Comparison of lattice QCD+QED predictions for radiative leptonic decays of light mesons with experimental data

We present a comparison of existing experimental data for the radiative leptonic decays $P\to\ell\nu_\ell\gamma$, where $P=K$ or $\pi$ and $\ell=e$ or $\mu$, from the KLOE, PIBETA, E787, ISTRA+ and OKA collaborations with theoretical predictions based on the recent non-perturbative determinations of the structure-dependent vector and axial-vector form factors, $F_V$ and $F_A$ respectively. These were obtained using lattice QCD+QED simulations at order $O(\alpha_{\mathrm{em}})$ in the electromagnetic coupling. We find good agreement with the KLOE data on $K\to e\nu_e\gamma$ decays from which the form factor $F^+=F_V+F_A$ can be determined. For $K\to\mu\nu_\mu\gamma$ decays we observe differences of up to 3 - 4 standard deviations at large photon energies between the theoretical predictions and the data from the E787, ISTRA+ and OKA experiments and similar discrepancies in some kinematical regions with the PIBETA experiment on radiative pion decays. A global study of all the kaon-decay data within the Standard Model results in a poor fit, largely because at large photon energies the KLOE and E787 data cannot be reproduced simultaneously in terms of the same form factor $F^+$. The discrepancy between the theoretical and experimental values of the form factor $F^-=F_V-F_A$ is even more pronounced. These observations motivate future improvements of both the theoretical and experimental determinations of the structure-dependent form factors $F^+$ and $F^-$, as well as further theoretical investigations of models of"new physics"which might for example, include possible flavor changing interactions beyond $V - A$ and/or non-universal corrections to the lepton couplings.


I. INTRODUCTION
The decays of charged pseudoscalar mesons into light leptons, P → ν [γ] where stands for an electron or a muon, represent an important contribution to flavor physics since they give access to fundamental parameters of the Standard Model (SM), in particular to the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1]. At tree level, i.e. without a photon in the final state, these decays are helicity suppressed in the SM due to the V −A structure of the leptonic weak charged current, while the helicity suppression can be overcome by the radiated photons. Therefore, radiative leptonic decays may provide sensitive probes of possible SM extensions inducing nonstandard currents and/or non-universal corrections to the lepton couplings.
Radiative leptonic decays also provide a powerful tool with which to investigate the internal structure of the decaying meson. In addition to the leptonic decay constant f P , there are other structure-dependent (SD) amplitudes describing the emission of real photons from hadronic states, usually parameterized in terms of the vector and axial-vector form factors, F V and F A respectively.
Thus, a first-principle calculation of radiative leptonic decays requires a non-perturbative accuracy, which can be provided by numerical QCD+QED simulations on the lattice. In Ref. [2] a strategy was proposed to enable lattice computations of QED radiative corrections to P + → + ν [γ] decay rates at order O(α em ). The strategy naturally obeys the Bloch-Nordsieck mechanism [3], in which the cancellation of infrared divergences occurs between contributions to the rate with real photons in the final state and those with virtual photons in the decay amplitude.
Within the RM123 expansion framework [4,5] the strategy of Ref. [2] was applied in Refs. [6,7] to provide the first non-perturbative model-independent calculation of the SD virtual contribution to the pion and kaon decay rates into muons. The contribution with a real photon in the final state was still evaluated in the point-like (pt) effective theory, which is only adequate for sufficiently soft photons (see Ref. [2]). This limitation has recently been removed in Ref. [8], where the pt and SD amplitudes for real photon emission have been determined non-perturbatively in numerical lattice QCD+QED simulations at order O(α em ) in the electromagnetic coupling. The calculations were performed in the electroquenched approximation in which the sea quarks are electrically neutral 1 .
The aim of this work is to carry out a comparison between the theoretical predictions based on the non-perturbative determination of the SD form factors F V and F A evaluated in Ref. [8] and the experimental data available on the leptonic radiative decay K → eν e γ from the KLOE Collaboration [9], on the decay K → µν µ γ from E787 [10], ISTRA+ [11] and OKA [12] collaborations and on the decay π + → e + ν e γ from the PIBETA Collaboration [13].
