Abstract
The amplitudes for the rare decay modes K± → π±ℓ+ℓ− and KS → π0ℓ+ℓ− are studied with the aim of obtaining predictions for them, such as to enable the possibility to search for violations of lepton-flavour universality in the kaon sector. The issue is first addressed from the perspective of the low-energy expansion, and a two-loop representation of the corresponding form factors is constructed, leaving as unknown quantities their values and slopes at vanishing momentum transfer. In a second step a phenomenological determination of the latter is proposed. It consists of the contribution of the resonant two-pion state in the P wave, and of the leading short-distance contribution determined by the operator-product expansion. The interpolation between the two energy regimes is described by an infinite tower of zero-width resonances matching the QCD short-distance behaviour. Finally, perspectives for future improvements in the theoretical understanding of these amplitudes are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Cirigliano, G. Ecker, H. Neufeld, A. Pich and J. Portolés, Kaon Decays in the Standard Model, Rev. Mod. Phys. 84 (2012) 399 [arXiv:1107.6001] [INSPIRE].
A. Ceccucci, Review and Outlook on Kaon Physics, Acta Phys. Polon. B 49 (2018) 1079 [INSPIRE].
A. Ceccucci on behalf of the NA62 collaboration, In-flight Search for \( {K}^{+}\to {\pi}^{+}\nu \overline{\nu} \) : First NA62 Results, PoS(ALPS2018)006.
T.K. Komatsubara, Experiments with K-Meson Decays, Prog. Part. Nucl. Phys. 67 (2012) 995 [arXiv:1203.6437] [INSPIRE].
KOTO collaboration, \( {K}_L^0\to {\pi}^0\nu \overline{\nu} \) at KOTO, in 8th International Workshop on the CKM Unitarity Triangle (CKM 2014) Vienna, Austria, September 8–12, 2014, arXiv:1411.4250 [INSPIRE].
KOTO collaboration, Search for the \( {K}_L^0\to {\pi}^0\nu \overline{\nu} \) and K L → π 0 X 0 decays at the J-PARC KOTO experiment, Phys. Rev. Lett. 122 (2019) 021802 [arXiv:1810.09655] [INSPIRE].
LHCb collaboration, Test of lepton universality with B 0 → K *0 ℓ + ℓ − decays, JHEP 08 (2017) 055 [arXiv:1705.05802] [INSPIRE].
LHCb collaboration, Test of lepton universality using B + → K + ℓ + ℓ − decays, Phys. Rev. Lett. 113 (2014) 151601 [arXiv:1406.6482] [INSPIRE].
C. Bobeth, G. Hiller and G. Piranishvili, Angular distributions of \( \overline{B}\to \overline{K}{\ell}^{+}{\ell}^{-} \) decays, JHEP 12 (2007) 040 [arXiv:0709.4174] [INSPIRE].
M. Bordone, G. Isidori and A. Pattori, On the Standard Model predictions for R K and \( {R}_{K^{*}} \), Eur. Phys. J. C 76 (2016) 440 [arXiv:1605.07633] [INSPIRE].
S. Descotes-Genon, T. Hurth, J. Matias and J. Virto, Optimizing the basis of B → K * ℓ + ℓ − observables in the full kinematic range, JHEP 05 (2013) 137 [arXiv:1303.5794] [INSPIRE].
S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, On the impact of power corrections in the prediction of B → K * μ + μ − observables, JHEP 12 (2014) 125 [arXiv:1407.8526] [INSPIRE].
S. Descotes-Genon, J. Matias and J. Virto, Understanding the B → K * μ + μ − Anomaly, Phys. Rev. D 88 (2013) 074002 [arXiv:1307.5683] [INSPIRE].
W. Altmannshofer and D.M. Straub, New physics in b → s transitions after LHC run 1, Eur. Phys. J. C 75 (2015) 382 [arXiv:1411.3161] [INSPIRE].
S. Jäger and J. Martin Camalich, Reassessing the discovery potential of the B → K * ℓ + ℓ − decays in the large-recoil region: SM challenges and BSM opportunities, Phys. Rev. D 93 (2016) 014028 [arXiv:1412.3183] [INSPIRE].
D. Aristizabal Sierra and A. Vicente, Explaining the CMS Higgs flavor violating decay excess, Phys. Rev. D 90 (2014) 115004 [arXiv:1409.7690] [INSPIRE].
