B D ( ) form factors from QCD sum rules

Transcription

1 B D ( ) form factors from QCD sum rules (Master Thesis, June 2008) Christoph Klein in collaboration with Sven Faller Alexander Khodjamirian Thomas Mannel Nils Offen Theoretische Physik 1 Universität Siegen Theorieseminar Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

2 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

3 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

4 Determination of V cb Current status of the determination of V cb [Kowalewski, Mannel 2008] Independent considerations of exclusive and inclusive decays (important cross-checking): Exclusive decays: Inclusive decays: V cb excl = (38,6 ± 1,3) 10 3 V cb incl = (41,6 ± 0,6) 10 3 Average: V cb = (41,2 ± 1,1) 10 3 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

5 Determination of V cb from exclusive decays Determination from exclusive decays Relative precision at about 3%. Consideration of the semileptonic decays B D and B D. B D more difficult to measure than B D in that edge of phase space (small rate, high background). Subject of this talk. See also talk by N. Uraltsev. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

6 Determination of V cb from inclusive decays Determination from inclusive decays Relative precision at below 2%. Most precise determination at the moment. Theoretical tool used here: Heavy Quark Expansion Uncertainty mainly comes from theoretical ignorance of higher order and nonperturbative corrections. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

7 Determination of V cb from inclusive decays Determination from inclusive decays Relative precision at below 2%. Most precise determination at the moment. Theoretical tool used here: Heavy Quark Expansion Uncertainty mainly comes from theoretical ignorance of higher order and nonperturbative corrections. Work in our group by B. Dassinger, R. Feger, Th. Mannel, S. Turczyk, N. Uraltsev,... See talk by S. Turczyk for more details. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

8 Exclusive decay B D ( ) Exclusive decay B D ( ) b B q W(q) V cb D ( ) ν l l c Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

9 Exclusive decay B D ( ) Exclusive decay B D ( ) b B q W(q) V cb D ( ) ν l l c Determination of V cb from the experimental measured spectrum: dγ (B Dlνl )(q 2 ) dq 2 V cb 2 D c γ µ (1 γ 5)b B 2 Here the important dynamics comes from the transition matrix elements beside V cb and kinematical factors. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

10 Form factors These are parametrized by form factors: D(p ) c γ µ b B(p) = (p µ + p µ ) f +(q 2 ) + (p µ p µ ) f (q 2 ) D (p,ɛ) c γ µ b B(p) = 2V (q2 ) ɛ µναβ ɛ ν p αp β (q µ = p µ p µ ) m B + m D D (p,ɛ) c γ µ γ 5 b B(p) = iɛ µ (m B + m D ) A 1 (q 2 ) i(ɛ q) (p µ + p µ A 2 (q 2 ) ) m B + m D i(ɛ q) (p µ p µ ) 2 m D q 2 (A 3 (q 2 ) A 0 (q 2 )) The complete expression for the decay rate (m l = 0): dγ (B Dlνl )(q 2 ) dq 2 = G2 F V cb 2 192π 3 m 3 B f +(q 2 ) 2 (q 2 m 2 B m2 D )2 4 m 2 3/2 B m2 D Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

11 Form factors and nonperturbative methods dγ (B Dlνl )(q 2 ) dq 2 = G2 F V cb 2 192π 3 m 3 B f +(q 2 ) 2 (q 2 m 2 B m2 D )2 4 m 2 3/2 B m2 D V cb and form factors always appear together. A theoretical determination of the form factors is necessary. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

12 Form factors and nonperturbative methods dγ (B Dlνl )(q 2 ) dq 2 = G2 F V cb 2 192π 3 m 3 B f +(q 2 ) 2 (q 2 m 2 B m2 D )2 4 m 2 3/2 B m2 D V cb and form factors always appear together. A theoretical determination of the form factors is necessary. B,D-Mesons are bound states of heavy b,c-quarks and light antiquarks (strong quark-gluon-interactions, confinement): Matrix elements are not perturbatively calculable! Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

13 Form factors and nonperturbative methods dγ (B Dlνl )(q 2 ) dq 2 = G2 F V cb 2 192π 3 m 3 B f +(q 2 ) 2 (q 2 m 2 B m2 D )2 4 m 2 3/2 B m2 D V cb and form factors always appear together. A theoretical determination of the form factors is necessary. B,D-Mesons are bound states of heavy b,c-quarks and light antiquarks (strong quark-gluon-interactions, confinement): Matrix elements are not perturbatively calculable! Use of nonperturbative methods: QCD-Sum rules, Lattice-QCD and Use of symmetries: Heavy Quark Symmetry (HQS) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

