Part I: Heavy Quark Effective Theory

Transcription

1 Part I: Thomas Mannel CERN-PH-TH and Theoretische Physik I, Siegen University KITPC, June 24th, 2008

2 Contents Motivation 1 Motivation Weak Interaction Reminder Peculiarties of the Standard Model 2 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group 3

3 Introduction and Motivation Weak Interaction Reminder Peculiarties of the Standard Model What is interesting about heavy quarks? Weak decays: Access to fundamental parameters CKM Matrix Elements CP violating parameters new physics?? Heavy mass makes QCD controllable: (Exclusive Decays) Heavy Quark Expansion (Inclusive Decays)

4 Weak Interaction Reminder Weak Interaction Reminder Peculiarties of the Standard Model Flavour Physics (of quarks): Transitions between different kinds of Quarks Its all about weak interactions... Strong interactions as a background

5 Weak Interaction Reminder Peculiarties of the Standard Model Quark Mixing in the Standard Model Mass term for the Up Quarks L u mass = Ū L M u U R + h.c. Mass term for the Down Quarks L d mass = D L M d D R + h.c. 3 3 Mass Matrices M u and M d : [ M u, M d] 0 relative Rotation between the two Eigenbases: M u = M u diag and M d = V CKM M d diagv CKM V CKM : CKM Matrix

6 Weak Interaction Reminder Peculiarties of the Standard Model Charged and Neutral Currents V CKM = V ud V us V ub V cd V cs V cb V td V ts V tb CKM Matrix appears only in the charged currents: L cc int = g 2 Ū L (γ W ± )V CKM D L Neutral currents remain flavour diagonal: (@ tree level) L nc int = g 2 Ū L (Γ Z 0 ) 1 U L + CP Violation through the irreducible phases of CKM

7 Weak Interaction Reminder Peculiarties of the Standard Model Peculiarities of SM Flavour Mixing Hierarchical structure of the CKM matrix Quark Mass spectrum ist widely spread m u 10 MeV to m t 170 GeV PMNS Matrix for lepton flavour mixing is not hierarchical Only the charged lepton masses are hierarchical m e 0.5 MeV to m τ 1772 MeV Up-type leptons Neutrinos have very small masses (Enormous) Suppression of Flavour Changing Neutral Currents: b s, c u, τ µ, µ e, ν 2 ν 1

8 Weak Interaction Reminder Peculiarties of the Standard Model Peculiarities of SM CP Violation Strong CP remains mysterious Flavour diagonal CP Violation is well hidden: e.g electric dipole moment of the neutron: At least three loops (Shabalin) d e e α s GF 2 mt 2 Im µ 3 π (16π 2 ) 2 MW e cm with µ 0.3 GeV d exp e cm

9 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Weak decays: Very different mass scales are involved: Λ QCD 200 MeV: Scale of strong interactions m c 1.5 GeV: Charm Quark Mass m b 4.5 GeV: Bottom Quark Mass m t 175 GeV and M W 81 GeV: Top Quark Mass and Weak Boson Mass Λ NP Scale of new physics At low scales the high mass particles / high energy degrees of freedom are irrelevant. Construct an effective field theory where the massive / energetic degrees of freedom are removed ( integrated out )

10 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Integrating out heavy degrees of freedom Z [j] = φ: light fields, Φ: heavy fields with mass Λ Generating functional as a functional integral Integration over the heavy degrees of freedom = ( [dφ][dφ] exp ( [dφ] exp ( exp d 4 x L eff (φ) ) d 4 x [L(φ, Φ) + jφ] ) with ( ) = [dφ] exp d 4 x L(φ, Φ) d 4 x [L eff (φ) + jφ] )