J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, Higgs → μτ in Abelian and non-Abelian flavor symmetry models, Nucl. Phys. B 896 (2015) 281 [arXiv:1412.3671] [INSPIRE].
A. Crivellin, G. D’Ambrosio and J. Heeck, Explaining h → μ ± τ ∓ , B → K * μ + μ − and B → Kμ + μ − /B → Ke + e − in a two-Higgs-doublet model with gauged L μ − L τ, Phys. Rev. Lett. 114 (2015) 151801 [arXiv:1501.00993] [INSPIRE].
I. Doršner, S. Fajfer, A. Greljo, J.F. Kamenik, N. Košnik and I. Nišandžic, New Physics Models Facing Lepton Flavor Violating Higgs Decays at the Percent Level, JHEP 06 (2015) 108 [arXiv:1502.07784] [INSPIRE].
Y. Omura, E. Senaha and K. Tobe, Lepton-flavor-violating Higgs decay h → μτ and muon anomalous magnetic moment in a general two Higgs doublet model, JHEP 05 (2015) 028 [arXiv:1502.07824] [INSPIRE].
I. de Medeiros Varzielas, O. Fischer and V. Maurer, \( \mathbb{A} \) 4 symmetry at colliders and in the universe, JHEP 08 (2015) 080 [arXiv:1504.03955] [INSPIRE].
A.J. Buras, F. De Fazio and J. Girrbach, 331 models facing new b → sμ + μ − data, JHEP 02 (2014) 112 [arXiv:1311.6729] [INSPIRE].
A. Crivellin, G. D’Ambrosio and J. Heeck, Addressing the LHC flavor anomalies with horizontal gauge symmetries, Phys. Rev. D 91 (2015) 075006 [arXiv:1503.03477] [INSPIRE].
S.L. Glashow, D. Guadagnoli and K. Lane, Lepton Flavor Violation in B Decays?, Phys. Rev. Lett. 114 (2015) 091801 [arXiv:1411.0565] [INSPIRE].
W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Quark flavor transitions in L μ − L τ models, Phys. Rev. D 89 (2014) 095033 [arXiv:1403.1269] [INSPIRE].
R. Gauld, F. Goertz and U. Haisch, An explicit Z’-boson explanation of the B → K * μ + μ − anomaly, JHEP 01 (2014) 069 [arXiv:1310.1082] [INSPIRE].
A.J. Buras and J. Girrbach, Left-handed Z′ and Z FCNC quark couplings facing new b → sμ + μ − data, JHEP 12 (2013) 009 [arXiv:1309.2466] [INSPIRE].
R. Gauld, F. Goertz and U. Haisch, On minimal Z′ explanations of the B → K * μ + μ − anomaly, Phys. Rev. D 89 (2014) 015005 [arXiv:1308.1959] [INSPIRE].
C. Niehoff, P. Stangl and D.M. Straub, Violation of lepton flavour universality in composite Higgs models, Phys. Lett. B 747 (2015) 182 [arXiv:1503.03865] [INSPIRE].
D. Aristizabal Sierra, F. Staub and A. Vicente, Shedding light on the b → s anomalies with a dark sector, Phys. Rev. D 92 (2015) 015001 [arXiv:1503.06077] [INSPIRE].
A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski and J. Rosiek, Lepton-flavour violating B decays in generic Z′ models, Phys. Rev. D 92 (2015) 054013 [arXiv:1504.07928] [INSPIRE].
A. Celis, J. Fuentes-Martin, M. Jung and H. Serodio, Family nonuniversal Z’ models with protected flavor-changing interactions, Phys. Rev. D 92 (2015) 015007 [arXiv:1505.03079] [INSPIRE].
A. Carmona and F. Goertz, Lepton Flavor and Nonuniversality from Minimal Composite Higgs Setups, Phys. Rev. Lett. 116 (2016) 251801 [arXiv:1510.07658] [INSPIRE].
Y. Sakaki, M. Tanaka, A. Tayduganov and R. Watanabe, Testing leptoquark models in \( \overline{B}\to {D}^{\left(\ast \right)}\tau \overline{\nu} \), Phys. Rev. D 88 (2013) 094012 [arXiv:1309.0301] [INSPIRE].
I. de Medeiros Varzielas and G. Hiller, Clues for flavor from rare lepton and quark decays, JHEP 06 (2015) 072 [arXiv:1503.01084] [INSPIRE].