14 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

15 Heavy Quark Effective Theory (HQET) Important tool for handling heavy quarks: Heavy Quark Effective Theory (HQET) Formulation of QCD as an effective theory for heavy quarks: m Q Λ QCD Definition: Meson has four-velocity v µ = pµ M mq Heavy Quark carries momentum p µ Q = m Qv µ + O(Λ QCD ) Parametrization and integrating out of the heavy degrees of freedom m Q of the meson = Expansion of the QCD-Lagrangian and the meson states as series in 1 : m Q Operators: Q(x) = e im Q v x h v (x) + O( 1 Meson-states: M = Hv + O( 1 m Q ) m Q ) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

16 Spin-flavour-symmetry for heavy quarks The light degrees of freedom in the meson interact with the heavy quark only at momentum scales Λ QCD m Q. 1 Beside of corrections of order O the spin-flavour-symmetry holds: m Q The light degrees of freedom in the meson neither notice the flavour nor the spin of the heavy quark. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

17 Spin-flavour-symmetry for heavy quarks The light degrees of freedom in the meson interact with the heavy quark only at momentum scales Λ QCD m Q. 1 Beside of corrections of order O the spin-flavour-symmetry holds: m Q The light degrees of freedom in the meson neither notice the flavour nor the spin of the heavy quark. In their dynamics the mesons B, B, D, D are not different. All form factors can be expressed by a single function, the Isgur-Wise-function ξ(w) w = v v = m2 B +m2 D ( ) q2 2 m B m D ( ) Normalization: ξ(1) = 1 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

18 Spin-flavour-symmetry for heavy quarks Alternative definition of the form factors more useful in HQET: D(v ) V µ B(v) p = h + (w)(v + v ) µ + h (w)(v v ) µ D (v,ɛ) V µ B(v) q mb m D m B m D = h V (w)ɛ µναβ ɛ ν v α v β D (v,ɛ) A µ B(v) q m B m D = iɛ µ (w + 1)h A1 (w) + i(ɛ q) v µ h A2 (w) + i(ɛ q) v µ h A3 (w) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

19 Spin-flavour-symmetry for heavy quarks Alternative definition of the form factors more useful in HQET: D(v ) V µ B(v) p = h + (w)(v + v ) µ + h (w)(v v ) µ D (v,ɛ) V µ B(v) q mb m D m B m D = h V (w)ɛ µναβ ɛ ν v α v β D (v,ɛ) A µ B(v) q m B m D = iɛ µ (w + 1)h A1 (w) + i(ɛ q) v µ h A2 (w) + i(ɛ q) v µ h A3 (w) The heavy-quark-symmetry now predicts at w = 1: «h +(1) = h V (1) = h A1 (1) = h A3 (1) = 1 + O ( 1mc 1mb ) 2 1 h (1) = h A2 (1) = 0 + O, 1 «m c m b Luke s theorem High theoretical precision for determination of V cb at w = 1. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

20 Determination of V cb But: phase space vanishes at w = 1 more difficult to compare to experiment. extrapolation of experimental data to w = 1 is done. It is important to know the shape of the form factor. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

21 Determination of V cb But: phase space vanishes at w = 1 more difficult to compare to experiment. extrapolation of experimental data to w = 1 is done. It is important to know the shape of the form factor. Certain parametrizations can be derived from fundamental principles. A widely used one is: [Caprini, Lellouch, Neubert 97] h A1 (w) = η A η QED 1 + δ 1/m ρ 2 z(w) + (53ρ 2 15)z(w) 2 (231ρ 2 91)z(w) 3 with: η A,η QED calculable QCD- and QED-radiative corrections δ 1/m 2 nonperturbative 1 -corrections estimated from HQET and inclusive decays m 2 w+1 z(w) = 2 w+1+ 2 ρ 2 a free parameter. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

22 Determination of V cb As alternative approaches, the form factor is also calculated in lattice QCD and with QCD sum rules. From experimental results, one gets: B D decays B D decays Most precise from B D V cb = (38,6 ± 0,9 exp ± 1,0 theo ) 10 3 V cb = (39,4 ± 4,2 exp ± 1,3 theo ) 10 3 V cb excl = (38,6 ± 1,3) 10 3 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

23 Discussion Good determination of V cb excl with the help of heavy quark symmetry. As it has been shown, the shape (esp. the slope) of the form factor is important for the evaluation. Lattice QCD and heavy quark symmetry only relyable in the region around w = 1. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

24 Discussion Good determination of V cb excl with the help of heavy quark symmetry. As it has been shown, the shape (esp. the slope) of the form factor is important for the evaluation. Lattice QCD and heavy quark symmetry only relyable in the region around w = 1. The QCD sum rule approach allows the calculation in other kinematical regions and determination of the shape. The light cone sum rule calculation done in this work gives the form factor in the region around w = w max. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

25 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

26 Method of QCD Sum rules Nonperturbative method developed by Shifman, Vainshtein and Zakharov ( SVZ sum rules ) in 1979 Widely used since then for treatment of nonperturbative hadron physics. Main principle: Usage of the analytical structure of QCD correlation functions, to relate nonperturbative quantities to calculable and/or observable quantities. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

