Joint lattice QCD - dispersion theory analysis confirms the top-row CKM unitarity deficit

Recently, the first ever lattice computation of the $\gamma W$-box radiative correction to the rate of the semileptonic pion decay allowed for a reduction of the theory uncertainty of that rate by a factor of $\sim3$. A recent dispersion evaluation of the $\gamma W$-box correction on the neutron also led to a significant reduction of the theory uncertainty, but shifted the value of $V_{ud}$ extracted from the neutron and superallowed nuclear $\beta$ decay, resulting in a deficit of the CKM unitarity in the top row. A direct lattice computation of the $\gamma W$-box correction for the neutron decay would provide an independent cross-check for this result but is very challenging. Before those challenges are overcome, we propose a hybrid analysis, converting the lattice calculation on the pion to that on the neutron by a combination of dispersion theory and phenomenological input. The new prediction for the universal radiative correction to free and bound neutron $\beta$-decay reads $\Delta_R^V=0.02477(24)$, in excellent agreement with the dispersion theory result $\Delta_R^V=0.02467(22)$. Combining with other relevant information, the top-row CKM unitarity deficit persists.

Recently, the first ever lattice computation of the γW -box radiative correction to the rate of the semileptonic pion decay allowed for a reduction of the theory uncertainty of that rate by a factor of ∼ 3. A recent dispersion evaluation of the γW -box correction on the neutron also led to a significant reduction of the theory uncertainty, but shifted the value of V ud extracted from the neutron and superallowed nuclear β decay, resulting in a deficit of the CKM unitarity in the top row. A direct lattice computation of the γW -box correction for the neutron decay would provide an independent cross-check for this result but is very challenging. Before those challenges are overcome, we propose a hybrid analysis, converting the lattice calculation on the pion to that on the neutron by a combination of dispersion theory and phenomenological input. The new prediction for the universal radiative correction to free and bound neutron β-decay reads ∆ V R = 0.02477 (24), in excellent agreement with the dispersion theory result ∆ V R = 0.02467 (22). Combining with other relevant information, the top-row CKM unitarity deficit persists.
Universality of the weak interaction, conservation of vector current and completeness of the Standard Model (SM) finds its exact mathematical expression in the requirement of unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Of various combinations of the CKM matrix elements constrained by unitarity, the toprow constraint is the best known both experimentally and theoretically. The 2018 value, ∆ u CKM ≡ |V ud | 2 + |V us | 2 + |V ub | 2 − 1 = −0.0006 (5) [1, 2] is in good agreement with zero required in the SM, putting severe constraints on Beyond Standard Model (BSM) physics.
Notably, the main source of the uncertainty in the ∆ u CKM constraint is theoretical: the γW -box radiative correction (RC), prone to effects of the strong interaction described by Quantum Chromodynamics (QCD), affects the value of |V ud | extracted from the free neutron and superallowed nuclear β decays. In a series of recent papers, this RC was reevaluated within the dispersion relation technique [3][4][5]. In particular, Ref. [3] observed that the universal, free-neutron correction received a significant shift, later confirmed qualitatively by Ref. [6]. This shift is the main cause of the current apparent unitarity deficit, ∆ u CKM = −0.0016(6) (using an average of V us from K 2 and K 3 decays [2]). The slight increase in the uncertainty is due to nuclear structure effects [4,5].
Since in superallowed β decays one aims for a 10 −4 precision, it is highly desirable to assess the uncertainty and possible, unaccounted for, systematic effects in the non-perturbative regime of QCD in a model-independent way. A common limitation of the studies above is the lack of experimental data to directly constrain the hadronic matrix element relevant to the RC. By means of isospin symmetry, Ref. [3] relates the input to the dispersion integral at low photon virtuality Q 2 to a very limited and imprecise set of data on neutrino scattering on light nuclei from the 1980s [7,8]. The analysis of Ref. [6] consists of pure model studies.