11 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group For length scales x 1/Λ: local effective Lagrangian Technically: (Operator Product) Expansion in inverse powers of Λ L eff (φ) = L (4) eff (φ) + 1 Λ L(5) eff (φ) + 1 Λ 2 L(6) eff (φ) + L eff is in general non-renormalizable, but... L (4) eff is the renormalizable piece For a fixed order in 1/Λ: Only a finite number of insertions of L (4) eff is needed! can be renormalized Renormalizability is not an issue here

12 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group

13 Example: Weak Hamiltonian EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Effektive Theorie II: Schwache Wechselwirkung Bei Massenskalen unterhalb der W-Boson bzw. Start out der from Top Masse the Standard sind diese Model Teilchen keine W ±, Z 0 dynamischen, top: muchfreiheitsgrade heavier thanmehr: any Sie hadron werden mass integrate ausintegriert out these particles at the scale µ M Hadron H eff = W has zero * Es entsteht range in eine this lokale limit: Wechselwirkung: 0 T [W 1 µ(x)weffektiver ν (y)] 0 Hamiltonoperator g µν δ 4 (x y) MW Fermi-Theorie der schwachen 2 Wechselwirkung Effective Interaction (Fermi Coupling) g2 2M 2 W V q q[ q γ µ (1 γ 5 )q][ ν l γ µ (1 γ 5 )l] = 4G F 2 V q q j µ,had j µ lep

14 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Renormalization Group Running H eff is defined at the scale Λ, where we integrated out the particles with mass Λ: General Structure H eff = 4G F λ CKM Ĉ k (Λ)O k (Λ) 2 O k (Λ): The matrix elements of O k have to be evaluated ( normalized ) at the scale Λ. Ĉ k (Λ): Short distance contribution, contains the information about scales µ > Λ Matrixelements of O k (Λ): Long Distance Contribution, contains the information about scales µ < Λ k

15 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group We could as well imagine a situation with a different definition of long and short distances, defined by a scale µ, in which case H eff = 4G F λ CKM C k (Λ/µ)O k (µ) 2 Key Observation: The matrix elements of H eff are physical Quantities, thus cannot depend on the arbitrary choice of µ compute this... k 0 = µ d dµ H eff

16 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group 0 = i ( µ d ) ( dµ C i(λ/µ) O i (µ) + C i (Λ/µ) µ d ) dµ O i(µ) Operator Mixing: Change in scale can turns the operator O i into a linear combination of operators (of the same dimension) i j µ d dµ O i(µ) = j γ ij (µ)o j (µ) and so ([ δ ij µ d ] ) dµ + γ ij(µ) C i (Λ/µ) O j (µ) = 0

17 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Assume: The operators O j from a basis, then [ δ ij µ d ] dµ + γt ij (µ) C j (Λ/µ) = 0 i QCD: Coupling constant α s depends on µ: β-function C j depend also on α s µ d dµ α s(µ) = β(α s (µ)) µ d dµ = ( µ µ + β(α s) α s In an appropriate scheme γ ij depend on µ only trough α s : γ ij (µ) = γ ij (α s (µ)) )

18 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Renormalization Group Equation (RGE) for the coefficients ( [δ ij µ µ + β(α s) ) ] + γij T (α s ) C j (Λ/µ, α s ) = 0 α s i This is a system of linear differential equations: Once the initial conditions are known, the solution is in general unique RGE Running: Use the RGE to relate the coefficients at different scales

19 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group The coefficients are at µ = Λ (at the matching scale ) C i (Λ/µ = 1, α s ) = n ( a (n) αs ) n i perturbative calculation 4π Perturbative calculation of the RG functions β and γ ij β(α s ) = α s n=0 β (n) ( αs 4π ) n+1 γ ij (α s ) = n=0 ( γ (n) αs ) n+1 ij 4π RG functions can be calculated from loop diagrams: β (0) = 2 3 (33 2n f) γ ij depends on the set of O i

