $g_{1}$ is the polarized structure function at Bjorken $x$. That prescription incorporates the leading $\mathcal{O}\left(\alpha\right)$ axial-current induced amplitude from the $\gamma W$ box diagram, given by

[TABLE]

where the asymptotic behavior of $F(Q^{2})$,

[TABLE]

will be called the Bjorken (Bj) function, and $\alpha_{g_{1}}$ is defined to be the sum of the four loop (or more if known) QCD corrections to the BjSR Baikov:2010je ,

[TABLE]

where $a_{s}={\alpha_{\overline{\scriptscriptstyle MS}}(Q^{2})\over\pi}$ and $n_{f}$ denotes the number of (effectively massless) quark flavors. That expression defines a coupling $\alpha_{g_{1}}$ which is valid perturbatively for $Q^{2}>Q_{0}^{2}$.

The behavior of $\alpha_{g_{1}}$ with $Q^{2}$ is shown in Fig. 2. Discontinuities in that plot are caused by changes in the number of active flavors in eq. (23): we change $n_{f}$ when $\sqrt{Q^{2}}$ crosses a quark decoupling threshold. Note however that eq. (23) is derived in massless QCD. There are very small singlet contributions Larin:2013yba ; Baikov:2015tea to the BjSR that enter the four loop QCD corrections. However, at the level of precision we consider their effect is negligible.

In our first LFHQCD approach, for the non-perturbative domain $Q^{2}<Q_{0}^{2}$, we employ the following prescription which is supported by low $Q^{2}$ experimental studies of the BjSR (down to about $Q^{2}=0.2\text{ GeV}^{2}$) Deur:2014vea . We continue to use the expression in eq. (22) but with Deur:2016tte

[TABLE]

and $F\left(Q_{0}^{2}=1.10(10)\text{ GeV}^{2}\right)=0.575(50)\text{GeV}^{-2}$, based on matching with the perturbative prediction obtained from $\alpha_{s}^{\overline{\text{MS}}}(m_{Z}^{2})$, evolved using a five loop beta function Herren:2017osy . Its functional exponential form is consistent with low $Q^{2}$ BjSR data and normalization $\alpha_{g_{1}}(Q^{2}=0)=\pi$ as suggested by AdS duality studies Deur:2016tte ; Deur:2016cxb . The Born (elastic) contributions to the hadronic corrections are computed separately using form factors for loop momenta $Q^{2}<Q_{0}^{2}$.

In 2006 Marciano:2005ec , when only three loop QCD corrections to the BjSR were known and considered, the transition $Q_{0}^{2}$ turned out to be close to $0.7\text{ GeV}^{2}$. The use of four loop based $\alpha_{g_{1}}$, charm and bottom threshold masses, and improved low $Q^{2}$ data have increased the transition $Q_{0}^{2}$ value (see eq. (12)) and better establish the AdS duality interpretation, features central to our update.

We can use the Bj function defined by eqs. (22-24) to evaluate the integral over $Q^{2}$ in eq. (21) for the different domains of the $\gamma W$ box diagram using $\alpha_{s}(m_{Z}^{2})=0.1181(10)$, $m_{c}=1.5\text{ GeV}$, $m_{b}=4.8\text{ GeV}$ (as decoupling thresholds of heavy quarks; see Chetyrkin:2017lif for an up-to-date discussion of quark masses) and $m_{t}=173.2\text{ GeV}$ with the results:

[TABLE]

where $I_{i}=2\times$the $\gamma W$ integrated box amplitude contribution, as appropriate for the radiative corrections. To those loop effects, we must add the $ZW$ box diagram contribution Sirlin:1974ni ,

[TABLE]

and the Born Marciano:2005ec  integral

[TABLE]

Contributions of QED vacuum polarization are incorporated via the coefficients in eq. (18) and (19).

Our first estimate which follows the 2006 evaluation Marciano:2005ec but with a four loop BjSR coupling definition and higher $Q_{0}^{2}$ value, does not include the contribution in eq. (25). The Born contribution in eq. (28) leads to

[TABLE]

while the sum of eqs. (26) and (27) gives

[TABLE]

Plugging those values into eq. (18) and (19) gives

[TABLE]

for our updated First Approximation radiative corrections to neutron and super-allowed nuclear beta decays.

