Abstract
Hadronic matrix elements of local four-quark operators play a central role in non-leptonic kaon decays, while vacuum matrix elements involving the same kind of operators appear in inclusive dispersion relations, such as those relevant in τ-decay analyses. Using an SU(3)L ⊗ SU(3)R decomposition of the operators, we derive generic relations between these matrix elements, extending well-known results that link observables in the two different sectors. Two relevant phenomenological applications are presented. First, we determine the electroweak-penguin contribution to the kaon CP-violating ratio ε′/ε, using the measured hadronic spectral functions in τ decay. Second, we fit our SU(3) dynamical parameters to the most recent lattice data on K → ππ matrix elements. The comparison of this numerical fit with results from previous analytical approaches provides an interesting anatomy of the ∆I = \( \frac{1}{2} \) enhancement, confirming old suggestions about its underlying dynamical origin.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179 (1969) 1499 [INSPIRE].
G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].
A, Pich, Effective field theory: Course, in Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, Les Houches France (1998), pg. 949
A.V. Manohar, Introduction to Effective Field Theories, Les Houches Lect. Notes 108 (2020) [arXiv:1804.05863].
M. Neubert, Renormalization Theory and Effective Field Theories, Les Houches Lect. Notes 108 (2020).
V. Cirigliano, G. Ecker, H. Neufeld, A. Pich and J. Portoles, Kaon Decays in the Standard Model, Rev. Mod. Phys. 84 (2012) 399 [arXiv:1107.6001] [INSPIRE].
RBC and UKQCD collaborations, Direct CP-violation and the ∆I = 1/2 rule in K → ππ decay from the standard model, Phys. Rev. D 102 (2020) 054509 [arXiv:2004.09440] [INSPIRE].
A. Pich, Precision physics with inclusive QCD processes, Prog. Part. Nucl. Phys. 117 (2021) 103846.
G. Kallen, On the definition of the Renormalization Constants in Quantum Electrodynamics, Helv. Phys. Acta 25 (1952) 417 [INSPIRE].
H. Lehmann, On the Properties of propagation functions and renormalization contants of quantized fields, Nuovo Cim. 11 (1954) 342.
E. Braaten, S. Narison and A. Pich, QCD analysis of the tau hadronic width, Nucl. Phys. B 373 (1992) 581 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147 (1979) 385 [INSPIRE].
J.F. Donoghue, E. Golowich and B.R. Holstein, The ∆S = 2 Matrix Element for K0 − \( {\overline{K}}^0 \) Mixing, Phys. Lett. B 119 (1982) 412 [INSPIRE].
J. Bijnens, H. Sonoda and M.B. Wise, On the Validity of Chiral Perturbation Theory for K0 − \( {\overline{K}}^0 \) Mixing, Phys. Rev. Lett. 53 (1984) 2367 [INSPIRE].
A. Pich and E. De Rafael, K − \( \overline{K} \) Mixing in the Standard Model, Phys. Lett. B 158 (1985) 477 [INSPIRE].
C.W. Bernard, T. Draper, A. Soni, H.D. Politzer and M.B. Wise, Application of Chiral Perturbation Theory to K → 2π Decays, Phys. Rev. D 32 (1985) 2343 [INSPIRE].
B. Guberina, A. Pich and E. de Rafael, The Decay K+ → π+π0 in the Standard Model, Phys. Lett. B 163 (1985) 198 [INSPIRE].
A. Pich, B. Guberina and E. de Rafael, Problem with the Delta I = 1/2 Rule in the Standard Model, Nucl. Phys. B 277 (1986) 197 [INSPIRE].
J. Kambor, J.H. Missimer and D. Wyler, The Chiral Loop Expansion of the Nonleptonic Weak Interactions of Mesons, Nucl. Phys. B 346 (1990) 17 [INSPIRE].
A. Pich and E. de Rafael, Four quark operators and nonleptonic weak transitions, Nucl. Phys. B 358 (1991) 311 [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. Kambor, J.F. Donoghue, B.R. Holstein, J.H. Missimer and D. Wyler, Chiral symmetry tests in nonleptonic K decay, Phys. Rev. Lett. 68 (1992) 1818 [INSPIRE].
