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Effective Field Theory Effective Field Theory Iain Stewart MIT The 19’th Taiwan Spring School on Particles and Fields April, 2006 Physics compartmentalized short distance String quantum gravity Theory? electro-weak Quantum General Relativity Field Theory QCD & quarks Special nuclei Quantum Relativity Mechanics Newtonian atoms Gravity Classical Classical chemistry Mechanics Electricity & Magnetism long distance us But, one doesn’t need nuclear physics to build a boat Generality vs. Precision Dynamics at long distance do not depend on the details of what happens at short distance λ ∼ 1 In the quantum realm, p , 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 Newton didn’t need quantum gravity calculations are simpler for projectile motion Bad: Phew! we have to work harder to probe the interesting physics at short distances Outline Top Lecture I 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 α Lecture II QCD, s matching, Heavy Quark Effective Theory (An Effective Theory for Static Sources) Lecture III Soft - Collinear Effective Theory Masses particle mass 17 γ < 6 10− eV gluon ×0 (theory) 2 5 2 νe, νµ, ντ ∆msun = 8 10− eV 2 × 3 2 ∆matm = 2 10− eV e ×0.511 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 mproton ∼ 1 GeV Lecture I 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 Example: Hydrogen non-relativistic quantum mechanics parameters: mass me charges Qe , Qp 1 coupling α = degrees of 137 freedom: → ∞ scales: mp = 938 MeV e- me = 0.511 MeV 1 proton p meα = 3.7 keV (aBohr)− ∼ ∼ m α2 13.6 eV E = e = + corrections n − 2n2 − n2 Why not quarks? QCD? b-quark charge? e + ? weak force? m p r o t o n ? spin? Equations for Taiwan School Iain W. Stewart1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 −1 meα ! (proton size) ∼ ΛQCD ∼ 200 MeV meα ! mp ∼ 1 GeV meα ! me p2 • L p, e−, γ, b α, m L p, e−, γ α" O ( ; b) = ( ; ) + ! 2 " (1) mb e suppressed coupling changed, iα2 1 E4quatioµn2 s for Tai4wka2n School b it rsuns: Aα(0) =¯A 0 = TF (T ⊗ T 13)7 ln M − . (21g) k2 3 m2 15m12 1 ! " Iai#n W. Stewart $ p α(mW ) = 128 In Eq. (21) the matrix elemen1tCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Insensitive to quarks2 in 2prot2 on: 2 2 q 0 p k S 3k Λ k T ∼ m αM! = 1 + −−1 ∼ −∼ − . (22) pγ e (proton smize2) 3mΛQ2CD 22m002MeV 12m2 q q Note that we disagreeshorwitht distancRef. [2e3] on thelongord distancer v0 sepin independent part of Eq. (21b). − 2 r = Λ 1 Ref. [23] has an additional non-logarithmic 1/m term. The differeQnCcDe arises because we 2 find a different value for the order k term in the non-Abelian F1 form-factor. Eqs. (21c) through (21g) agree with Ref. [23]. The diagrams in Eq. (19) and (21) have contributions from several different scales. In particular, in the language of the threshold expansion [13], the hard regime gives ln(µ/m)’s, the soft regime gives ln(µ/k)’s, and the ultrasoft regime gives ln(µ/λ)’s. There are also i ln(λ/p) and i ln(λ/k) terms from the Coulomb divergence in the potential regime. In addition to logarithms, all regimes can give constant factors. In the effective theory, terms from the hard regime are absorbed into matching coefficients such as the four-quark potential operators and the remaining terms correspond to graphs involving modes in the effective theory. These graphs are discussed in more detail below. Next consider the one-loop annihilation diagrams in QCD. Since the intermediate gluons are hard we expect these graphs to include a factor of 1/m2, thus giving hard order v0 contributions to the potential. However, the graph in Eq. (23c) also has terms enhanced by a factor of m/p, which are order 1/v contributions. 2 2 iαs C1 Cd(Nc −1) Cd A ¯A + = − 2 + 2 (1 ⊗ 1) + −2C1 (T ⊗ T ) m % Nc 8N 4Nc !" c # " # $ (N 2 −1) 1 S2 c A ¯A × ( −2)(iπ+2−2 ln 2) − CA 2 (1 ⊗ 1)+ (T ⊗ T ) ! 