20 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Structure of the perturbative expansion of the coefficient at some other scale c i (Λ/µ, α s ) = b 00 i ( + bi 11 αs 4π + b 22 i + b 33 i ( αs 4π ( αs 4π ) ln Λ µ ) 2 ln 2 Λ ( + bi 10 αs 4π µ + b21 i ) 3 ln 3 Λ µ + b32 i ( αs 4π ( αs 4π ) ) 2 ln Λ µ + b20 i ) 3 ln 2 Λ µ + b31 i ( αs ) 2 4π ( αs 4π ) 3 ln Λ µ +,

21 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group LLA (Leading Log Approximation): Resummation of the bi nn terms C i (Λ/µ, α s ) = n=0 b nn i ( αs ) n ln n Λ 4π µ leading terms in the expansion of the RG functions NLLA (Next-to-leading log approximation): C i (Λ/µ, α s ) = n=0 [ ( bi nn + b n+1,n αs )] ( αs ) n i ln n Λ 4π 4π µ next-to-leading terms of the RG functions

22 EFT in a nutshell Example: Weak Hamiltonian Introduction to Renormalization Group Typical Proceedure: Matching at the scale µ = M W Running to a scale of the order µ = m b includes operator mixing Resummation of the large logs ln(m 2 W /m2 b ) Matching at the scale µ = m b Running to the scale m c Resummation of the large logs ln(m 2 b /m2 c)... Untli α s (µ) becomes too large...

23 Heavy Quark Limit Isgur, Wise, Voloshin, Shifman, Georgi, Grinstein,... 1/m Q Expansion: Substantial Theoretical Progress! Static Limit: m b, m c with fixed (four)velocity In this limit we have v Q = p Q m Q, m Hadron = m Q p Hadron = p Q Q = b, c } v Hadron = v Q For m Q the heavy quark does not feel any recoil from the light quarks and gluons (Cannon Ball) This is like the H-atom in Quantum Mechanics I!

24 Heavy Quark Symmetries The interaction of gluons is identical for all quarks Flavour enters QCD only through the mass terms m 0: (Chiral) Flavour Symmetry (Isospin) m Heavy Flavour Symmetry Consider b and c heavy: Heavy Flavour SU(2) Coupling of the heavy quark spin to gluons: H int = g 2m Q Q( σ B)Q m Q 0 Spin Rotations become a symmetry Heavy Quark Spin Symmetry: SU(2) Rotations Spin Flavour Symmetry Multiplets

25 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

26 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

27 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

28 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

29 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

30 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

31 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

32 Mesonic Ground States Bottom: Charm: (bū) J=0 = B (bū) J=1 = B (b d) J=0 = B 0 (b d) J=1 = B 0 (b s) J=0 = B s (b s) J=1 = B s (cū) J=0 = D 0 (cū) J=1 = D 0 (c d) J=0 = D + (c d) J=1 = D + (c s) J=0 = D s (c s) J=1 = D s

33 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

34 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

35 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

36 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

37 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

38 Baryonic Ground States [(ud) 0 Q] 1/2 = Λ Q [(uu) 1 Q] 1/2, [(ud) 1 Q] 1/2, [(dd) 1 Q] 1/2 = Σ Q [(uu) 1 Q] 3/2, [(ud) 1 Q] 3/2, [(dd) 1 Q] 3/2 = Σ Q [(us) 0 Q] 1/2, [(ds) 0 Q] 1/2 = Ξ Q [(us) 1 Q] 1/2, [(ds) 1 Q] 1/2 = Ξ Q [(us) 1 Q] 3/2, [(ds) 1 Q] 3/2 = Ξ Q [(ss) 1 Q] 1/2 = Ω Q [(ss) 1 Q] 3/2 = Ω Q

39 Wigner Eckart Theorem for HQS HQS imply a Wigner Eckart Theorem H ( ) (v) Qv ΓQ v H ( ) (v ) = C Γ (v, v )ξ(v v ) with H ( ) (v) = D ( ) (v) or B ( ) (v) C Γ (v, v ): Computable Clebsh Gordan Coefficient ξ(v v ): Reduced Matrix Element ξ(v v ): universal non-perturbative Form Faktor: Isgur Wise Funktion Normalization of ξ at v = v : ξ(v v = 1) = 1