A. Pich and E. de Rafael, Weak K amplitudes in the chiral and 1/NC expansions, Phys. Lett. B 374 (1996) 186 [hep-ph/9511465] [INSPIRE].
M. Knecht, S. Peris and E. de Rafael, Matrix elements of electroweak penguin operators in the 1/NC expansion, Phys. Lett. B 457 (1999) 227 [hep-ph/9812471] [INSPIRE].
J.F. Donoghue and E. Golowich, Dispersive calculation of \( {B}_7^{3/2} \) and \( {B}_8^{3/2} \) in the chiral limit, Phys. Lett. B 478 (2000) 172 [hep-ph/9911309] [INSPIRE].
J. Bijnens, E. Gamiz and J. Prades, Matching the electroweak penguins Q7, Q8 and spectral correlators, JHEP 10 (2001) 009 [hep-ph/0108240] [INSPIRE].
M. Knecht, S. Peris and E. de Rafael, A critical reassessment of Q7 and Q8 matrix elements, Phys. Lett. B 508 (2001) 117 [hep-ph/0102017] [INSPIRE].
V. Cirigliano, J.F. Donoghue, E. Golowich and K. Maltman, Determination of \( \left\langle {\left(\pi \pi \right)}_{I=2}\left|{\mathcal{Q}}_{7,8}\right|{K}^0\right\rangle \) in the chiral limit, Phys. Lett. B 522 (2001) 245 [hep-ph/0109113] [INSPIRE].
V. Cirigliano, J.F. Donoghue, E. Golowich and K. Maltman, Improved determination of the electroweak penguin contribution to ϵ′/ϵ in the chiral limit, Phys. Lett. B 555 (2003) 71 [hep-ph/0211420] [INSPIRE].
O. Catà and S. Peris, Long distance dimension eight operators in BK, JHEP 03 (2003) 060 [hep-ph/0303162] [INSPIRE].
T. Hambye, S. Peris and E. de Rafael, ∆I = 1/2 and ϵ′/ϵ in large Nc QCD, JHEP 05 (2003) 027 [hep-ph/0305104] [INSPIRE].
H. Gisbert and A. Pich, Direct CP violation in K0 → ππ: Standard Model Status, Rep. Prog. Phys. 81 (2018) 076201.
V. Cirigliano, H. Gisbert, A. Pich and A. Rodríguez-Sánchez, Isospin-violating contributions to ϵ′/ϵ, JHEP 02 (2020) 032 [arXiv:1911.01359] [INSPIRE].
T. Blum et al., K → ππ ∆I = 3/2 decay amplitude in the continuum limit, Phys. Rev. D 91 (2015) 074502 [arXiv:1502.00263] [INSPIRE].
T. Blum et al., Lattice determination of the K → (ππ)I=2 Decay Amplitude A2, Phys. Rev. D 86 (2012) 074513 [arXiv:1206.5142] [INSPIRE].
A. Pich, Effective Field Theory with Nambu-Goldstone Modes, Les Houches Lect. Notes 108 (2020).
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
A. Pich, Chiral perturbation theory, Rept. Prog. Phys. 58 (1995) 563 [hep-ph/9502366] [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
C. Lehner and C. Sturm, Matching factors for ∆S = 1 four-quark operators in RI/SMOM schemes, Phys. Rev. D 84 (2011) 014001 [arXiv:1104.4948] [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].
R.J. Crewther, Chiral Reduction of K → 2π Amplitudes, Nucl. Phys. B 264 (1986) 277 [INSPIRE].
F.J. Gilman and M.B. Wise, Effective Hamiltonian for ∆s = 1 Weak Nonleptonic Decays in the Six Quark Model, Phys. Rev. D 20 (1979) 2392 [INSPIRE].
J.A. Cronin, Phenomenological model of strong and weak interactions in chiral U(3) × U(3), Phys. Rev. 161 (1967) 1483 [INSPIRE].
