Applications of QCD Sum Rules to Heavy Quark Physics

Transcription

1 Applications of QCD Sum Rules to Heavy Quark Physics Alexander Khodjamirian UNIVERSITÄT SIEGEN Theoretische Physik 1 RESEARCH UNIT q et f 3 lectures at Helmholtz International School "Physics of Heavy Quarks and Hadrons", Dubna, July 2013

2 Preface QCD sum rule, the three key elements Correlation function of quark-antiquark currents Operator Product Expansion in terms of quark-gluon diagrams and universal QCD parameters = Dispersion Relation, a sum over hadronic amplitudes sum rules are analytical relations in continuum QCD as opposed to numerical simulation of the correlation functions in the lattice QCD the origin of the method: Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 2 / 30

3 major development of the QCD SR method in 1980 s: standard 2-point sum rules; modifications: 3-,4-point SR s; Finite Energy Sum Rules; method of external fields QCD Light-cone sum rules (LCSR) [V. Braun, I. Balitsky et al; V. Chernyak, I.Zhitnisky, 1989,...] a more advanced technique to calculate hadronic transition form factors These lectures concentrate on the QCD sum rule applications to Heavy Quark Physics from ABC of the method to recent applications Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 3 / 30

4 Outline of the lectures Lecture 1: Introducing the method of QCD Sum Rules calculation of B-meson decay constant B τν τ, B s µ + µ Lecture 2: Light-cone Sum Rules for Heavy-Light Form Factors calculation of B π form factor V ub from B πlν l Lecture 3: Hadronic effects in B K ( ) l + l, B K γ from LCSR s Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 4 / 30

5 Lecture 1: Introducing the method of QCD Sum Rules Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 5 / 30

6 Our study object: B τν τ decay the decay amplitude in the Standard Model A(B τ ν τ ) SM = G F V 2 ub τγ µ (1 γ 5 )ν τ 0 ūγ µ γ 5 b B {b u flavour-changing transition } {QCD colour forces} hadronic matrix element decay constant: 0 ūγ µ γ 5 b B(p B ) = ip µ B f B, p 2 B = m2 B partial width: (suppressed for l = µ, e ) ( BR(B τ ν τ ) SM = G2 F V ub 2 mτ 2 m B 8π 1 m2 τ m 2 B ) 2 f 2 B τ B, Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 6 / 30

7 Rare leptonic decays: B s,d l + l b B s W t Z µ t s µ recently detected by LHCb in SM t, W, Z -loops, sensitive to V ts V tb, realistic chances to find/constrain new physics after integrating out the heavy particle loops: the hadronic matrix element in decay amplitude reduced to 0 sγ µ γ 5 b B s (p B ) = ip µ B f B s, or f Bd for B d µ + µ f Bd f Bu f B (isospin symmetry), but f Bs f B, (SU(3) fl violation) Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 7 / 30

8 The task is to calculate the hadronic matrix element 0 ūγ µ γ 5 b B f B in QCD Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 8 / 30

9 Quantum Chromodynamics (QCD) Quantum field theory of quarks, gluons and their interactions L QCD (x) = 1 4 Ga µν(x)g a µν (x) + q i (x)(id µ γ µ m q )q i (x) q=u,d,s,c,b,t D µ = µ ig s λ a 2 Aa µ, G a µν = µ A a ν ν A a µ + g s f abc A b µa c ν, colour charges i = 1, 2, 3, a = 1,.., 8, α s = g 2 s /4π flavour neutrality basic elements of Feynman graphs: q i g a Propagators g a g b g a g b V ertices q i q j g a g c g c g d Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 9 / 30

10 Asymptotic Freedom of QCD Quark-quark scattering, energy/momentum transfer Q +... the quantum loop corrections play a crucial role: α s α s (Q), effective, scale-dependent coupling α s (Q) small for processes with Q 1 GeV perturbative expansion in powers of α s applicable Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 10 / 30

11 QCD at long distances α s (Q) at small momenta long distances α s (Q) confinement ~0.5 asympt. freedom Λ QCD hadronization 1 GeV perturbative QCD an intrinsic scale emerges: Λ QCD MeV at Q Λ QCD quarks/gluons strongly interact, hadronization, confinement Q Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 11 / 30