The plan of the remainder of this paper is as follows. In Sec. II we recall the basic formulae for the double and single differential decay rates d 2 Γ 1 /dE γ dE and dΓ 1 /dE γ for real photon emission, where E γ (E ) is the photon (lepton) energy in the rest frame of the decaying meson. The subscript 1 indicates that there is a single photon in the final state. In Sec. III the impact of the SD contributions to the total rates of π e2[γ] , π µ2[γ] , K e2[γ] and K µ2[γ] decays is evaluated. We confirm the expectation that the SD contributions to Γ 1 are negligible for decays into muons, but find that they are a very large contribution to the totally inclusive rate for K e2 [γ] decays. In Sections IV-VI the experimental data of Refs. [9][10][11][12][13] are briefly described and compared with our theoretical results and with the predictions of Chiral Perturbation Theory (ChPT) at order O(e 2 p 4 ) [14][15][16][17][18].
For kaon decays, we show that there is a good agreement between our determination of the form factor F + and the KLOE data on K → eν e γ decays, but we find discrepancies of up to 3 -4 standard deviations at large photon energies between our predictions and the E787, ISTRA+ and OKA data on K → µν µ γ decays. We also find similar discrepancies in some kinematical regions of the PIBETA experiment on the radiative pion decay. In Sec. VII a simultaneous fit of the KLOE, E787, ISTRA+ and OKA experimental data on radiative kaon decays is performed within the SM and adopting a linear dependence of the SD form factors F ± ≡ F V ± F A on the photon energy, as suggested by the lattice results of Ref. [8]. The quality of the fit is poor because the KLOE and E787 data cannot be reproduced simultaneously in terms of the same form factor F + . There is also a particularly pronounced discrepancy between the theoretical and experimental determinations of the form factor F − . These observation motivate future improved theoretical and experimental determinations of the structure-dependent form factors F + and F − , as well as further theoretical investigations of theories "Beyond the Standard Model" which might for example, include possible flavor changing interactions beyond V −A and/or non-universal corrections to the lepton couplings.
Our conclusions are summarized in Sec. VIII.

II. DIFFERENTIAL RATES FOR RADIATIVE LEPTONIC DECAYS
Following Refs. [2,8] the double differential rate for the radiative leptonic decay of a charged pseudoscalar meson, P + → + ν γ, can be written as the sum of three contributions: where the subscript 1 denotes the number of photons in the final state, while x γ and x are the photon and lepton kinematical variables, defined as where P is the four-momentum of the decaying meson with mass m P , p is the four-momentum of the final-state lepton with mass m , k is the four-momentum of the photon and r ≡ m /m P . In the rest frame of the decaying meson one has x γ = 2E γ /m P and x = 2E /m P − r 2 , where E γ and E are the photon and lepton energies respectively.
In Eq. (1) the quantity Γ (0) is the leptonic decay rate at tree level, given explicitly by where G F is the Fermi constant, V CKM the relevant CKM matrix element and f P the leptonic decay constant of the P -meson.
The other entries on the right-hand side of Eq. (1) are where the superscripts ± correspond to the two photon helicities and the three terms in Eqs. (4)-(6) represent respectively the contribution of the pt approximation of the decaying meson, the SD contribution and the contribution from the interference (INT) between the pt and SD terms. Note that in the literature the pt contribution is often referred to as the inner-bremsstrahlung term.
The kinematical functions appearing in Eqs. (4)- (6) are given by and the quantities F ± (x γ ) are the simple combinations of the vector F V (x γ ) and axial-vector F A (x γ ) form factors which, together with f P , describe the emission of a real photon in the leptonic decay of the P -meson.
