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Mass insertions vs. mass eigenstates calculations in flavour physics

A preprint version of the article is available at arXiv.


We present and prove a theorem of matrix analysis, the Flavour Expansion Theorem (or FET), according to which, an analytic function of a Hermitian matrix can be expanded polynomially in terms of its off-diagonal elements with coefficients being the divided differences of the analytic function and arguments the diagonal elements of the Hermitian matrix. The theorem is applicable in case of flavour changing amplitudes. At one-loop level this procedure is particularly natural due to the observation that every loop function in the Passarino-Veltman basis can be recursively expressed in terms of divided differences. FET helps to algebraically translate an amplitude written in mass eigenbasis into flavour mass insertions, without performing diagrammatic calculations in flavour basis. As a non-trivial application of FET up to a third order, we demonstrate its use in calculating strong bounds on the real parts of flavour changing mass insertions in the up- squark sector of the MSSM from neutron Electric Dipole Moment (nEDM) measurements, assuming that CP-violation arises only from the CKM matrix.


  1. S. Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967) 1264 [INSPIRE].

    ADS  Article  Google Scholar 

  2. S. Glashow, Partial symmetries of weak interactions, Nucl. Phys. 22 (1961) 579.

    Article  Google Scholar 

  3. A. Salam, Weak and electromagnetic Interactions in Proceedings of the Eighth Nobel Symposium, N. Svartholm ed., Wiley, New York U.S.A. (1968).

  4. N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10 (1963) 531 [INSPIRE].

    ADS  Article  Google Scholar 

  5. M. Kobayashi and T. Maskawa, CP violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49 (1973) 652 [INSPIRE].

    ADS  Article  Google Scholar 

  6. B. Pontecorvo, Mesonium and anti-mesonium, Sov. Phys. JETP 6 (1957) 429 [INSPIRE].

    ADS  Google Scholar 

  7. Z. Maki, M. Nakagawa and S. Sakata, Remarks on the unified model of elementary particles, Prog. Theor. Phys. 28 (1962) 870 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  8. F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model, Nucl. Phys. B 477 (1996) 321 [hep-ph/9604387] [INSPIRE].

    ADS  Article  Google Scholar 

  9. M. Misiak, S. Pokorski and J. Rosiek, Supersymmetry and FCNC effects, Adv. Ser. Direct. High Energy Phys. 15 (1998) 795 [hep-ph/9703442] [INSPIRE].

    ADS  Article  Google Scholar 

  10. J. Hisano and D. Nomura, Solar and atmospheric neutrino oscillations and lepton flavor violation in supersymmetric models with the right-handed neutrinos, Phys. Rev. D 59 (1999) 116005 [hep-ph/9810479] [INSPIRE].

    ADS  Google Scholar 

  11. R. Bhatia, Matrix analysis, Springer, Germany (1997).

    Book  MATH  Google Scholar 

  12. R.A. Horn and C.R. Johnson, Matrix analysis, Cambridge University Press, Cambridge U.K. (1990).

    MATH  Google Scholar 

  13. G. Passarino and M.J.G. Veltman, One loop corrections for e + e annihilation into μ + μ in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].

    ADS  Article  Google Scholar 

  14. C. de Boor, Divided differences, Surv. Approx. Theory 1 (2005) 46 [math/0502036].

    MathSciNet  MATH  Google Scholar 

  15. A.J. Buras, A. Romanino and L. Silvestrini, Kπ neutrino anti-neutrino: a model independent analysis and supersymmetry, Nucl. Phys. B 520 (1998) 3 [hep-ph/9712398] [INSPIRE].

    ADS  Google Scholar 

  16. G.F. Giudice, M. Nardecchia and A. Romanino, Hierarchical soft terms and flavor physics, Nucl. Phys. B 813 (2009) 156 [arXiv:0812.3610] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  17. A. Crivellin and J. Girrbach, Constraining the MSSM sfermion mass matrices with light fermion masses, Phys. Rev. D 81 (2010) 076001 [arXiv:1002.0227] [INSPIRE].

    ADS  Google Scholar 

  18. A. Dedes, M. Paraskevas, J. Rosiek, K. Suxho and K. Tamvakis, Rare top-quark decays to Higgs boson in MSSM, JHEP 11 (2014) 137 [arXiv:1409.6546] [INSPIRE].

    ADS  Article  Google Scholar 

  19. H.P. Nilles, Supersymmetry, supergravity and particle physics, Phys. Rept. 110 (1984) 1 [INSPIRE].

    ADS  Article  Google Scholar 

  20. H.E. Haber and G.L. Kane, The search for supersymmetry: probing physics beyond the standard model, Phys. Rept. 117 (1985) 75 [INSPIRE].

    ADS  Article  Google Scholar 

  21. S.P. Martin, A supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 21 (2010) 1 [hep-ph/9709356] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  22. K. Fuyuto, J. Hisano, N. Nagata and K. Tsumura, QCD corrections to quark (chromo)electric dipole moments in high-scale supersymmetry, JHEP 12 (2013) 010 [arXiv:1308.6493] [INSPIRE].

    ADS  Article  Google Scholar 

  23. A. Manohar and H. Georgi, Chiral quarks and the nonrelativistic quark model, Nucl. Phys. B 234 (1984) 189 [INSPIRE].

    ADS  Article  Google Scholar 

  24. S. Pokorski, J. Rosiek and C.A. Savoy, Constraints on phases of supersymmetric flavor conserving couplings, Nucl. Phys. B 570 (2000) 81 [hep-ph/9906206] [INSPIRE].

    ADS  Article  Google Scholar 

  25. J. Rosiek, Complete set of Feynman rules for the MSSM: erratum, hep-ph/9511250 [INSPIRE].

  26. J. Rosiek, Complete set of feynman rules for the minimal supersymmetric extension of the standard model, Phys. Rev. D 41 (1990) 3464 [INSPIRE].

    ADS  Google Scholar 

  27. C.A. Baker et al., An improved experimental limit on the electric dipole moment of the neutron, Phys. Rev. Lett. 97 (2006) 131801 [hep-ex/0602020] [INSPIRE].

    ADS  Article  Google Scholar 

  28. J. Rosiek, P. Chankowski, A. Dedes, S. Jager and P. Tanedo, SUSY FLAVOR: a computational tool for FCNC and CP-violating processes in the MSSM, Comput. Phys. Commun. 181 (2010) 2180 [arXiv:1003.4260] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  29. A. Crivellin et al., SUSY FLAVOR v2: a computational tool for FCNC and CP-violating processes in the MSSM, Comput. Phys. Commun. 184 (2013) 1004 [arXiv:1203.5023] [INSPIRE].

    ADS  Article  Google Scholar 

  30. J. Rosiek, SUSY FLAVOR library and constraints on B s μ + μ decay rate, arXiv:1212.0032 [INSPIRE].

  31. A. Crivellin and J. Rosiek, SUSY FLAVOR library for rare decays in the MSSM, PoS(EPS-HEP 2013)081 [arXiv:1308.6299] [INSPIRE].

  32. J. Rosiek, SUSY FLAVOR v2.5: a computational tool for FCNC and CP-violating processes in the MSSM, Comput. Phys. Commun. 188 (2014) 208 [arXiv:1410.0606] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. W. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics/Readings in Mathematics. Springer (1991).

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Correspondence to M. Paraskevas.

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ArXiv ePrint: 1504.00960v2

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Dedes, A., Paraskevas, M., Rosiek, J. et al. Mass insertions vs. mass eigenstates calculations in flavour physics. J. High Energ. Phys. 2015, 151 (2015).

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  • Supersymmetry Phenomenology