K0 decays: l3γ branching ratios and T-odd momenta correlations 1 A.S.Rudenko Budker Institute of Nuclear Physics and Novosibirsk State University, 630090, Novosibirsk, Russia Abstract The branching ratios of the K0 π−l+ν γ (l = e,µ) decays, and the T-odd l → triple momenta correlations ξ = ~q [p~ ~p ]/M3, due to the electromagnetic final · l × π K stateinteraction, intheseprocessesarecalculated. Thecontributions ontheorderof 3 ω−1 and ω0 to the corresponding amplitudes are treated exactly. For the branching 1 0 ratios and T-odd correlation in K0 π−e+νeγ decay, the corrections on the order → 2 of ω are estimated and demonstrated to be small. The results for the branching n ratios are in good agreement with the previous ones. The T-odd triple momenta a correlations in K0 decays are calculated for the first time. The values of the ξ- J l3γ odd asymmetry constitute 1 10−4 and 4.5 10−4 in the K0 π−µ+ν γ and 7 µ − × − × → 1 K0 π−e+ν γ decays, respectively. e → ] h p 1. The K0 π−l+ν γ (l = e,µ) decays were previously studied theoretically in l - → p Refs. [1, 2, 3, 4]. Therein the branching ratios of these decays were calculated. As for the e T-odd triple momenta correlations ξ = ~q [p~ p~ ]/M3, as induced by the electromagnetic h · l× π K [ final state interaction, they were considered only in the K+ π0l+νlγ decays [5, 6, 7]; → here and below M is the kaon mass and ~q, p~ , p~ are the momenta of γ, l, π, respectively. 3 K l π v In principle, these triple correlations can be used to probe new CP-odd effects beyond 9 the Standard Model, which could also contribute to them. 5 4 In the theoretical analysis of radiative effects in the discussed processes, the treatment 5 of the accompanying radiation, which gives the effects on the order of ω−1 and ω0 (the last . 4 ones originate from the radiation due to the lepton magnetic moment), is straightforward 0 1 (here and below ω is the photon energy). As to the structure radiation contribution on 1 the order of ω0, it is also under control, in fact due to the gauge invariance [8]. The : v contributions on the order of ω (and higher) depend directly on the photon field strength i X F (and its derivatives) and cannot be fixed in a model-independent way. We assume µν r that the corrections on the order of ω and higher are relatively small. Indeed, more a quantitative arguments presented below demonstrate that such contributions into the discussed branching ratios and into the T-odd momenta correlation in K0 decay do not e3γ exceed 20%. 2. At the tree level, the K0 π−l+ν γ decays are described by the Feynman graphs l → in Fig. 1. The matrix elements for diagrams 1a and 1b look as follows: G p e∗ qˆeˆ∗ M = sinθ e[f (t)(p + p ) + f (t)(p p ) ]u¯ γα(1 + γ ) l + v , 1a c + K π α − K π α ν 5 l √2 − p q 2p q (cid:18) l l (cid:19) (1) [email protected] 1 a b c Figure 1: The tree diagrams of K0 π−l+ν γ decays l → G p e∗ M = sinθ e[f (t′)(p +p +q) +f (t′)(p p q) ]u¯ γα(1+γ )v π ; (2) 1b c + K π α − K π α ν 5 l −√2 − − p q π here G is the Fermi coupling constant, θ is the Cabibbo angle, e is the elementary charge c (e > 0), t = (p p )2, t′ = (p p q)2; here and below the lower indices attached to K π K π − − − the matrix elements match the corresponding Feynman diagrams in the figures. Usually the dependence of the form factors f and f on the momentum transfer t is + − described by formula t f (t) = f (0) 1+λ . (3) ± ± ± m2 (cid:18) π(cid:19) The experimental data are adequately described by Eq. (3) with λ 0.03 for l = µ and + ≈ l = e; λ = 0 for l = µ [9]; λ for l = e is unknown, but one may assume that it is also − − close to zero. In the K0 decays the ratio λ t/m2 is small, λ t/m2 . 