27 Method of QCD Sum rules Nonperturbative method developed by Shifman, Vainshtein and Zakharov ( SVZ sum rules ) in 1979 Widely used since then for treatment of nonperturbative hadron physics. Main principle: Usage of the analytical structure of QCD correlation functions, to relate nonperturbative quantities to calculable and/or observable quantities. Method with in principle good, but in the end limited accuracy (as perturbation series is). today lattice QCD gets competitable results (thanks to modern computers and more elaborate theoretical methods). Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

28 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

29 Calculation of the B-Meson decay constant Starting-point of QCD sum rules: Consideration of a correlation function. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

30 Calculation of the B-Meson decay constant Starting-point of QCD sum rules: Consideration of a correlation function. Example: Calculation of the B-Meson decay constant f B defined by: m b 0 q i γ 5 b B(q) = m 2 B f B 0 q γ µ γ 5 b B(q) = i q µ f B Here we consider the (two-point) correlation function: Z Π(q 2 ) = i q q x 0 d 4 x e iqx 0 T q(x) i γ5 b(x), b(0) i γ 5 q(0) 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

31 Quark-hadron duality Now this can be evaluated in two different languages : Perturbative calculation Hadronic representation Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

32 Quark-hadron duality Now this can be evaluated in two different languages : Perturbative calculation Hadronic representation This refers to the quark-hadron duality: Each hadronic quantity (like a correlation function) is determined by quark interactions, but can only be seen from outside in the different Hilbert-space of hadrons. These descriptions have to be equal. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

33 Quark-hadron duality Now this can be evaluated in two different languages : Perturbative calculation Hadronic representation This refers to the quark-hadron duality: Each hadronic quantity (like a correlation function) is determined by quark interactions, but can only be seen from outside in the different Hilbert-space of hadrons. These descriptions have to be equal. In practice this is limited by the truncation of the perturbative series (as well as general problems with its large order behaviour...). But a certain estimate can be made. That will be used below: semi-local quark-hadron duality More to come on the next slides... Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

34 Hadronic representation Consider the correlation function: Z Π(q 2 ) = i d 4 x e iqx 0 T q(x) i γ5 b(x), b(0) i γ 5 q(0) 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

35 Hadronic representation Consider the correlation function: Z Π(q 2 ) = i d 4 x e iqx 0 T q(x) i γ5 b(x), b(0) i γ 5 q(0) 0 and put in a complete sum over hadronic intermediate states 1 = P h h h : Π(q 2 ) = X h Z i d 4 x e iqx 0 Z q(x) i γ 5 b(x) h h b(0) i γ5 q(0) 0 =... = ds ρ(s) s q 2 m 2 B Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

36 Hadronic representation Consider the correlation function: Z Π(q 2 ) = i d 4 x e iqx 0 T q(x) i γ5 b(x), b(0) i γ 5 q(0) 0 and put in a complete sum over hadronic intermediate states 1 = P h h : h Π(q 2 ) = X Z i d 4 x e iqx 0 Z q(x) i γ 5 b(x) h h b(0) i γ5 q(0) 0 =... = ds ρ(s) s q 2 h m B 2 Use the definition of the decay constant f B to write the first term seperately: Π (hadr) (q 2 m 4 B ) = f 2 Z B mb 2(m2 B q2 ) + ds ρ(s) s q 2 s h 0 Hadronic spectral density: ρ(s) B ρ(s) This is the hadronic dispersion relation. m 2 B s h 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

37 Perturbative calculation q q x 0 The correlation function can be evaluated perturbatively, by calculating the corresponding loop diagrams: One gets an expression with the known two-point-functions: Π (pert) (q 2 ) = 3 8π 2 `B0 (0,m 2 b,m 2 b)m 2 b + m 2 b B 0 (q 2,0,m 2 b)(m 2 b q 2 ) + O (α s) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

38 Perturbative calculation q q x 0 The correlation function can be evaluated perturbatively, by calculating the corresponding loop diagrams: One gets an expression with the known two-point-functions: Π (pert) (q 2 ) = 3 8π 2 `B0 (0,m 2 b,m 2 b)m 2 b + m 2 b B 0 (q 2,0,m 2 b)(m 2 b q 2 ) + O (α s) But this is not the whole part of the calculation. Since we re in QCD, we have to take into account nonperturbative effects occuring from small internal loop momenta. The above part is the first order of an operator-product-expansion, which takes into acount these contributions in form of the so-called vacuum condensates. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

39 OPE and vacuum condensates Π(q 2 ) = X d C d (q 2 ) 0 Od 0 with: = Π (pert) (q 2 ) + m b qq +... q 2 mb 2 O d Operators with the quantum numbers of the vacuum and dimension d 0 O d 0 The corresponding condensates, which are nonperturbative quantities, derivable by other sum rules and methods C d (q 2 ) Perturbatively calculable Wilson-coefficients qq The quark-condensate (next-leading term) The first terms in the OPE are: Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

