Effective Field Theory

Transcription

1 Effective Field Theory Iain Stewart MIT The 19 th Taiwan Spring School on Particles and Fields April, 2006

2 Physics compartmentalized Quantum Field Theory String Theory? General Relativity short distance quantum gravity electro-weak QCD & quarks Quantum Mechanics Classical Mechanics Special Relativity Classical Electricity & Magnetism Newtonian Gravity long distance nuclei atoms chemistry us But, one doesn t need nuclear physics to build a boat Generality vs. Precision

3 Dynamics at long distance do not depend on the details of what happens at short distance In the quantum realm, λ p1, wavelength and momentum are related, so Low energy interactions do not depend on the details of high energy interactions Good: we can focus on the relevant interactions & degrees of freedom calculations are simpler Bad: Newton didn t need quantum gravity for projectile motion Phew! we have to work harder to probe the interesting physics at short distances

4 Lecture I Outline Top Principles, Operators, Power Counting, Bottom NRQED, Why the Sky is Blue, Weak Decays Operator Counting, Matching, Using Equations of Motion Quantum Loops, Renormalization and Decoupling Summing Large Logarithms Lecture II Lecture III QCD, α s matching, Heavy Quark Effective Theory (An Effective Theory for Static Sources) Soft - Collinear Effective Theory

5 Masses particle mass γ < ev gluon 0 (theory) ν e, ν µ, ν τ m 2 sun = ev 2 m 2 atm = ev 2 e MeV u quark 4 MeV d quark 7 MeV µ 106 MeV s quark 120 MeV c quark 1.4 GeV τ 1.78 GeV b quark 4.5 GeV W boson 80.4 GeV Z boson 91.2 GeV higgs > 114 GeV t quark 174 GeV m proton 1 GeV

6 Lecture I

7 EFT Concepts 1) The ingredients in any EFT: Relevant degrees of freedom, symmetries, scales & power counting 2) Renormalization: Meaning of parameters 3) Decoupling: Effects from Heavy Particles are suppressed 4) Matching: How we can encode dynamics of one theory into another 5) Running: Connecting physics at different momentum scales

8 Example: Hydrogen non-relativistic quantum mechanics parameters: degrees of freedom: mass charges coupling m e Q e, Q p α = e- proton scales: m p = 938 MeV m e = MeV p m e α = 3.7 kev E n = m eα 2 2n 2 = 13.6 ev n 2 (a Bohr ) 1 + corrections Why not quarks? QCD? b-quark charge?? spin? m proton e +? weak force?

9 ( p L(p, e, γ, b; α, m b ) = L(p, e, γ; α 2 ) + O m 2 b b e p coupling changed, it runs: α(0) = α(m W ) = ) suppressed Insensitive to quarks in proton: q p γ m e α (proton size) 1 Λ QCD 200 MeV q q short distance long distance r = Λ 1 QCD

10 Insensitive to proton mass: m e α m p 1 GeV hyperfine splitting (static proton suffices) m2 e m p µ e µ p α 4 from QCD but universal Experiment for m p, µ p is more accurate that QCD computations

11 Non-relativistic Lagrangian m e α m e (no e + ) expand in p m e m p m e,, α QED NRQED short distance theory is more general L NRQED = L 0 + long distance theory where its easier to compute exact answer is irrelevant, work to the desired level of precision n=1 ɛ n L n Leading Order L 0 = ψ ( i 0 2 2m e +... ) ψ + Ψ i 0 Ψ + V (ψ ψ)(ψ Ψ)

12 Non-relativistic Lagrangian m e α m e (no e + ) expand in p m e m p m e,, α QED NRQED short distance theory is more general L NRQED = L 0 + long distance theory where its easier to compute exact answer is irrelevant, work to the desired level of precision n=1 ɛ n L n Symmetries of QED constrain the form of NRQED: Charge conjugation ( ) Parity ( x x ) Time-Reversal ( ) t t Gauge Symmetry Spin-Statistics Theorem e + e constrain the s L n

13 Why L QED = ψ(id/ m)ψ 1 4 F µν F µν? Why L SM? Use: Observed d.o.f. Symmetries Lorentz Invariance Gauge Invariance Unitarity (Hermitian L ) SU(3) SU(2) U(1) Renormalizability ( ) (absorb divergences from loops in finite # of parameters) this was not in our list! Modern Definition: L = L 0 + renormalizable order by order in power expansion ɛ n L n n=1 Ken Wilson

14 Power Counting in Mass Dimension Dimension of Operators (Marginal, Irrelevant, Relevant Operators) Consider S[φ] = d d x ( 1 2 µφ µ φ 1 2 m2 φ 2 λ 4! φ4 τ 6! φ6) (board)

15 ( ) eg. Standard Model C = iγ 2 γ 0 L = L dim [ ] Λ new L T L i Cɛ ijh j H l ɛ lk L k L Left-handed Lepton doublet + L dim Higgs SU(2) doublet ( νl e L ) Exercise 1: Show that this is most general dim-5 operator that is consistent with the symmetries of the Standard Model When the Higgs gets a vev this gives a small Majorana Neutrino Mass

