Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 049, 29 pages      arXiv:1001.0428      https://doi.org/10.3842/SIGMA.2010.049
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

Finite Unification: Theory and Predictions

Sven Heinemeyer a, Myriam Mondragón b and George Zoupanos c, d
a) Instituto de Física de Cantabria (CSIC-UC), Santander, Spain
b) Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, México 01000, México
c) Theory Group, Physics Department, CERN, Geneva, Switzerland
d) Physics Department, National Technical University, 157 80 Zografou, Athens, Greece

Received January 03, 2010, in final form May 25, 2010; Published online June 11, 2010

Abstract
All-loop Finite Unified Theories (FUTs) are very interesting N=1 supersymmetric Grand Unified Theories (GUTs) which not only realise an old field theoretic dream but also have a remarkable predictive power due to the required reduction of couplings. The reduction of the dimensionless couplings in N=1 GUTs is achieved by searching for renormalization group invariant (RGI) relations among them holding beyond the unification scale. Finiteness results from the fact that there exist RGI relations among dimensionless couplings that guarantee the vanishing of all beta-functions in certain N=1 GUTs even to all orders. Furthermore developments in the soft supersymmetry breaking sector of N=1 GUTs and FUTs lead to exact RGI relations, i.e. reduction of couplings, in this dimensionful sector of the theory too. Based on the above theoretical framework phenomenologically consistent FUTS have been constructed. Here we present FUT models based on the SU(5) and SU(3)3 gauge groups and their predictions. Of particular interest is the Higgs mass prediction of one of the models which is expected to be tested at the LHC.

Key words: unification; gauge theories; finiteness; reduction of couplings.

pdf (718 kb)   ps (509 kb)   tex (510 kb)

References

  1. Connes A., Douglas M.R., Schwarz A.S., Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 35 pages, hep-th/9711162.
  2. Maldacena J.M., The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252, hep-th/9711200.
  3. Mandelstam S., Light-cone superspace and the ultraviolet finiteness of the N=4 model, Nuclear Phys. B 213 (1983), 149-168.
  4. Brink L., Lindgren O., Nilsson B.E.W., The ultraviolet finiteness of the N=4 Yang-Mills theory, Phys. Lett. B 123 (1983), 323-328.
  5. Bern Z., Carrasco J.J., Dixon L.J., Johansson H., Roiban R., Ultraviolet behavior of N=8 supergravity at four loops, Phys. Rev. Lett. 103 (2009), 081301, 4 pages, arXiv:0905.2326.
  6. Kallosh R., On UV finiteness of the four loop N=8 supergravity, J. High Energy Phys. 2009 (2009), no. 9, 116, 9 pages, arXiv:0906.3495.
  7. Bern Z., Carrasco J.J., Dixon L.J., Johansson H., Kosower D.A., Roiban R., Cancellations beyond finiteness in N=8 supergravity at three loops, Phys. Rev. Lett. 98 (2007), 161303, 4 pages, hep-th/0702112.
  8. Bern Z., Dixon L.J., Roiban R., Is N=8 supergravity ultraviolet finite?, Phys. Lett. B 644 (2007), 265-271, hep-th/0611086.
  9. Green M.B., Russo J.G., Vanhove P., Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98 (2007), 131602, 4 pages, hep-th/0611273.
  10. Pati J.C., Salam A., Is baryon number conserved?, Phys. Rev. Lett. 31 (1973), 661-664.
  11. Georgi H., Glashow S.L., Unity of all elementary particle forces, Phys. Rev. Lett. 32 (1974), 438-441.
  12. Georgi H., Quinn H.R., Weinberg S., Hierarchy of interactions in unified gauge theories, Phys. Rev. Lett. 33 (1974), 451-454.
  13. Fritzsch H., Minkowski P., Unified interactions of leptons and hadrons, Ann. Physics 93 (1975), 193-266.
  14. Georgi H., The state of the art - gauge theories, in Particles and Fields (Williamsburg, 1974), Editor C.E. Carlson, AIP Conference Proceedings, Vol. 23, American Institute of Physics, New York, 1974, 575-582.
  15. Amaldi U., de Boer W., Furstenau H., Comparison of grand unified theories with electroweak and strong coupling constants measured at LEP, Phys. Lett. B 260 (1991), 447-455.