40 Calculation of the B-Meson decay constant One can derive a dispersion form for Π (pert) (q 2 ) and show: Π (pert) (q 2 ) = 1 π Z m 2 b ds Im`Π (pert) (s) s q 2 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

41 Calculation of the B-Meson decay constant One can derive a dispersion form for Π (pert) (q 2 ) and show: Π (pert) (q 2 ) = 1 π Z m 2 b ds Im`Π (pert) (s) s q 2 Now we can in principle compare the hadronic part and the perturbative part, to get: mb 4 f B 2 Z mb 2(m2 B q2 ) + ds ρ(s) s q 2 = 1 Z ds Im`Π (pert) (s) π s q 2 + m b qq +... q 2 mb 2 s h 0 m 2 b Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

42 Calculation of the B-Meson decay constant One can derive a dispersion form for Π (pert) (q 2 ) and show: Π (pert) (q 2 ) = 1 π Z m 2 b ds Im`Π (pert) (s) s q 2 Now we can in principle compare the hadronic part and the perturbative part, to get: mb 4 f B 2 Z mb 2(m2 B q2 ) + ds ρ(s) s q 2 = 1 Z ds Im`Π (pert) (s) π s q 2 + m b qq +... q 2 mb 2 s h 0 m 2 b Here we make use of the semi-local quark-hadron duality: ρ(s) Z ρ(s) ds s 0 h s q 2 1 Z ds Im`Π (pert) (s) π s 0 B s q B ρ(s) 1 π Im Π(pert) (s) New method-inherent parameter s 0 m 2 b s h 0 s B 0 s Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

43 Calculation of the B-Meson decay constant We get the result: mb 4 f B 2 mb 2(m2 B q2 ) 1 π Z sb 0 m 2 b ds Im`Π (pert) (s) s q 2 + m b qq q 2 mb 2 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

44 Calculation of the B-Meson decay constant We get the result: mb 4 f B 2 mb 2(m2 B q2 ) 1 π Z sb 0 m 2 b ds Im`Π (pert) (s) s q 2 + m b qq q 2 mb 2 To improve the convergence of the series and suppress higher order condensates, we apply a Borel transformation: Π(M 2 ) = B M 2 Π(q 2 ) = lim q 2,n, q2 n = M2 ( q 2 ) (n+1) n! «d n dq 2 Π(q )! 2 where the momentum q 2 is transformed to the ( method-inherent ) Borel-parameter M 2. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

45 Calculation of the B-Meson decay constant In the end, we get the final sum rule: fb 2 = 3m2 b 8π 2 mb 4 Z sb 0 m 2 b ds (s m2 b )2 m B 2 s e M 2 s m2 m B b `mb qq e 2 m2 b mb 4 M 2 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

46 Calculation of the B-Meson decay constant In the end, we get the final sum rule: fb 2 = 3m2 b 8π 2 mb 4 Z sb 0 m 2 b ds (s m2 b )2 m B 2 s e M 2 s m2 m B b `mb qq e 2 m2 b mb 4 M In dependence of M 2 one gets with the corresponding input values: f B GeV Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

47 Calculation of the B-Meson decay constant In the end, we get the final sum rule: fb 2 = 3m2 b 8π 2 mb 4 Z sb 0 m 2 b ds (s m2 b )2 m B 2 s e M 2 s m2 m B b `mb qq e 2 m2 b mb 4 M In dependence of M 2 one gets with the corresponding input values: f B GeV We get an estimate of f B 150 MeV. More elaborate studies calculate: [Jamin, Lange 2001] f B = (210 ± 19) MeV Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

48 Summary Here have a method to translate perturbative calculations and universal nonperturbative parameters (condensates) into hadronic observables. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

49 Summary Here have a method to translate perturbative calculations and universal nonperturbative parameters (condensates) into hadronic observables. This is the main method of the QCD sum rule approach. All other calculations go the same way. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

50 Summary Here have a method to translate perturbative calculations and universal nonperturbative parameters (condensates) into hadronic observables. This is the main method of the QCD sum rule approach. All other calculations go the same way. Now the method shall be applied to B D ( ) decays in the framework of light cone sum rules (LCSR). This work is based on the corresponding approach to the B π decay, done by A. Khodjamirian, Th. Mannel and N. Offen. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

51 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

52 Three point sum rules Three point sum rules in principle can be used to calculate meson decay form factors: q b c Consider a correlator of three currents. Nonperturbative contributions in form of vacuum condensates. Disadvantage/uncertainty: Difficult theoretical treatment Double dispersion relation / hadronic spectral density Problems with applicability of OPE... Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

