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What do we know about the Standard Model? Sally Dawson Lecture 4 TASI, 2006

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

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Unitarity Consider 2 2 elastic scattering Partial wave decomposition of amplitude a l are the spin l partial waves

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

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More on Unitarity Optical theorem Unitarity requirement: Optical theorem derived assuming only conservation of probability

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

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Example 1: W + W - W + W - Recall scalar potential (Include Goldstone Bosons in Unitarity gauge) Consider Goldstone boson scattering: + - +

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+ - + - Two interesting limits: s, t >> M h 2 s, t << M h 2

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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

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Electroweak Equivalence Theorem This is a statement about scattering amplitudes, NOT individual Feynman diagrams

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Plausibility argument for Electroweak Equivalence Theorem Compute (h W L + W L - ) for M h >>M W (h WW) M h for M h 1.4 TeV

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Landau Pole 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 in a calculable way

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Consider hh hh Real scattering, s+t+u=4M h 2 Consider momentum space-like and off-shell: s=t=u=Q 2 <0 Tree level: iA 0 =-6i

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hh hh, #2 One loop: A=A 0 +A s +A t +A u

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hh hh, #3 Sum the geometric series to define running coupling (Q) blows up as Q (called Landau pole)

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hh hh, #4 This is independent of starting point BUT…. Without 4 interactions, theory is non- interacting Require quartic coupling be finite

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hh hh, #5 Use =M h 2 /(2v 2 ) and approximate log(Q/M h ) log(Q/v) Requirement for 1/ (Q)>0 gives upper limit on M h Assume theory is valid to 10 16 GeV Gives upper limit on M h < 180 GeV Can add fermions, gauge bosons, etc.

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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

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Does Spontaneous Symmetry Breaking Happen? SM requires spontaneous symmetry This requires For small Solve

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Does Spontaneous Symmetry Breaking Happen? (#2) ( ) >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

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Bounds on SM Higgs Boson If SM valid up to Planck scale, only a small range of allowed Higgs Masses

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Problems with the Higgs Mechanism We often say that the SM cannot be the entire story because of the quadratic divergences of the Higgs Boson mass

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

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Masses at one-loop, #2 Calculate mass renormalization for

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Renormalized fermion mass Do integral in Euclidean space

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Renormalized fermion mass, #2 Renormalization of fermion mass:

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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 threoy Yukawa coupling (proportional to mass) breaks symmetry and so corrections m F

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

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Scalars (#2) M h diverges quadratically! Requires large cancellations (hierarchy problem) Can do this in Quantum Field Theory 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

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M h 200 GeV requires large cancellations Higgs mass grows with scale of new physics, No additional symmetry for M h =0, no protection from large corrections hh Light Scalars are Unnatural

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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

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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

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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

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