J. Bijnens and M.B. Wise, Electromagnetic Contribution to ϵ′/ϵ, Phys. Lett. B 137 (1984) 245 [INSPIRE].
B. Grinstein, S.-J. Rey and M.B. Wise, CP Violation in Charged Kaon Decay, Phys. Rev. D 33 (1986) 1495 [INSPIRE].
A.J. Buras, M. Jamin and M.E. Lautenbacher, The Anatomy of ϵ′/ϵ beyond leading logarithms with improved hadronic matrix elements, Nucl. Phys. B 408 (1993) 209 [hep-ph/9303284] [INSPIRE].
E. Pallante, A. Pich and I. Scimemi, The Standard model prediction for ϵ′/ϵ, Nucl. Phys. B 617 (2001) 441 [hep-ph/0105011] [INSPIRE].
F.J. Gilman and M.B. Wise, K0 − \( {\overline{K}}^0 \) Mixing in the Six Quark Model, Phys. Rev. D 27 (1983) 1128 [INSPIRE].
J. Brod, M. Gorbahn and E. Stamou, Standard-Model Prediction of ϵK with Manifest Quark-Mixing Unitarity, Phys. Rev. Lett. 125 (2020) 171803 [arXiv:1911.06822] [INSPIRE].
A.J. Buras, M. Jamin and P.H. Weisz, Leading and Next-to-leading QCD Corrections to t Parameter and B0 − \( {\overline{B}}^0 \) Mixing in the Presence of a Heavy Top Quark, Nucl. Phys. B 347 (1990) 491 [INSPIRE].
S. Herrlich and U. Nierste, Enhancement of the KL − KS mass difference by short distance QCD corrections beyond leading logarithms, Nucl. Phys. B 419 (1994) 292 [hep-ph/9310311] [INSPIRE].
S. Herrlich and U. Nierste, The Complete |∆S| = 2 Hamiltonian in the next-to-leading order, Nucl. Phys. B 476 (1996) 27 [hep-ph/9604330] [INSPIRE].
J. Brod and M. Gorbahn, ϵK at Next-to-Next-to-Leading Order: The Charm-Top-Quark Contribution, Phys. Rev. D 82 (2010) 094026 [arXiv:1007.0684] [INSPIRE].
J. Bijnens, E. Gamiz and J. Prades, The BK kaon parameter in the chiral limit, JHEP 03 (2006) 048 [hep-ph/0601197] [INSPIRE].
J. Bijnens and J. Prades, The BK parameter in the 1/NC expansion, Nucl. Phys. B 444 (1995) 523 [hep-ph/9502363] [INSPIRE].
S. Peris and E. de Rafael, K0 − \( {\overline{K}}^0 \) mixing in the 1/NC expansion, Phys. Lett. B 490 (2000) 213 [hep-ph/0006146] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Order \( {\alpha}_s^4 \) QCD Corrections to Z and tau Decays, Phys. Rev. Lett. 101 (2008) 012002 [arXiv:0801.1821] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order \( {\alpha}_s^4 \) in a General Gauge Theory, Phys. Rev. Lett. 104 (2010) 132004 [arXiv:1001.3606] [INSPIRE].
F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On Higgs decays to hadrons and the R-ratio at N4LO, JHEP 08 (2017) 113 [arXiv:1707.01044] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn and J. Rittinger, Adler Function, Sum Rules and Crewther Relation of Order \( \mathcal{O}\left({\alpha}_s^4\right) \): the Singlet Case, Phys. Lett. B 714 (2012) 62 [arXiv:1206.1288] [INSPIRE].
V. Cirigliano and E. Golowich, Analysis of O(p2) corrections to 〈ππ|Q7, 8|K〉, Phys. Lett. B 475 (2000) 351 [hep-ph/9912513] [INSPIRE].
L.V. Lanin, V.P. Spiridonov and K.G. Chetyrkin, Contribution of Four Quark Condensates to Sum Rules for ρ and A1 Mesons, (in Russian), Yad. Fiz. 44 (1986) 1372 [INSPIRE].
L.E. Adam and K.G. Chetyrkin, Renormalization of four quark operators and QCD sum rules, Phys. Lett. B 329 (1994) 129 [hep-ph/9404331] [INSPIRE].
