Skip to main content
Log in

Flavour issues for string-motivated heavy scalar spectra with a low gluino mass: the G 2-MSSM case

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal C Aims and scope Submit manuscript

Abstract

In recent years it has been learned that scalar superpartner masses and trilinear couplings should both generically be larger than about 20 TeV at the short-distance string scale if our world is described by a compactified string or M-theory with supersymmetry breaking and stabilized moduli (Acharya et al. in arXiv:1006.3272 [hep-ph], 2010). Here we study implications of this, somewhat generally and also in detail for a particular realization (compactification of M-theory on a G 2 manifold) where there is significant knowledge of the superpotential and gauge kinetic function, and a light gluino. In a certain sense this yields an ultraviolet completion of minimal flavour violation. Flavour violation stems from off-diagonal and non-universal diagonal elements of scalar mass matrices and trilinear couplings, and from renormalization group running. We also examine stability bounds on the scalar potential. While heavy scalars alone do not guarantee the absence of flavour problems, our studies show that models with heavy scalars and light gluinos can be free from such problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. If this inequality is violated, there arises too large an off-diagonal term in the super-CKM basis.

  2. The values used here are those from [28]. The functions S(x i ,x j ) are the well-known Inami–Lim functions [29] entering in the SM box contributions, S(x) is listed also in Appendix A.2. \(G_{\tilde{g}}(x_{\tilde{g}})\) is the loop function of the box diagram involving internal gluinos and squarks and it is defined in Appendix A.2.

  3. That is, \(Y^{f}_{\mathrm{diag}}=\hat{Y}^{f}=U^{f}_{\mathrm{R}}Y^{f} U^{f\dagger}_{\mathrm{L}}\) and consequently trilinear terms are rotated as \(\hat{a}^{f}=U^{f}_{\mathrm{R}}a^{f} U^{f\dagger}_{\mathrm{L}}\) and soft mass-squared matrices as \({\hat{m}}^{2}_{\tilde{f}\mathrm{LL}}=U^{f}_{\mathrm{L}} m^{2}_{\tilde{Q}} U^{f\dagger}_{\mathrm{L}}\) and \({\hat{m}}^{2}_{\tilde{f}\mathrm{RR}}=U^{f}_{\mathrm{R}} m^{2}_{\tilde{f}} U^{f\dagger}_{\mathrm{R}}\).

  4. ϵ′ being the parameter measuring the direct CP violation in the decay amplitude of K→2π, \(\epsilon^{\prime i(\delta_{2}-\delta_{0})} \mathrm{Re}[A_{2}](\mathrm{Re} [A_{0}] \mathop {\mathrm {Im}}[A_{2}]/\mathrm{Re}[A_{2}]-\mathop {\mathrm {Im}}[A_{0}])/(\sqrt{2} \mathrm{Re}[A_{0}])\), \(A_{I} e^{i\delta_{I}}=\langle\pi\pi(I)| H^{\Delta S=1}_{\mathrm{eff}}| K^{0} \rangle\), I=0,2.

  5. The simplified expression (16) is derived considering the \(\mathcal{D}\)-flat direction \(\alpha^{2}=|H_{d}^{0}|^{2}+|\tilde{\nu}_{m}|^{2}=|\tilde{e}_{L_{i}}|^{2}=|\tilde{e}_{R_{j}}|^{2}\) (mi,j) in the limit \(\alpha\gg[m^{2}_{H_{d}}+|\mu|^{2}-(\hat{m}^{2}_{\tilde{\nu}})_{mm}]/[(\hat{Y}_{ii}^{e})^{2} +(\hat{Y}_{jj}^{e})^{2}]\) with \(\alpha^{2}>|H_{d}^{0}|^{2}\) [42].

  6. Neither program is completely suited for precisely the calculation required here.

  7. We express ϵ as \(\epsilon= \epsilon^{\mathrm{SM}} + \delta\epsilon^{\mathrm{SUSY}}, \quad\delta\epsilon^{\mathrm{SUSY}}= \delta\epsilon^{H^{\pm}} + \delta \epsilon^{\tilde{\chi}^{\pm}}+ \delta\epsilon^{\tilde{\chi}^{0}} + \delta\epsilon^{\tilde{\chi}^{0} \tilde{g}} + \delta\epsilon^{\tilde{g}}\), where δϵ SUSY is the total SUSY contribution and the individual terms refer to the charged Higgs, the chargino, the neutralino, the neutralino–gluino, and the gluino contribution, respectively.

