What do we know about the Standard Model?

Sally Dawson Lecture 4 TASI, 2006 The Standard Model Works

‰Any discussion of the Standard Model has to start with its success

‰This is unlikely to be an accident! Unitarity

„ Consider 2 → 2 elastic scattering dσ 1 2 = A dΩ 64π 2s

„ Partial wave decomposition of amplitude ∞ A =16π ∑(2l +1)Pl (cosθ )al l=0

„ al are the spin l partial waves Unitarity

„ Pl(cosθ) are Legendre polynomials:

1 2δl,l′ dxPl (x)Pl′ (x) = ∫−1 2l +1

∞ ∞ 1 8π * σ = (2l +1) (2l′ +1)al al′ d cosθPl (cosθ )Pl′ (cosθ ) ∑ ∑ ∫−1 s l=0 l′=0 ∞ 16π 2 = ∑(2l +1) al s l=0 Sum of positive definite terms More on Unitarity

∞ 1 16π 2 „ Optical theorem σ = Im[]A(θ = 0) = ∑(2l +1) al s s l=0

2 Optical theorem derived Im(al ) = al assuming only conservation of probability

„ Unitarity requirement: 1 Re(a ) ≤ l 2 More on Unitarity

„ Idea: Use unitarity to limit parameters of theory

Cross sections which grow with energy always violate unitarity at some energy scale Example 1: W+W-→W+W-

„ Recall scalar potential (Include Goldstone Bosons in Unitarity gauge) M 2 M 2 V = h h2 + h h()h2 + z 2 + 2ω +ω − 2 2v 2 M 2 + h ()h2 + z 2 + 2ω +ω − 8v2

„ Consider Goldstone boson scattering: 2 + - + M 2 ⎛ M 2 ⎞ i ω ω →ω ω iA(ω +ω → ω +ω − ) = −2i h + ⎜− i h ⎟ 2 ⎜ ⎟ 2 v ⎝ v ⎠ t − M h 2 ⎛ M 2 ⎞ i +⎜− i h ⎟ ⎜ ⎟ 2 ⎝ v ⎠ s − M h ω+ω-→ω+ω-

„ Two interesting limits: 2 ‰ s, t >> Mh 2 2 0 M h + − + − M a → − A(ω ω → ω ω ) → −2 h 0 2 v2 8πv

2 ‰ s, t << Mh s + − + − u 0 A(ω ω → ω ω ) → − a0 → − 2 v2 32πv Use Unitarity to Bound Higgs 1 Re(a ) ≤ l 2 „ High energy limit:

2 0 M h a → − Mh < 800 GeV 0 8πv2

„ Heavy Higgs limit

s Ec ∼1.7 TeV a0 → − 0 32πv2 → New physics at the TeV scale

Can get more stringent bound from coupled channel analysis Electroweak Equivalence Theorem

1 N 1 N′ N N′ A(VL ...VL →VL ...VL ) = (i) (−i) A(ω1...ωN → ω1...ωN′ ) ⎛ M 2 ⎞ +O⎜ W ⎟ ⎜ 2 ⎟ ⎝ E ⎠

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

+ - „ Compute Γ(h→WL WL ) for Mh>>MW

µν iA = − igMW g ε µεν

p+ ⋅ p− = − igMW 2 MW 2 1 r p M h ε L = ( p ,0,0, p0 ) ≈ → − ig MW MW 2MW

G M 3 Γ(h→WW) ≈ M Γ(h →W +W − ) ≈ F h h L L 8π 2 for Mh ≈1.4 TeV = Γ(h → ω +ω − ) Landau Pole

„ Mh is a free parameter in the Standard Model „ Can we derive limits on the basis of consistency?

„ Consider a scalar potential: M 2 λ V = h h2 + h4 2 4

„ This is potential at electroweak scale

„ Parameters evolve with energy in a calculable way Consider hh→hh

2 „ Real scattering, s+t+u=4Mh „ Consider momentum space-like and off-shell: s=t=u=Q2<0

„ Tree level: iA0=-6iλ hh→hh, #2

„ One loop: 1 d nk i i iA = (−6iλ)2 s ∫ 2 2 2 2 2 2 (2π ) k − M h (k + p + q) − M h 2 9λ −ε = (4πµ 2 )Γ(ε )()M 2 − Q2 x(1− x) 8π 2 h

„ A=A0+As+At+Au

⎛ 9λ 2 2 2 −ε ⎞ A = −6λ⎜1+ (4πµ )Γ(ε )()M h − Q x(1− x) +...⎟ ⎝ 16π 2 ⎠ hh→hh, #3

„ Sum the geometric series to define running coupling ⎛ 9λ Q2 ⎞ A = −6λ⎜1+ log ⎟ +... ⎜ 2 2 ⎟ ⎝ 16π M h ⎠ 6λ A= ≡ 6λ(Q) 9λ ⎛ Q ⎞ 1− log⎜ ⎟ 2 ⎜ 2 ⎟ 8π ⎝ M h ⎠ „ λ(Q) blows up as Q→∞ (called Landau pole) hh→hh, #4

„ This is independent of starting point

„ BUT…. Without λφ4 interactions, theory is non- interacting

„ Require quartic coupling be finite

1 > 0 λ(Q) hh→hh, #5

2 2 „ Use λ=Mh /(2v ) and approximate log(Q/Mh) → log(Q/v)

„ Requirement for 1/λ(Q)>0 gives upper limit on Mh 32π 2v2 M 2 < h ⎛ Q2 ⎞ 9log⎜ ⎟ ⎜ 2 ⎟ ⎝ v ⎠ „ Assume theory is valid to 1016 GeV

‰ Gives upper limit on Mh< 180 GeV „ Can add fermions, gauge bosons, etc. High Energy Behavior of λ

1 1 ⎛ Q ⎞ „ Renormalization group scaling = + (...)log⎜ ⎟ λ(Q) λ(µ) ⎝ µ ⎠ dλ 16π 2 =12λ2 +12λg 2 −12g 4 + (gauge) dt t t

2 ⎛ Q ⎞ M t t ≡ log⎜ ⎟ g = ⎜ 2 ⎟ t ⎝ µ ⎠ v „ Large λ (Heavy Higgs): self coupling causes λ to grow with scale „ Small λ (Light Higgs): coupling to top quark causes λ to become negative Does Spontaneous Symmetry Breaking Happen?