16 eg. Why the Sky is Blue Low energy scattering of photons from neutral atoms in their ground state E γ E m e α 2 a 1 0 m e α M atom Let v µ = (1, 0, 0, 0), atoms are static L = φ v i 0 φ v = φ v iv φ v φ v is field which destroys an atom, mass dim. [φ v ] = 3/2 Interactions are constrained by gauge invariance, parity & charge conjugation. Consider even number of F µν s L int = τ 1 φ v φ vf µν F µν + τ 2 φ v φ vv λ F λµ v σ F σµ even no F µν here [F µν ] = 2 so [τ 1 ] = [τ 2 ] = 3 µ F µν = 0, v µ µ φ v = 0 Very low energy photons do not probe inside the atom, so expect cross section to depend on size of atom: τ 1 τ 2 a 3 0 [σ] = 2 σ A 2 τ 2 i so σ E 4 γ a 6 0 blue light is scattered stronger than red light

17 Exercise 2: Consider QED for photon momenta much less than the mass of the electron. By constructing an appropriate operator in a low energy EFT, estimate the cross section for γγ γγ with 10 kev photons. Include factors of e in your estimate, by considering which QED graphs generate your operator.

18 Can we really use equations of motion to simplify operators? µ F µν = 0, v µ µ φ v = 0 eg. p 2 τ 1 L = 1 2 µφ µ φ 1 2 m2 φ 2 λφ 4 + τg 1 φ 6 + τg 2 φ 3 2 φ { e.o.m. 2 φ = m 2 φ 4λφ 3 L = 1 2 µφ µ φ 1 2 m2 φ 2 λ φ 4 + τg 1φ 6 λ = λ + m 2 g 2 τ g 1 = g 1 4λg 2 (board) Exercise 3: Demonstrate the equivalence for tree level 6-pt functions at, pre- and post- the use of the equations of motion. O(τ)

19 Regularization and Renormalization Regularization: How we cutoff UV infinities in loop integrals Renormalization: How we pick a scheme to give definite meaning to parameters of a theory Computations are easier if our regulator preserve symmetries and preserves power counting by not mixing up terms of different order in the expansion Dimensional Regularization Linearity: Scaling: Translation: d d p(p 2 ) α = { d d p d d p [af(p) + bf(p)] = a d d p f(p) + b d d p g(p) d d p f(sp) = s d d d p f(p) d d p f(p + q) = d d p f(p) = dp p d 1 dω d d = 4 2ɛ 0 for α < 4 and α > 4 no power divergences i 16π 2 ( 1 ɛ UV 1 ɛ IR ) = 0 for α = 4 be careful!

20 Rescale couplings to keep them dimensionless µ eg. g (0) = Z g µ ɛ g(µ) is dim.reg. parameter and acts like a resolution in MS scheme Mass independent regulator eg. L = ψ(i/ m)ψ a M 2 ( ψψ) m M, a 1 gives δm i a M 2 d 4 k (2π) 4 /k + m k 2 m 2 = i a M 2 d 4 k (2π) 4 m k 2 m 2 δm cutoff δm dim.reg. a M 2 Λ a M 2 m2 the cutoff indep. terms are hidden small as expected, power counting manifest

21 Decoupling Theorem If remaining low energy theory is renormalizable then all effects due to heavy particles appear as changes in the coupling constants or are suppressed by 1/M requires a physical renormalization scheme, not true in Solution: MS (board) e-!! We must implement decoupling by hand at µ m e β QED = e3 12π 2 1 α E = m e resolution µ = E β QED = hydrogen E (MeV )

22 Wilsonian vs. Continuum EFT Wilson e S Λ δλ = effective action for soft modes δλ dφ e S Λ e S Λ removing modes with Λ δλ < E < Λ E Λ hard modes soft modes Continuum operators for soft modes Wilson coefficients for hard modes L EFT = C(µ)O(µ) < E O C < Λ (µ) < Λ (µ) Λ E hard modes soft modes Sending Λ to double counts the hard region in matrix < Λ elments of our operators, but we fix (µ) to correct for this. µis the scale where this matching is done. C

23 Top Theory 1 is understood, but it is useful to have the simpler theory 2 at low energies. theory1 Integrate out heavier particles in 1 and match onto 2 L theory1 n Theory 1 & theory 2 agree in the IR, differ in UV eg. NRQED; Remove t,w,z: H weak ; L (n) theory2 Heavy Quark Effective Theory; Soft Collinear Effective Theory theory2 Bottom Theory 1 is unknown or matching is too difficult to carry out analytically Construct n L (n) theory2 by writing down most general set of interactions consistent with symmetries theory1? eg. Standard Model; Chiral Perturbation theory for low energy pion and Kaon interactions theory2

24 EFT Principles 1) Dynamics at low E does not depend on details of dynamics at high E 2) Build an EFT using the relevant d.o.f. and known symmetries. 3) EFT has an infinite number of operators, but only a finite number are needed for a given precision as determined by the power counting. With this precision this set closes under renormalization. 4) EFT has same infrared but different ultraviolet than the more fundamental theory. 5) Nature of high energy theory shows up as couplings and symmetries in the low energy EFT.

25 Effective Field Theories of QCD unstable particles * NRQCD cc states spectrum top quark SCET jets HQET * * bd states ChPT pions * QCD nuclear forces HTL * finite T finite density energetic hadrons perturbative QCD NNEFT * HDET SCET * Lattice QCD