12 QCD Vacuum the lowest energy state, no hadrons contains fluctuating quark-antiquark and gluon fields: vacuum condensates e.g., 0 qq 0 0, q = u, d, s -spontaneous breaking of chiral symmetry 0 G µν G µν 0 0, 0 qσ µν G µν q 0 0,... universal set of vacuum condensate densities with dimension d = 3, 4, 5,.. Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 12 / 30

13 B-meson annihilation in QCD energy scale of quark-gluon interactions binding b and ū inside B: Λ m B m b MeV, (m B 5.3 GeV, m b = GeV ( pole quark mass) ) α s (Q) confinement ~0.5 asympt. freedom +... Λ QCD hadronization 1 GeV perturbative QCD Q no perturbative expansion in α s ( Λ) can be used: nonperturbative QCD Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 13 / 30

14 B-meson annihilation in QCD m B, f B and other B-meson observables are described by long-distance (soft) quark-gluon interactions in QCD in addition to "valence" quarks, partonic components with soft gluons and qq -pairs in the B-meson state B = bū būg bū qq... the QCD vacuum state 0, (the lowest energy state, no hadrons) is populated by fluctuating quark-antiquark and gluon fields even for the simlpest hadronic matrix element 0 ū..b B f B there is no exact solution in QCD alternatives numerical simulation on the lattice use of QCD sum rules (SVZ method): the main idea: construct an object calculable in QCD and simultaneously related to f B Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 14 / 30

15 Correlation function of ub currents formal definition of the vacuum correlation function: Π µν (q 2 ) = d 4 x e iqx 0 T {j W µ (x)j W ν (0)} 0, a quantum amplitude of emission and absorbtion of ub pair in vacuum by the external current: j W µ = ūγ µ γ 5 b, j W µ = bγ µ γ 5 u Π µν (q 2 ) = q j W b j W u Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 15 / 30

16 Correlation function far below the B hreshold q j W b u j W 4-momentum of the bū pair: q = (q 0, q), q 2 = q 2 0 q 2, rest frame: q = 0, q 2 = q 2 0, fix the energy q 0 m b, m B the bū-pair is virtual: E t 1, the energy deficit E m b, t 1/m b m b Λ QCD : t 1/Λ QCD virtual quarks propagate during short times, are asymptotically free, at q 2 m 2 b, m2 B, Π µν (q 2 ) simple loop diagram { calculable QCD corrections} Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 16 / 30

17 Transforming to pseudoscalar currents simplifying trick: multiply the correlation function, q µ q ν Π µν (q 2 ) Π 5 (q 2 ), scalar function of q 2 note that q µ ūγ µ γ 5 b = (p b µ + p u µ)ūγ µ γ 5 b = (m b + m u )ūiγ 5 b j 5 (apply Dirac equation for both quark fields) ) Π 5 (q 2 ) = d 4 x e iqx 0 T {j 5 (x)j 5 (0)} 0, equivalent definition of the decay constant: at q = p B (q 2 = m 2 B ), q µ 0 ūγ µ γ 5 b B(p B ) = 0 j 5 B(p B ) = m 2 B f B Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 17 / 30

18 Calculating the correlation function at q 2 m 2 B adding perturbative gluon exchanges to the simple loop, α s (m B ) 1 including nonperturbative effects due to condensates typical diagams b q u loop gluon exchange quark condensate gluon condensate technically, using Feynman rules of QCD and considering the vacuum quark-antiquarks and gluons as external static fields. The result: analytical expression for Π 5 (q 2 ) in terms of m b, m u and universal QCD parameters α s, qq,... Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 18 / 30

