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 a l are the spin l partial waves s=center of mass energy-squared

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

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

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 - A(W L + W L - →W L + W L - ) =A(  +  - →  +  - )+O(M W 2 /s) Electroweak Equivalence theorem:  are Goldstone bosons which become the longitudinal components of massive W and Z gauge bosons

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

10.  +  -  +  - Two interesting limits: –s, t >> M H 2 –s, t << M H 2

11. Use Unitarity to Bound Higgs High energy limit: Heavy Higgs limit M H < 800 GeV E c  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) (q) e(k) k=p-p + =p - -q W + (p + ) 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 Contributions which grow with energy cancel between t- and s- channel diagrams Depends on special form of 3-gauge boson couplings Z(k) W + (p + ) W  - (p - ) (p) (q)

15. No deviations from SM at LEP2 LEP EWWG, hep-ex/0312023 No evidence for Non-SM 3 gauge boson vertices Contribution which grows like m e 2 s cancels between Higgs diagram and others

16. Limits on Scalar Potential M H 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 M H If Standard Model valid to 10 16 GeV For any given scale, , there is a theoretically consistent range for M H

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 H 4 interactions, theory is non- interacting Require quartic coupling be finite Requirement for 1/ (Q)>0 gives upper limit on M h Assume theory is valid to 10 16 GeV –Gives upper limit of M H < 180 GeV

22. Bounds on SM Higgs Boson If SM valid up to Planck scale, only a small range of allowed Higgs Masses  (GeV) M H (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  M F  M F –M F =0, then quantum corrections vanish –When M F =0, Lagrangian is invariant under  L  e i  L  L  R  e i  R  R –M F  0 increases the symmetry of the theory –Yukawa coupling (proportional to mass) breaks symmetry and so corrections  M F

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

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

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

30. What’s the problem? Compute M h in dimensional regularization and absorb infinities into definition of M H 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)