In our next more complete AdS Deur:2016tte motivated approach, we retain the non-perturbative low $Q^{2}$ contribution from eq. (25) and find $A_{\scriptscriptstyle NP}=2.44\times 10^{-3}$ which leads to

[TABLE]

The two methods differ by 0.00047 with the latter about midway between our First Approximation and the DR results Seng:2018yzq . The generic error attached to eqs. (33) and (34) as well as to later alternative approaches, $\pm 3.2\times 10^{-4}$ corresponds to $\pm 2.5\times 10^{-4}$ from a 10% non-perturbative uncertainty combined in quadrature with a $\pm 2.0\times 10^{-4}$ perturbative error that includes QCD effects as well as uncalculated two loop electroweak corrections and other small effects. Further reduction of that error is likely to require a first principle’s lattice calculation along with a more complete two loop electroweak comparison between neutron beta decay and muon decay.

The values and uncertainties given above should be compared with the DR results Seng:2018yzq ,

[TABLE]

It is interesting to contrast the AdS (34) based value,

[TABLE]

with

[TABLE]

We note that the value of $V_{\text{ud}}$ in eq. (37) has moved closer to unitarity expectations ($\sim 0.9742$). An additional shift of about $-0.0006$ in the universal radiative corrections to super-allowed decays or an equivalent change in another part of those studies would fully restore unitarity.

To examine the sensitivity of our estimate to the specific Bj function interpolator used to integrate through the non-perturbative $Q^{2}<Q_{0}^{2}$ region, we consider the resonance sum interpolator approach introduced in 2006 Marciano:2005ec but with somewhat modified matching conditions used to determine $F(Q^{2})$ in the low momentum domain. The new conditions allow us to better specify the non-perturbative constraints implied by the very precise perturbative requirements. In that way we can match the non-perturbative and perturbative values of $F(Q^{2})$.

The underlying model of our next interpolator is large $N$ for SU($N$) QCD, which predicts $F(Q^{2})$ should correspond to an infinite sum of vector and axial-vector resonances. As an approximation to that model, we can use a finite sum of resonances with residues set by the boundary and matching conditions. We can then integrate over $Q^{2}$ between [math] and $Q_{0}^{2}$ i.e. including the non-perturbative domain in an approximation to that model. For that purpose, we chose the sum of three resonances with residues determined using three matching or boundary conditions. We impose conditions:

We require that in the domain $Q_{0}^{2}\leq Q^{2}<\infty$ the three resonance interpolator lead to the same perturbative corrections to the decay rates as the BjSR approach. This implies that the three resonance $F(Q^{2})$ function satisfies the integral condition $\int_{Q_{0}^{2}}^{\infty}{m^{2}_{W}\over Q^{2}+m^{2}_{W}}F(Q^{2})\text{d}Q^{2}=7.86$, four times the coefficient of $\alpha/\pi$ in $I_{2}$ (cf. eq. (26)). We apply a condition on the integral rather than asymptotic matching in order to better reflect the effect of perturbative QCD; 2. 2.

No $1/Q^{4}$ terms in expansion of $F(Q^{2})$ for $Q^{2}$ large or (see eq. (39)) $m_{1}^{2}A+m_{2}^{2}B+m_{3}^{2}C=0$. That condition enforces chiral symmetry asymptotically; 3. 3.

We employ $F(0)=A/m_{1}^{2}+B/m_{2}^{2}+C/m_{3}^{2}$ with $F(0)$ arbitrary until we consider two possible ways to fix its value, i.e. using either the perturbative value of $F(Q_{0}^{2})$ or the AdS value of $F(0)$ as a normalization condition for the three resonance interpolator.

These conditions are similar to those imposed in 2006 Marciano:2005ec with some improvements. Because of the larger $Q_{0}^{2}$ employed, the integral in condition 1 is extended down to $Q_{0}^{2}$. More importantly, as pointed out in the DR analysis, the condition $F(0)=0$ used in 2006 was not justified. Instead, we use the perturbative value of $F(Q_{0}^{2})$ to normalize the three resonance interpolator and determine its underlying uncertainty.