19 interpreting the calculation as an operator-product expansion: T {j 5 (x)j 5 (0)} = C d (x 2, m b, m u, α s )O d (0) d=0,3,4,.. in local operators with the quantum numbers of vacuum (Lorentz-scalar, C-,P-,T-invariant, colorless) and growing dimensions: O 0 = 1, O 3 = qq, O 4 = G µν G µν,... (no operator of dimension 2 in QCD!) vacuum average, integrating over x Π5 (q 2 ) = d 4 x e iqx 0 T {j 5 (x)j 5 (0)} 0 = C d (q 2, m b, m u, α s ) 0 O d 0 d=0,3,4,.. perturbative loops Wilson coefficeints C d as series in α s, d 0, 0 O d 0 (Λ QCD ) d - vacuum condensate densities, at q 2 m 2 b, high-d terms suppressed by O[(Λ QCD/m b ) d ] the OPE can safely be truncated Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 19 / 30

20 Correlation function above B threshold Hypothetical neutrino-electron scattering, varying c.m. energy s = q 2, Π 5 (q 2 ) is the part of the scattering amplitude ν e e s 0 W b u W ν e e ν e e s = mb W b u B Meson W ν e e highly virtual quark pair, B-meson, resonance ν e e s > mb W b u Excited B state W ν e e ν e e s mb W b u W ν e e excited B mesons multiple hadrons (continuum) Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 20 / 30

21 Hadronic representation of the correlation function at q 2 mb 2 : a short-distance, short-lived bū -fluctuation, Π 5 (q 2 ) Π (OPE) 5 (q 2 ) OPE region Π 5 (q 2 ) at q 2 mb 2, describes propagation of B meson and excited and multiparticle B states the hadronic representation (dispersion relation): B B* π Π 5 (q 2 ) = 0 j 5 B B j 5 0 mb j 5 B exc B exc j 5 0 q2 m 2 B exc B exc q 2... q^2 rigorous derivation: analyticity of Π 5 (q 2 ) Cauchy theorem unitarity As a result: at q 2 mb 2 there is a relation between Π (OPE) 5 (q 2 ) and a hadronic sum containing f B Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 21 / 30

22 Deriving the sum rule for f 2 B isolating the ground-state B-state and introducing the spectral density of excited hadronic states Π 5 (q 2 ) = f 2 B m4 B m 2 B q2 + s h ρ h (s) ds s q 2 expressing the OPE result as a dispersion relation Π(q 2 ) (OPE) = 1 π s 0 m 2 b ds ImΠ(OPE) 5 (s) s q π s 0 ImΠ (OPE) 5 (s) ds s q 2 equating the two representations at q 2 m 2 b - global quark-hadron duality at sufficiently large s the local duality is also valid: ρ h (s) 1 π ImΠ(OPE) 5 (s), Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 22 / 30

23 Deriving the sum rule for f 2 B semilocal quark-hadron duality is used, the effective threshold s 0 ρ h (s) ds s q 2 1 ImΠ (OPE) 5 (s) ds π s q 2 s h this yields approximate analytical relation for decay constant: f 2 B m4 B Borel transformation m 2 B q2 = 1 π Π 5 (M 2 ) B M 2 Π(q 2 ) = s 0 s 0 m 2 b lim q 2,n q 2 /n=m 2 ds ImΠ(OPE) 5 (s) s q 2 ( q 2 ) (n+1) n! «d n dq 2 Π 5 (q 2 ). suppresses the higher-state contributions to the hadronic sum, the sum rule less sensitive to the duality approximation B M 2( 1 m 2 q 2 ) = exp( m 2 /M 2 ) Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 23 / 30

24 The resulting QCD sum rule fb 2 s 0 m4 B e m2 B /M2 = dse s/m2 mb 2 ImΠ (OPE) 5 (s, m b, m u, α s, 0 qq 0,...) current accuracy of Π (OPE) 5 (q 2 ) at q 2 m 2 b : vacuum condensates with d 6 loop O(α s ) O(α 2 s) [K.Chetyrkin, M.Steinhauser (2001)] standard way to fix s 0 : calculate the mass of B-meson from the same sum rule: m 2 B = d d(1/m 2 ) [SR] SR Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 24 / 30