  8. For the uncertainties in the hadronic matrix element calculations, we have used the bag parameters of [50].

  9. One could start working in the basis where Y d is diagonal and Y u is not. Recall that it is not possible to work in a basis where both are diagonal precisely due to the CKM matrix.

References

  1. B.S. Acharya, G. Kane, E. Kuflik, String theories with moduli stabilization imply non-thermal cosmological history, and particular dark matter. arXiv:1006.3272 [hep-ph]

  2. A.G. Cohen, D. Kaplan, A. Nelson, The more minimal supersymmetric standard model. Phys. Lett. B 388, 588–598 (1996). arXiv:hep-ph/9607394

    Article  ADS  Google Scholar 

  3. J.D. Wells, PeV-scale supersymmetry. Phys. Rev. D 71, 015013 (2005). arXiv:hep-ph/0411041

    Article  ADS  Google Scholar 

  4. M.K. Gaillard, B.D. Nelson, On quadratic divergences in supergravity, vacuum energy and the supersymmetric flavor problem. Nucl. Phys. B 751, 75–107 (2006). arXiv:hep-ph/0511234

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. G. Coughlan, W. Fischler, E.W. Kolb, S. Raby, G.G. Ross, Cosmological problems for the Polonyi potential. Phys. Lett. B 131, 59 (1983)

    Article  ADS  Google Scholar 

  6. G. Kane, P. Kumar, J. Shao, CP-violating phases in M theory and implications for electric dipole moments. Phys. Rev. D 82, 055005 (2010). arXiv:0905.2986 [hep-ph]

    Article  ADS  Google Scholar 

  7. B.S. Acharya, K. Bobkov, P. Kumar, An M theory solution to the strong CP problem and constraints on the axiverse. J. High Energy Phys. 1011, 105 (2010). arXiv:1004.5138 [hep-th]

    Article  ADS  Google Scholar 

  8. B.S. Acharya, K. Bobkov, G.L. Kane, P. Kumar, J. Shao, Explaining the electroweak scale and stabilizing moduli in M theory. Phys. Rev. D 76, 126010 (2007). arXiv:hep-th/0701034

    Article  MathSciNet  ADS  Google Scholar 

  9. N. Arkani-Hamed, H. Murayama, Can the supersymmetric flavor problem decouple? Phys. Rev. D 56, 6733–6737 (1997). arXiv:hep-ph/9703259

    Article  ADS  Google Scholar 

  10. G.F. Giudice, M. Nardecchia, A. Romanino, Hierarchical soft terms and flavor physics. Nucl. Phys. B 813, 156–173 (2009). arXiv:0812.3610 [hep-ph]

    Article  MATH  ADS  Google Scholar 

  11. N. Arkani-Hamed, S. Dimopoulos, G. Giudice, A. Romanino, Aspects of split supersymmetry. Nucl. Phys. B 709, 3–46 (2005). arXiv:hep-ph/0409232

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. G. Kane, S. King, I. Peddie, L. Velasco-Sevilla, Study of theory and phenomenology of some classes of family symmetry and unification models. J. High Energy Phys. 0508, 083 (2005). arXiv:hep-ph/0504038

    Article  MathSciNet  ADS  Google Scholar 

  13. B.S. Acharya, K. Bobkov, G.L. Kane, J. Shao, P. Kumar, G 2-MSSM: an M theory motivated model of particle physics. Phys. Rev. D 78, 065038 (2008). arXiv:0801.0478 [hep-ph]

    Article  ADS  Google Scholar 

  14. B.S. Acharya, K. Bobkov, Kahler independence of the G 2-MSSM. J. High Energy Phys. 09, 001 (2010). arXiv:0810.3285 [hep-th]

    Article  MathSciNet  ADS  Google Scholar 

  15. A. Brignole, L.E. Ibañez, C. Muñoz, Soft supersymmetry-breaking terms from supergravity and superstring models. arXiv:hep-ph/9707209

  16. K.A. Olive, L. Velasco-Sevilla, Constraints on supersymmetric flavour models from b. J. High Energy Phys. 05, 052 (2008). arXiv:0801.0428 [hep-ph]