D. Boito, D. Hornung and M. Jamin, Anomalous dimensions of four-quark operators and renormalon structure of mesonic two-point correlators, JHEP 12 (2015) 090 [arXiv:1510.03812] [INSPIRE].
C. Vafa and E. Witten, Restrictions on Symmetry Breaking in Vector-Like Gauge Theories, Nucl. Phys. B 234 (1984) 173.
W.J. Marciano and A. Sirlin, Electroweak Radiative Corrections to tau Decay, Phys. Rev. Lett. 61 (1988) 1815 [INSPIRE].
E. Braaten and C.-S. Li, Electroweak radiative corrections to the semihadronic decay rate of the tau lepton, Phys. Rev. D 42 (1990) 3888 [INSPIRE].
J. Erler, Electroweak radiative corrections to semileptonic tau decays, Rev. Mex. Fis. 50 (2004) 200 [hep-ph/0211345] [INSPIRE].
M. Davier, A. Höcker, B. Malaescu, C.-Z. Yuan and Z. Zhang, Update of the ALEPH non-strange spectral functions from hadronic τ decays, Eur. Phys. J. C 74 (2014) 2803 [arXiv:1312.1501] [INSPIRE].
F. Le Diberder and A. Pich, Testing QCD with τ decays, Phys. Lett. B 289 (1992) 165 [INSPIRE].
M.S. Dubovikov and A.V. Smilga, On nonperturbative qcd effects in imaginary part of polarization operator of quark and gluon currents, (in Russian), Yad. Fiz. 37 (1983) 984 [INSPIRE].
B. Chibisov, R.D. Dikeman, M.A. Shifman and N. Uraltsev, Operator product expansion, heavy quarks, QCD duality and its violations, Int. J. Mod. Phys. A 12 (1997) 2075 [hep-ph/9605465] [INSPIRE].
M.A. Shifman, Quark hadron duality, in 8th International Symposium on Heavy Flavor Physics, Southampton U.K. (1999), World Scientific, Singapore (2000), pg. 1447 [hep-ph/0009131] [INSPIRE].
O. Catà, M. Golterman and S. Peris, Duality violations and spectral sum rules, JHEP 08 (2005) 076 [hep-ph/0506004] [INSPIRE].
M. Gonzalez-Alonso, A. Pich and J. Prades, Violation of Quark-Hadron Duality and Spectral Chiral Moments in QCD, Phys. Rev. D 81 (2010) 074007 [arXiv:1001.2269] [INSPIRE].
M. Gonzalez-Alonso, A. Pich and J. Prades, Pinched weights and Duality Violation in QCD Sum Rules: a critical analysis, Phys. Rev. D 82 (2010) 014019 [arXiv:1004.4987] [INSPIRE].
D. Boito, I. Caprini, M. Golterman, K. Maltman and S. Peris, Hyperasymptotics and quark-hadron duality violations in QCD, Phys. Rev. D 97 (2018) 054007 [arXiv:1711.10316] [INSPIRE].
D. Boito, M. Golterman, K. Maltman, J. Osborne and S. Peris, Strong coupling from the revised ALEPH data for hadronic τ decays, Phys. Rev. D 91 (2015) 034003 [arXiv:1410.3528] [INSPIRE].
A. Pich and A. Rodríguez-Sánchez, Determination of the QCD coupling from ALEPH τ decay data, Phys. Rev. D 94 (2016) 034027 [arXiv:1605.06830] [INSPIRE].
M. González-Alonso, A. Pich and A. Rodríguez-Sánchez, Updated determination of chiral couplings and vacuum condensates from hadronic τ decay data, Phys. Rev. D 94 (2016) 014017 [arXiv:1602.06112] [INSPIRE].
S. Weinberg, Precise relations between the spectra of vector and axial vector mesons, Phys. Rev. Lett. 18 (1967) 507 [INSPIRE].
B. Blok, M.A. Shifman and D.-X. Zhang, An Illustrative example of how quark hadron duality might work, Phys. Rev. D 57 (1998) 2691 [Erratum ibid. 59 (1999) 019901] [hep-ph/9709333] [INSPIRE].