0.1, so one can put f (t) = l3γ ± π ± π ± f (0). Since the ratio ξ(0) = f (0)/f (0) . 0.1 is also small [10], one can neglect f (0) ± − + − with the same accuracy. Thus, our expressions (1) and (2) simplify to G p e∗ qˆeˆ∗ M = sinθ ef (0)(p +p ) u¯ γα(1+γ ) l + v , (4) 1a c + K π α ν 5 l √2 p q 2p q (cid:18) l l (cid:19) G p e∗ M = sinθ ef (0)(p +p +q) u¯ γα(1+γ )v π . (5) 1b c + K π α ν 5 l −√2 p q π However, the sum of diagrams 1a and 1b is not gauge invariant: it does not vanish under the substitution e∗ q. To restore the gauge invariance, one should add the third → diagram where a photon is directly emitted from the vertex (see Fig. 1c). This contact amplitude has no single-particle intermediate states and therefore is on the order of ω0 and higher. The contribution ω0, as derived with the Low technique [8], is ∼ G M = sinθ ef (0)e∗u¯ γα(1+γ )v . (6) 1c √2 c + α ν 5 l Thus, the model-independent gauge invariant tree amplitude of K0 decays, including l3γ only terms on the order of ω−1 and ω0 (but all of them!), is G p e∗ p e∗ M = M +M +M = sinθ ef (0) (p +p ) u¯ γα(1+γ )v l π tree 1a 1b 1c c + K π α ν 5 l √2 p q − p q (cid:26) (cid:18) l π (cid:19) 2 qˆeˆ∗ p e∗ +(p +p ) u¯ γα(1+γ ) v + e∗ π q u¯ γα(1+γ )v . (7) K π α ν 5 2p q l α − p q α ν 5 l l (cid:18) π (cid:19) (cid:27) This expression agrees with the corresponding formulas in Ref. [3] (if our f (0) is set to + its SU(3) value f (0) = 1). + It is convenient to present amplitude (7) as a sum of gauge-invariant contributions. They arethe “infrared” term M corresponding to the sum ofthe amplitudes of accompa- IR nying radiation by the pion and lepton (independent of the lepton magnetic moment), the magnetic term M which is the amplitude of spin-dependent accompanying radiation mag of the lepton magnetic moment, and the Low term M : Low G p e∗ p e∗ M = sinθ ef (0)(p +p ) u¯ γα(1+γ )v l π , (8) IR c + K π α ν 5 l √2 p q − p q (cid:18) l π (cid:19) G qˆeˆ∗ M = sinθ ef (0)(p +p ) u¯ γα(1+γ ) v , (9) mag c + K π α ν 5 l √2 2p q l G p e∗ M = sinθ ef (0) e∗ π q u¯ γα(1+γ )v . (10) Low √2 c + α − p q α ν 5 l (cid:18) π (cid:19) In the lowest order in G the decay width Γ(K0 π−l+ν γ) is equal to the sum l → Γ(K0 π−l+ν γ)+Γ(K0 π+l−ν¯γ), which is measured experimentally (see also [4]). L → l L → l The results of calculation and experimental values for K0 π±l∓ν γ branching ratios are L → l presented in Table 1; here the following cuts in the kaon rest frame are used: ω > 30 MeV and θ > 20◦ (except the experimental results for l = µ, where only restriction ω > 30 lγ MeV is imposed). We use in calculation the particle masses m = 497.6 MeV, m = 139.57 MeV, K π m = 105.658MeV, andm = 0.511MeV,theFermicoupling constant G = 1.1664 10−11 µ e × MeV−2, the fine-structure constant e2/4π = α = 1/137, the Planck constant ~ = 6.582 × 10−22 MeV s, K0 mean life τ = 5.1 10−8 s, sinθ f (0) = 0.217. · L × c + l = µ l = e Bijnens et al. [3] 5.2 10−4 3.6 10−3 × × present work 5.00 10−4 3.45 10−3 × × experimental values [9] (5.65 0.23) 10−4 (3.79 0.06) 10−3 ± × ± × ω > 30 MeV Table 1: Branching ratios of K0 π±l∓ν γ decays L → l The accuracy of our results can be estimated as follows. The leading