A complete change of landscape is expected following the first direct application of the lattice QCD to RC in leptonic meson decays, K → µν µ and π → µν µ [9]. Very recently, the first ever direct lattice calculation of the RC in semi-leptonic β decay was presented, where the relevant hadronic matrix element responsible for the γW -box diagram in the pion is calculated to high precision as a function of Q 2 [10]. As a result, the theory uncertainty of the π e3 (π − → π 0 eν e ) decay rate is reduced by a factor of 3. While theoretically very clean, π e3 is not the easiest avenue to extract V ud due to its tiny branching ratio ∼ 10 −8 . Nonetheless, it provides useful information about the involved nonperturbative dynamics, especially its low-Q 2 behavior and its smooth transition to the perturbative regime. Using the same method or other approaches such as Feynman-Hellmann theorem [11], a first-principle calculation of the RC to the free neutron β decay, while very challenging, is expected to be performed in the near future. In this Letter, we perform a combined lattice QCD -phenomenological analysis. Making use of a body of hadron-hadron scattering data, known meson decay widths and the guidance of Regge theory and vector dominance, along with constraints from isospin symmetry, analyticity and unitarity, we are able to unambiguously relate the input into the dispersion integral for the γWbox RC on the pion and on the neutron. Fixing the strength of the pion matrix element from the lattice, we thus obtain an estimate of an analogous matrix element on the neutron, in accord with all the aforementioned physics constraints.
We start by writing down the dispersive representation of the contribution of the γW box diagram (see Fig.1) to the rate of the Fermi part of a semileptonic β decay process of H i → H f eν e [3,4]: where α is the fine-structure constant. The above definition of the γW -box correction corresponds to a shift |V ud | 2 → |V ud | 2 (1 + δ V A γW,H ), affecting the apparent value of V ud extracted from an experiment. The function stands for the first Nachtmann moment of the (spinindependent) parity-odd structure function F 3H (x, Q 2 ), resulting from the product between the axial charged weak current and the isoscalar electromagnetic current: Above, M H is the average mass of H i , H f , Q 2 = −q 2 , x = Q 2 /2p·q, and r H = 1 + 4M 2 H x 2 /Q 2 , and the factor F H + defines the normalization of the tree-level hadronic matrix element of the vector charged weak current: By isospin symmetry, F n + = 1 and F π − + = √ 2. The quantity δ V A γW,H is the source of the largest theory uncertainty of the RC in the π e3 , free neutron β decay,  Figure 2: Comparison between the lattice calculation of M s ) pQCD corrections (red curve) and the low-Q 2 CCFR data [14,15] (green points). and the universal RC in superallowed nuclear β decays, and has long been the limiting factor for the precise determination of V ud . To obtain δ V A γW,H we need to know the Nachtmann moment M   [3] for the explanation). Also, the transition point between perturbative and non-perturbative regime is a priori unknown, or uncertain.
The first calculation of M (0) 3π (1, Q 2 ) on the lattice in Ref. [10] serves as an important step in addressing the questions above. Its result is presented in Fig.2 as a function of Q 2 . At low Q 2 where the integral (1) is strongly weighted, lattice provides an extremely precise description of M (0) 3π (1, Q 2 ), but its uncertainty increases at large Q 2 due to the discretization error. Fortunately, at Q 2 > 2GeV 2 there exists very precise data for the first Nachtmann moment of the parity-odd structure function F νp+νp 3 measured in the ν/ν scattering on light nuclei by the CCFR Collaboration [14,15]. Their good agreement with pQCD prediction indicates a smooth transition to the perturbative regime at Q 2 > 2 GeV 2 , which also implies that these data, upon simple rescaling, can be converted to M  low 2 GeV 2 effects of generic higher-twist terms start to show up, and the LO OPE+pQCD prediction disagrees significantly with the lattice result.
We shall describe how the lattice result for δ V A γW on the pion can be used to improve our understanding of δ V A γW on the neutron. First, for the neutron we parametrized the structure function F [3,4]: where Q 2 0 ≈ 2 GeV 2 is the scale above which the LO OPE + pQCD description is valid. Above, we isolated the contributions from the elastic intermediate state (el) fixed by the nucleon magnetic [16,17] and axial elastic form factor [18], from the non-resonance πN continuum (πN ) in the low-energy region, from the N * resonances (res) 2 , and the Regge contribution (R) that allow to economically describe the multi-hadron continuum.