25 Input parameters sensitivity to the b-quark mass (in MS scheme) independent determination of m b - quarkonium sum rules: the correlation function of two Qγ µ Q currents (Q = b, c) calculated in QCD up to O(α 3 s) and matched to the sum over J PC = 1 heavy quarkonia levels measured in e + e Υ, Υ(2S),... or e + e J/ψ, ψ(2s),... heavy quark masses in MS: m b ( m b ) = (4.18 ± 0.03) GeV, m c ( m c ) = (1.275 ± 0.025) GeV, PDG values including QCD SR for quarkonia [K.Chetyrkin et al. (2012)] light quark masses and quark condensate density: (QCD SR for strange meson channels) [ A.K., K. Chetyrkin (2005); M.Jamin, Oller, A.Pich (2006)] m s (µ = 2 GeV) = (98 ± 16) MeV, m s ChPT m u,d qq (2GeV ) = ( MeV)3 Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 25 / 30

26 Universality of the method qq currents with various flavour and J P in the correlation functions sum rules for decay constants of B s, D, D s, π ρ, K, K, also baryonic, gluonic currents (any Lorentz-covariant and colour-invariant local operator ) the coefficients in the OPE depend on the currents, inputs are universal (quark masses, α s, condensates) QCD (SVZ) sum rules address the question: why are the hadrons not alike? SU(3) flavour and heavy-quark symmetry violations can be estimated (finite quark masses, strange/nonstrange condensates) Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 26 / 30

27 B (s) and D (s) decay constants, recent update P.Gelhausen, AK, A.A.Pivovarov, D.Rosenthal, hep/ph] Decay constant Lattice QCD [ref.] this work ± 9.1 [1] f B [MeV] 186 ± 4 [2] ± 10.0 [1] f Bs [MeV] 224 ± 5 [2] ± [1] f Bs /f B ± [2] ± 11.3 [1] f D [MeV] 213 ± 4 [2] ± 10.8 [1] f Ds [MeV] ± 2.5 [2] ± [1] f Ds /f D ± [2] [1]-Fermilab/MILC, [2]-HPQCD Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 27 / 30

28 Can we trust QCD sum rule estimates? The story of Γ(J/ψ η c γ) QCD sum rules from three-point correlator with cc currents Γ(J/ψ η c γ) = 2.2 ± 0.8 KeV (m c = 1.25GeV ) [ A.K. (1980); including gluon condensate correction (1984)] use of disp. relation for η c 2γ amplitude: Γ(J/ψ η c γ) = 2.6 ± 1.1KeV [M. Shifman, 1979 ] three-point sum rules including O(α s ) corrections: Γ(J/ψ η c γ) = 2.6 ± 0.5KeV [V.Beilin, A.Radyushkin (1985)] the only measurement in 1977 [Crystal Ball] PDG: Γ(J/ψ η c γ) = 1.3 ± 0.4 kev for many years considered a serious problem for QCD sum rule approach QCD will not survive if m ηc = GeV and J/ψ η cγ rate is lower than 2 kev... I expect that the experimental result will turn out close to 2 kev [M. Shifman, Z. Phys. (1980)] Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 28 / 30

29 after thirty (!) years the CLEO data (2008): BR(J/ψ η c γ) = 1.98 ± 0.09 ± 0.3% Γ(J/ψ η c γ) = 1.84 ± 0.29 kev (later confirmed by Novosibirsk experiment) the agreement of QCD sum rule prediction with experiment is restored. The accuracy of the latter can be improved but demands two-loop, three-point, multi-scale perturbative diagrams, still a technical challenge... Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 29 / 30

30 Concluding: QCD sum rules: a method based on the twofold treatment of correlation functions: 1) OPE 2) hadronic dispersion relations B,D decay constants calculated with an accuracy, comparable to the one in lattice QCD further reading: reviews with more details A. Khodjamirian and R. Rückl, [hep-ph/ ]. "Snapshots of hadrons or the story of how the vacuum medium determines the properties of the classical mesons which are produced, live and die in the QCD vacuum" Mikhail A. Shifman [hep-ph/ ] P. Colangelo and A. Khodjamirian, [hep-ph/ ]. several interesting applications ahead Alexander Khodjamirian Applications of QCD Sum Rules to Heavy Quark Physics 30 / 30