  16. Dimopoulos S., Georgi H., Softly broken supersymmetry and SU(5), Nuclear Phys. B 193 (1981), 150-162.
  17. Sakai N., Naturalness in supersymmetric GUTs, Zeit. Phys. C 11 (1981), 153-157.
  18. Buras A.J., Ellis J.R., Gaillard M.K., Nanopoulos D.V., Aspects of the grand unification of strong, weak and electromagnetic interactions, Nuclear Phys. B 135 (1978), 66-92.
  19. Kubo J., Mondragón M., Olechowski M., Zoupanos G., Testing gauge-Yukawa-unified models by Mt, Nuclear Phys. B 479 (1996), 25-45, hep-ph/9512435.
  20. Kubo J., Mondragón M., Zoupanos G., Unification beyond GUTs: gauge Yukawa unification, Acta Phys. Polon. B 27 (1997), 3911-3944, hep-ph/9703289.
  21. Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Exact finite and gauge-Yukawa unified theories and their predictions, Acta Phys. Polon. B 30 (1999), 2013-2027.
  22. Fayet P., Spontaneous generation of massive multiplets and central charges in extended supersymmetric theories, Nuclear Phys. B 149 (1979), 137-169.
  23. Decker R., Pestieau J., Lepton self-mass, Higgs scalar and heavy-quark masses, Nuovo Cimento Lett. 29 (1980), 560-564.
  24. Veltman M.J.G., The infrared-ultraviolet connection, Acta Phys. Polon. B 12 (1981), 437-457.
  25. Ferrara S., Girardello L., Palumbo F., A general mass formula in broken supersymmetry, Phys. Rev. D 20 (1979), 403-408.
  26. Chaichian M., Gonzalez Felipe R., Huitu K., On quadratic divergences and the Higgs mass, Phys. Lett. B 363 (1995), 101-105, hep-ph/9509223.
  27. Abbiendi G. et al., Search for the standard model Higgs boson at LEP, Phys. Lett. B 565 (2003), 61-75, hep-ex/0306033.
  28. Pendleton B., Ross G.G., Mass and mixing angle predictions from infrared fixed points, Phys. Lett. B 98 (1981), 291-294.
  29. Zimmermann W., Infrared behavior of the coupling parameters in the standard model, Phys. Lett. B 308 (1993), 117-122.
  30. Hill C.T., Quark and lepton masses from renormalization group fixed points, Phys. Rev. D 24 (1981), 691-703.
  31. Tevatron Electroweak Working Group, Combination of CDF and D0 Results on the mass of the top quark, arXiv:0903.2503.
  32. Schael S. et al., Search for neutral MSSM Higgs bosons at LEP, Eur. Phys. J. C 47 (2006), 547-587, hep-ex/0602042.
  33. Kapetanakis D., Mondragón M., Zoupanos G., Finite unified models, Z. Phys. C 60 (1993), 181-185, hep-ph/9210218.
  34. Mondragón M., Zoupanos G., Finite unified theories and the top quark mass, Nuclear Phys. B Proc. Suppl. 37 (1995), 98-105.
  35. Kubo J., Mondragón M., Zoupanos G., Reduction of couplings and heavy top quark in the minimal SUSY GUT, Nuclear Phys. B 424 (1994), 291-307.
  36. Kubo J., Mondragón M., Tracas N.D., Zoupanos G., Gaug-Yukawa unification in asymptotically nonfree theories, Phys. Lett. B 342 (1995), 155-162, hep-th/9409003.
  37. Kubo J., Mondragón M., Shoda S., Zoupanos G., Gauge-Yukawa unification in SO(10) SUSY GUTs, Nuclear Phys. B 469 (1996), 3-20, hep-ph/9512258.
  38. Kubo J., Mondragón M., Zoupanos G., Perturbative unification of soft supersymmetry-breaking terms, Phys. Lett. B 389 (1996), 523-532, hep-ph/9609218.
  39. Zimmermann W., Reduction in the number of coupling parameters, Comm. Math. Phys. 97 (1985), 211-225.
  40. Oehme R., Zimmermann W., Relation between effective couplings for asymptotically free models, Comm. Math. Phys. 97 (1985), 569-582.
  41. Ma E., Modified quantum chromodynamics: exact global color symmetry and asymptotic freedom, Phys. Rev. D 17 (1978), 623-628.
  42. Ma E., Fixing the Higgs boson mass, Phys. Rev. D 31 (1985), 1143-1145.
  43. Lucchesi C., Piguet O., Sibold K., Necessary and sufficient conditions for all order vanishing β-functions in supersymmetric Yang-Mills theories, Phys. Lett. B 201 (1988), 241-244.