D. Bečirević, S. Fajfer and N. Košnik, Lepton flavor nonuniversality in b → sℓ + ℓ − processes, Phys. Rev. D 92 (2015) 014016 [arXiv:1503.09024] [INSPIRE].
B. Gripaios, M. Nardecchia and S.A. Renner, Composite leptoquarks and anomalies in B-meson decays, JHEP 05 (2015) 006 [arXiv:1412.1791] [INSPIRE].
L. Calibbi, A. Crivellin and T. Ota, Effective Field Theory Approach to b → sℓℓ (′) , \( B\to {K}^{\left(*\right)}\nu \overline{\nu} \) and B → D (*) τν with Third Generation Couplings, Phys. Rev. Lett. 115 (2015) 181801 [arXiv:1506.02661] [INSPIRE].
R. Alonso, B. Grinstein and J. Martin Camalich, Lepton universality violation and lepton flavor conservation in B-meson decays, JHEP 10 (2015) 184 [arXiv:1505.05164] [INSPIRE].
M. Bauer and M. Neubert, Minimal Leptoquark Explanation for the \( {R}_{D^{\left(\ast \right)}} \) , R K and (g − 2)μ Anomalies, Phys. Rev. Lett. 116 (2016) 141802 [arXiv:1511.01900] [INSPIRE].
R. Barbieri, G. Isidori, A. Pattori and F. Senia, Anomalies in B-decays and U(2) flavour symmetry, Eur. Phys. J. C 76 (2016) 67 [arXiv:1512.01560] [INSPIRE].
S. Fajfer and N. Košnik, Vector leptoquark resolution of R K and \( {R}_{D^{\left(\ast \right)}} \) puzzles, Phys. Lett. B 755 (2016) 270 [arXiv:1511.06024] [INSPIRE].
A. Greljo, G. Isidori and D. Marzocca, On the breaking of Lepton Flavor Universality in B decays, JHEP 07 (2015) 142 [arXiv:1506.01705] [INSPIRE].
A. Crivellin, G. D’Ambrosio, M. Hoferichter and L.C. Tunstall, Violation of lepton flavor and lepton flavor universality in rare kaon decays, Phys. Rev. D 93 (2016) 074038 [arXiv:1601.00970] [INSPIRE].
E. Goudzovski, New results and prospects in kaon physics from the NA62 experiment, talk at the XIIIth Quark Confinement and Hadron Spectrum Conference, University of Maynooth, Ireland, 31 July — 6 August 2018, https://indico.cern.ch/event/648004/contributions/2987967/.
LHCb collaboration, Physics case for an LHCb Upgrade II — Opportunities in flavour physics and beyond, in the HL-LHC era, arXiv:1808.08865 [INSPIRE].
A.A. Alves Junior et al., Prospects for Measurements with Strange Hadrons at LHCb, arXiv:1808.03477 [INSPIRE].
G. Isidori, G. Martinelli and P. Turchetti, Rare kaon decays on the lattice, Phys. Lett. B 633 (2006) 75 [hep-lat/0506026] [INSPIRE].
RBC and UKQCD collaborations, Prospects for a lattice computation of rare kaon decay amplitudes: K → πℓ + ℓ − decays, Phys. Rev. D 92 (2015) 094512 [arXiv:1507.03094] [INSPIRE].
N.H. Christ, X. Feng, A. Juttner, A. Lawson, A. Portelli and C.T. Sachrajda, First exploratory calculation of the long-distance contributions to the rare kaon decays K → πℓ + ℓ −, Phys. Rev. D 94 (2016) 114516 [arXiv:1608.07585] [INSPIRE].
C. Alliegro et al., Study of the decay K + → π + e + e −, Phys. Rev. Lett. 68 (1992) 278 [INSPIRE].
E787 collaboration, Observation of the decay K + → π + μ + μ −, Phys. Rev. Lett. 79 (1997) 4756 [hep-ex/9708012] [INSPIRE].
E865 collaboration, A new measurement of the rare decay K + → π + μ + μ −, Phys. Rev. Lett. 84 (2000) 2580 [hep-ex/9910047] [INSPIRE].
HyperCP collaboration, Observation of the decay K − → π − μ + μ − and measurements of the branching ratios for K ± → π ± μ + μ −, Phys. Rev. Lett. 88 (2002) 111801 [hep-ex/0110033] [INSPIRE].