Recently the vector and axial-vector form factors have been determined on the lattice for decaying pions, kaons, D and D s mesons for a wide range of values of x γ , adopting the electroquenched approximation in which the sea quarks are electrically neutral [8]. In this work we adopt the definition of the vector (F V ) and axial-vector (F A ) form factors given in Section II of Ref. [8] (see also Appendix B of Ref. [2]). For the decays of the pion and the kaon (P = π, K) we make use of the linear parameterization of the physical results for F V and F A provided in Section V of Ref. [8], which is an excellent representation of our lattice data throughout the physical region, i.e. we write and where the uncertainties include statistical errors as well as the various sources of systematic errors, except for the QED quenching effect [8]. The impact of the latter is expected to be mild as it is an SU (3)-breaking effect. The full correlation matrices of the parameters in Eqs. (14) - (17) are collected in Tables I and II for pion and the kaon decays respectively. In the following the uncertainties and correlations of the two form factors are taken into account adopting multivariate gaussian distributions for the parameters in Eqs. (14) -(17) with 10,000 events.  (14) and (16)) of the linear parameterization (13) provided in Ref. [8] for the decays of the pion.   (15) and (17)) of the linear parameterization (13) provided in Ref. [8] for the decays of the kaon.
The experimental data from the KLOE, E787, ISTRA+, OKA and PIBETA, collaborations [9][10][11][12][13] correspond to radiative decay rates integrated over the lepton variable x and including specific kinematical cuts on the lepton momentum and/or on the emission angle θ γ between the lepton and the photon. We therefore introduce the (partially) integrated kinematical functions where x 0 and x 1 depend on the specific experimental conditions (see later Sections IV-VI). Thus, the partially integrated radiative decay rate for x ∈ [x 0 , x 1 ] is given by where dR pt In the absence of kinematical cuts x varies between x 0 = 1 − x γ + x γ r 2 /(1 − x γ ) and x 1 = 1, so that in this case For real photon emissions the knowledge of the SD vector and axial form factors, F V (x γ ) and F A (x γ ), and of the meson decay constant f P is sufficient to compute the partially integrated decay rate (23) for any choice of the range of integration [x 0 , x 1 ] over the lepton variable x . In this section we consider inclusive decay rates with no kinematical cuts on x and after integration over the photon variable x γ in its full kinematical range.
From Eqs. (27) -(31) it can readily be checked that as x γ → 0 one has dR SD 1 /dx γ ∝ x 3 γ and dR INT 1 /dx γ ∝ x γ , while dR pt 1 /dx γ ∝ 1/x γ . Therefore, the inclusive SD and INT contributions are infrared safe, while the pt contribution exhibits a logarithmic, structure-independent infrared divergence. This divergence cancels the corresponding logarithmic infrared divergence of the virtual photon contribution (Γ 0 ) to the inclusive decay rate [3] where ∆E γ is the maximum detected energy of the emitted real photon (in the meson rest-frame).
Thus, in the intermediate steps of the calculation of Eq. (32) it is necessary to introduce an infrared regulator. To this end, a strategy to work only with quantities that are finite when the infrared regulator is removed, has been developed in Ref. [2] and applied to pion and kaon leptonic decays in Refs. [6,7]. The inclusive rate Γ(∆E γ ) is reorganized as follows with the length of the lattice L and µ γ (for example, a photon mass) acting as infrared regulators in the first two terms on the right-hand side. The exchange of a virtual photon depends on the structure of the meson since all momentum modes are included, and Γ 0 (L) must therefore be computed non-perturbatively. We will now explain that on the right-hand side of Eq. (33), each of the two terms on the top-line are infrared finite, as are separately the two terms on the second line.
In the first term on the right-hand side of Eq. (33) the quantities Γ 0 (L) and Γ pt 0 (L) can be evaluated on the lattice using the lattice size L as the intermediate infrared regulator. Both Γ 0 (L) and Γ pt 0 (L) have the same infrared divergences which therefore cancel in the difference. In our papers we use the lattice size L as the infrared regulator by working in the QED L formulation of QED in a finite volume [19], but any other consistent formulation of QED in a finite volume could equally well be used. The difference Γ 0 − Γ pt 0 is independent of the regulator as this is removed [20]. Γ 0 (L) depends on the structure of the decaying meson and is computed non-perturbatively on the lattice [7,20].
In the second term on the right-hand side of Eq. (33) the decaying meson is taken to be a pointlike charged particle and both Γ pt 0 (µ γ ) and Γ pt 1 (∆E γ , µ γ ) are computed directly in infinite volume, in perturbation theory, using some infrared regulator, for example a photon mass µ γ . Each of the two terms is infrared divergent, but the sum is convergent and independent of the regulator [3].
