High-precision determination of the $K_{e3}$ radiative corrections

We report a high-precision calculation of the Standard Model electroweak radiative corrections in the $K\to \pi e^+\nu(\gamma)$ decay as a part of the combined theory effort to understand the existing anomaly in the determinations of $V_{us}$. Our new analysis features a chiral resummation of the large infrared-singular terms in the radiative corrections and a well-under-control strong interaction uncertainty based on the most recent lattice QCD inputs. While being consistent with the current state-of-the-art results obtained from chiral perturbation theory, we reduce the existing theory uncertainty from $10^{-3}$ to $10^{-4}$. Our result suggests that the Standard Model electroweak effects cannot account for the $V_{us}$ anomaly.

|V us | = 0.2252(5) (K µ2 ) , = 0.2231 (7) which may hint to the existence of physics beyond the Standard Model (BSM). The value obtained from the K l3 decay is particularly interesting because it also leads to a violation of the top-row CKM unitarity at (3 − 5)σ upon combining with the most recent updates of V ud [4][5][6][7], depending on the amount of nuclear uncertainties assigned to the latter [8,9]. However, despite of an active discussion about the possible BSM origin of the K µ2 -K l3 discrepancy [10][11][12][13][14][15][16][17][18], the current significance level is not yet sufficient to claim a discovery. One of the main obstacles is the large hadronic uncertainty in the electroweak radiative corrections (EWRC), which are the focus of this work.
Among the many studies of the EWRC in K l3 [19][20][21][22][23][24][25][26][27][28][29], the standard inputs in global analyses [30,31] are based on chiral perturbation theory (ChPT) which is the low-energy effective field theory of Quantum Chromodynamics (QCD). Within this framework, the "shortdistance" electroweak corrections are isolated as a constant factor, while the "long-distance" electromagnetic corrections are calculated up to O(e 2 p 2 ) [32][33][34], with e the electric charge and p a small momentum/meson mass. The estimated theory uncertainties in these calculations are of the order 10 −3 , and originate from: (1) the neglected contributions at O(e 2 p 4 ), and (2) the contri-butions from non-perturbative QCD at the chiral symmetry breaking scale Λ χ 4πF π that exhibit themselves as the poorly-constrained low-energy constants (LECs) in the theory [35,36]. These natural limitations prohibit further improvements of the precision level within the original framework.
In this letter we report a new calculation of the EWRC in K e3 . Based on a newly-proposed computational framework [37,38] that hybridizes the classical approach by Sirlin [39] and modern ChPT, we effectively resum the numerically largest terms in the EWRC to all orders in the chiral expansion and significantly reduce the O(e 2 p 4 ) uncertainty. Also, we utilize the high-precision lattice QCD calculations of the forward axial γW -box diagrams [40,41] to constrain the physics from the nonperturbative QCD. With these improvements, we reduce the theory uncertainty in the EWRC to K e3 to an unprecedented level of 10 −4 . We will outline here the most important steps that lead to the final results, while the full detail of the calculation will appear in a longer paper [42].
Our primary goal is to study the fractional correction to the K e3 decay rate due to EWRC: up to the precision level of 10 −4 . The denominator in Eq.
(2) comes from the tree-level amplitude for K(p) → π(p )e + (p e )ν(p ν ): where G F is the Fermi constant, F λ (p , p) = V * us f + (t)(p + p ) λ + f − (t)(p − p ) λ is the charged weak matrix element and f ± (t) are the charged weak form factors, with t = (p − p ) 2 . We restrict ourselves to K e3 for which the contribution from f − to the decay rate is suppressed by m 2 e /M 2 K ≈ 10 −6 and can be neglected.  The full EWRC includes both the virtual corrections and the bremsstrahlung contributions, and we shall start with the former. A generic one-loop correction to the decay amplitude reads: where the loop integrals are contained in I λ . It results in a shift of the form factors: f ± → f ± + δf ± , except that δf ± can also depend on s = (p + p e ) 2 or u = (p − p e ) 2 . Again, in K e3 only δf + is relevant.
We follow the categorization of the different components of the O(G F α) virtual corrections in Refs. [37,38], with α = e 2 /4π. First, there are pieces in which the loop integrals are independent of the hadron properties and can be computed analytically. They are contained in Eqs.(2.4) and (2.13) in Ref. [38], which combine to give: whereã g = −0.083 and δ QED HO = 0.0010(3) come from perturbative QCD corrections and the resummation of large QED logarithms, respectively. Notice also that we have introduced a small photon mass M γ to regularize the infrared (IR)-divergence.