53 Method of light cone sum rules Description of heavy-meson decays: 90 s: Three-point sum rules [Ball,Colangelo,Ovchinnikov, Slobodenyuk,Radyushkin et. al.] Today more advanced method: light cone sum rules q b c Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

54 Method of light cone sum rules Description of heavy-meson decays: 90 s: Three-point sum rules [Ball,Colangelo,Ovchinnikov, Slobodenyuk,Radyushkin et. al.] Today more advanced method: light cone sum rules q b c Considered correlation function ν l B(p+q) b B q W(q) l b V cb c D ( ) q Γ b c q Γ a p F ab (p,q) = i d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

55 Hadronic representation Choose e.g. Γ a = iγ 5, Γ b = γ µ: Z F µ(p,q) = i d 4 x e ipx 0 T { q(x)iγ 5 c(x), c(0)γ µ b(0)} B Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

56 Hadronic representation Choose e.g. Γ a = iγ 5, Γ b = γ µ: Z F µ(p,q) = i d 4 x e ipx 0 T { q(x)iγ 5 c(x), c(0)γ µ b(0)} B Put in a complete sum over hadronic intermediate states 1 = P h F µ(p,q) = Hadronic spectral density h h : 0 qiγ5 c D D cγµ b B m 2 D p2 + Z s h 0 ds ρ(s) s p 2 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

57 Hadronic representation Choose e.g. Γ a = iγ 5, Γ b = γ µ: Z F µ(p,q) = i d 4 x e ipx 0 T { q(x)iγ 5 c(x), c(0)γ µ b(0)} B Put in a complete sum over hadronic intermediate states 1 = P h F µ(p,q) = Hadronic spectral density h h : 0 qiγ5 c D D cγµ b B m 2 D p2 + Z s h 0 ds ρ(s) s p 2 Access to form factor-definition: F µ(p,q) f D `(2 p µ + q µ)f + (q 2 ) + q µf (q 2 ) m 2 D p2 + Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

58 Calculation of the correlation function The correlation function can by evaluated using the operator-product-expansion (OPE): Z F ab (p,q) = i d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B B(p+q) b q c Γb Γa q p Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

59 Calculation of the correlation function The correlation function can by evaluated using the operator-product-expansion (OPE): Z F ab (p,q) = i d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B b Γb q B(p+q) B-meson: Transition to HQET: B(p + q)> B v > +O(1/m b ), b(0) h v (0) + O(1/m b ) m b -dependence is neglected, but later comes in with kinematic m B -terms c q Γa p Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

60 Calculation of the correlation function The correlation function can by evaluated using the operator-product-expansion (OPE): Z F ab (p,q) = i d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B b Γb q B(p+q) B-meson: Transition to HQET: B(p + q)> B v > +O(1/m b ), b(0) h v (0) + O(1/m b ) m b -dependence is neglected, but later comes in with kinematic m B -terms One can prove light-cone-dominance: Only contributions from x 2 0 Contraction of c(x) c(0) to the free quark-propagator (OPE: light-cone-expansion for q 2, p 2 m 2 b ) The last is only valid for q 2 far enough from q 2 max = (m B m D ) 2, i.e. w far enough from w = 1. c q Γa p Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

61 Calculation of the correlation function The correlation function can by evaluated using the operator-product-expansion (OPE): Z F ab (p,q) = i d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B b Γb q B(p+q) B-meson: Transition to HQET: B(p + q)> B v > +O(1/m b ), b(0) h v (0) + O(1/m b ) m b -dependence is neglected, but later comes in with kinematic m B -terms One can prove light-cone-dominance: Only contributions from x 2 0 Contraction of c(x) c(0) to the free quark-propagator (OPE: light-cone-expansion for q 2, p 2 m 2 b ) The last is only valid for q 2 far enough from q 2 max = (m B m D ) 2, i.e. w far enough from w = 1. c q Γa p Z F ab (p,q) = i Z d 4 x d 4» k (2π) 4 ei(p k)x (/k + m c) Γ a k 2 mc 2 Γ b 0 T q α(x)h vβ (0) B v βα Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

62 Perturbative calculation Z F ab (p,q) = i Z d 4 x d 4» k (2π) 4 ei(p k)x The residual matrix element is the nonperturbative quantity. (/k + m c) Γ a k 2 mc 2 Γ b 0 T q α(x)h vβ (0) B v βα It is parametrized by two B-meson light-cone distribution amplitudes φ +(ω), φ (ω): Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

63 Perturbative calculation Z F ab (p,q) = i Z d 4 x d 4» k (2π) 4 ei(p k)x The residual matrix element is the nonperturbative quantity. (/k + m c) Γ a k 2 mc 2 Γ b 0 T q α(x)h vβ (0) B v βα It is parametrized by two B-meson light-cone distribution amplitudes φ +(ω), φ (ω): D 0 qα(x) h vβ (0) Bv E = if B m Z ( B dωe "(1 iω v x + /v ) 4 0 φ + (ω) φ + (ω) φ (ω) ) # /x γ5 2 v x βα These are dependent of a dimensionful variable ω: Projection of light-quark momentum onto the light-cone. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