After solving the three coupled condition equations, one finds (for the three resonance form of $F(Q^{2})$ with given vector and axial vector masses)

[TABLE]

That interpolator, integrated over $0<Q^{2}<Q_{0}^{2}$, leads to

[TABLE]

With that change in $I_{1}$ the radiative corrections become a function of $F(0)$,

[TABLE]

If we match the interpolator in eq. (39) with the perturbative value $F(Q_{0}^{2})=0.575(50)$, we find $F(0)=1.42(15)$ and the radiative corrections which we adopt as the three resonance solution,

[TABLE]

where a small uncertainty, $\pm 4\times 10^{-5}$, is accounted for in the $\pm 32\times 10^{-5}$ overall errors. To test the sensitivity of our results to the specific resonance mass scales employed, we have redone the three resonance interpolator with each of $m_{1,2,3}$ reduced by 5%, one $m_{i}$ at a time. Although the values of $A,\,B$ and $C$ are significantly modified, the different interpolators, value of $I_{1}$ and radiative corrections are essentially unchanged, as illustrated in Fig. 3. Indeed, our results are rather insensitive to reasonable changes in the $m_{i}$ values.

As an alternative prescription, we evaluate the radiative corrections resulting from the three resonance interpolator for the AdS boundary condition $F(0)={1\over Q_{0}^{2}}=0.91\text{ GeV}^{-2}$ and find

[TABLE]

Those values are in very good agreement with the AdS BjSR results in eqs. (33) and (34). They provide a nice consistency check on the AdS BjSR approach. We do not consider them as independent since both employ the same $F(0)$ boundary condition.

The $Q^{2}$ dependences of the various interpolators are illustrated in Fig. 4. The band surrounding the $F(0)=0.575(50)$ curve corresponds to the uncertainty associated with the error in $\alpha_{s}(m_{Z}^{2})=0.1181(10)$. Similar bands (not shown) exist for the other curves as well but all are small in comparison with our overall uncertainty, $\pm 32\times 10^{-5}$ for the radiative corrections. The good agreement between the AdS and three resonance solution for $F(0)=0.91\text{ GeV}^{-2}$ helps validate the AdS approach. In all cases the radiative corrections are proportional to areas under the curves.

The dashed curve in Fig. 4 corresponds to an example of a two resonance interpolating function given by

[TABLE]

which exhibits the following features,

[TABLE]

It roughly represents our approximation of an effective DR interpolator for $0<Q^{2}<2\text{ GeV}^{2}$. Integrating ${\alpha\over 4\pi}F_{2}(Q^{2})$ over that domain leads to $0.47\alpha/\pi$, in good agreement with the $0.48(7)\alpha/\pi$ found in the DR study Seng:2018yzq . Those contributions are to be compared with the roughly $0.35\alpha/\pi$ coming from our three resonance interpolator when integrated over that same $Q^{2}$ domain. That $0.13\alpha/\pi$ difference combined with the $0.06\alpha/\pi$ Born difference $=0.19\alpha/\pi$ is responsible for about a $4\times 10^{-4}$ difference between the DR result and our three resonance interpolator finding.

The $1.1\sigma$ difference between the DR and our results may therefore be traced primarily to our use of a larger $\alpha_{s}(m_{Z}^{2})$, four loops rather than three in the QCD sum rule corrections, different perturbative-nonperturbative matching and three rather than two vector/axial vector poles in the interpolator. More specifically, to reproduce the DR low $Q^{2}$ contribution requires an interpolator with an $F(0)$ central value near $2.3\text{ GeV}^{-2}$ while our matching conditions and interpolator imply $F(0)=1.42(15)\text{ GeV}^{-2}$. From that perspective it would be interesting if a more first principles method, such as Lattice QCD, could be employed to directly compute the value of $F(0)$.

In Table 1 we compare the universal and neutron specific radiative corrections obtained from a dispersion relation approach (line 1) with an earlier calculation from 2006 (line 2) as well as the AdS BjSR result (line 3), three resonance interpolator (line 4), and the average of 3 and 4 in line 5.

