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Vacuum stability conditions from copositivity criteria

  • Regular Article - Theoretical Physics
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Abstract

A scalar potential of the form \(\lambda_{ab} \varphi_{a}^{2} \varphi_{b}^{2}\) is bounded from below if its matrix of quartic couplings λ ab is copositive—positive on non-negative vectors. Scalar potentials of this form occur naturally for scalar dark matter stabilised by a ℤ2 symmetry. Copositivity criteria allow us to derive analytic necessary and sufficient vacuum stability conditions for the matrix λ ab . We review the basic properties of copositive matrices and analytic criteria for copositivity. To illustrate these, we re-derive the vacuum stability conditions for the inert doublet model in a simple way, and derive the vacuum stability conditions for the ℤ2 complex singlet dark matter, and for the model with both a complex singlet and an inert doublet invariant under a global U(1) symmetry.

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Notes

  1. Thus random generation of copositive matrices is rather easy for n≤4, since by Cholesky decomposition every positive semidefinite matrix can be factorised as S=LL †, where L is a lower triangular matrix.

  2. Had we attempted to find the vacuum stability conditions from copositivity via real components of the fields, we would get, besides (18), redundant inequalities like λ 2+|λ 2|⩾0, due to the SU(2) symmetry. In the basis of the eight real component fields, it would be much more troublesome to show that the conditions (18) are not just necessary, but also sufficient.

  3. The previously given conditions in [27] pertaining to H 1 and S are necessary conditions for positivity, not copositivity.

  4. The conditions given in [27] are necessary conditions for positivity, not copositivity; in [6] sufficient conditions were used.

References

  1. J. Casas, J. Espinosa, M. Quiros, Phys. Lett. B 342, 171 (1995). hep-ph/9409458

    Article  ADS  Google Scholar 

  2. J. Casas, J. Espinosa, M. Quiros, Phys. Lett. B 382, 374 (1996). hep-ph/9603227

    Article  ADS  Google Scholar 

  3. G. Isidori, G. Ridolfi, A. Strumia, Nucl. Phys. B 609, 387 (2001). hep-ph/0104016

    Article  ADS  MATH  Google Scholar 

  4. G. Isidori, V.S. Rychkov, A. Strumia et al., Phys. Rev. D 77, 025034 (2008). 0712.0242

    Article  ADS  Google Scholar 

  5. J. Elias-Miro, J.R. Espinosa, G.F. Giudice et al., Phys. Lett. B 709, 222 (2012). 1112.3022

    Article  ADS  Google Scholar 

  6. M. Kadastik, K. Kannike, A. Racioppi et al., J. High Energy Phys. 1205, 061 (2012). 1112.3647

    Article  ADS  Google Scholar 

  7. M. Gonderinger, H. Lim, M.J. Ramsey-Musolf, 1202.1316 (2012)

  8. J. Elias-Miro, J.R. Espinosa, G.F. Giudice et al., J. High Energy Phys. 1206, 031 (2012). 1203.0237

    Article  ADS  Google Scholar 

  9. O. Lebedev, Eur. Phys. J. C 72, 1 (2012). 1203.0156

    Article  Google Scholar 

  10. C.-S. Chen, Y. Tang, J. High Energy Phys. 1204, 019 (2012). 1202.5717

    Article  ADS  Google Scholar 

  11. C. Cheung, M. Papucci, K.M. Zurek, 1203.5106 (2012)

  12. T.S. Motzkin, in National Bureau of Standards Report 1818 (1952), pp. 11–22

    Google Scholar 

  13. A.J. Quist, E. de Klerk, C. Roos et al., Optim. Methods Softw. 9, 9 (1998)

    Article  Google Scholar 

  14. M. Dür, in Recent Advances in Optimization and Its Applications in Engineering, ed. by M. Diehl, F. Glineur, E. Jarlebring et al. (Springer, Berlin, 2010), pp. 3–20

    Chapter  Google Scholar 

  15. K. Murty, S. Kabadi, Math. Program. 39, 117 (1987). doi:10.1007/BF02592948

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Ikramov, N. Savel’eva, J. Math. Sci. 98, 1 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.-B. Hiriart-Urruty, A. Seeger, SIAM Rev. 52(4), 593 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Bundfuss, Copositive Matrices, Copositive Programming, and Applications. Ph.D. thesis, TU Darmstadt, 2009

  19. I.M. Bomze, Eur. J. Oper. Res. 216(3), 509 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. N.G. Deshpande, E. Ma, Phys. Rev. D 18, 2574 (1978)