64 Perturbative calculation Z F ab (p,q) = i Z d 4 x d 4» k (2π) 4 ei(p k)x The residual matrix element is the nonperturbative quantity. (/k + m c) Γ a k 2 mc 2 Γ b 0 T q α(x)h vβ (0) B v βα It is parametrized by two B-meson light-cone distribution amplitudes φ +(ω), φ (ω): D 0 qα(x) h vβ (0) Bv E = if B m Z ( B dωe "(1 iω v x + /v ) 4 0 φ + (ω) φ + (ω) φ (ω) ) # /x γ5 2 v x βα These are dependent of a dimensionful variable ω: Projection of light-quark momentum onto the light-cone. F ab (p,q) = f [ φ + (ω),φ (ω), p, q ] Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

65 Summary Z F ab (p,q) = i Correlation function: d 4 x e ipx 0 T { q(x)γ a c(x), c(0)γ b b(0)} B hadronic dispersion relation F ab (p,q) = 0 qγa c D D cγ b b B m 2 D p2 + OPE & light-cone-expansion F ab (p,q) 0 q(x) b(0) B f +, f form factors φ +, φ distribution amplitudes quark-hadron-duality (parameter s 0 ) Borel-transformation (parameter M 2 ) = Sum rule: f +(q 2 ), f (q 2 ) = fˆφ +(ω),φ (ω), q 2,... Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

66 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

67 Distribution amplitudes (DAs) The important nonperturbative input parameters are the light-cone distribution amplitudes (DAs) φ +(ω), φ (ω). Generally: Light-cone distribution amplitudes are important (universal) quantities in the description of heavy meson decays and describe the momentum distribution of the quarks/gluons in the meson. (Here in the 2-particle Fock state.) Central role in factorization theorems of hadronic decays. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

68 Distribution amplitudes (DAs) The important nonperturbative input parameters are the light-cone distribution amplitudes (DAs) φ +(ω), φ (ω). Generally: Light-cone distribution amplitudes are important (universal) quantities in the description of heavy meson decays and describe the momentum distribution of the quarks/gluons in the meson. (Here in the 2-particle Fock state.) Central role in factorization theorems of hadronic decays. In principle one has access to the DAs with two-point sum rules. But: Theoretical problems in the condensate expansion. No direct calculation of the DAs, but constraints and implications. Use of models. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

69 Models for distribution amplitudes The models are suggested under some general aspects: Consistent behaviour of the momentum distribution with respect to QCD equations of motion. Right behaviour under renormalization. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

70 Models for distribution amplitudes The models are suggested under some general aspects: Consistent behaviour of the momentum distribution with respect to QCD equations of motion. Example: Right behaviour under renormalization. The light meson distribution amplitudes (Pion DAs) can be modelled as an expansion in orthogonal Gegenbauer-polynomials. For example the leading DA of the corresponding OPE (twist expansion) has the form: 0 1 φ π(u,µ) = 6u(1 + X a n(µ)c 3/2 n (u (1 u)) A n=2,4,... The coefficients a n(µ) can be determined from two-point sum rules. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

71 LCSR with light meson DAs The pion DAs are very well studied and know in a good accuracy. The B π - form factors have been studied extensively with these DAs and light cone sum rules. Correlation functions B(p+q) π(q) b q u d Γ b c Γ a iγ 5 b γ µ q p p+q p Contributions by A. Khodjamirian, N. Offen, R. Rückl, P. Ball and many others... Determination of V ub from exclusive B π decays. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

72 B-meson-2-particle-distribution amplitudes The B-meson DAs cannot be expanded in that way... But: Two-point sum rules suggest a behaviour: φ +(ω 0) ω, φ (ω 0) const Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

73 B-meson-2-particle-distribution amplitudes The B-meson DAs cannot be expanded in that way... But: Two-point sum rules suggest a behaviour: φ +(ω 0) ω, φ (ω 0) const We use an exponential model: [Grozin and Neubert (1997)] φ +(ω) = ω e ω ω 0 ω λ B = φ (ω) = 1 ω 0 e ω ω 0 Z 0 dω φ+(ω) ω ω 0 = λ B = (460 ± 110) MeV [Braun, Ivanov, Korchemsky (2003)] Φ Ω Φ Ω Yet the knowlegde about these DAs is much less precise than the light meson DAs. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