In a similar way, we parametrize the respective pion structure function as We note the absence of the elastic and the low-energy continuum contributions. The former is identically zero because the axial current does not couple to the spin-0 pion ground state. The latter would correspond to the non-resonant part of the ππ continuum in the p-wave; however, this partial wave is known to be entirely dominated by the ρ 0 resonance up to the KK threshold.
Comparing the parameterizations of Eqs. (5,6), we make an important observation. Among the various contributions there are the process-specific ones that reside in the lower part of the spectrum (elastic, resonance and that of F 3N,pQCD . The Regge universality is among the central predictions of Regge theory. In fact, it dictates that the upper and lower vertices in the Regge ρ-exchange amplitudes T ρ (W + + π − → γ + π 0 ) and T ρ (W + + n → γ + p) in Fig.  3 factorize, so that, e.g., where T ρ ππ→ππ , T ρ πN →πN , T ρ N N →N N stand for the amplitudes in elastic ππ, πN, N N scattering in the channel that corresponds to an exchange of the quantum numbers of the ρ meson in the t-channel. Regge factorization has been tested on global data sets for elastic pion, pionnucleon and nucleon-nucleon scattering.
This leads to a prediction based on Regge universality, with α ρ 0 = 0.477 [20]. Here we define the threshold func- ( 1 x − 1) and Λ = 1 GeV 2 [21]. The threshold parameter W th,H characterizes the threshold for the multi-hadron contributions. In Ref. [3] we fixed W th,N = m N + 2M π , such that the threshold function f N th ≈ 1 for W 2.5GeV. In the pion sector, one expects W th,π to lie between M ρ and 1.2 GeV, the scale above which Regge description is valid [19]. In this work we choose W th,π ≈ 1 GeV, and account for the uncertainty due to its variation between the two boundaries. The function A(Q 2 ) describes the interaction at the upper half of Fig.3 and is, within the Regge framework, common for neutron and pion. It is in principle a complicated function of Q 2 , but is now completely fixed by the lattice curve in Fig.2 upon subtracting the resonance contribution. With these, we obtain the following ratio between the first Nachtmann moments of the Regge contributions: To fully specify the parametrization of F (0) 3π we turn now to the resonance contribution depicted in Fig. 4. Its strength is derived from the following effective Lagrangian densities [22], The respective couplings are obtained as follows: |g ργπ | = 0.645(43) from the ρ → γπ decay width, |g a1ρπ | = 5.7(1.3) from the total width of a 1 assuming that a 1 → ρπ is the dominant channel [2], and |w a1 /g ρ | = 0.133 from the τ − → a − 1 ν τ decay width [23]. They give, upon the narrow-width approximation of ρ 0 , The overall sign is fixed by requiring that it matches the sign of the ππ contribution calculated in Chiral Perturbation Theory at small Q 2 . Numerically, the size of M (0) 3π,res is rather small, ≤ 10% of the total, as can be seen in the bottomright subview of Fig.5 where the resonance estimate of Eq. (11) (red dashed curves and band) is plotted along with the full lattice calculation (blue curves and band). This smallness guarantees that the removal of the nonuniversal resonance contribution does not introduce an uncontrolled systematic uncertainty in our analysis.
With Eq.(9), one could now obtain M (0) 3N,R (1, Q 2 ) directly from the lattice results and the rescaling factor R π/N . On the one hand, the most recent analysis of ππ scattering [19] made the factorization test with respect to πN analysis and found (omitting the isospin factor F π − + /F n + ), 3N,R (1, Q 2 ) (blue band with solid boundaries) is compared to the result of Ref. [3] (orange band with dashed boundaries), the pQCD prediction (red curve) and the CCFR data [14,15] (green points). In the bottom-right subview, the resonance contribution to M On the other hand, the OPE suggests that R π/N = 1 in the perturbative regime (note also the ρ coupling universality hypothesis in the hidden local symmetry [24]). Therefore, to ensure a continuous matching at all Q 2 values we allow R π/N to slightly depend on Q 2 , where R π/N (0) is fixed by Eq.