    Article  ADS  Google Scholar 

  17. G. D’Ambrosio, G. Giudice, G. Isidori, A. Strumia, Minimal flavor violation: an effective field theory approach. Nucl. Phys. B 645, 155–187 (2002). arXiv:hep-ph/0207036

    Article  ADS  Google Scholar 

  18. S. Abel, S. Khalil, O. Lebedev, Additional stringy sources for electric dipole moments. Phys. Rev. Lett. 89, 121601 (2002). arXiv:hep-ph/0112260

    Article  ADS  Google Scholar 

  19. G.G. Ross, O. Vives, Yukawa structure, flavor changing, and CP violation in supergravity. Phys. Rev. D 67, 095013 (2003). arXiv:hep-ph/0211279

    Article  ADS  Google Scholar 

  20. G.G. Ross, L. Velasco-Sevilla, O. Vives, Spontaneous CP violation and non-Abelian family symmetry in SUSY. Nucl. Phys. B 692, 50–82 (2004). arXiv:hep-ph/0401064

    Article  MATH  ADS  Google Scholar 

  21. S. Antusch, S.F. King, M. Malinský, Solving the SUSY flavour and CP problems with SU(3) family symmetry. J. High Energy Phys. 06, 068 (2008). arXiv:0708.1282 [hep-ph]

    Article  ADS  Google Scholar 

  22. S. Antusch, S.F. King, M. Malinský, G.G. Ross, Solving the SUSY flavour and CP problems with non-Abelian family symmetry and supergravity. Phys. Lett. B 670, 383–389 (2009). arXiv:0807.5047 [hep-ph]

    Article  ADS  Google Scholar 

  23. L. Calibbi, J. Jones-Perez, A. Masiero, J.-h. Park, W. Porod, O. Vives, FCNC and CP violation observables in a SU(3)-flavoured MSSM. Nucl. Phys. B 831, 26–71 (2010). arXiv:0907.4069 [hep-ph]

    Article  MATH  ADS  Google Scholar 

  24. L. Calibbi, E.J. Chun, L. Velasco-Sevilla, Bridging flavour violation and leptogenesis in SU(3) family models. J. High Energy Phys. 1011, 090 (2010). arXiv:1005.5563 [hep-ph]

    Article  ADS  Google Scholar 

  25. K. Kadota, J. Kersten, L. Velasco-Sevilla, Supersymmetric musings on the predictivity of family symmetries. Phys. Rev. D 82, 085022 (2010). arXiv:1007.1532 [hep-ph]

    Article  ADS  Google Scholar 

  26. S. Antusch, L. Calibbi, V. Maurer, M. Spinrath, From flavour to SUSY flavour models. arXiv:1104.3040 [hep-ph]

  27. W. Altmannshofer, A.J. Buras, S. Gori, P. Paradisi, D.M. Straub, Anatomy and phenomenology of FCNC and CPV effects in SUSY theories. Nucl. Phys. B 830, 17–94 (2010). arXiv:0909.1333 [hep-ph]

    Article  MATH  ADS  Google Scholar 

  28. Particle Data Group, K. Nakamura et al., Review of particle physics. J. Phys. G 37, 075021 (2010)

    Article  ADS  Google Scholar 

  29. T. Inami, C. Lim, Effects of superheavy quarks and leptons in low-energy weak processes \(K_{L} \to \mu \bar{\mu}\), \(K^{+} \to \pi^{+} \nu \bar{\nu}\) and \(K^{0} \leftrightarrow \bar{K}^{0}\). Prog. Theor. Phys. 65, 297 (1981)

    Article  ADS  Google Scholar 

  30. S. Bertolini, F. Borzumati, A. Masiero, G. Ridolfi, Effects of supergravity induced electroweak breaking on rare B decays and mixings. Nucl. Phys. B 353, 591–649 (1991)

    Article  ADS  Google Scholar 

  31. R. Contino, I. Scimemi, The supersymmetric flavor problem for heavy first-two generation scalars at next-to-leading order. Eur. Phys. J. C 10, 347–356 (1999). arXiv:hep-ph/9809437

    Article  ADS  Google Scholar 

  32. L.J. Hall, V.A. Kostelecky, S. Raby, New flavor violations in supergravity models. Nucl. Phys. B 267, 415 (1986)

    Article  ADS  Google Scholar 

  33. F. Gabbiani, E. Gabrielli, A. Masiero, L. Silvestrini, A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model. Nucl. Phys. B 477, 321–352 (1996). arXiv:hep-ph/9604387