M.A. Shifman, Snapshots of hadrons or the story of how the vacuum medium determines the properties of the classical mesons which are produced, live and die in the QCD vacuum, Prog. Theor. Phys. Suppl. 131 (1998) 1 [hep-ph/9802214] [INSPIRE].
O. Catà, M. Golterman and S. Peris, Unraveling duality violations in hadronic tau decays, Phys. Rev. D 77 (2008) 093006 [arXiv:0803.0246] [INSPIRE].
D. Boito et al., Low-energy constants and condensates from ALEPH hadronic τ decay data, Phys. Rev. D 92 (2015) 114501 [arXiv:1503.03450] [INSPIRE].
V. Cirigliano, G. Ecker, H. Neufeld and A. Pich, Isospin breaking in K → ππ decays, Eur. Phys. J. C 33 (2004) 369 [hep-ph/0310351] [INSPIRE].
J. Bijnens and J. Prades, \( {\upepsilon}_K^{\prime }/{\upepsilon}_K \) epsilon in the chiral limit, JHEP 06 (2000) 035 [hep-ph/0005189] [INSPIRE].
S. Narison, New QCD estimate of the kaon penguin matrix elements and ϵ′/ϵ, Nucl. Phys. B 593 (2001) 3 [hep-ph/0004247] [INSPIRE].
G. Ecker, J. Kambor and D. Wyler, Resonances in the weak chiral Lagrangian, Nucl. Phys. B 394 (1993) 101 [INSPIRE].
G. Ecker, G. Isidori, G. Muller, H. Neufeld and A. Pich, Electromagnetism in nonleptonic weak interactions, Nucl. Phys. B 591 (2000) 419 [hep-ph/0006172] [INSPIRE].
E. Pallante and A. Pich, Strong enhancement of ϵ′/ϵ through final state interactions, Phys. Rev. Lett. 84 (2000) 2568 [hep-ph/9911233] [INSPIRE].
G. Colangelo, J. Gasser and H. Leutwyler, ππ scattering, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088] [INSPIRE].
P.A. Zyla et al. Review of Particle Physics, PTEP 2020 (2020) 083C01.
J. Aebischer, C. Bobeth and A.J. Buras, ε′/ε in the Standard Model at the Dawn of the 2020s, Eur. Phys. J. C 80 (2020) 705 [arXiv:2005.05978] [INSPIRE].
G. Altarelli and L. Maiani, Octet Enhancement of Nonleptonic Weak Interactions in Asymptotically Free Gauge Theories, Phys. Lett. B 52 (1974) 351 [INSPIRE].
M.K. Gaillard and B.W. Lee, ∆I = 1/2 Rule for Nonleptonic Decays in Asymptotically Free Field Theories, Phys. Rev. Lett. 33 (1974) 108 [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Effective Hamiltonians for ∆S = 1 and ∆B = 1 nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B 370 (1992) 69 [Addendum ibid. 375 (1992) 501] [INSPIRE].
A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Two loop anomalous dimension matrix for ∆S = 1 weak nonleptonic decays I: \( \mathcal{O}\left({\alpha}_s^2\right) \), Nucl. Phys. B 400 (1993) 37 [hep-ph/9211304] [INSPIRE].
A.J. Buras, M. Jamin and M.E. Lautenbacher, Two loop anomalous dimension matrix for ∆S = 1 weak nonleptonic decays. 2. \( \mathcal{O} \)(ααs), Nucl. Phys. B 400 (1993) 75 [hep-ph/9211321] [INSPIRE].
M. Ciuchini, E. Franco, G. Martinelli and L. Reina, ϵ′/ϵ at the Next-to-leading order in QCD and QED, Phys. Lett. B 301 (1993) 263 [hep-ph/9212203] [INSPIRE].
M. Ciuchini, E. Franco, G. Martinelli and L. Reina, The ∆S = 1 effective Hamiltonian including next-to-leading order QCD and QED corrections, Nucl. Phys. B 415 (1994) 403 [hep-ph/9304257] [INSPIRE].
W.A. Bardeen, A.J. Buras and J.M. Gerard, A Consistent Analysis of the ∆I = 1/2 Rule for K Decays, Phys. Lett. B 1920 (1987) 138.