The remaining loop diagrams in the EWRC, in which the entire dependence on hadronic structure is contained, are depicted in Fig.1. They depend on the following quantities: which are both functions of the momenta {q , p , p}. In particular, we may split the tensor T µν into two pieces:  With these, the first relevant integral can be written as: where the first two lines come from Eq.(2.13) of Ref. [38], and the third line is a part of δM A γW in Eq. (2.10) of the same paper.
The operator product expansion (OPE) shows that the |q | > Λ χ region does not contribute to the integral I λ A , therefore only the low-energy expressions of T µν and Γ µ are needed. To this end, we find it useful to split them into the "pole" and "seagull" terms respectively, as depicted in Fig. 2: Furthermore, we can obtain the so-called "convection term" by setting q → 0 in both the electromagnetic form factor and the charged weak vertex of the pole term [43]. It represents the minimal expression that satisfies the exact electromagnetic Ward identity, and thus gives the full IR-divergent structures in the loop integrals. The seagull term receives contributions from resonances and the many-particle continuum. An estimate operating with low-lying resonances [44][45][46] suggests that its contribution to δ Ke3 is at most 10 −4 . Note that tchannel exchanges that still retain some sensitivity to the long-range effects do not contribute to Eq.(7). To stay on the conservative side, we assign to it a generic uncertainty of 2 × 10 −4 . Therefore, δf + derived from I λ A is dominated by the pole contribution which is fully determined by the K and π electromagnetic and charged weak form factors. The result splits into two pieces: where (δf + ) II is a model-independent IR-divergent piece. The IR-finite piece, (δf + ) fin A , on the other hand, is evaluated numerically by adopting a monopole parameterization of the hadronic form factors [47][48][49]. Notice that the integral I λ A only probes the region q ∼ p e ∼ p − p ∼ M K − M π , where different parameterizations of the form factors are practically indistinguishable. In particular, we find that the main source of the uncertainty is the K + mean-square charge radius and the experimental uncertainty thereof, r 2 K = 0.34(5) fm 2 [47]. The second relevant integral is: which picks up the remaining part of δM A γW in Eq.(2.10) of Ref. [38]. It is IR-finite, but probes the physics from |q | = 0 all the way up to |q | ∼ M W . A significant amount of theoretical uncertainty thus resides in the region |q | ∼ Λ χ where non-perturbative QCD takes place, and has been an unsettled issue for decades. The situation is changed following the recent lattice QCD calculations of the so-called "forward axial γW -box": where T if µν is just T µν except that the initial and final states are now is the form factor f if + (0) multiplied by the appropriate CKM matrix element. Following the existing literature, we split it into two pieces: which come from the loop integral at Q 2 ≡ −q 2 > Q 2 cut and Q 2 < Q 2 cut respectively, where Q 2 cut = 2 GeV 2 is a scale above which perturbative QCD works well. The ">" term is flavor-and mass-independent, and was calculated to O(α 4 s ): V A> γW = 2.16 × 10 −3 [40]. In the meantime, direct lattice calculations of the "<" term were performed in two channels [40,41]: from which we can also obtain V A< γW (K + , π 0 , M π ) = 1.06(7) lat × 10 −3 through a ChPT matching [38].
The only difference between the integrals in Eq. (10) and (11) is the non-forward (NF) kinematics in the former (i.e. p = p and p e = 0), which only affect the integral in the Q 2 < Q 2 cut region. Therefore one could similarly split (δf + ) B into two pieces: (δf + ) B = (δf + ) > B + (δf + ) < B , where the ">" piece matches trivially to the forward axial γW -box: On the other hand, the matching between the "<" components is not exact due to the NF effects. We characterize the latter by an energy scale E that could be either M K − M π , (s − M 2 π ) 1/2 or (u − M 2 π ) 1/2 . The matching then reads: where O(E 2 /Λ 2 χ ) represents the NF corrections. Numerically, since E < M K , we may multiply the right-hand side of Eq. (15) by M 2 K /Λ 2 χ as a conservative estimation of the NF uncertainty.