  44. Lucchesi C., Piguet O., Sibold K., Vanishing β-functions in N=1 supersymmetric gauge theories, Helv. Phys. Acta 61 (1988), 321-344.
  45. Lucchesi C., Zoupanos G., All-order finiteness in N=1 SYM theories: criteria and applications, Fortschr. Phys. 45 (1997), 129-143, hep-ph/9604216.
  46. Ermushev A.V., Kazakov D.I., Tarasov O.V., Finite N=1 supersymmetric grand unified theories, Nuclear Phys. B 281 (1987), 72-84.
  47. Kazakov D.I., Finite N=1 SUSY gauge field theories, Modern Phys. Lett. A 2 (1987), 663-674.
  48. Schrempp B., Schrempp F., A renormalization group invariant line and infrared attractive top-Higgs mass relation, Phys. Lett. B 299 (1993), 321-328.
  49. Schrempp B., Infrared fixed points and fixed lines in the top-bottom-tau sector in supersymmetric grand unification, Phys. Lett. B 344 (1995), 193-200, hep-ph/9411241.
  50. Schrempp B., Wimmer M., Top quark and Higgs boson masses: interplay between infrared and ultraviolet physics, Prog. Part. Nuclear Phys. 37 (1996), 1-90, hep-ph/9606386.
  51. Jack I., Jones D.R.T., Renormalization-group invariance and universal soft supersymmetry-breaking, Phys. Lett. B 349 (1995), 294-299, hep-ph/9501395.
  52. Hisano J., Shifman M.A., Exact results for soft supersymmetry-breaking parameters in supersymmetric gauge theories, Phys. Rev. D 56 (1997), 5475-5482, hep-ph/9705417.
  53. Jack I., Jones D.R.T., The gaugino β-function, Phys. Lett. B 415 (1997), 383-389, hep-ph/9709364.
  54. Avdeev L.V., Kazakov D.I., Kondrashuk I.N., Renormalizations in softly broken SUSY gauge theories, Nuclear Phys. B 510 (1998), 289-312, hep-ph/9709397.
  55. Kazakov D.I., Exploring softly broken SUSY theories via Grassmannian Taylor expansion, Phys. Lett. B 449 (1999), 201-206, hep-ph/9812513.
  56. Kazakov D.I., Finiteness of soft terms in finite N = 1 SUSY gauge theories, Phys. Lett. B 421 (1998), 211-216, hep-ph/9709465.
  57. Jack I., Jones D.R.T., Pickering A., Renormalisation invariance and the soft β-functions, Phys. Lett. B 426 (1998), 73-77, hep-ph/9712542.
  58. Kobayashi T., Kubo J., Zoupanos G., Further all-loop results in softly-broken supersymmetric gauge theories, Phys. Lett. B 427 (1998), 291-299, hep-ph/9802267.
  59. Yamada Y., Two-loop renormalization group equations for soft supersymmetry-breaking scalar interactions: supergraph method, Phys. Rev. D 50 (1994), 3537-3545, hep-ph/9401241.
  60. Delbourgo R., Superfield perturbation theory and renormalization, Nuovo Cimento A 25 (1975), 646-656.
  61. Salam A., Strathdee J.A., Feynman rules for superfields, Nuclear Phys. B 86 (1975), 142-152.
  62. Fujikawa K., Lang W., Perturbation calculations for the scalar multiplet in a superfield formulation, Nuclear Phys. B 88 (1975), 61-76.
  63. Grisaru M.T., Siegel W., Rocek M., Improved methods for supergraphs, Nuclear Phys. B 159 (1979), 429-450.
  64. Girardello L., Grisaru M.T., Soft breaking of supersymmetry, Nuclear Phys. B 194 (1982), 65-76.
  65. Jones D.R.T., Mezincescu L., Yao Y.P., Soft breaking of two loop finite N=1 supersymmetric gauge theories, Phys. Lett. B 148 (1984), 317-322.
  66. Jack I., Jones D.R.T., Soft supersymmetry breaking and finiteness, Phys. Lett. B 333 (1994), 372-379, hep-ph/9405233.
  67. Ibáñez L.E., Lüst D., Duality-anomaly cancellation, minimal string unification and the effective low-energy Lagrangian of 4D strings, Nuclear Phys. B 382 (1992), 305-364, hep-th/9202046.
  68. Kaplunovsky V.S., Louis J., Model-independent analysis of soft terms in effective supergravity and in string theory, Phys. Lett. B 306 (1993), 269-275, hep-th/9303040.