    Article  ADS  Google Scholar 

  34. J.S. Hagelin, S. Kelley, T. Tanaka, Supersymmetric flavor changing neutral currents: exact amplitudes and phenomenological analysis. Nucl. Phys. B 415, 293–331 (1994)

    Article  ADS  Google Scholar 

  35. G. Eyal, A. Masiero, Y. Nir, L. Silvestrini, Probing supersymmetric flavor models with ϵ′/ϵ. J. High Energy Phys. 9911, 032 (1999). arXiv:hep-ph/9908382

    Article  ADS  Google Scholar 

  36. R. Barbieri, R. Contino, A. Strumia, ϵ′ from supersymmetry with nonuniversal A terms? Nucl. Phys. B 578, 153–162 (2000). arXiv:hep-ph/9908255

    Article  ADS  Google Scholar 

  37. E. Gabrielli, A. Masiero, L. Silvestrini, Flavor changing neutral currents and CP violating processes in generalized supersymmetric theories. Phys. Lett. B 374, 80–86 (1996). arXiv:hep-ph/9509379

    Article  ADS  Google Scholar 

  38. S. Abel, J. Frere, Could the MSSM have no CP violation in the CKM matrix? Phys. Rev. D 55, 1623–1629 (1997). arXiv:hep-ph/9608251

    Article  ADS  Google Scholar 

  39. A. Masiero, H. Murayama, Can ϵ′/ϵ be supersymmetric? Phys. Rev. Lett. 83, 907–910 (1999). arXiv:hep-ph/9903363

    Article  ADS  Google Scholar 

  40. S. Khalil, T. Kobayashi, O. Vives, EDM-free supersymmetric CP violation with non-universal soft terms. Nucl. Phys. B 580, 275–288 (2000). arXiv:hep-ph/0003086

    Article  ADS  Google Scholar 

  41. H. Murayama, Can ϵ′/ϵ be supersymmetric? arXiv:hep-ph/9908442

  42. J. Casas, S. Dimopoulos, Stability bounds on flavor violating trilinear soft terms in the MSSM. Phys. Lett. B 387, 107–112 (1996). arXiv:hep-ph/9606237

    Article  ADS  Google Scholar 

  43. E. Witten, Deconstruction, G 2 holonomy, and doublet-triplet splitting. arXiv:hep-ph/0201018

  44. B.S. Acharya, F. Denef, R. Valandro, Statistics of M theory vacua. J. High Energy Phys. 0506, 056 (2005). arXiv:hep-th/0502060

    Article  MathSciNet  ADS  Google Scholar 

  45. B.C. Allanach, SOFTSUSY: a program for calculating supersymmetric spectra. Comput. Phys. Commun. 143, 305–331 (2002). arXiv:hep-ph/0104145

    Article  MATH  ADS  Google Scholar 

  46. W. Porod, SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e + e colliders. Comput. Phys. Commun. 153, 275–315 (2003). arXiv:hep-ph/0301101

    Article  ADS  Google Scholar 

  47. W. Porod, F. Staub, SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM. arXiv:1104.1573 [hep-ph]

  48. S. Khalil, O. Lebedev, Chargino contributions to ϵ and ϵ′. Phys. Lett. B 515, 387–394 (2001). arXiv:hep-ph/0106023

    Article  ADS  Google Scholar 

  49. Y.G. Kim, P. Ko, J.S. Lee, Possible new-physics signals in b and bsl + l . Nucl. Phys. B 544, 64–88 (1999). arXiv:hep-ph/9810336

    Article  ADS  Google Scholar 

  50. L. Conti, C. Allton, A. Donini, V. Gimenez, L. Giusti et al., B-parameters for ΔS=2 supersymmetric operators. Nucl. Phys. B, Proc. Suppl. 73, 315–317 (1999). arXiv:hep-lat/9809162

    Article  ADS  Google Scholar 

  51. O. Gedalia, G. Isidori, G. Perez, Combining direct and indirect Kaon CP violation to constrain the warped KK scale. Phys. Lett. B 682, 200–206 (2009). arXiv:0905.3264 [hep-ph]

    Article  ADS  Google Scholar 

  52. S. Abel, S. Khalil, O. Lebedev, EDM constraints in supersymmetric theories. Nucl. Phys. B 606, 151–182 (2001). arXiv:hep-ph/0103320