The last virtual correction is the so-called "three-point function" contribution to the charged weak form factors, which was derived within ChPT to O(e 2 p 2 ) in Ref. [37]. However, it contains an IR-divergent piece that comes from the convection term contribution, and can be resummed to all orders in the chiral expansion by simply adding back the charged weak form factors. This leads to the following partially-resummed ChPT expression: where the IR-divergent piece (δf + ) III is exact, i.e. resummed to all orders in ChPT. It combines with (δf + ) II in Eq. (9) to give: where M i is the mass of the charged meson (K + in K + e3 and π − in K 0 e3 ) and β i is the speed of the positron in the rest frame of the charged meson. Meanwhile, the IR-finite pieces, (δf +,3 ) Next we switch to the bremsstrahlung contributions, as depicted in Fig. 3. Its amplitude is given by: in which the tensor T µν appears again, except that now it deals with an on-shell photon momentum k whose size is restricted by phase space. Similar to δf +,3 , we find that  the most efficient way to calculate the bremsstrahlung contributions is to adopt a partially-resummed ChPT expression for T µν : where the full convection term T µν conv is explicitly singled out, while the remaining terms in the curly bracket are expanded to O(p 2 ). Consequently, one can split M brems into two separately gauge-invariant pieces: where the terms in the curly bracket of Eq. (19) reside in M B . The contribution to the decay rate from |M A | 2 contains the full IR-divergent structure (which cancels with the virtual corrections), is numerically the largest and does not associate to any chiral expansion uncertainty. The term 2Re {M * A M B } + |M B | 2 , on the other hand, is subject to O(e 2 p 4 ) corrections. We find that its contribution to δ Ke3 is 10 −3 , so the associated chiral expansion uncertainty, which is obtained by multiplying the central value with M 2 K /Λ 2 χ , is of the order 10 −4 . With the above, we have calculated all EWRC to 10 −4 and may compare with existing results. The standard parameterization of the fully-inclusive K e3 decay rate reads [3]: among which S EW = 1.0232(3) describes the shortdistance EWRC [50] (the uncertainty comes from δ QED HO [51]) and δ Ke EM describes the long-distance electromagnetic corrections respectively. We also realize that in the existing ChPT treatment a residual component of the electromagnetic corrections, which corresponds exactly to (δf +,3 ) fin e 2 p 2 in our language, is redistributed into I (0) Ke (λ i ) and δ Kπ SU(2) that describe the t-dependence of the charged weak form factors and the isospin breaking correction, respectively [32][33][34]. Therefore, the correspondence between δ Ke EM in the ChPT calculation and δ Ke3 in our approach reads: Our results of the different components of δ Ke3 are summarized in Table I, from which we obtain: The uncertainties are explained as follows: "sg" is our estimate of the seagull contribution to I λ A , " r 2 K " comes from the experimental uncertainty of the K + meansquare charge radius that enters I λ A , "lat" and "NF" are the uncertainties in (δf + ) B from lattice QCD and the NF effects, respectively, and "e 2 p 4 " represents the chiral expansion uncertainty in the 2Re {M * A M B } + |M 2 B | term from the bremsstrahlung contribution. We should compare Eq. (23) to the ChPT result [34]: They are consistent within error bars, but Eq. (23) shows a reduction of the total uncertainty by almost an order of magnitude, which can be easily understood as follows. First, in ChPT the O(e 2 p 4 ) uncertainty is obtained by multiplying the full result, including the IRsingular pieces that are numerically the largest, with M 2 K /Λ 2 χ ; meanwhile, within the new formalism those pieces can be evaluated exactly by simply isolating the pole/convection term in T µν and Γ µ . The remainders are generically an order of magnitude smaller, so their associated O(e 2 p 4 ) uncertainty is also suppressed. Secondly, in ChPT the LECs {X i } were estimated within resonance models [52,53] and were assigned a 100% uncertainty. On the other hand, some of us pointed out in Ref. [38] that these LECs are associated with the forward axial γW -box diagram, and promoted first-principle calculations with lattice QCD. This effectively transforms the LEC uncertainties in ChPT into the lattice and NF uncertainties in (δf + ) B which are much better under control.
To conclude, we performed a significantly improved calculation of the EWRC in the K e3 channel. We observe no large systematic corrections with respect to previous analyses. Although the error analysis in the K µ3 channel is somewhat more complicated, we deem such large corrections in this channel unlikely. Hence, it is safe to conclude that the EWRC in K l3 cannot be responsible for the K µ2 -K l3 discrepancy in V us . One should then switch to other SM inputs, such as the lattice calculation of |f K 0 π − + (0)| and the theory inputs of I (0) Kl (λ i ) and δ Kπ SU (2) . Finally, our improvement in δ Ke EM also opens a new pathway for the precise measurement of V us /V ud through the ratio between the semileptonic kaon and pion decay rate [54]. For instance, we may define: Since both the RC uncertainties in K 0 e3 and π e3 are now at the 10 −4 level, the dominant theory uncertainty (apart from lattice inputs) of R V comes from the K 0 e3 phase space (PS) integral. We compare this to: which is currently used to extract V us /V ud . We see that R V possesses a much smaller theoretical uncertainty than R A , and hence represents a more promising avenue in the future. Our work thus provides a strong motivation for experimentalists to measure the π e3 branching ratio with an order-of-magnitude increase in precision [55]. We thank Vincenzo Cirigliano for many inspiring discussions. This work is supported in part by the DFG