74 B-meson-3-particle-distribution amplitudes Additional: Consideration of the 3-particle-contribution 0 q 2α (x) G λρ (ux)h vβ (0) Bv = f Bm B 4 Z Z dω dξe "(1 i(ω+uξ) v x + /v) 0 0 j (v λ γ ρ v ργ λ )(Ψ A (ω, ξ) Ψ V (ω, ξ)) iσ λρ Ψ V (ω, ξ) «# xλ v ρ x ρv λ xλ v ρ x ρv λ X A (ω,ξ) Y A (ω,ξ) «ffγ 5 v x v x B(p+q) q Γ b c Γ a q βα b p Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

75 B-meson-3-particle-distribution amplitudes Additional: Consideration of the 3-particle-contribution 0 q 2α (x) G λρ (ux)h vβ (0) Bv = f Bm B 4 Z Z dω dξe "(1 i(ω+uξ) v x + /v) 0 0 j (v λ γ ρ v ργ λ )(Ψ A (ω, ξ) Ψ V (ω, ξ)) iσ λρ Ψ V (ω, ξ) «# xλ v ρ x ρv λ xλ v ρ x ρv λ X A (ω,ξ) Y A (ω,ξ) «ffγ 5 v x v x B(p+q) q Γ b c Γ a q βα b p Applying the same method as for 2-particle-DAs: Exponential model: [A. Khodjamirian, Th. Mannel, N. Offen (2007)] Ψ A (ω, ξ) = Ψ V (ω, ξ) = λ2 E 6ω 0 4 ξ 2 (ω+ξ) e ω 0 X A (ω, ξ) = λ2 (ω+ξ) E 6ω 0 4 ξ(2ω ξ)e ω 0 YA (ω, ξ) = λ2 (ω+ξ) E 24ω 0 4 ξ(7ω 0 13ω + 3ξ)e ω 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

76 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

77 Results for the form factors Example: sum rule for f + (q 2 ) M 2 : Borel-parameter, s 0 : duality-parameter f + (q 2 ) = f B mc Z ω0 (q 2 0,s 0 ) 2f D m D 2 dω s ω,q m 2 D 0 M 2 A ( `mb ω + mc φ+ (ω) Φ ± (ω) (1 ω m B ) 1 2m B mc (m B ω) `mb ω + mc (m B ω) 2 + mc 2 q2 (m B ω) 2 + mc 2 Φ ± (ω) q2 + m B mc `mb «!) ω + mc φ + (ω) φ (ω) Φ ± (ω) + f + (q 2 ) 3-particle s(ω,q 2 ) = ωm B + m B m2 c ωq2 m B ω Z Φ ± (ω) = dτ φ + (τ) φ (τ) 0 ω 0 (q 2,s 0 ) = 1 2m B m 2 B q2 + s 0 r 4! mc 2 s 0 m B 2 + m B 2 2 q2 + s 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

78 Results B D-form factors (straight) and only 2-particle-contribution (dashed) f q f q 2 f q f +(q 2 ) with uncertainty (left) and M 2 -dependence (right) 1.4 f q f Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

79 Results B D -form factors (straight line) and only 2-particle-contribution (dashed) 1.5 V q A 1 q A 1 (q 2 ) with uncertainty (left) and M 2 -dependence (right) A 1 q A Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

80 Results Comparison with experiment (BABAR 08) Definition of the form factors in HQET: = v v m B 2 +m2 D( ) q2 1 = A 2 m B m D( ) D(v ) V µ B(v) p = h + (w)(v + v ) µ + h (w)(v v ) µ D (v,ɛ) V µ B(v) q mb m D m B m D = h V (w)ɛ µναβ ɛ ν v α v β D (v,ɛ) A µ B(v) q m B m D = iɛ µ (w + 1)h A1 (w) + i(ɛ q) v µ h A2 (w) + i(ɛ q) v µ h A3 (w) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

81 Results Comparison with experiment (BABAR 08) Definition of the form factors in HQET: = v v m B 2 +m2 D( ) q2 1 = A 2 m B m D( ) D(v ) V µ B(v) p = h + (w)(v + v ) µ + h (w)(v v ) µ D (v,ɛ) V µ B(v) q mb m D m B m D = h V (w)ɛ µναβ ɛ ν v α v β D (v,ɛ) A µ B(v) q m B m D = iɛ µ (w + 1)h A1 (w) + i(ɛ q) v µ h A2 (w) + i(ɛ q) v µ h A3 (w) HQET-prediction for m b, m c : h + (w) = h V (w) = h A1 (w) = h A3 (w) = ξ(w) Isgur-Wise-function h (w) = h A2 (w) = 0 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