  69. Brignole A., Ibáñez L.E., Muñoz C., Towards a theory of soft terms for the supersymmetric standard model, Nuclear Phys. B 422 (1994), 125-171, hep-ph/9308271.
  70. Casas J.A., Lleyda A., Muñoz C., Problems for supersymmetry breaking by the dilaton in strings from charge and color breaking, Phys. Lett. B 380 (1996), 59-67, hep-ph/9601357.
  71. Kawamura Y., Kobayashi T., Kubo J., Soft scalar-mass sum rule in gauge-Yukawa unified models and its superstring interpretation, Phys. Lett. B 405 (1997), 64-70, hep-ph/9703320.
  72. Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Constraints on finite soft supersymmetry-breaking terms, Nuclear Phys. B 511 (1998), 45-68, hep-ph/9707425.
  73. Novikov V.A., Shifman M.A., Vainshtein A.I., Zakharov V.I., Instanton effects in supersymmetric theories, Nuclear Phys. B 229 (1983), 407-420.
  74. Novikov V.A., Shifman M.A., Vainshtein A.I., Zakharov V.I., The beta function in supersymmetric gauge theories. Instantons versus traditional approach, Phys. Lett. B 166 (1986), 329-333.
  75. Shifman M.A., Little miracles of supersymmetric evolution of gauge couplings, Internat. J. Modern Phys. A 11 (1996), 5761-5784, hep-ph/9606281.
  76. Oehme R., Reduction and reparametrization of quantum field theories, Progr. Theoret. Phys. Suppl. (1986), no. 86, 215-237.
  77. Kubo J., Sibold K., Zimmermann W., Higgs and top mass from reduction of couplings, Nuclear Phys. B 259 (1985), 331-350.
  78. Kubo J., Sibold K., Zimmermann W., New results in the reduction of the standard model, Phys. Lett. B 220 (1989), 185-190.
  79. Piguet O., Sibold K., Reduction of couplings in the presence of parameters, Phys. Lett. B 229 (1989), 83-88.
  80. Zimmermann W., Reduction of couplings in massive models of quantum field theory, in Theoretical Physics. Fin de siècle (Wroclaw, 1998), Lecture Notes in Phys., Vol. 539, Springer, Berlin, 2000, 304-314.
  81. Wess J., Zumino B., A Lagrangian model invariant under supergauge transformations, Phys. Lett. B 49 (1974), 52-54.
  82. Iliopoulos J., Zumino B., Broken supergauge symmetry and renormalization, Nuclear Phys. B 76 (1974), 310-332.
  83. Parkes A., West P.C., Finiteness in rigid supersymmetric theories, Phys. Lett. B 138 (1984), 99-104.
  84. Rajpoot S., Taylor J.G., On finite quantum field theories, Phys. Lett. B 147 (1984), 91-95.
  85. Rajpoot S., Taylor J.G., Toward finite quantum field theories, Int. J. Theor. Phys. 25 (1986), 117-138.
  86. West P., The Yukawa β-function in N=1 rigid supersymmetric theories, Phys. Lett. B 137 (1984), 371-373.
  87. Jones D.R.T., Parkes A.J., Search for a three-loop-finite chiral theory, Phys. Lett. B 160 (1985), 267-270.
  88. Jones D.R.T., Mezincescu L., The chiral anomaly and a class of two-loop finite supersymmetric gauge theories, Phys. Lett. B 138 (1984), 293-295.
  89. Parkes A.J., Three loop finiteness conditions in N=1 super-Yang-Mills, Phys. Lett. B 156 (1985), 73-79.
  90. O'Raifeartaigh L., Spontaneous symmetry breaking for chiral scalar superfields, Nuclear Phys. B 96 (1975), 331-352.
  91. Fayet P., Iliopoulos J., Spontaneously broken supergauge symmetries and goldstone spinors, Phys. Lett. B 51 (1974), 461-464.
  92. Ferrara S., Zumino B., Transformation properties of the supercurrent, Nuclear Phys. B 87 (1975), 207-220.
  93. Piguet O., Sibold K., The supercurrent in N=1 supersymmetrical Yang-Mills theories. I. The classical case, Nuclear Phys. B 196 (1982), 428-446.
  94. Piguet O., Sibold K., The supercurrent in N=1 supersymmmetrical Yang-Mills theories. II. Renormalization, Nuclear Phys. B 196 (1982), 447-460.