    Article  ADS  Google Scholar 

  21. R. Barbieri, L.J. Hall, V.S. Rychkov, Phys. Rev. D 74, 015007 (2006). hep-ph/0603188

    Article  ADS  Google Scholar 

  22. E. Ma, Phys. Rev. D 73, 077301 (2006). hep-ph/0601225

    Article  ADS  Google Scholar 

  23. L. Lopez Honorez, E. Nezri, J.F. Oliver et al., J. Cosmol. Astropart. Phys. 0702, 028 (2007). hep-ph/0612275

    Article  ADS  Google Scholar 

  24. J. McDonald, Phys. Rev. D 50, 3637 (1994). hep-ph/0702143

    Article  ADS  Google Scholar 

  25. V. Barger, P. Langacker, M. McCaskey et al., Phys. Rev. D 79, 015018 (2009). 0811.0393

    Article  ADS  Google Scholar 

  26. M. Kadastik, K. Kannike, M. Raidal, Phys. Rev. D 81, 015002 (2010). 0903.2475

    Article  ADS  Google Scholar 

  27. M. Kadastik, K. Kannike, M. Raidal, Phys. Rev. D 80, 085020 (2009). 0907.1894

    Article  ADS  Google Scholar 

  28. G.T. Gilber, Am. Math. Mon. 98(1), 44 (1991)

    Article  Google Scholar 

  29. P.H. Diananda, Math. Proc. Camb. Philos. Soc. 58(01), 17 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. W. Kaplan, Linear Algebra Appl. 337(1–3), 237 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. K. Hadeler, Linear Algebra Appl. 49, 79 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Chang, T.W. Sederberg, Comput. Aided Geom. Des. 11(1), 113 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Ping, F.Y. Yu, Linear Algebra Appl. 194, 109 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. L.-E. Andersson, G. Chang, T. Elfving, Linear Algebra Appl. 220, 9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. R. Cottle, G. Habetler, C. Lemke, Linear Algebra Appl. 3(3), 295 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Kaplan, Linear Algebra Appl. 313(1–3), 203 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. I.M. Bomze, in Proc. 12th SOR (Methods in OR 58), ed. by P. Kleinschmidt, F. Radermacher (Atheneum, Frankfurt/Main, 1989), pp. 27–35

    Google Scholar 

  38. I.M. Bomze, Linear Algebra Appl. 248, 161 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Bundfuss, M. Dür, Linear Algebra Appl. 428(7), 1511 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. T.D. Lee, Phys. Rev. D 8, 1226 (1973)

    Article  ADS  Google Scholar 

  41. G.C. Branco, Phys. Rev. D 22, 2901 (1980)

    Article  ADS  Google Scholar 

  42. G. Branco, P. Ferreira, L. Lavoura et al., 1106.0034 (2011)

  43. I.F. Ginzburg, M. Krawczyk, Phys. Rev. D 72, 115013 (2005). hep-ph/0408011

    Article  ADS  Google Scholar 

  44. K. Klimenko, Theor. Math. Phys. 62, 58 (1985)

    Article  Google Scholar 

  45. S. Nie, M. Sher, Phys. Lett. B 449, 89 (1999). hep-ph/9811234

    Article  ADS  Google Scholar 

  46. S. Kanemura, T. Kasai, Y. Okada, Phys. Lett. B 471, 182 (1999). hep-ph/9903289

    Article  ADS  Google Scholar 

  47. M. Maniatis, A. von Manteuffel, O. Nachtmann et al., Eur. Phys. J. C 48, 805 (2006). hep-ph/0605184

    Article  ADS  Google Scholar 

  48. I. Ivanov, Phys. Rev. D 75, 035001 (2007). hep-ph/0609018

    Article  ADS  Google Scholar 

  49. P.A. Parrilo, Math. Program. 96, 293 (2003). doi:10.1007/s10107-003-0387-5

    Article  MathSciNet  MATH  Google Scholar 

  50. V. Silveira, A. Zee, Phys. Lett. B 161, 136 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  51. C.P. Burgess, M. Pospelov, T. ter Veldhuis, Nucl. Phys. B 619, 709 (2001). hep-ph/0011335

    Article  ADS  Google Scholar 

  52. V. Barger, P. Langacker, M. McCaskey et al., Phys. Rev. D 77, 035005 (2008). 0706.4311

    Article  ADS  Google Scholar 

  53. M. Gonderinger, Y. Li, H. Patel et al., J. High Energy Phys. 1001, 053 (2010). 0910.3167

    Article  ADS  Google Scholar 

  54. V. Barger, M. McCaskey, G. Shaughnessy, Phys. Rev. D 82, 035019 (2010). 1005.3328

    Article  ADS  Google Scholar 

  55. M. Kadastik, K. Kannike, A. Racioppi et al., Phys. Rev. Lett. 104, 201301 (2010). 0912.2729

    Article  ADS  Google Scholar 

  56. M. Kadastik, K. Kannike, A. Racioppi et al., Phys. Lett. B 694, 242 (2010). 0912.3797

    Article  ADS  Google Scholar 

  57. G. Belanger, K. Kannike, A. Pukhov et al., 1202.2962 (2012)

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Acknowledgements

We thank Martti Raidal for comments and suggestions and Julia Polikarpus for consultation. This work was supported by the ESF grants 8090, 8943, MTT8, MTT60, MJD140 by the recurrent financing SF0690030s09 project and by the European Union through the European Regional Development Fund.

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  1. Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Kristjan Kannike

  2. National Institute of Chemical Physics and Biophysics, Rävala 10, Tallinn, Estonia

    Kristjan Kannike

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  1. Kristjan Kannike
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Correspondence to Kristjan Kannike.

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Kannike, K. Vacuum stability conditions from copositivity criteria. Eur. Phys. J. C 72, 2093 (2012). https://doi.org/10.1140/epjc/s10052-012-2093-z

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