We take the average on line 5 as representative of our study and use it in discussing implications. Although it is somewhat smaller than the earlier DR result Seng:2018yzq , they are fairly consistent. In fact, the agreement can be viewed as a validation of the LFHQCD and three resonance interpolator approaches.

Although we are consistent with the DR results at about the $1.1\sigma$ level, i.e. $\sim 4\times 10^{-4}$, the remaining difference is important for interpreting CKM unitarity and making predictions for neutron decay. In that regard, we speculate on the basis of our analysis that the central value difference may decrease if the DR approach is extended to include four loop QCD corrections, low $Q^{2}$ corrections are parametrized using three rather than two vector meson mass scales and perturbation theory matching is extended below $2\text{ GeV}^{2}$.

We also note that the radiative corrections on lines 3, 4 and 5 of Table 1 are reduced somewhat if larger values of $\alpha_{s}(m_{Z}^{2})$ are employed as input. For example, using $\alpha_{s}(m_{Z}^{2})=0.1200$, a preferred value for some experimental inputs into the world average Tanabashi:2018oca , leads to a reduction by roughly $1\times 10^{-4}$ which increases our $V_{\text{ud}}$, as currently extracted from super-allowed Fermi decays Hardy:2016vhg , from $0.97389(18)$ to $0.97394(18)$.

IV Implications of larger radiative corrections

For our discussion of implications from larger radiative corrections, we employ our averages given in line 5 of Table 1, $\Delta^{V}_{R}=0.02426(32)$ and $\text{RC}=0.03947(32)$. That scenario leads to $V_{\text{ud}}=0.97389(18)$ which comes closer to CKM unitarity expectations than the DR value $V_{\text{ud}}=0.97370(14)$. Combined with $|V_{\text{us}}|=0.2243(9)$ from eq. (4), they correspond to roughly 2.3 and $3.3\,\sigma$ deviations respectively. The 3.3 $\sigma$ deviation is large enough to start taking “New Physics” extensions of the SM seriously Belfatto:2019swo while the $2.3\,\sigma$ effect is more suggestive of missing SM effects. For example, nuclear physics quenching of the Born corrections to super-allowed beta decays has been suggested as a way of increasing $V_{\text{ud}}$ by about 0.00022 Seng:2018qru .

More specifically, if the quenching correction, as evaluated in ref. Seng:2018qru , is applied to our $V_{\text{ud}}=0.97389(18)$ result, it leads to $V_{\text{ud}}^{Q}=0.97414(28)$ where the increased error is due to a nuclear quenching uncertainty. Using it together with $V_{\text{us}}=0.2243(9)$, one finds $|V_{\text{ud}}^{Q}|^{2}+|V_{\text{us}}|^{2}+|V_{\text{ub}}|^{2}-1=-0.00074(68)$, so that the first CKM row sum is consistent with unity at close to the $1\sigma$ level.

If instead we employ the relation Aoki:2019cca ; Marciano:2004uf

[TABLE]

the deviation from unity is further reduced to $-0.00028(62)$, or $-0.45\sigma$, in excellent agreement with CKM unitarity.

As a further application, we consider the RC to neutron decay. Using $1+\text{RC}=1.03947(32)$ in the neutron lifetime formula Czarnecki:2018okw , one finds a master formula relating $|V_{\text{ud}}|$, $\tau_{n}$ and $g_{A}$

[TABLE]

Employing $\tau_{n}^{\text{trap}}=879.4(6)$ s and post 2002 average $g_{A}$=1.2762(5) leads to

[TABLE]

The uncertainty in $V_{\text{ud}}$ from neutron decay measurements is starting to become competitive in accuracy with super-allowed beta decay determinations. In addition, its central value may also be indicating a deviation from unitarity. A central value shift to unitarity and $V_{\text{ud}}\sim 0.9742$ would require a reduction in either $\tau_{n}$ or $g_{A}$. Given the recent precision of Perkeo III, we consider $g_{A}$ fixed at the new post 2002 average 1.2762(5) which then suggests a $\tau_{n}<879$ s.