E865 collaboration, A new measurement of the properties of the rare decay K + → π + e + e −, Phys. Rev. Lett. 83 (1999) 4482 [hep-ex/9907045] [INSPIRE].
NA48/2 collaboration, Precise measurement of the K ± → π ± e + e − decay, Phys. Lett. B 677 (2009) 246 [arXiv:0903.3130] [INSPIRE].
NA48/2 collaboration, New measurement of the K ± → π ± μ + μ − decay, Phys. Lett. B 697 (2011) 107 [arXiv:1011.4817] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
NA48/1 collaboration, Observation of the rare decay K S → π 0 e + e −, Phys. Lett. B 576 (2003) 43 [hep-ex/0309075] [INSPIRE].
NA48/1 collaboration, Observation of the rare decay K S → π 0 μ + μ −, Phys. Lett. B 599 (2004) 197 [hep-ex/0409011] [INSPIRE].
G. Ecker, A. Pich and E. de Rafael, K → πℓ + ℓ − Decays in the Effective Chiral Lagrangian of the Standard Model, Nucl. Phys. B 291 (1987) 692 [INSPIRE].
G. Ecker, A. Pich and E. de Rafael, Radiative Kaon Decays and CP-violation in Chiral Perturbation Theory, Nucl. Phys. B 303 (1988) 665 [INSPIRE].
G. D’Ambrosio, G. Ecker, G. Isidori and J. Portolés, The decays K → πℓ + ℓ − beyond leading order in the chiral expansion, JHEP 08 (1998) 004 [hep-ph/9808289] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Gasser and H. Leutwyler, On the Low-energy Structure of QCD, Phys. Lett. 125B (1983) 321 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
S. Friot, D. Greynat and E. De Rafael, Rare kaon decays revisited, Phys. Lett. B 595 (2004) 301 [hep-ph/0404136] [INSPIRE].
E. Coluccio Leskow, G. D’Ambrosio, D. Greynat and A. Nath, K → πℓ + ℓ − form factor in the large-N c and cut-off regularization method, Phys. Rev. D 93 (2016) 094031 [arXiv:1603.09721] [INSPIRE].
A.Z. Dubničkovà, S. Dubnicka, E. Goudzovski, V.N. Pervushin and M. Secansky, Kaon decay probe of the weak static interaction, Phys. Part. Nucl. Lett. 5 (2008) 76 [hep-ph/0611175] [INSPIRE].
A.J. Buras, The 1/n Approach To Nonleptonic Weak Interactions, in CP Violation, C. Jarlskog ed., Adv. Ser. Direct. High Energy Phys. 3 (1989) 575 [INSPIRE].
J. Portolés, K → πℓ + ℓ − : Status and update, J. Phys. Conf. Ser. 800 (2017) 012030 [arXiv:1611.07195] [INSPIRE].
B. Ananthanarayan and I. Sentitemsu Imsong, The 27-plet contributions to the CP-conserving K → πℓ + ℓ − decays, J. Phys. G 39 (2012) 095002 [arXiv:1207.0567] [INSPIRE].
M.K. Gaillard and B.W. Lee, ΔI = 1/2 Rule for Nonleptonic Decays in Asymptotically Free Field Theories, Phys. Rev. Lett. 33 (1974) 108 [INSPIRE].
G. Altarelli and L. Maiani, Octet Enhancement of Nonleptonic Weak Interactions in Asymptotically Free Gauge Theories, Phys. Lett. 52B (1974) 351 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Light Quarks and the Origin of the ΔI = 1/2 Rule in the Nonleptonic Decays of Strange Particles, Nucl. Phys. B 120 (1977) 316 [INSPIRE].
E. Witten, Short Distance Analysis of Weak Interactions, Nucl. Phys. B 122 (1977) 109 [INSPIRE].
M.B. Wise and E. Witten, A Diagrammatic Analysis of Some Contributions to the ΔI = 1/2 Rule, Phys. Rev. D 20 (1979) 1216 [INSPIRE].
F.J. Gilman and M.B. Wise, Effective Hamiltonian for ΔS = 1 Weak Nonleptonic Decays in the Six Quark Model, Phys. Rev. D 20 (1979) 2392 [INSPIRE].