82 Results Comparison with experiment (BABAR 08) Definition of the form factors in HQET: = v v m B 2 +m2 D( ) q2 1 = A 2 m B m D( ) D(v ) V µ B(v) p = h + (w)(v + v ) µ + h (w)(v v ) µ D (v,ɛ) V µ B(v) q mb m D m B m D = h V (w)ɛ µναβ ɛ ν v α v β D (v,ɛ) A µ B(v) q m B m D = iɛ µ (w + 1)h A1 (w) + i(ɛ q) v µ h A2 (w) + i(ɛ q) v µ h A3 (w) HQET-prediction for m b, m c : h + (w) = h V (w) = h A1 (w) = h A3 (w) = ξ(w) Isgur-Wise-function h (w) = h A2 (w) = 0 Experimentally measurable: G(w) = h + (w) m B m D m B + m D h (w) R 1 (w) = h V (w) h A1 (w) h A1 (w) R 2 (w) = m D h m A2 (w) + h A3 (w) B h A1 (w) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

83 Comparison with experiment (BABAR 08) Form factors G(w), h A1 (w) [Sum rules: (w = 1.3 w max ), exp.: (w = 1 w max )] G(w) h A1 (w) w w Sum rules only applicable for small momentum transfer: w 1 w 1.3 w max Fit to experimental data with the parametrization: [Caprini, Lellouch, Neubert 97] h A1 (w) = h A1 (1) 1 8ρ 2 z(w) + (53ρ 2 15)z(w) 2 (231ρ 2 91)z(w) 3 G(w) = G(1) 1 8ρ 2 z(w) + (51ρ 2 10)z(w) 2 (252ρ 2 84)z(w) 3 Normalization h A1 (1), G(1) from Lattice-QCD z(w) = w+1 2 w+1+ 2 Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

84 Comparison with experiment (BABAR 08) Ratios R 1 (w), R 2 (w) [Sum rules: (w = ), exp.: (w = 1 1.5)] R 1 w R 2 w R 1 (w) = R 1 (1) 0.12(w 1) (w 1) 2 R 2 (w) = R 2 (1) (w 1) 0.06(w 1) 2 In the ratios the normalizations cancel: better prediction Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

85 Limit of infinitely heavy quarks Taking the limit: m c, m b, κ := mc = const for the sum rules: m b h +(w) = h V (w) = h A1 (w) = h A3 (w) : = ξ(w) = β 0 /w Z 0 dρ e Λ ρ w τ h 1 2w φb (ρ) + `1 1 φb 2w + (ρ)i h (w) = h A2 (w) = 0 Scaling of the parameter: m B = m b + Λ, m D = m c + Λ, s D 0 = m2 c + 2mcβ 0, M 2 = 2m cτ Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

86 Limit of infinitely heavy quarks Taking the limit: m c, m b, κ := mc = const for the sum rules: m b h +(w) = h V (w) = h A1 (w) = h A3 (w) : = ξ(w) = β 0 /w Z 0 dρ e Λ ρ w τ h 1 2w φb (ρ) + `1 1 φb 2w + (ρ)i h (w) = h A2 (w) = 0 Scaling of the parameter: m B = m b + Λ, m D = m c + Λ, s D 0 = m2 c + 2mcβ 0, M 2 = 2m cτ 1.2 Independent of the quark mass ratio κ Numerical one gets the right value (taking into account the uncertainty): ξ(1) Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

87 Outline 1 Theory of B D ( ) decays Introduction: Determination of V cb Use of Heavy Quark Symmetry 2 QCD Sum rules Basic idea of QCD sum rules Light cone sum rules Distribution amplitudes 3 Results and Discussion Numerical results Summary and Outlook Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

88 Summary Sum rules derived for all B D ( ) -form factors. Only applicable for small momentum transfer q 2 qmax 2 bzw. w (1,3 wmax ). But this is the region, where HQS and Lattice QCD have difficulties. Two- and three-particle-contributions considered. Good agreement with experimental data. Right behaviour in the limit m c, m b. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

89 Summary Sum rules derived for all B D ( ) -form factors. Only applicable for small momentum transfer q 2 qmax 2 bzw. w (1,3 wmax ). But this is the region, where HQS and Lattice QCD have difficulties. Two- and three-particle-contributions considered. Good agreement with experimental data. Right behaviour in the limit m c, m b. Relative big uncertainty, mainly due to errors in the input quantities: B-meson-DAs and decay constants. Yet, in accurracy the result cannot compete with the HQS-determination, but it is an important cross-check and allows more insight into the form factor shape and the heavy quark limit. Further studies of 1/m-corrections in HQET are possible here. Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

90 Outlook In the future... : Closer examination of 1 m Q -expansion Determination of V ub V cb using the sum rules smaller uncertainty Examination/calculation of α s -corrections (currently at work) Further project: Evaluation of light cone sum rules with π-meson-distribution amplitudes for D π- and D K -form factors with A. Khodjamirian and N. Offen...? Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52

91 Ich habe Jahre gebraucht, um sagen zu können, dass alles ganz einfach ist. It took me years to get able to, but now I can say that all this is simple. - N.Offen Special thanks again to A. Khodjamirian and Th. Mannel! Christoph Klein (Univ. Siegen) B D ( ) formf. from QCD sum rules / 52