V. Antonelli, S. Bertolini, M. Fabbrichesi and E.I. Lashin, The ∆I = 1/2 selection rule, Nucl. Phys. B 469 (1996) 181 [hep-ph/9511341] [INSPIRE].
V. Antonelli, S. Bertolini, J.O. Eeg, M. Fabbrichesi and E.I. Lashin, The ∆S = 1 weak chiral lagrangian as the effective theory of the chiral quark model, Nucl. Phys. B 469 (1996) 143 [hep-ph/9511255] [INSPIRE].
S. Bertolini, J.O. Eeg, M. Fabbrichesi and E.I. Lashin, The ∆I = 1/2 rule and BK at O(p4) in the chiral expansion, Nucl. Phys. B 514 (1998) 63 [hep-ph/9705244] [INSPIRE].
J. Bijnens and J. Prades, The ∆I = 1/2 rule in the chiral limit, JHEP 01 (1999) 023 [hep-ph/9811472] [INSPIRE].
T. Hambye, G.O. Kohler and P.H. Soldan, New analysis of the ∆I = 1/2 rule in kaon decays and the \( {\hat{B}}_K \) parameter, Eur. Phys. J. C 10 (1999) 271 [hep-ph/9902334] [INSPIRE].
A.J. Buras, J.-M. Gérard and W.A. Bardeen, Large N Approach to Kaon Decays and Mixing 28 Years Later: ∆I = 1/2 Rule, \( {\hat{B}}_K \) and ∆MK, Eur. Phys. J. C 74 (2014) 2871 [arXiv:1401.1385] [INSPIRE].
A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, A Possible mechanism for the ∆T = 1/2 rule in nonleptonic decays of strange particles, JETP Lett. 22 (1975) 55 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Light Quarks and the Origin of the ∆I = 1/2 Rule in the Nonleptonic Decays of Strange Particles, Nucl. Phys. B 120 (1977) 316 [INSPIRE].
E. Pallante and A. Pich, Final state interactions in kaon decays, Nucl. Phys. B 592 (2001) 294 [hep-ph/0007208] [INSPIRE].
RBC, UKQCD collaboration, Emerging understanding of the ∆I = 1/2 Rule from Lattice QCD, Phys. Rev. Lett. 110 (2013) 152001 [arXiv:1212.1474] [INSPIRE].
A. Donini, P. Hernández, C. Pena and F. Romero-López, Nonleptonic kaon decays at large Nc, Phys. Rev. D 94 (2016) 114511 [arXiv:1607.03262] [INSPIRE].
A. Donini, P. Hernández, C. Pena and F. Romero-López, Dissecting the ∆I = 1/2 rule at large Nc, Eur. Phys. J. C 80 (2020) 638 [arXiv:2003.10293] [INSPIRE].
A. Pich, QCD-Duality Approach to Nonleptonic Weak Transitions: Towards an Understanding of the ∆I = 1/2 Rule, Nucl. Phys. B Proc. Suppl. 7 (1989) 194.
M. Jamin and A. Pich, QCD corrections to inclusive ∆S = 1, 2 transitions at the next-to-leading order, Nucl. Phys. B 425 (1994) 15 [hep-ph/9402363] [INSPIRE].
A. Pich and E. de Rafael, Bounds on the Strength of ∆I = 1/2 Weak Amplitudes, Phys. Lett. B 189 (1987) 369 [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG Review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020) 113 [arXiv:1902.08191] [INSPIRE].
V. Cirigliano, A. Falkowski, M. González-Alonso and A. Rodríguez-Sánchez, Hadronic τ Decays as New Physics Probes in the LHC Era, Phys. Rev. Lett. 122 (2019) 221801 [arXiv:1809.01161] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2102.09308
Rights and permissions
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.
About this article
Cite this article
Pich, A., Rodríguez-Sánchez, A. SU(3) analysis of four-quark operators: K → ππ and vacuum matrix elements. J. High Energ. Phys. 2021, 5 (2021). https://doi.org/10.1007/JHEP06(2021)005
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2021)005