    Article  ADS  Google Scholar 

  53. I. Masina, C.A. Savoy, Sleptonarium: constraints on the CP and flavor pattern of scalar lepton masses. Nucl. Phys. B 661, 365–393 (2003). arXiv:hep-ph/0211283

    Article  ADS  Google Scholar 

  54. T. Moroi, The muon anomalous magnetic dipole moment in the minimal supersymmetric standard model. Phys. Rev. D 53, 6565–6575 (1996). arXiv:hep-ph/9512396

    Article  ADS  Google Scholar 

  55. G.-C. Cho, K. Hagiwara, Y. Matsumoto, D. Nomura, The MSSM confronts the precision electroweak data and the muon g−2. arXiv:1104.1769 [hep-ph]

  56. Muon (g−2) Collaboration, G. Bennett et al., Final report of the muon E821 anomalous magnetic moment measurement at BNL. Phys. Rev. D 73, 072003 (2006). arXiv:hep-ex/0602035

    Article  ADS  Google Scholar 

  57. ALEPH Collaboration, R. Barate et al., A measurement of the inclusive b branching ratio. Phys. Lett. B 429, 169–187 (1998)

    Article  ADS  Google Scholar 

  58. F. Borzumati, C. Greub, T. Hurth, D. Wyler, Gluino contribution to radiative B decays: organization of QCD corrections and leading order results. Phys. Rev. D 62, 075005 (2000). arXiv:hep-ph/9911245

    Article  ADS  Google Scholar 

  59. C. Greub, T. Hurth, D. Wyler, Indirect search for supersymmetry in rare B decays. arXiv:hep-ph/9912420

  60. D. Wyler, F. Borzumati, Gluino contribution to radiative B decays: new operators, organization of QCD corrections and leading order results. arXiv:hep-ph/0104046

  61. P. Gambino, M. Misiak, Quark mass effects in \(\bar{B} \to X_{s} \gamma\). Nucl. Phys. B 611, 338–366 (2001). arXiv:hep-ph/0104034

    Article  ADS  Google Scholar 

  62. T. Hurth, E. Lunghi, W. Porod, Untagged \(\bar{B} \to X_{s+d} \gamma\) CP asymmetry as a probe for new physics. Nucl. Phys. B 704, 56–74 (2005). arXiv:hep-ph/0312260

    Article  ADS  Google Scholar 

  63. M. Ciuchini, E. Franco, D. Guadagnoli, V. Lubicz, M. Pierini et al., D\(\bar{D}\) mixing and new physics: general considerations and constraints on the MSSM. Phys. Lett. B 655, 162–166 (2007). arXiv:hep-ph/0703204

    Article  ADS  Google Scholar 

  64. P. Paradisi, Constraints on SUSY lepton flavor violation by rare processes. J. High Energy Phys. 0510, 006 (2005). arXiv:hep-ph/0505046

    Article  ADS  Google Scholar 

  65. K. Agashe, M. Graesser, Supersymmetry breaking and the supersymmetric flavor problem: an analysis of decoupling the first two generation scalars. Phys. Rev. D 59, 015007 (1999). arXiv:hep-ph/9801446

    Article  ADS  Google Scholar 

  66. J.A. Bagger, K.T. Matchev, R.-J. Zhang, QCD corrections to flavor-changing neutral currents in the supersymmetric standard model. Phys. Lett. B 412, 77–85 (1997). arXiv:hep-ph/9707225

    Article  ADS  Google Scholar 

  67. J. Ellis, R.N. Hodgkinson, J.S. Lee, A. Pilaftsis, Flavour geometry and effective Yukawa couplings in the MSSM. J. High Energy Phys. 1002, 016 (2010). arXiv:0911.3611 [hep-ph]

    Article  ADS  Google Scholar 

  68. K. Blum, Y. Grossman, Y. Nir, G. Perez, Combining K 0\(\bar{K}^{0}\) mixing and D 0\(\bar{D}^{0}\) mixing to constrain the flavor structure of new physics. Phys. Rev. Lett. 102, 211802 (2009). arXiv:0903.2118 [hep-ph]

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We particularly thank Bobby Acharya for encouragement and participating in the initial stages of the analysis, and we also thank Francesca Borzumati for clarifications on issues regarding b. We thank Luca Silvestrini for clarifications on the loop functions entering into the Wilson coefficients of the ΔS=2 Hamiltonian. We thank Brent Nelson and Werner Porod for useful discussions. J.K. would like to thank CINVESTAV in Mexico City for hospitality during stages of this work. This work was supported by the Michigan Center for Theoretical Physics, and by the German Science Foundation (DFG) via the Junior Research Group “SUSY Phenomenology” within the Collaborative Research Centre 676 “Particles, Strings and the Early Universe”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Kadota.

