What do we know about the Standard Model?

1. What do we know about the Standard Model? Sally Dawson Lecture 2 SLAC Summer Institute

2. The Standard Model Works • Any discussion of the Standard Model has to start with its success • This is unlikely to be an accident

3. Theoretical Limits on Higgs Sector • Unitarity • Really we mean perturbative unitarity • Violation of perturbative unitarity leads to consideration of strongly interacting models of EWSB such as technicolor, Higgless • Consistency of Standard Model • Triviality (What happens to couplings at high energy?) • Does spontaneous symmetry breaking actually happen? • Naturalness • Renormalization of Higgs mass is different than renormalization of fermion mass • One motivation for supersymmetric models

4. Unitarity • Consider 2  2 elastic scattering • Partial wave decomposition of amplitude • al are the spin l partial waves s=center of mass energy-squared

5. Unitarity • Pl(cos) are Legendre polynomials: Sum of positive definite terms

6. More on Unitarity • Optical theorem • Unitarity requirement: Optical theorem derived assuming only conservation of probability Im(al) Re(al)

7. More on Unitarity • Idea: Use unitarity to limit parameters of theory Cross sections which grow with energy always violate unitarity at some energy scale

8. Example: W+W-W+W- Electroweak Equivalence theorem: A(WL+WL -→WL+WL-) =A(+ -→ +-)+O(MW2/s)  are Goldstone bosons which become the longitudinal components of massive W and Z gauge bosons

9. W+W-W+W- • Consider Goldstone boson scattering: +-+ • Recall scalar potential

10. +-+- • Two interesting limits: • s, t >> MH2 • s, t << MH2

11. Use Unitarity to Bound Higgs • High energy limit: • Heavy Higgs limit MH < 800 GeV Ec 1.7 TeV  New physics at the TeV scale Can get more stringent bound from coupled channel analysis

12. Consider W+W- pair production Example: W+W- • t-channel amplitude: • In center-of-mass frame: (p) W+(p+) k=p-p+=p--q e(k) (q) W-(p-)

13. W+W- pair production, 2 • Interesting physics is in the longitudinal W sector: • Use Dirac Equation: pu(p)=0 Grows with energy

14. W+W- pair production, 3 • SM has additional contribution from s-channel Z exchange • For longitudinal W’s (p) W+(p+) Z(k) (q) W-(p-) Contributions which grow with energy cancel between t- and s- channel diagrams Depends on special form of 3-gauge boson couplings

15. No deviations from SM at LEP2 No evidence for Non-SM 3 gauge boson vertices Contribution which grows like me2s cancels between Higgs diagram and others LEP EWWG, hep-ex/0312023

16. Limits on Scalar Potential • MH is a free parameter in the Standard Model • Can we derive limits on the basis of consistency? • Consider a scalar potential: • This is potential at electroweak scale • Parameters evolve with energy

17. High Energy Behavior of  • Renormalization group scaling • Large  (Heavy Higgs): self coupling causes  to grow with scale • Small  (Light Higgs): coupling to top quark causes  to become negative

18. Does Spontaneous Symmetry Breaking Happen? • SM requires spontaneous symmetry • This requires • For small  • Solve

19. Does Spontaneous Symmetry Breaking Happen? • () >0 gives lower bound on MH • If Standard Model valid to 1016 GeV • For any given scale, , there is a theoretically consistent range for MH

20. What happens for large ? • Consider HH→HH • (Q) blows up as Q (called Landau pole)

21. Landau Pole • (Q) blows up as Q, independent of starting point • BUT…. Without H4 interactions, theory is non-interacting • Require quartic coupling be finite • Requirement for 1/(Q)>0 gives upper limit on Mh • Assume theory is valid to 1016 GeV • Gives upper limit of MH< 180 GeV

22. Bounds on SM Higgs Boson • If SM valid up to Planck scale, only a small range of allowed Higgs Masses MH (GeV)  (GeV)

23. Naturalness • We often say that the SM cannot be the entire story because of the quadratic divergences of the Higgs Boson mass • Renormalization of scalar and fermion masses are fundamentally different

24. Masses at one-loop • First consider a fermion coupled to a massive complex Higgs scalar • Assume symmetry breaking as in SM:

25. Masses at one-loop • Calculate mass renormalization for  To calculate with a cut-off, see my Trieste notes

26. Symmetry and the fermion mass • MF  MF • MF=0, then quantum corrections vanish • When MF=0, Lagrangian is invariant under • LeiLL • ReiRR • MF0 increases the symmetry of the theory • Yukawa coupling (proportional to mass) breaks symmetry and so corrections  MF

27. Scalars are very different • MH diverges quadratically! • This implies quadratic sensitivity to high mass scales

28. Scalars • MH diverges quadratically • Requires large cancellations (hierarchy problem) • H does not obey decoupling theorem • Says that effects of heavy particles decouple as M • MH0 doesn’t increase symmetry of theory • Nothing protects Higgs mass from large corrections

29. Light Scalars are Unnatural • Higgs mass grows with  • No additional symmetry for MH=0, no protection from large corrections H H MH 200 GeV requires large cancellations

30. What’s the problem? • Compute Mh in dimensional regularization and absorb infinities into definition of MH • Perfectly valid approach • Except we know there is a high scale

31. Try to cancel quadratic divergences by adding new particles • SUSY models add scalars with same quantum numbers as fermions, but different spin • Little Higgs models cancel quadratic divergences with new particles with same spin

32. We expect something at the TeV scale • If it’s a SM Higgs then we have to think hard about what the quadratic divergences are telling us • SM Higgs mass is highly restricted by requirement of theoretical consistency • Expect that Tevatron or LHC will observe SM Higgs (or definitively exclude it)