  95. Piguet O., Sibold K., Nonrenormalization theorems of chiral anomalies and finiteness in supersymmetric Yang-Mills theories, Internat. J. Modern Phys. A 1 (1986), 913-942.
  96. Piguet O., Sibold K., Non-renormalization theorems of chiral anomalies and finiteness, Phys. Lett. B 177 (1986), 373-376.
  97. Ensign P., Mahanthappa K.T., The supercurrent and the Adler-Bardeen theorem in coupled supersymmetric Yang-Mills theories, Phys. Rev. D 36 (1987), 3148-3171.
  98. Piguet O., Supersymmetry, ultraviolet finiteness and grand unification, hep-th/9606045.
  99. Alvarez-Gaumé L., Ginsparg P.H., The topological meaning of non-abelian anomalies, Nuclear Phys. B 243 (1984), 449-474.
  100. Bardeen W.A., Zumino B., Consistent and covariant anomalies in gauge and gravitational theories, Nuclear Phys. B 244 (1984), 421-453.
  101. Zumino B., Wu Y.S., Zee A., Chiral anomalies, higher dimensions, and differential geometry, Nuclear Phys. B 239 (1984), 477-507.
  102. Leigh R.G., Strassler M.J., Exactly marginal operators and duality in four dimensional N=1 supersymmetric gauge theory, Nuclear Phys. B 447 (1995), 95-133, hep-th/9503121.
  103. Mondragón M., Zoupanos G., Higgs mass prediction in finite unified theories, Acta Phys. Polon. B 34 (2003), 5459-5468.
  104. Hamidi S., Patera J., Schwarz J.H., Chiral two-loop-finite supersymmetric theories, Phys. Lett. B 141 (1984), 349-352.
  105. Jiang X.D., Zhou X.J., Finite N=1 supersymmetric theories of classical groups, Phys. Lett. B 216 (1989), 160-166.
  106. Jiang X.D., Zhou X.J., Finite N=1 supersymmetric theories of SU(N), Phys. Lett. B 197 (1987), 156-160.
  107. Jones D.R.T., Raby S., A two-loop finite supersymmetric SU(5) theory: towards a theory of fermion masses, Phys. Lett. B 143 (1984), 137-141.
  108. León J., Pérez-Mercader J., Quirós M., Ramírez-Mittelbrunn J., A sensible finite SU(5) SUSY GUT?, Phys. Lett. B 156 (1985), 66-72.
  109. Kazakov D.I., Kalmykov M.Y., Kondrashuk I.N., Gladyshev A.V., Softly broken finite supersymmetric grand unified theory, Nuclear Phys. B 471 (1996), 389-408, hep-ph/9511419.
  110. Yoshioka K., Finite SUSY GUT revisited, Phys. Rev. D 61 (2000), 055008, 14 pages, hep-ph/9705449.
  111. Hamidi S., Schwarz J.H., A realistic finite unified theory?, Phys. Lett. B 147 (1984), 301-306.
  112. Babu K.S., Enkhbat T., Gogoladze I., Finite grand unified theories and the quark mixing matrix, Phys. Lett. B 555 (2003), 238-247, hep-ph/0204246.
  113. Heinemeyer S., Mondragón M., Zoupanos G., Confronting finite unified theories with low-energy phenomenology, J. High Energy Phys. 2008 (2008), no. 7, 135, 8 pages, arXiv:0712.3630.
  114. Carena M.S., Garcia D., Nierste U., Wagner C.E.M., Effective Lagrangian for the tbH+ interaction in the MSSM and charged Higgs phenomenology, Nuclear Phys. B 577 (2000), 88-120, hep-ph/9912516.
  115. Amsler C. et al., Review of particle physics, Phys. Lett. B 667 (2008), 1-6.
  116. Kobayashi T., Kubo J., Mondragón M., Zoupanos G., Finite unification, Surveys High Energ. Phys. 16 (2001), 87-129.
  117. Barate R. et al., A measurement of the inclusive bsγ branching ratio, Phys. Lett. B 429 (1998), 169-187.
  118. Chen S. et al., Branching fraction and photon energy spectrum for bsγ, Phys. Rev. Lett. 87 (2001), 251807, 5 pages, hep-ex/0108032.
  119. Koppenburg P. et al., An inclusive measurement of the photon energy spectrum in bsγ decays, Phys. Rev. Lett. 93 (2004), 061803, 6 pages, hep-ex/0403004.