Appendices

Appendix A: Notation

1.1 A.1 Wilson coefficients

We follow various references [30, 31, 66] for the extraction of the effective Hamiltonians. The ΔS=2 operators involved in (5) are

(29)

where P L and P R are the left- and right-handed projection operators, respectively, \(\tilde{O}_{i}= O_{i} (L\leftrightarrow R)\), and \(\langle O_{i}\rangle =\langle\tilde{O}_{i}\rangle \).

1.2 A.2 Loop functions

We collect in this appendix the loop functions that we have used in our analysis.

(30)
(31)
(32)
(33)
(34)

Appendix B: Details of the running from M G down to \(\protect \mu_{\tilde{f}}\)

We take the scalar soft-squared mass matrices to be proportional to the unit matrix at M G. Their running to \(\mu_{\tilde{f}} \sim m_{3/2} \), the scale at which the scalars decouple, will produce off-diagonal entries. We require these off-diagonal elements to be significantly smaller than the diagonal elements, since otherwise the mass-insertion approximation would not be justified. To be concrete and conservative, let us consider m 3/2=m 0=20 TeV and demand

(35)

at \(\mu_{\tilde{f}}\). As we consider small values of tanβ, we can neglect the contributions of Y d and a d to the running. Furthermore, we consider CKM-like matrices diagonalizing Y u, which implies \(Y^{u\dagger} Y^{u} \sim Y^{u} Y^{u\dagger} \sim y_{t}^{2} \mathop {\mathrm {diag}}(0,0,1)\). Consequently, Y u does not affect the running of the off-diagonal elements of \(m^{2}_{\tilde{f}}\). Of course, the same is true of the terms in the renormalization group equation (RGE) of \(m^{2}_{\tilde{f}}\) that involve gauge couplings. Thus, the only relevant terms in the RGE are those proportional to a u a u and a u a u. Approximating the right-hand side of the RGE by a constant value (leading-log approximation), we then obtain at \(\mu_{\tilde{f}}\)

(36)
(37)
(38)

Using (2, 23), \(U^{u}_{\mathrm{L}} \sim U^{u}_{\mathrm{R}} \sim V_{\mathrm{CKM}}\) as well as \(Y^{u}_{\mathrm{diag}} \sim \mathop {\mathrm {diag}}(\lambda^{8},\lambda^{4},1)\), where λ≈0.23 is the sine of the Cabibbo angle, and assuming no accidental cancellations, we obtain

(39)

where x max is the maximum value of \(x^{f}_{ij}\). Thus, the strongest constraint stems from \(\tilde{f} = \tilde{u}\) and ij=23 in (35),

(40)

With \(A_{\tilde{f}} = 1.5 \frac{m_{0}^{2}}{m_{3/2}} \approx30~\mbox {TeV}\), this yields x max∼1.8. To be conservative, we have chosen \(x_{\mathrm{max}} = \sqrt{2}\) for our numerical analysis.

Appendix C: Comments on MFV

Trilinear terms

The term MFV [17] refers to scenarios where all higher-dimensional operators, constructed from SM and fields with Yukawa interactions, are invariant under CP and under the flavour group G F . Here \(G_{F} = \mathrm {SU}(3)_{q_{L}}\ {\otimes}\ \mathrm {SU}(3)_{u_{R}}\ {\otimes}\ \mathrm {SU}(3)_{d_{R}} \ {\otimes}\ \mathrm {SU}(3)_{l_{L}}\ {\otimes}\ \mathrm {SU}(3)_{e_{R}}\ {\otimes} \mathrm {U}(1)_{B} \ {\otimes}\ \mathrm {U}(1)_{L} \ {\otimes}\ \mathrm {U}(1)_{Y}\ {\otimes}\ \mathrm {U}(1)_{PQ} \ {\otimes}\ \mathrm {U}(1)_{e_{R}}\), and the Yukawa couplings are formally regarded as auxiliary fields that transform under G F . As a consequence, MFV requires that the dynamics of flavour violation is completely determined by the structure of the ordinary Yukawa couplings and in particular, all CP violation originates from the CKM phase.