corrections to them are due to the structure radiation from the hadronic vertex. They are proportional to the photon field strength, i.e. are on the order of ω. There are good reasons to believe thatthese corrections areless thanthe Lowstructure amplitudes which areontheorder of ω0. The Low contributions (including of course their interference with the accompanying radiation) to the discussed K0 π±µ∓ν γ and K0 π±e∓ν γ branching ratios, accord- L → µ L → e ing to our calculations, constitute 0.79 10−4 and 0.27 10−3, respectively. Let us note × × 3 also that corrections to the quoted results derived in Ref. [3] in the chiral perturbation theory are of similar magnitude. As mentioned, additional corrections on the level of 20% to the branching ratios orig- inate from our neglect of the form factor f (t) and of the t-dependence of f (t) (see also − + Table I in Ref. [2]). As to the relative accuracy of our numerical integration over phase space of final particles, it is about 10−3. To compare properly our results with those of Ref. [3] one should keep in mind that now the experimental values of some quantities are known with better accuracy. Indeed, we use sinθ f (0) = 0.217 in our calculation and, as far as we can see, in Ref. [3] the c + corresponding value is 0.22. Substitution of one of these values for another alters the results by about 3%. Thus, our results for the branching ratios agree reasonably well with those of Ref. [3]. 3. The T-odd triple momenta correlations ξ = ~q [p~ ~p ]/M3 in the K0 π−l+ν γ · l × π K → l decays arise from the interference term 2Re(M† A ) in the decay rate; here M is tree loop tree the tree amplitude and A is the anti-Hermitian part of the loop diagrams presented loop below. As we are interested in the effect due to the electromagnetic final state interaction we consider only the one-loop diagrams generated from the tree ones by attaching the virtual photon. The on-mass-shell intermediate particles on the diagrams are marked by crosses. The anti-Hermitian part of the sum of one-loop diagrams is written as i A = M M∗ , (11) loop 8π2 fn in n X where the sum over n includes the summation over the polarizations and the integration over the phase space of intermediate particles. It is natural to divide the loop diagrams into four groups according to the type of amplitudes M and M∗ . fn in In the first group (see Fig. 2) the amplitude M depicts the Compton scattering off fn the intermediate lepton (see Fig. 3): Pˆ m pˆ kˆ m M = M +M = e2v¯ eˆ − leˆ∗v +e2v¯ eˆ∗ l − − leˆ v . (12) fn 3a 3b kl k 2p q l kl 2p k k l l l − As to M∗ , it is the same tree amplitude (7), up to the change of some notations: in G k e∗ p e∗ M∗ = M = M = sinθ ef (0) (p +p ) u¯ γα(1+γ )v l k π k in ni tree|pql→→kkl,, √2 c + (cid:26) K π α ν 5 kl (cid:18) plq − pπk (cid:19) e→ek kˆeˆ∗ p e∗ +(p +p ) u¯ γα(1+γ ) k v + e∗ π kk u¯ γα(1+γ )v . (13) K π α ν 5 2p q kl kα − p k α ν 5 kl l (cid:18) π (cid:19) ) The element of phase space is d3k d3k dρ = lδ(4)(k +k P), (14) l 2ω 2ω − k l 4 Figure 2: The one-loop diagrams (group I – the Compton scattering off the lepton) a b Figure 3: The Compton scattering off the lepton where P = p +q. l In the second group of one-loop diagrams (see Fig. 4) the amplitude M corresponds fn to the Compton scattering off the π-meson (see Fig. 5): (p e∗)(k e ) (p e )(k e∗) M = M +M +M = 2e2 π π k + π k π +(e e∗) . (15) fn 5a 5b 5c k − p q p k (cid:26) π