2Nc Nc $ iπ 1 ln 2 λ × S2 + − −ln , (23a) 12 6 6 m & ! " #$ 2 2 iαs A ¯A 1 (Nc − 1) + = − 2 CA (T ⊗ T ) + (1 ⊗ 1) m ! Nc 2Nc $ 3 µ 4 2 ln 2 iπ × S2 ln + + − , (23b) 2 m 3 3 3 ! " # $ 2 2 iαs A ¯A 1 (Nc − 1) + = − 2 (2CF − CA) (T ⊗ T ) + (1 ⊗ 1) 4m ! Nc 2Nc $ π2m 2iπm 2p µ λ × S2 + ln − 4 + 2 ln + 4 ln , (23c) p p λ m m ! " # " # " #$ 11 Equations for Taiwan School Iain W. Stewart1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 • m α ! −1 ∼ ∼ Insensitivee (pr ottoo nprotsizeon) mass:ΛQCD 200 MeV Equations for Taiwan School meα ! mp ∼ 1 GeV (static proton suffices) 2 Iain W. Stewart1 me 4 hyperfine splitting ∼ µ µ α 1 e p Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 mp from QCD ! −1 ∼ ∼ meα but(p universalroton size) ΛQCD 200 MeV meα ! mp ∼ 1 GeV Experimentmeα ! m fore m p , µ p is more accurate that QCD computations 2 L − L − " O p (p, e , γ, b; α, mb) = (p, e , γ; α ) + 2 (1) !mb " ∞ NRQED n L = L0 + # Ln (2) n#=1 2 † 0 ∇ † 0 † † L0 = ψ i∂ − + . ψ + Ψ i∂ Ψ + “V”(ψ ψ)(Ψ Ψ) (3) ! 2me " L ¯ − − 1 µν Why QED = ψ(iD/ m)ψ 4 F Fµν ? Why LSM ? SU(3) × SU(2) × U(1) L d 1 µ 1 2 2 λ 4 τ 6 S[φ] = d x ∂µφ∂ φ − m φ − φ − φ (4) $ !2 2 4! 6! " L Ldim-4 1 T i j " k Ldim-6 = + new LL C#ijH H #"kLL + + . (5) Λ % & C = iγ2γ0 ν L (6) ' eL ( Equations for Taiwan School Equations for Taiwan School Iain W. Stewart1 Iain W. Stewart1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 m α ! (proton size)−1 ∼ Λ ∼ 200 MeV e EquationQs CfDor Taiwan School − ! ∼ m α ! 1 ∼ ∼ meα mp 1 GeV Iain W. Stewaret1 (proton size) ΛQCD 200 MeV 1 ! Cenmtereαfor Thmeoeretical Physics, Massachusetts Instmituteeαof !Techmnolpog∼y, C1amGbreidVge, MA 02139 • −1 m α ! m meα ! (proton sizNe)on∼-relativisticΛQCD ∼ 200 M LaeV grangian e e (no ) 2 − − " p m α ! m ∼ L(p, e , γ, b; α, mb) = L(p, e , γ; α ) + O (1) e p 1 GeV !m2 " QED short distance theory b meα ! me is more general expand in ∞ p m , e , α LNRQED L n L 2 = 0 + # n (2) me mp p # L(p, e−, γ, b; α, m ) = L(p, e−, γ; α") + O n=1 (1) b !m2 " NRQED long distance theory whereb its easier to compute ∞ exact answer is irrelevant, work to the desired level of precision LNRQED = L + #n L (2) 0 # n n=1 Leading Order 2 † 0 ∇ † 0 † † L0 = ψ i∂ − + . ψ + Ψ i∂ Ψ + “V”(ψ ψ)(Ψ Ψ) (3) ! 2me " e+ Equations for Taiwan School Equations for Taiwan School Iain W. Stewart1 Iain W. Stewart1 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Equations for Taiwan School −1 meα ! (proton size) ∼ ΛQCD ∼ 200 MeV 1 Imainα W. Stewart −1 meα ! mp ∼ 1 GeV e ! (proton size) ∼ ΛQCD ∼ 200 MeV 1Center !for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 meα me meα ! mp ∼ 1 GeV • m α ! m Non−-relativistic Lagrangian e e (no ) ! 1 ∼ ∼ 2 meα (proton size) ΛQCD 200 MeV − − " p L(p, e , γ, b; α, m ) = L(p, e , γ; α ) + O (1) b !m2 " QED short distance theory b meα ! mp ∼ 1 GeV is more general expand in m ∞ meα ! me p e α NRQED n , , L = L0 + # Ln (2) e mp # m n=1 NRQED long distance theory where its easier to compute p2 exact answL per, ise− irrele, γ,vant,b α w, mork to theL desiredp, e− le, velγ αof" precisionO ( ; b) = ( ; ) + ! 2 " (1) mb Symmetries of QED constrain the form of NRQED: + − Charge conjugation ( e ↔ e ) ∞ Parity ( ! x → − !x ) LNRQED = L + #n L (2) Time-Reversal ( t → − t ) constrain0 # the n ’s + n=1 Gauge Symmetry e Spin-Statistics Theorem Equations for Taiwan School Equations for Taiwan School Equations for Taiwan School Equations for Taiwan School Iain W. Stewart1 1 Iain W. Stewart 1 Iain W. Stewart1 Iain W. Stewart 1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 1 1 1Center for TheoreticaClePnhteyrsicfso,rMTahsesoacrehtuisceatltsPIhnCysetsinitctuestr,e fMorfaTssehcaehocnrheoutlisoceagltyt,PsChIyansmisctsbi,rtiuMdtgaees,soaMfchTAuesec0th2ts1n3oI9nlosgtiytu,teCoafmTbecrhidngoelo,gyM, CAam0b2r1id3g9e, MA 02139 −1 meα ! (proton size) ∼ ΛQCD ∼ 200 MeV ! −1 ∼ ∼ ! −1 !∼ ∼ −1 ∼meα (∼proton size) ΛQCD 200 MeV meα ! mp ∼ 1 GeV meα (proton sizme)eα Λ(pQrCoDton2s0i0zeM) eV ΛQCD 200 MeV Equations for Taiwan School meα ! mp ∼ 1 GeV m α ! m m , µ meα ! mp ∼ 1 GeV e e p p meα ! mp ∼ 1 GeV Iain W.
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