G. Altarelli, G. Curci, G. Martinelli and S. Petrarca, QCD Nonleading Corrections to Weak Decays as an Application of Regularization by Dimensional Reduction, Nucl. Phys. B 187 (1981) 461 [INSPIRE].
A.J. Buras and P.H. Weisz, QCD Nonleading Corrections to Weak Decays in Dimensional Regularization and ’t Hooft-Veltman Schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Effective Hamiltonians for ΔS = 1 and ΔB = 1 nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B 370 (1992) 69 [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Two loop anomalous dimension matrix for ΔS = 1 weak nonleptonic decays I: \( \mathcal{O}\left({\alpha}_s^2\right) \), Nucl. Phys. B 400 (1993) 37 [hep-ph/9211304] [INSPIRE].
M. Ciuchini, E. Franco, G. Martinelli and L. Reina, The ΔS = 1 effective Hamiltonian including next-to-leading order QCD and QED corrections, Nucl. Phys. B 415 (1994) 403 [hep-ph/9304257] [INSPIRE].
C. Dib, I. Dunietz and F.J. Gilman, CP Violation in the K L → π 0 ℓ + ℓ − Decay Amplitude for Large M t, Phys. Lett. B 218 (1989) 487 [INSPIRE].
C. Dib, I. Dunietz and F.J. Gilman, K L → π 0 ℓ + ℓ − Decays for Large m t, Phys. Rev. D 39 (1989) 2639 [INSPIRE].
F.J. Gilman and M.B. Wise, K → πe + e − in the Six Quark Model, Phys. Rev. D 21 (1980) 3150 [INSPIRE].
J. Flynn and L. Randall, The CP Violating Contribution to the Decay K L → π 0 e + e −, Nucl. Phys. B 326 (1989) 31 [Erratum ibid. B 334 (1990) 580] [INSPIRE].
A.J. Buras, M.E. Lautenbacher, M. Misiak and M. Münz, Direct CP-violation in K L → π 0 e + e − beyond leading logarithms, Nucl. Phys. B 423 (1994) 349 [hep-ph/9402347] [INSPIRE].
M.S. Chanowitz, M. Furman and I. Hinchliffe, The Axial Current in Dimensional Regularization, Nucl. Phys. B 159 (1979) 225 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
P. Breitenlohner and D. Maison, Dimensional Renormalization and the Action Principle, Commun. Math. Phys. 52 (1977) 11 [INSPIRE].
G. Ecker, J. Kambor and D. Wyler, Resonances in the weak chiral Lagrangian, Nucl. Phys. B 394 (1993) 101 [INSPIRE].
G. Ecker, A. Pich and E. de Rafael, Vector Meson Exchange in Radiative Kaon Decays and Chiral Perturbation Theory, Phys. Lett. B 237 (1990) 481 [INSPIRE].
A. Pich and E. de Rafael, Four quark operators and nonleptonic weak transitions, Nucl. Phys. B 358 (1991) 311 [INSPIRE].
G. D’Ambrosio and J. Portolés, Spin 1 resonance contributions to the weak chiral Lagrangian: The vector field formulation, Nucl. Phys. B 533 (1998) 494 [hep-ph/9711211] [INSPIRE].
L. Cappiello, O. Catà and G. D’Ambrosio, A holographic approach to low-energy weak interactions of hadrons, Phys. Rev. D 85 (2012) 015003 [arXiv:1106.0467] [INSPIRE].
G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Chiral Lagrangians for Massive Spin 1 Fields, Phys. Lett. B 223 (1989) 425 [INSPIRE].
G. Ecker, J. Gasser, A. Pich and E. de Rafael, The Role of Resonances in Chiral Perturbation Theory, Nucl. Phys. B 321 (1989) 311 [INSPIRE].
L. Cappiello, O. Catà and G. D’Ambrosio, Closing in on the radiative weak chiral couplings, Eur. Phys. J. C 78 (2018) 265 [arXiv:1712.10270] [INSPIRE].
B.R. Holstein, Current algebra, Glashow’s model of CP nonconservation and K → 3π, Phys. Rev. 177 (1969) 2417 [INSPIRE].
L.-F. Li and L. Wolfenstein, Current Algebra Analysis of CP Violations in K → 3π Decay in the Six Quark Weinberg-Salam Model, Phys. Rev. D 21 (1980) 178 [INSPIRE].