The result reads b = −0.076 +0.100 −0.072 GeV −2 . With the prescription above we fully fix M (0) 3N,R (1, Q 2 ) at low Q 2 using the lattice curve of M (0) 3π (1, Q 2 ). The result is shown in Fig.5, with the uncertainties from R π/N (Q 2 ) and W th,π added in quadrature. Integrating over Q 2 gives an updated estimate of the Regge contribution to δ V A γW,N : in excellent agreement with the previous determination δ V A γW,N R = 1.02(16) × 10 −3 [3]. One can also study the effect of varying the perturbative matching point by evaluating the Q 2 -integral in Eq. (1) between 2 GeV 2 and 3 GeV 2 using the CCFR data instead of the pQCD expression. That gives an insignificant extra uncertainty of 1 × 10 −5 , confirming the robustness of our error analysis.
We next discuss the impact of this result on the extraction of V ud . From superallowed nuclear β decay, we have [25]: , (superallowed) (16) where Ft is the f t-value corrected by nuclear effects, ∆ V R = δ V A γW,N + ... is the nucleus-independent RC that contains the largest theoretical error. In this paper we update the Regge contribution to δ V A γW,N according to Eq. (15). Meanwhile, we also update the pQCD contribution above 2 GeV 2 from O(α 3 s ) to O(α 4 s ) [12,13], which reduces ∆ V R by mere 1 × 10 −5 . As a result we obtain a slight shift upward with respect to the result of Ref. [3]: The recent Ref. [6] estimated a lower value, ∆ V R = 0.02426 (32), based on the assumption that the full Nachtmann moment should follow the perturbative curve down to as far as Q 2 = 1 GeV 2 , and only afterwards highertwist effects (estimated in a holographic QCD model) become important. The lattice calculation on the pion [10] suggests that already at Q 2 ≤ 2 GeV 2 the higher twist contributions are non-negligible.
Finally, we discuss the current situation of the top-row CKM unitarity. There are two different measurements of V us , using K 2 [2] and K 3 [33] decay separately: They disagree with each other at 2σ level, K 3 giving a smaller |V us | which leads to a larger unitarity violation. This, however, depends critically on the existing lattice calculation of the Kπ vector form factor f K 0 π − + (0) which is recently questioned by theory [34] and a new lattice paper [35]. Another possible issue is the electromagnetic RC in K 3 , which may be re-analyzed in a dispersive approach [36]. We summarize the resulting ∆ u CKM from different combinations in Table I. In short, we observe a (3 − 5)σ unitarity violation excluding the NNC, and (1.7 − 3)σ violation with the NNC. (15) -0.0012(4) -0.0021(4) w/ NNC 0.97366 (33) -0.0012 (7) -0.0021(7) To conclude, we devise the value of the universal, hadronic structure-dependent electroweak RC to the free neutron and superallowed nuclear β decays from the value of that correction evaluated in lattice QCD for the semileptonic pion decay. To connect the two, we combine the available phenomenological input. The lattice evaluation of the first Nachtmann moment M (0) 3π (1, Q 2 ) of the parity-violating spin-independent structure function of the pion, F (0) 3π provides the full control of its low-Q 2 behavior, overcoming the main deficiencies of previous works. The proposed method offers an independent assessment of systematic uncertainties of the theory of RC, and confirms the previously found deficit of the top-row CKM unitarity. In our new analysis the main source of uncertainty comes from the rescale factor R π/N , the relative strength of the Regge ρ trajectory in high-energy ππ and πN scattering. If future studies of ππ and πN scattering will allow one to improve the uncertainty of R π/N , our prediction of ∆ V R will also become more precise. However, a more straightforward solution is expected from a direct lattice calculation of the γW -box on the neutron. Either way, it will shift the emphasis to a reassessment of the nuclear structure corrections that enter the analysis of superallowed nuclear decay. On the other hand, with upcoming more precise measurements of the neutron lifetime and of neutron β decay asymmetries, neutron decay will become competitive as the source of our precise knowledge of V ud and CKM unitarity.
We appreciate Guido Martinelli and Ulf-G. Meißner for for inspiring discussions. The work of C.Y.S.