Each of the two terms on the second line of Eq. (33) are infrared finite and can be computed directly in infinite volume limit requiring only the knowledge of the structure dependent form factors, F A (x γ ) and F V (x γ ), and of the meson decay constant f P [8].
Using the decomposition (33), the infrared-finite inclusive decay rate Γ(∆E γ ) can be written as where In Eqs. (34) -(35) δR 0 represents the SD virtual contribution (including also the universal short- is the (infrared-safe) sum of the point-like contributions of a virtual and a real photon with energy up to ∆E γ , evaluated within the W -regularization scheme for the ultraviolet divergences which was calculated in Ref. [2] to be where r E ≡ 2∆E γ /m P and Li 2 (x) = − x 0 du log(1 − u)/u. Using the vector and axial form factors given in Eqs. (13) - (17) we have calculated the (totally where ∆E max γ = m P (1 − r 2 )/2. Our non-perturbative results are shown in Table III  and K, respectively 2 .  , are also shown for each decay process.
In the same Table we also show the values of the SD virtual contributions δR 0 (π µ2 ) and δR 0 (K µ2 ), which can be derived from the results of Ref. [7]. There, the combination δR 0 + For decays into a final-state electron, the lattice determinations of the SD virtual contributions δR 0 (π e2 ) and δR 0 (K e2 ), which are currently missing in Table III, are in progress.
From The situation is very different for radiative kaon decays into electrons where the relative SD contribution is very large and even exceeds 1. This is related to the presence of the factor r 2 in the denominator of Eq. (37), which compensates the factor r 2 present in the tree-level rate Γ (0) because of helicity suppression (see Eq. (3)). In the next Section we will compare our nonperturbative predictions with results from the KLOE experiment on the radiative kaon decay K e2γ , which is devoted to the investigation of this large SD contribution [9].

IV. COMPARISON WITH THE EXPERIMENTAL RESULTS FROM THE KLOE COLLABORATION
In Ref. [9] the KLOE Collaboration has measured the differential decay rate dΓ(K e2γ )/dE γ for photon energies in the range 10 MeV < E γ < E max γ 250 MeV with the constraint p e > 200 MeV.
More precisely, they have measured the differential branching ratio integrated in five different bins of photon energies: with E i γ = {10, 50, 100, 150, 200, 250} MeV. Since we work at first order in α em , we can replace Γ(K µ2[γ] ) with its tree-level expression (3) in the denominator of Eq. (42) 3 . Thus, the theoretical prediction ∆R th,i can be decomposed into the sum of three terms where 3 The results shown in Table III imply that the difference between the total rate Γ(K µ2[γ] ) and its tree-level expression and r e = m e /m K and r µ = m µ /m K .
The presence of a constraint of the type p e > p e,min implies that x e > x min , where x min is given by We therefore obtain where x 0 is given by Using our form factors (13) with the parameters given in Eqs. (15) and (17) Table IV In Table IV with L r 9 + L r 10 = 0.0017 (7) [22] and taking m K /f K = 493.7 MeV/156.1 MeV. Such predictions are in good agreement with the experimental points within about 1 standard deviation.

V. COMPARISON WITH THE E787, ISTRA+ AND OKA EXPERIMENTS
In this Section we compare our lattice predictions with the experimental data on the leptonic radiative decays of kaons into muons, K µ2γ , obtained by the E787 [10], ISTRA+ [11] and OKA [12] collaborations. The kinematical regions in terms of photon and lepton energies were suitably chosen in order to enhance the contributions of the SD + term in the case of the E787 experiment and of the INT − term in the case of the ISTRA+ and OKA experiments. We remind the reader that the SD + and INT − terms are related to the square of the form factor F + and to the form factor F − , respectively.