Because of the running of all couplings of a theory, this scenario can only be realized at one particular scale, usually a low energy scale. Starting with parameters defined at M G, MFV can only be a good approximation at M EW, if

(41)

where A f is a universal mass parameter for all families and kind of fermions that is small in comparison with other soft masses of the theory and if Yukawa couplings are small. This can be analyzed by studying the dependence of the RGEs of Yukawa and trilinear couplings on Y f [16, 67].Footnote 9 If just the third-family Yukawa couplings are evolved, of course the size of A f does not matter because no off-diagonal terms are produced. With a full RG evolution of complex 3×3 Yukawa and trilinear matrices with small off-diagonal values, MFV can be emulated, albeit never reproduced, for sufficiently small values of A f [16].

Soft-squared masses

The one-loop running of the soft-squared parameters \((m^{2}_{\tilde{f}})_{ij}\) in the SCKM basis is governed by the β functions

(42)

where f∈{u,d} and the functions \(G_{m^{2}_{\tilde{f}}}\) contain flavour-diagonal contributions to the running involving gauge couplings and gaugino masses. Note that at an arbitrary scale μM G, the terms which contain

are not diagonal because of the different running of the diagonal elements in \(m^{2}_{\tilde{Q}}\) and \(m^{2}_{\tilde{f}}\). Therefore, off-diagonal terms will necessarily be induced.

Recall that even if we consider only the running of the Yukawa couplings of the third family, this will produce a split in the masses of \(m^{2}_{\tilde{f}}\). We can always choose to go to the basis where one of the Yukawa couplings is diagonal at M G, but this does not guarantee diagonal soft mass-squared matrices in the SCKM basis because the fact that

(43)

necessarily implies that not all of the matrices \(\hat{m}^{2}_{\tilde{f}\mathrm{LL}} = U^{f}_{L}m^{2}_{\tilde{Q}}U^{f\dagger}_{L}\) and \(\hat{m}^{2}_{\tilde{f}\mathrm{RR}} = U^{f}_{R}m^{2}_{\tilde{f}}U^{f\dagger}_{R}\) are diagonal. As is known ϵ is very sensitive to this [68]. If the coupling of the particles beyond the SM was of the same order as that of the SM particles, this would push the limit on the scale of new physics entering into the ΔS=2 processes up to

(44)

In the lepton sector, we assume heavy right-handed neutrinos that decouple close to the GUT scale. Therefore only the superpartners of right- and left-handed charged leptons as well as left-handed neutrinos can induce flavour violation. Considering the structure of fermion masses we are using, see Sect. 5.4 and references therein, the Yukawa coupling matrix for neutrinos is the same as that for the up-quark sector at M G, therefore the flavour violation induced in this scenario is relatively small. Table 4 shows the MI parameters relevant for the observables i j γ.

In the G 2-MSSM case, where all the examples that are known [6] correspond to the case that trilinear couplings are proportional to Yukawa couplings, we can have a theory, depending on the choice of Yukawa couplings, for which at low energy, all flavour violation present is below the experimental bounds. It is only in this sense that we can say that we have an ultraviolet version of MFV but not in the sense in which MFV is defined. For the case of the relation (2) with \(c^{f}_{ij}\) as in (22), that is, O(1) real random numbers between 0 and \(\sqrt{2}\), at low energies CP violating phases in addition to the CKM phase appear but also flavour violation is below experimental bounds. With \(c^{f}_{ij}\) as in (23), i.e., with random numbers between 0 and \(\sqrt{2}\) and explicit CP phases at M G, flavour violation is more difficult to neglect but still below the experimental bounds. This means that even with more CP phases than in the SM and large mixing present due to the choice of Yukawa couplings, after the running to low energy we obtain a theory which satisfies flavour-violation constraints. Not surprisingly, the reason are the large scalar masses. What is not a trivial result of the analysis is that still bounds on the size of trilinear terms can be obtained.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kadota, K., Kane, G., Kersten, J. et al. Flavour issues for string-motivated heavy scalar spectra with a low gluino mass: the G 2-MSSM case. Eur. Phys. J. C 72, 2004 (2012). https://doi.org/10.1140/epjc/s10052-012-2004-3

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1140/epjc/s10052-012-2004-3

Keywords

Navigation