  120. Heavy Flavor Analysis Group, see http://www.slac.stanford.edu/xorg/hfag/.
  121. Heinemeyer S., Hollik W., Weiglein G., FeynHiggs: a program for the calculation of the masses of the neutral CP-even Higgs bosons in the MSSM, Comput. Phys. Comm. 124 (2000), 76-89, hep-ph/9812320.
  122. Heinemeyer S., Hollik W., Weiglein G., The masses of the neutral CP-even Higgs bosons in the MSSM: accurate analysis at the two-loop level, Eur. Phys. J. C 9 (1999), 343-366, hep-ph/9812472.
  123. Degrassi G., Heinemeyer S., Hollik W., Slavich P., Weiglein G., Towards high-precision predictions for the MSSM Higgs sector, Eur. Phys. J. C 28 (2003), 133-143, hep-ph/0212020.
  124. Frank M. et al., The Higgs boson masses and mixings of the complex MSSM in the Feynman-diagrammatic approach, J. High Energy Phys. 2007 (2007), no. 2, 047, 56 pages, hep-ph/0611326.
  125. Davier M., Hoecker A., Malaescu B., Yuan C.Z., Zhang Z., Reevaluation of the hadronic contribution to the muon magnetic anomaly using new e+e → π+π cross section data from BABAR, Eur. Phys. J. C 66 (2010), 1-9, arXiv:0908.4300.
  126. Goldberg H., Constraint on the photino mass from cosmology, Phys. Rev. Lett. 50 (1983), 1419-1422.
  127. Ellis J.R., Hagelin J.S., Nanopoulos D.V., Olive K.A., Srednicki M., Supersymmetric relics from the big bang, Nuclear Phys. B 238 (1984), 453-477.
  128. Valle J.W.F., Neutrino mass: theory, data and interpretation, hep-ph/9907222.
  129. Dreiner H.K., An introduction to explicit R-parity violation, Pramana 51 (1998), 123-133, hep-ph/9707435.
  130. Bhattacharyya G., A brief review of R-parity-violating couplings, hep-ph/9709395.
  131. Allanach B.C., Dedes A., Dreiner H.K., Bounds on R-parity violating couplings at the weak scale and at the GUT scale, Phys. Rev. D 60 (1999), 075014, 10 pages, hep-ph/9906209.
  132. Romão J.C., Valle J.W.F., Neutrino masses in supersymmetry with spontaneously broken R-parity, Nuclear Phys. B 381 (1992), 87-108.
  133. Lyth D.H., Stewart E.D., Thermal inflation and the moduli problem, Phys. Rev. D 53 (1996), 1784-1798, hep-ph/9510204.
  134. Gelmini G.B., Gondolo P., Neutralino with the right cold dark matter abundance in (almost) any supersymmetric model, Phys. Rev. D 74 (2006), 023510, 5 pages, hep-ph/0602230.
  135. Belanger G., Boudjema F., Pukhov A., Semenov A., micrOMEGAs: a program for calculating the relic density in the MSSM, Comput. Phys. Comm. 149 (2002), 103-120, hep-ph/0112278.
  136. Belanger G., Boudjema F., Pukhov A., Semenov A., MicrOMEGAs: Version 1.3, Comput. Phys. Comm. 174 (2006), 577-604, hep-ph/0405253.
  137. Ma E., Mondragón M., Zoupanos G., Finite SU(N)k unification, J. High Energy Phys. 2004 (2004), no. 12, 026, 17 pages, hep-ph/0407236.
  138. Glashow S.L., Trinification of all elementary particle forces, in Proceedings of Fifth Workshop on Grand Unification, Editors K. Kang, H. Fried and P. Frampton, World Scientific, Singapore, 1984, 88-94.
  139. Lazarides G., Panagiotakopoulos C., Shafi Q., Supersymmetric unification without proton decay, Phys. Lett. B 315 (1993), 325-330, hep-ph/9306332.
  140. Lazarides G., Panagiotakopoulos C., MSSM from SUSY trinification, Phys. Lett. B 336 (1994), 190-193, hep-ph/9403317.
  141. Ma E., Particle dichotomy and left-right decomposition of E6 superstring models, Phys. Rev. D 36 (1987), 274-276.
  142. Heinemeyer S., Ma E., Mondragón M., Zoupanos G., Finite SU(3)3 unification, in preparation.
  143. Dirac P.A.M., Lectures on quantum field theory, Academic Press Inc., 1967.
  144. Dyson F.J., Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85 (1952), 631-632.
  145. Weinberg S., Living with infinities, arXiv:0903.0568 (and references therein).

Previous article   Next article   Contents of Volume 6 (2010)