A. The E787 experiment
In Ref. [10] the E787 Collaboration has investigated the K µ2γ decay for photon energies in the range 90 MeV < E γ < E max γ 235 MeV with the constraint that the muon kinetic energy is larger than 137 MeV (i.e. E µ > m µ + 137 MeV 243 MeV). In such kinematical regions the radiated photons come mainly from the pt contribution and the SD + terms [10]. In order to compare their results with those from other experiments, the E787 data are integrated over the small allowed range of muon energies 243 MeV < E µ ≤ E max µ 258 MeV, assuming a constant acceptance, to obtain the differential branching ratio as a function of the emission angle θ µγ between the muon and the photon in the kaon rest-frame.
At leading order, O(α em ), the theoretical prediction for dR th /d cos(θ µγ ) can be written as the sum of the following five terms where Since the pt contribution is a purely kinematical factor, it can be subtracted from the experimental data without introducing any uncertainty. The corresponding subtracted data are compared with our theoretical predictions in Table V  Note that, though generally small, the relative contribution of SD − +INT − , which depends on the form factor F − (x γ ), becomes more important as cos(θ µγ ) increases (i.e. as x γ decreases), reaching about 20 -30% of the term SD + +INT + at the lowest available values of x γ .
We remind the reader that, as shown in Sec. IV, our lattice form factor F + (x γ ) leads to a good description of the KLOE data [9]. A consequence of this is that the tension between our theoretical predictions and the E787 data which is visible at large x γ in Fig. 2 is not unexpected because of a tension between the two experiments. The KLOE collaboration has estimated F + (x γ = 1) to be equal to 0.125 ± 0.007 stat ± 0.001 syst [9], while the estimate of E787, assuming a constant form factor, is 0.165 ± 0.007 stat ± 0.011 syst [10]. The difference is at the level of about 3 standard deviations (see also the discussion in Sec. VII below). Our theoretical prediction for this quantity is F + (x γ = 1) = 0.1362 ± 0.0096.
Thus, further experimental investigations of the form factor F + (x γ ) in radiative kaon decays into electrons and muons are required. In particular, an investigation of the decay K e2γ at large electron energies will provide the opportunity for an accurate determination of |F + (x γ )| for a wide range of values of x γ . This is illustrated in Fig. 3, where the pt, SD + , SD − , INT + and INT − contributions to the differential branching ratio

B. The ISTRA+ and OKA experiments
In Refs. [11] and [12] the ISTRA+ and OKA collaborations have selected appropriate kinematical regions (strips) in order to determine the contribution of the interference term INT − . For each strip, specific bins are selected in the photon and muon variables x γ and y µ ≡ 2E µ /m K = x µ + r 2 µ , where E µ is the muon energy in the kaon rest frame. A further constraint cos(θ µγ ) > cos(θ cut ) is imposed on the emission angle θ µγ between the muon and the photon. The kinematical cuts are collected in Tables VI and VII and can be taken into account by using the kinematical functions f pt,SD,INT (x γ ; x 0 , x 1 ), given in Eqs. (18)- (22), with where the index i labels the strip and x + is equal to and cos(θ cut ) given in Tables VI and VII for each strip.  In both experiments the measured observable is the ratio N exp /N pt of the number of observed photons in each strip to the number of pt (or inner-bremsstrahlung) events. N pt is estimated using the Geant3 package [26].
The comparison of the experimental results with our predictions, and also with those obtained using ChPT at order O(e 2 p 4 ) based on the vector and axial-vector form factors given in Eq. (52) with m K /f K = 493.7 MeV/156.1 MeV, is presented in Table VIII and in Fig. 4. It can clearly be seen that at large photon energies there is a significant tension between the experimental data and our non-perturbative results (and also those obtained using ChPT  In Ref. [13] the PIBETA Collaboration has investigated the radiative pion decay into electrons π e2γ and has measured the following branching ratios As was the case for K e2γ decays in Eq. (44), at order O(α em ) the theoretical prediction for each bin for π e2γ decays, ∆R th,i , can be decomposed into the sum of three terms where in this case with and r e = m e /m π and r µ = m µ /m π .