π (cid:27) The amplitude M∗ is the tree amplitude (7) up to the change of notations: in G p e∗ k e∗ M∗ = M = M = sinθ ef (0) (p +k ) u¯ γα(1+γ )v l k π k in ni tree|pπq→→kk,π, √2 c + (cid:26) K π α ν 5 l(cid:18) plk − pπq (cid:19) e→ek kˆeˆ∗ k e∗ +(p +k ) u¯ γα(1+γ ) k v + e∗ π kk u¯ γα(1+γ )v . (16) K π α ν 5 2p k l kα − p q α ν 5 l l (cid:18) π (cid:19) ) The element of phase space is d3k d3k dρ′ = πδ(4)(k +k P′), (17) π 2ω 2ω − k π where P′ = p +q. π 5 Figure 4: The one-loop diagrams (group II – the Compton scattering off the π-meson) Inthethirdgroupofone-loopdiagrams(seeFig.6)theamplitudeM isπ-l scattering fn amplitude (see Fig. 7): 1 M = M = e2 F (k2) (p +k ) v¯ γαv . (18) fn 7 | π |k2 π π α kl l The amplitude M∗ is also the tree amplitude (7) up to the change of notations: in G k e∗ k e∗ M∗ = M = M = sinθ ef (0) (p +k ) u¯ γα(1+γ )v l π in ni tree|ppπl→→kklπ, √2 c + (cid:26) K π α ν 5 kl(cid:18) klq − kπq (cid:19) qˆeˆ∗ k e∗ +(p +k ) u¯ γα(1+γ ) v + e∗ π q u¯ γα(1+γ )v . (19) K π α ν 5 2k q kl α − k q α ν 5 kl l (cid:18) π (cid:19) (cid:27) The element of phase space is d3k d3k dρ′′ = l πδ(4)(k +k P′′), (20) l π 2ω 2ω − l π where P′′ = p +p . l π And finally, in the fourth group of one-loop diagrams (see Fig. 8) the amplitude M fn is π-l scattering with emission of a photon (see Fig. 9): 6 a b c Figure 5: The Compton scattering off the π-meson Figure 6: The one-loop diagrams (group III – π-l scattering) 1 p e∗ qˆeˆ∗ M = M +M +M +M +M = e3 F (k2) (p +k ) v¯ γα l + v fn 9a 9b 9c 9d 9e | π |k2 π π α kl p q 2p q l (cid:26) (cid:18) l l (cid:19) k e∗ qˆeˆ∗ p e∗ (p +k ) v¯ l + γαv (p +k +q) v¯ γαv π − π π α kl k q 2k q l − π π α kl l p q (cid:18) l l (cid:19) π k e∗ +(p +k q) v¯ γαv π +2e∗v¯ γαv . (21) π π − α kl l k q α kl l π (cid:27) As to M∗ , it is the amplitude of K0 decays (see Fig. 10): in l3 G M∗ = M = M = sinθ ef (0)(p +k ) u¯ γα(1+γ )v . (22) in ni 10 √2 c + K π α ν 5 kl The element of phase space is d3k d3k dρ′′′ = l πδ(4)(k +k P′′′), (23) l π 2ω 2ω − l π where P′′′ = p +p +q. l π In the discussed decays k2 < 0 and √ k2 < m (1 m2/m2 ) . 450 MeV. At these − K − π K energies the pion electromagnetic form factor can be approximated as F (k2) 1+ < r2 > k2/6, (24) | π | ≈ π here < r2 > is the pion mean square charge radius. According to our calculations (we π use the PDG value < r2 >= (0.67fm)2 [9]), the contribution of term < r2 > k2/6 to π π the momenta correlation is small, namely, it does not exceed 5% of the contribution of 1. Therefore, we can put F (k2) = 1. Moreover, keeping < r2 > term itself is beyond the | π | π accuracy of our calculation, because we have already neglected the form factor f (t) and − the t-dependence of f (t). + 7 Figure 7: The π-l scattering diagram Figure 8: The one-loop diagrams (group IV – π-l scattering with emission of a photon) The one-loop diagrams from group III contain infrared divergent terms, which can be omitted because infrared corrections to the decay width are factorized and, therefore, do not contribute to the effect. The details of the calculation of the T-odd correlation ξ = ~q [p~ p~ ]/M3 can be · l × π K found in Appendices. In fact, what is really measured experimentally is not the T-odd triple momenta correlation ξ by itself but the asymmetry N N ( M 2 + M 2 )dΦ ( M 2 + M 2 )dΦ A = + − − = | |even | |odd ξ>0 − | |even | |odd ξ<0 ξ N +N ( M 2 + M 2 )dΦ + ( M 2 + M 2 )dΦ + − R | |even | |odd ξ>0 R | |even | |odd ξ<0 (25) M 2 dΦ M 2 dΦ = | |odd ξ>R0 | |odd ξ>0 R M 2 dΦ ≈ M 2dΦ R| |even ξ>0 R| tree| ξ>0 induced byRthis correlation; hRere N and N are the numbers of events with ξ > 0 and + − ξ < 0, M 2 and M 2 are the ξ-even and ξ-odd terms in the amplitude squared, and | |even | |odd integration is performed over the phase space of the final particles. The results for the asymmetry A in K0 π−l+ν γ decays are presented in Table 2 ξ l → (here K0 means both K0 and K0, and the following cuts in the kaon rest frame are used: L S ω > 30 MeV and θ > 20◦). lγ The contributions of the Low term M to the asymmetry A in K0 π−µ+ν γ Low ξ µ → and K0 π−e+ν γ decays, according to our calculations, constitute 0.9 10−4 and e → × 0.9 10−4,respectively. Therefore, thecontributionofM forK0 decayisabout20%. − × Low e3γ 8 a b c d e Figure 9: π-l scattering with emission of a photon Figure 10: K0 decays l3 For K0 decay, however, it is large, comparable to the contributions of the accompanying µ3γ radiation which are on the order of ω−1 and ω0. So, it is difficult to estimate reliably the relative magnitude of the structure radiation contribution proportional to ω, i.e. to estimate reliably the true accuracy of thus-derived result for A in K0 π−µ+ν γ decay. ξ µ → However, we note here that corrections to the discussed asymmetry in the K+ π0l+ν γ l → decays, derived in Ref. [6] within the chiral perturbation theory, are small, on the order of 1%. The asymmetry A was measured only in the K+ decays [11, 12]. The experimental ξ l3γ errors are on the level of 10−1 for l = µ and 10−2 for l = e. Therefore, at present it is impossible to test the discussed theoretical predictions that are on the order of 10−4 [5, 7]. In any case experimental study of the T-odd asymmetry A in the K decays would ξ l3γ be very important. It would allow one either to test CP-odd effects beyond the Standard Model or to set limits on them. Acknowledgements I am grateful to I.B.Khriplovich for his interest in the work and useful discussions as well as for critical reading of the text, to L.V.Kardapoltsev for his advice concerning numerical calculations, and to R.N.Lee and V.S.Fadin for pointing out the necessity to check the initial results. 9 l = µ l = e group I (l γ) 0.54 10−4 1.32 10−4 − − × − × group II (π γ) 3.6 10−4 3.2 10−4 − − × − × group III (π l) 1.73 10−3 8.6 10−4 − × × group IV (π l γ) 1.41 10−3 8.6 10−4 − − − × − × total 1 10−4 4.5 10−4 − × − × Table 2: A in K0 π−l+ν γ decays ξ l → (ω > 30 MeV and θ > 20◦) lγ TheworkwassupportedinpartbytheRussianFoundationforBasicResearchthrough Grant No.11-02-00792-а, by the Federal Program “Personnel of Innovational Russia” through Grant No.14.740.11.0082, and by Dmitry Zimin’s Dynasty Foundation. Appendix A In this Appendix the list of the integrals that contribute to A is given. loop In formulas below we use the following notations: I = (p p )2 m2m2, lπ l π − l π q I = (p P)2 m2P2, πP π − π I′ = p(p P′)2 m2P′2, lP l − l q I = (p p +p q+p q)2 m2m2. lπq l π l π − l π q We start with the integrals that contribute to group I of the one-loop diagrams: π(p q) l dρ = a = ; (A.1) 0 P2 Z p q l k dρ = a P , where a = a ; (A.2) µ P µ P P2 0 Z dρ π P2 = b = ln ; (A.3) p k 0 2p q m2 Z l l (cid:18) l (cid:19) k µ dρ = B = b p +b P , (A.4) 1µ l lµ P µ p k l Z where b and b are the solutions of the set of equations l P b m2 +b (p P) = a , l l P l 0 b (p P)+b P2 = b (p q); l l P 0 l 10