40 Representations for ground state meson Spin 1/2 of heavy quark Spin 1/2 of light anti-quark 0 state C CG (0; s h, s l )uα heavy (v, s h ) v light β (v, s l ) = N [(/v + 1)γ 5 ] αβ s h,s l 1 state, Polarization ɛ (v ɛ = 0) C CG (0; s h, s l )uα heavy (v, s h ) v light β (v, s l ) = N [(/v + 1)/ɛ] αβ s h,s l

41 Calculation of the Coefficients: H(v) Q v ΓQ v H(v ) = N 2 Tr{γ 5 (/v +1)Γ(/v +1)γ 5 }ξ(v v ) H (v, ɛ) Q v ΓQ v H(v ) = N 2 Tr{/ɛ(/v+1)Γ(/v +1)γ 5 }ξ(v v ) H (v, ɛ) Q v ΓQ v H(v, ɛ ) = N 2 Tr{/ɛ(/v+1)Γ(/v +1)/ɛ }ξ(v v )

42 The heavy mass limit can be formulated as an effective field theory Expansion in inverse powers of m Q Define the static field h v for the velocity v h v (x) = e im Qv x 1 (1 + /v)b(x) 2 p Q = m Q v + k HQET Lagrangian L = h v (iv D)h v + 1 2m Q hv (i /D) 2 h v + Dim-4 Term: Feynman rules, loops, renormalization...

43 Example: b c current At scales above m b : Bottom and charm dynamical: Full QCD No anomalous dimension At scales between m b and m c : Bottom static q v p γ = αs π At scales below m c : Bottom and charm static q v v γ = αs π f (vv )

44 : B D ( ) l ν l Kinematic variable for a heavy quark: Four Velovity v Differential Rates dγ dω (B D l ν l ) = dγ dω (B Dl ν l) = G2 F 48π 3 V cb 2 m 3 D (ω2 1) 1/2 P(ω)(F(ω)) 2 G2 F 48π 3 V cb 2 (m B + m D ) 2 m 3 D(ω 2 1) 3/2 (G(ω)) 2 with ω = vv and P(ω): Calculable Phase space factor F and G: Form Factors

45 Heavy Quark Symmetries Normalization of the Form Factors is known at vv = 1: (both initial and final meson at rest) Corrections can be calculated / estimated [ F(ω) = η QED η A 1 + δ1/µ 2 + ] (ω 1)ρ 2 + O((ω 1) 2 ) ( )] mb m D G(1) = η QED η V [1 + O m B + m D Parameter of HQS breaking: 1 µ = 1 m c 1 m b η A = ± 0.007, η V = ± 0.004, δ 1/µ 2 = (8 ± 4)%, η QED = 1.007

46 B D ( ) Form Factors from the Lattice Unquenched Calculations become available! Heavy Mass Limit is not used Lattice Calculations of the deviation from unity F(1) = G(1) = ± ± A. Kronfeld et al.

47 B D l ν l 3 "10 cb F(1) V ALEPH OPAL (part.reco.) OPAL (excl.) DELPHI (part.reco.) BELLE CLEO BABAR B0 DELPHI (excl.) BABAR B+ Average ! 2 A 1

48 B Dl ν l ALEPH ± ± 6.20 CLEO ± 5.90 ± 3.45 BELLE ± 4.40 ± 5.15 ] -3 G(1) V cb [ χ = 1 ALEPH BELLE CLEO AVERAGE Average ± HFAG LP χ /dof = 0.26/ 4 (CL = 99 %) G(1) V cb [10 ] 20 HFAG LP χ /dof = 0.26/ ρ

49 V cb,excl = (38.6 ± 1.3) 10 3 Potentially large 1/m 2 c terms Hints that F(1) 0.89 (see below)