„ SM requires spontaneous symmetry

„ This requires V (v) < V (0)

„ For small λ dλ 16π 2 ≈ −16g 4 dt t

3g 4 ⎛ Λ2 ⎞ „ Solve λ(Λ) ≈ λ(v) − t log⎜ ⎟ 2 ⎜ 2 ⎟ 4π ⎝ v ⎠ Does Spontaneous Symmetry Breaking Happen? (#2)

„ λ(Λ) >0 gives lower bound on Mh 3v2 ⎛ Λ2 ⎞ M 2 > log⎜ ⎟ h 2 ⎜ 2 ⎟ 2π ⎝ v ⎠ „ If Standard Model valid to 1016 GeV

M h > 130 GeV

„ For any given scale, Λ, there is a theoretically consistent range for Mh Bounds on SM Higgs Boson

„ If SM valid up to Planck scale, only a small range of allowed Higgs Masses 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 Masses at one-loop

„ First consider a fermion coupled to a massive complex Higgs scalar

2 2 L = Ψ(i∂)Ψ + ∂ µφ − ms φ − (λF ΨLΨRφ + h.c.)

„ Assume symmetry breaking as in SM: (h + v) λ v φ = m = F 2 F 2 Masses at one-loop, #2

„ Calculate mass renormalization for Ψ

2 ⎛ − iλ ⎞ d 4k k + m − iΣ ( p) = ⎜ F ⎟ (i)2 F F ∫ 4 2 2 2 2 ⎝ 2 ⎠ (2π ) [k − mF ][(k − p) − ms ] Renormalized fermion mass

δmF = ΣF ( p) p=mF λ2 1 m (1+ x) = i F dx d 4k′ F 4 ∫ ∫ 2 2 2 2 2 32π 0 [k′ − mF x − ms (1− x)]

„ Do integral in Euclidean space

k0 → ik4 4 4 d k′ → id kE 2 2 2 2 r 2 r 2 k′ = k0 − k → k4 − k = −kE

Λ2 d 4k f (k 2 ) = π 2 y dy f (y) ∫∫E E 0 Renormalized fermion mass, #2

„ Renormalization of fermion mass:

2 λ2 m 1 Λ y dy δm = − F F dx (1+ x) F 2 ∫∫2 2 2 2 32π 00[y + mF x + ms (1− x)]

3λ2 m ⎛ Λ2 ⎞ = − F F log⎜ ⎟ +..... 2 ⎜ 2 ⎟ 32π ⎝ mF ⎠ Symmetry and the fermion mass

„ δmF ≈ mF

‰ mF=0, then quantum corrections vanish

‰ When mF=0, Lagrangian is invariant under iθL „ ΨL→e ΨL iθR „ ΨR→e ΨR

‰ mF→0 increases the symmetry of the threoy ‰ Yukawa coupling (proportional to mass) breaks

symmetry and so corrections ≈ mF Scalars are very different 4 2 d k Tr[(k + mF )((k − p) + mF )] 2 ⎛ − iλF ⎞ 2 4 2 2 2 2 − iΣs ( p ) = −⎜ ⎟ (i) (2π ) (k − mF )[(k − p) − mF ] ⎝ 2 ⎠ ∫ λ2 Λ2 ⎛ Λ ⎞ δM 2 = Σ (m2 ) = − F + ()m2 − m2 log⎜ ⎟ h S s 2 s F ⎜ ⎟ 8π ⎝ mF ⎠ 2 2 m ⎛ ⎛ m ⎞⎞ ⎛ 1 ⎞ 1 2 s ⎜ ⎜ s ⎟⎟ +(2mF − ) 1+ I1 2 + O⎜ 2 ⎟ I (a) = dx log()1− ax(1− x) 2 ⎜ ⎜ m ⎟⎟ Λ 1 ∫ ⎝ ⎝ F ⎠⎠ ⎝ ⎠ 0

„ Mh diverges quadratically! „ This implies quadratic sensitivity to high mass scales Scalars (#2)

„ Mh 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→∞

„ Mh→0 doesn’t increase symmetry of theory ‰ Nothing protects Higgs mass from large corrections Light Scalars are Unnatural • Higgs mass grows with scale of new physics, Λ

• No additional symmetry for Mh=0, no protection from large corrections

hh

G 2 δ M 2 = F (6M 2 +3M 2 + M 2 −12M 2 ) h 4 2π 2 Λ W Z h t 2 ⎛ Λ ⎞ =−⎜ 200 GeV⎟ ⎝ 0.7 TeV ⎠

Mh ≤ 200 GeV requires large cancellations What’s the problem?

„ Compute Mh in dimensional regularization and absorb infinities into definition of Mh

1 M 2 = M 2 + (...) h h0 ε

„ Perfectly valid approach

„ Except we know there is a high scale 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 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