J. Kambor, J.H. Missimer and D. Wyler, K → 2π and K → 3π decays in next-to-leading order chiral perturbation theory, Phys. Lett. B 261 (1991) 496 [INSPIRE].
J. Bijnens, P. Dhonte and F. Borg, K → 3π decays in chiral perturbation theory, Nucl. Phys. B 648 (2003) 317 [hep-ph/0205341] [INSPIRE].
G. D’Ambrosio, G. Isidori, A. Pugliese and N. Paver, Strong rescattering in K → 3π decays and low-energy meson dynamics, Phys. Rev. D 50 (1994) 5767 [Erratum ibid. D 51 (1995) 3975] [hep-ph/9403235] [INSPIRE].
L. Maiani and N. Paver, CP conserving nonleptonic K → 3π decays, in L. Maiani et al. eds., The second DAPHNE physics handbook, vol. 1, (1995), 239–264.
https://hepdata.net/record/ins815724, for K ± → π ± e + e −.
https://hepdata.net/record/ins878312, for K ± → π ± μ + μ −.
J. Gasser and U.G. Meissner, Chiral expansion of pion form-factors beyond one loop, Nucl. Phys. B 357 (1991) 90 [INSPIRE].
S. Descotes-Genon and M. Knecht, Two-loop representations of low-energy pion form factors and ππ scattering phases in the presence of isospin breaking, Eur. Phys. J. C 72 (2012) 1962 [arXiv:1202.5886] [INSPIRE].
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press (1966).
S. Descotes-Genon and B. Moussallam, Analyticity of ηπ isospin-violating form factors and the τ → ηπν second-class decay, Eur. Phys. J. C 74 (2014) 2946 [arXiv:1404.0251] [INSPIRE].
J.B. Bronzan and C. Kacser, Khuri-Treiman Representation and Perturbation Theory, Phys. Rev. 132 (1963) 2703 [INSPIRE].
C. Kacser, Analytic Structure of Partial-Wave Amplitudes for Production and Decay Processes, Phys. Rev. 132 (1963) 2712 [INSPIRE].
M. Zdráhal, Construction of pseudoscalar meson amplitudes in chiral perturbation theory using a dispersive approach, Ph.D. Thesis, Charles University, Prague (2011), https://is.cuni.cz/webapps/zzp/download/140012164.
K. Kampf, M. Knecht, J. Novotný and M. Zdráhal, in preparation.
J. Kennedy and T.D. Spearman, Singularities in spatial wave amplitudes for two ingoing and two outgoing particles, Phys. Rev. 126 (1961) 1596.
F. Guerrero and A. Pich, Effective field theory description of the pion form-factor, Phys. Lett. B 412 (1997) 382 [hep-ph/9707347] [INSPIRE].
T.N. Truong, Chiral Perturbation Theory and Final State Theorem, Phys. Rev. Lett. 61 (1988) 2526 [INSPIRE].
A. Dobado, M.J. Herrero and T.N. Truong, Unitarized Chiral Perturbation Theory for Elastic Pion-Pion Scattering, Phys. Lett. B 235 (1990) 134 [INSPIRE].
L. Beldjoudi and T.N. Truong, ππ scattering and pion form-factors, hep-ph/9403348 [INSPIRE].
T. Hannah, Unitarity, chiral perturbation theory and meson form-factors, Phys. Rev. D 54 (1996) 4648 [hep-ph/9611307] [INSPIRE].
NA48/2 collaboration, Measurement of the Dalitz plot slopes of the K ± → π ± π + π − decay, Phys. Lett. B 649 (2007) 349 [hep-ex/0702045] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
E. Witten, Baryons in the 1/n Expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
M.A. Shifman, Quark hadron duality, in At the frontier of particle physics. Handbook of QCD. Vol. 1-3, World Scientific, (2001), pp. 1447–1494, hep-ph/0009131 [INSPIRE].
E. de Rafael, Large Nc QCD and Harmonic Sums, Pramana 78 (2012) 927.
D. Greynat, E. de Rafael and G. Vulvert, Asymptotic behaviour of pion-pion total cross-sections, JHEP 03 (2014) 107 [arXiv:1312.2881] [INSPIRE].
J. Gasser and M.E. Sainio, Two loop integrals in chiral perturbation theory, Eur. Phys. J. C 6 (1999) 297 [hep-ph/9803251] [INSPIRE].
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22, F.W.J. Olver et al. eds.