The constraint on the electron energies E e > E i e implies x e > x i min , where while, using momentum conservation, the constraint θ eγ > θ cut = 40  50), with m K , f K now replaced by m π , f π , and with x 0 equal to Using the form factors (13) with the parameters given in Eqs. (14) and (16) Table IX and shown in Fig. 5. In Table IX  Possible contributions in the PIBETA kinematics arising from tensor interactions beyond the SM have been discussed in the literature (see e.g. Refs. [16,24] and references therein). In Ref. [17] 5 A tension of about 2.8 standard deviations is also present between our predictions and the older experimental data from ISTRA Collaboration [23]. There the kinematical cuts Eγ > 21 MeV and Ee > (70 − 0.8Eγ) MeV were applied, which implies that θeγ > 60 • .   OKA data can be fitted simultaneously since they concern kaon decays, while only the PIBETA experiment measures the pion decay rates. We stress that the discussion in this section assumes the validity of the SM in general, and lepton-flavour universality in particular, allowing us to combine data from kaon decays into electrons and muons.
For radiative kaon decays we observe that: • In fitting the kaon data we adopt a simple linear parameterization of the form factors F ± (x γ ), suggested by our lattice results, namely where the four quantities C ± and D ± are now treated as free parameters.
A total of 51 experimental data points (5 points from KLOE, 25 points from E787, 11 points from ISTRA+ and 10 points from OKA) are then fitted using the form factors (70) adopting a standard χ 2 -minimization procedure with a bootstrap sample of 5000 events generated to propagate the uncertainties of the experimental data and giving the same weight to each of the four experiments.
The quality of the best fit is poor: the optimal value of χ 2 /(no. of points) is equal to 1.3, 5.3, 3.1 and 2.2 for the KLOE, E787, ISTRA+ and OKA data, respectively. The comparison of the results of the global SM fit with all the experimental data is shown in Fig. 6. The largest tension occurs for the E787 data and is a consequence of the simultaneous presence of the KLOE data, as will be explained below.  6. Results of the global SM fit (black diamonds) applied to the KLOE [9], E787 [10], ISTRA+ [11] and OKA [12] data (red circles) adopting the linear parameterization (70) for the form factors F + (x γ ) and F − (x γ ). The blue squares represent the theoretical SM predictions evaluated with the lattice form factors determined in Ref. [8].
The values found for the four parameters appearing in Eq. (70) are determined to be C + = 0.134 ± 0.012 , D + = −0.002 ± 0.019 , while for comparison the values of the same parameters corresponding to the lattice form factors (15) and (17) are The corresponding correlation matrices are presented in Tables X and XI.  F + (x γ ) and F − (x γ ) determined in Ref. [8].
Note that the dependence on the form factor F + (x γ ) in the global fit to all the data is dominated by the SD + term and hence by |F + (x γ )|. We are therefore unable to determine the sign of C + from the global fit alone. Given that both our lattice results and ChPT yield a positive value of C + , we have started our minimization procedure with a positive value and subsequently always obtained positive final values of C + for all the bootstrap events.
In Fig. 7 the "optimal" form factors (obtained from Eqs. The difficulty in performing a global fit within the SM is partly due to the inconsistent results in the form factor F + (x γ ) from the KLOE and E787 experiments, as discussed in Sec. V A. This is further illustrated in Fig. 8 where the results for the form factors from the best fits are plotted omitting either the E787 data or the KLOE data and compared to the lattice results. The best separate fits to the KLOE and E787 data result in significantly different values of the form factor F + (x γ ). It can also be seen that the optimal form factor F − (x γ ) always deviates significantly from our lattice results and its slope is also sensitive to the inclusion of either the KLOE or the E787 data or both. At low values of x γ the KLOE data prefer smaller values of the form factor F − (x γ ), while the E787 data are compatible with larger ones. This is again related to the different values of the form factor F + (x γ ) from the KLOE and E787 experiments shown in the left panel of Fig. 8.
At large values of x γ the form factor F − (x γ ) is mainly governed by the ISTRA+ and OKA data 6 .