An alternate interpretation of the apparent violation of CKM unitarity in eq. (9) resulting from larger universal radiative corrections, consistent with $|V_{\text{us}}|/|V_{\text{ud}}|=0.2313$ from $K_{\mu 2}/\pi_{\mu 2}$, suggests the solution $V_{\text{ud}}=0.9735$ and $V_{\text{us}}=0.2252$ which requires “new physics.” One possible explanation could be the existence of a $0.1\%$ increase in the muon decay rate from “new physics” which shifts $G_{\mu}$ to a value larger than the real $G_{F}$. Alternatively, it could stem from an opposite sign effect in nuclear beta decay. That solution agrees with the current central value in eq. (50). Of course, such a scenario would be very exciting. It will also be well tested by the next generation of precise $\tau_{n}$ and $g_{A}$ measurements.

Recently, we discussed Czarnecki:2018okw  a resolution of the neutron lifetime problem (the beam $\tau_{n}^{\text{beam}}=888.0(2.0)$ s and trap $\tau_{n}^{\text{trap}}=879.4(0.6)$ s lifetime discrepancy) based on a precise connection between $\tau_{n}$ and $g_{A}$, the axial coupling measured in neutron decay asymmetries. We note that a shift in the universal beta decay radiative corrections alone makes a negligible change in the relationship

[TABLE]

used for that study due to a cancelation of uncertainties and common shifts between super-allowed and neutron beta decay rates. Similarly, a change in $G_{F}$ will not change eq. (51). However, a shift in the nuclear theory corrections as suggested in Seng:2018qru will modify it. For example, a shift in $V_{\text{ud}}$ by $+0.0002$ by further quenching of the Born contribution would lower the 5172.0 s in eq. (51) to 5169.9 s. More important is the recent increase in the post 2002 $g_{A}^{\text{average}}$ from 1.2755(11) to 1.2762(5) with the addition of the Perkeo III result Markisch:2018ndu ; Dubbers:2018kgh  in eq. (13). That shift reduces the predicted neutron lifetime from 879.5(1.3) s to

[TABLE]

That prediction is further reduced if $V_{\text{ud}}$ were to increase to respect CKM unitarity. Indeed, one would expect $\tau_{n}$ closer to 878 s.

We conclude by noting that the new post 2002 $g_{A}$ average in eq. (14) can be used in the analysis of Czarnecki:2018okw to reduce the bound on exotic neutron decays (such as $n\to\text{dark particles}$ Fornal:2018eol ; Berezhiani:2018udo ) from $<0.27\%$ to

[TABLE]

where we have not allowed for negative exotic branching ratios in the statistical distribution. That bound leaves little chance for a 1% dark particle decay as the solution to the neutron lifetime problem (unless one modifies the neutron asymmetry with new physics, e.g. ref. Ivanov:2018vit ).

We have presented an updated analysis of the radiative corrections to neutron and super-allowed nuclear beta decays. It extends the BjSR function into the non-perturbative low loop momentum region, incorporating four loop QCD effects as well as LFHQCD ideas and their confirmation by low energy experimental data. The value obtained was averaged with a slightly larger three resonance result. On the basis of our considerations, we advocate the universal value $\Delta^{V}_{R}=+2.426(32)\%$ as a competitive result about midway between earlier estimates Marciano:2005ec and the recent dispersion relation result Seng:2018yzq . Further study of the remaining small difference is warranted. Tests of both approaches will result from the next generation of neutron lifetime and $g_{A}$ asymmetry measurements that aim for $10^{-4}$ sensitivity. Lattice calculations of $F(Q^{2})$ may be possible Seng:2019plg . Will CKM unitarity be violated and “New Physics” uncovered? Time will tell.

Acknowledgement:

We thank Konstantin Chetyrkin for helpful remarks on the manuscript. W. J. M. thanks C. Y. Seng, M. Gorchtein and M. J. Ramsey-Musolf for discussions. The work of A. C. was supported by the Natural Sciences and Engineering Research Council of Canada. The work of W. J. M. was supported by the U.S. Department of Energy under grant DE-SC0012704. The work of A. S. was supported in part by the National Science Foundation under Grant PHY-1620039.