P. Flajolet, X. Gourdon and P. Dumas, Mellin Transforms and Asymptotics: Harmonic Sums, Theor. Comput. Sci. 144 (1995) 3.
S. Friot, D. Greynat and E. De Rafael, Asymptotics of Feynman diagrams and the Mellin-Barnes representation, Phys. Lett. B 628 (2005) 73 [hep-ph/0505038] [INSPIRE].
G.J. Gounaris and J.J. Sakurai, Finite width corrections to the vector meson dominance prediction for ρ → e + e −, Phys. Rev. Lett. 21 (1968) 244 [INSPIRE].
J.A. Oller, E. Oset and J.E. Palomar, Pion and kaon vector form-factors, Phys. Rev. D 63 (2001) 114009 [hep-ph/0011096] [INSPIRE].
A. Pich and J. Portolés, The vector form-factor of the pion from unitarity and analyticity: A model independent approach, Phys. Rev. D 63 (2001) 093005 [hep-ph/0101194] [INSPIRE].
H. Leutwyler, Electromagnetic form-factor of the pion, in Continuous advances in QCD. Proceedings, Conference, Minneapolis, U.S.A., May 17–23, 2002, pp. 23–40, hep-ph/0212324 [INSPIRE].
C. Bruch, A. Khodjamirian and J.H. Kühn, Modeling the pion and kaon form factors in the timelike region, Eur. Phys. J. C 39 (2005) 41 [hep-ph/0409080] [INSPIRE].
H. Czyż, A. Grzelinska and J.H. Kühn, Narrow resonances studies with the radiative return method, Phys. Rev. D 81 (2010) 094014 [arXiv:1002.0279] [INSPIRE].
E.L. Lomon and S. Pacetti, Analytic pion form factor, Phys. Rev. D 94 (2016) 056002 [arXiv:1603.09527] [INSPIRE].
N.N. Khuri and S.B. Treiman, Pion-Pion Scattering and K ± → 3π Decay, Phys. Rev. 119 (1960) 1115 [INSPIRE].
J. Kambor, C. Wiesendanger and D. Wyler, Final state interactions and Khuri-Treiman equations in η → 3π decays, Nucl. Phys. B 465 (1996) 215 [hep-ph/9509374] [INSPIRE].
A.V. Anisovich and H. Leutwyler, Dispersive analysis of the decay η → 3π, Phys. Lett. B 375 (1996) 335 [hep-ph/9601237] [INSPIRE].
M. Albaladejo and B. Moussallam, Extended chiral Khuri-Treiman formalism for η → 3π and the role of the a 0(980), f 0(980) resonances, Eur. Phys. J. C 77 (2017) 508 [arXiv:1702.04931] [INSPIRE].
F. Mescia and C. Smith, Improved estimates of rare K decay matrix-elements from K l3 decays, Phys. Rev. D 76 (2007) 034017 [arXiv:0705.2025] [INSPIRE].
S. Descotes-Genon, N.H. Fuchs, L. Girlanda and J. Stern, Analysis and interpretation of new low-energy ππ scattering data, Eur. Phys. J. C 24 (2002) 469 [hep-ph/0112088] [INSPIRE].
J. Stern, H. Sazdjian and N.H. Fuchs, What π-π scattering tells us about chiral perturbation theory, Phys. Rev. D 47 (1993) 3814 [hep-ph/9301244] [INSPIRE].
M. Knecht, B. Moussallam, J. Stern and N.H. Fuchs, The low-energy ππ amplitude to one and two loops, Nucl. Phys. B 457 (1995) 513 [hep-ph/9507319] [INSPIRE].
M. Zdráhal and J. Novotný, Dispersive Approach to Chiral Perturbation Theory, Phys. Rev. D 78 (2008) 116016 [arXiv:0806.4529] [INSPIRE].
K. Kampf, M. Knecht, J. Novotný and M. Zdráhal, Analytical dispersive construction of η → 3π amplitude: first order in isospin breaking, Phys. Rev. D 84 (2011) 114015 [arXiv:1103.0982] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1812.00735
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
D’Ambrosio, G., Greynat, D. & Knecht, M. On the amplitudes for the CP-conserving K±(KS) → π±(π0)ℓ+ℓ− rare decay modes. J. High Energ. Phys. 2019, 49 (2019). https://doi.org/10.1007/JHEP02(2019)049
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)049