Finally note that the extraction of the form factor F + (x γ ) from the KLOE data is affected by the simultaneous inclusion of the ISTRA+ and OKA data at low values of x γ (compare the red circles 6 The inclusion of the E787 data has two main consequences on the optimal form factors corresponding to the KLOE+ISTRA+OKA analysis: 1) at large xγ the form factor F + (xγ) increases and correspondingly the form factor F − (xγ) should decrease to keep unchanged the sum SD + + SD − governed by the KLOE data; 2) at low xγ the E787 data for cos(θµγ) −0.9 (see Fig. 2) require larger values of F − (xγ) to compensate the SD + + INT + contribution. The above features produce the flattening of F − (xγ) observed in Fig. 8  in the right panel of Fig. 1 with those of the left panel of Fig. 8). obtained by the fitting either KLOE [9], ISTRA+ [11] and OKA [12] data (red circles) or E787 [10], IS-TRA+ [11] and OKA [12] data (green diamonds). The black shaded areas correspond to the simultaneous fit of all the experimental data from KLOE [9], E787 [10], ISTRA+ [11] and OKA [12]. The blue squares represent our lattice results from Ref. [8]. The errors represent uncertainties at the level of 1 standard deviation.

VIII. CONCLUSIONS
We have presented a comparison of our theoretical predictions with the existing experimental data on the radiative leptonic decays K → eν e γ from the KLOE collaboration [9], K → µν µ γ from the E787, ISTRA+ and OKA collaborations [10][11][12] and π → eν e γ from the PIBETA experiment [13]. The theoretical predictions are based on our recent non-perturbative determinations of the vector and axial-vector form factors corresponding to the emission of a real photon, using lattice QCD+QED simulations at leading order in the electromagnetic coupling, O(α em ), in the electroquenched approximation [8].
We find good consistency between our theoretical predictions and the experimental results from the KLOE experiment on K → eνγ decays [9], but a discrepancy at the level of about 2 standard deviations for the data at large x γ from the E787 experiment on K → µνγ decays. Indeed the results from the two experiments do not agree. We also find differences of up to 3 -4 standard deviations at large photon energies in the comparison of our predictions with the E787, ISTRA+ and OKA data on radiative kaon decays as well as for some kinematical regions of the PIBETA experiment on the radiative pion decay.
We have also performed a simultaneous fit of the KLOE, E787, ISTRA+ and OKA experimental data on the radiative kaon decays staying within the SM and adopting the linear ansatz in Eq. (70) for the SD form factors F ± (x γ ), as suggested by the lattice results of Ref. [8]. The quality of the fit is poor because, as mentioned above, the KLOE and E787 data cannot be reproduced simultaneously in terms of the same form factor F + (x γ ). We find a particularly significant discrepancy between our predictions and the experimental data for the form factor F − (x γ ).
These conclusions call for improvements in the determination of the structure-dependent form factors F + (x γ ) and F − (x γ ) from both experiment and theory. In this respect, we look forward to the results from the analysis of the NA62 experiment on the K e2γ decay, which is in progress and which is expected to provide the most precise determination of |F + (x γ )| [25]. If the results from NA62 confirm that there is a discrepancy between the form factors obtained from decays into electrons and those obtained from decays into muons from the E787 experiment, this would provide a motivation for better determinations also of the form factors from K → µν µ γ decays.
On the theoretical side is should be noted that the values of F V,A in Ref. [8] are the first lattice results of these quantities, so it can be expected that the precision will be improved in the next generation of computations.
We end by repeating that it is also conceivable that the tensions observed above between the experimental data and our lattice predictions are due to the presence of new physics, such as flavor changing interactions beyond the V − A couplings of the Standard Model and/or non-universal corrections to the lepton couplings. This possibility deserves further theoretical investigations.

ACKNOWLEDGMENTS
The authors warmly thanks D. Giusti for a careful check of the implementation of the kinematical cuts. We gratefully acknowledge discussions with members of the experimental collaborations about their data and results and we thank in particular B. Sciascia and T. Spadaro from KLOE [9], M.R. Convery from E787 [10] and M. Bychkov from PIBETA [13]. We also thank V. Kravtsov, V. Duk and V. Obraztsov for supplying us with the kinematical cuts of the OKA experiment [12] and A. Romano for discussions about the status of the ongoing analysis by the NA62 experiment. We acknowledge PRACE for awarding us access to Marconi at CINECA, Italy under the grant Pra17-4394 and CINECA for the provision of CPU time under the specific initiative INFN-