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Why Does the Standard Model Need the Higgs Boson ?

Why Does the Standard Model Need the Higgs Boson ?. I. Augusto Barroso. Sesimbra 2007. Laws of Nature are invariant under: Lorentz Transformations Space x Time Translations. Poincaré Group. The building blocks of any QFT must be Irreducible Representations of the Poincaré Group.

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Why Does the Standard Model Need the Higgs Boson ?

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  1. Why Does the Standard Model Need the Higgs Boson ? I Augusto Barroso Sesimbra 2007

  2. Laws of Nature are invariant under: Lorentz Transformations Space x Time Translations Poincaré Group The building blocks of any QFT must be Irreducible Representations of the Poincaré Group

  3. Representations of the Poincaré group are classified according to the values of and Eigenvalues:

  4. Rotations Boosts Lorentz Transformations Generators

  5. Representations Two SU(2) Algebras (0,0) Scalar (1/2,0) two dimensional (0,1/2) two dimensional Parity Because QED conserves parity we have to stack up both representations. Electron belongs to (1/2,0)+(0,1/2). But we obtain 4 degrees of freedom! Discover the Positron!

  6. The Standard Model • SU(3)xSU(2)xU(1) QFT • This means: 1 - We know the symmetries of the building blocks; 2 - We have a general principle to obtain the equations of motion; 3 - We don’t know how to solve these equations. But we know how to use perturbation theory.

  7. Example: QED • - Building Blocks • Symmetries • Lagrangian • Equations of Motion

  8. LEP tested: Perturbative QCD Weak loop corrections Praise the SM I (LEP working group) Direct

  9. Praise the SM II • MH < 144 GeV (95%CL) • Probability MH>114 GeV:15%

  10. Difference between and • Spin ½ • Spin 1 Two degrees of freedom Two degrees of freedom Three degrees of freedom Two degrees of freedom

  11. Frame S Frame S’(Rest frame) Lorentz boost T L

  12. Scattering W- W- W+ q Incoming W+ Outgoing W- W+

  13. Scattering

  14. Examine the amplitudes and take the Limit S-channel Photon + Z

  15. T-channel Photon+Z

  16. Examine the amplitudes in the Limit Summary 4 W vertex The gauge structure cancels the quadratic term. But the sum grows as s! Amplitudes can grow at most as Froissart bound

  17. Let us save the theory. Invent a Scalar H. H s-channel H t-channel

  18. The scalar Boson H couples to W with lH Then, there are two additional amplitudes: In the high energy limit the sum is:

  19. The “normal” contributions give: The scalar contributions are: The divergent terms cancel if A massive spin 1 particle “cries out” for a spin zero companion coupled with a strength proportional to g.

  20. Summary • Massive Spin 1 particles have a bad High Energy Behaviour; • The introduction of a Spin 0 companion can cure the disease; • There is no known alternative medicine.

  21. References • LEP and SLD working groups, “Precison Electroweak Measurements on the Z Resonance”, Phys. Reports, 2007. • A. Djouadi, “The Anatomy of Electro-Weak Symmetry Breaking” Tome I: The Higgs Boson in the Standard Model”, Phys. Rep, December 2007 or January 2008.

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