B-meson form factors

Transcription

1 B-meson form factors Mikhail A. Ivanov JINR Dubna HQP 08, Dubna

2 Contents Introduction Relativistic Quark Model of Hadrons B c-meson Heavy baryons B- to light-meson transition form factors Conclusion

3 Introduction The weak decays of B-meson provide us by an information on parameters of the standard model (SM) L int = g2 2 ` ū, c, t γ µ 1 γ 5 V 2 0 V V ud V us V ub V cd V cs V cb V td V ts V tb 1 A d s b 1 AW µ + h.c. Semileptonic and nonleptonic B-decays induced by charged-current quark transitions b c, u give a direct access to the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements V cb and V ub because of they proceed on the tree-level. Rare B-decays induced by flavor-changing neutral-current (FCNC) quark transitions b s, d are forbidden at the tree-level in the SM and proceed via loop diagrams only. It provides an interesting hunting ground to look for new physics beyond the SM.

4 A theoretical framework for description of the B-decays is based on the factorization of short- and long- distance dynamics. The short-distance effects can be treated perturbatively. The long-distance effects are encoded in hadronic matrix elements of the relevant QCD-operators. Their calculation requires information about the structure of hadrons and therefore cannot be done in perturbation theory. A variety of theoretical approaches have been applied to this problem.

5 Example: b cūs-quark transitions The W-exchange tree-level amplitude:» g2 2 8 V usv cb ( so µ gµν u) M 2 W ( co ν b) k2 The momentum transfer k << M W: g µν M 2 W M 2 W k2 gµν 8 g 2 2 «GF 2 «g µν Thus we arrive at effective Hamiltonian H tree eff = GF 2 V usv cb Q 2, Q 2 ( s αo µ u α)( c βo µb β) By taking into account QCD corrections: H eff = GF 2 V usv cb [C 1(µ)Q 1 + C 2(µ)Q 2], Q 1 ( s αo µ u β)( c βo µb α)

6 Effective Hamiltonian Using the operator product expansion (OPE) formalism and renormalization group techniques, the effective Hamiltonian of the weak decays is derived. A(f i) = f H eff i = GF X λ CKM C k(µ) 2 {z } k SD SD = Short-Distance contributions LD = Long-Distance contributions f Q k(µ) i {z } LD The Wilson coefficients C i(µ) are calculated by using matching the full and effective theories, and the renormalization group. Q k(µ) are the local operators generated by electroweak interactions and QCD The problem is to evaluate the matrix elements f Q k(µ) i

7 Relativistic Quark Model of Hadrons Main assumption: hadrons interact via quark exchange only Interaction Lagrangian L int = g H H(x) J H(x) Efimov, Ivanov 1993

8 Relativistic Quark Model of Hadrons Main assumption: hadrons interact via quark exchange only Interaction Lagrangian L int = g H H(x) J H(x) Efimov, Ivanov 1993 Quark currents Z Z J M(x) = dx 1 dx 2 F M(x; x 1, x 2) q a f 1 (x 1) Γ M q a f 2 (x 2) Meson J B(x) = Z Z Z dx 1 dx 2 dx 3 F B(x; x 1, x 2, x 3) Γ 1 q a 1 f 1 (x 1) q a 2 f 2 (x 2)C Γ 2 q a 3 f 3 (x 3) ε a 1a 2 a 3 Baryon Z Z J µ T (x) = dx 1... dx 4 F B(x; x 1,..., x 4) Tetraquark u a 1 (x 1) Cγ 5 c a 2 (x 2) ū a 3 (x 3) γ µ C c a 4 (x 4) ε a 1a 2 c ε a 3a 4 c

9 The vertex functions and quark propagators The vertex functions F B(x, x 1,..., x n) = δ (4) x nx X «w ix i Φ H (x i x j) 2 i<j i=1 where w i = m i/ P i m i. The quark propagators S q(x 1 x 2) = Z d 4 k e ik(x 1 x 2 ) (2π) 4 i m q k Confinement restriction m H < P i m qi Adjustable parameters: constituent quark masses m q size parameters Λ H from the vertex functions Φ H p 2 /Λ 2 H

10 Compositeness condition Z H = 0 Salam 1962; Weinberg 1963 A composite field and its constituents are introduced as elementary particles The transition of a composite field to its constituents is provided by the interaction Lagrangian The renormalization constant Z 1/2 is the matrix element between a physical state and the corresponding bare state. If there is a stable bound state which we wish to represent by introducing a quasi-particle H, then elementary particle must have renormalization factor Z equal to zero Z 1/2 H = < H bare H dressed > = 0 We use the compositeness condition to determine the hadron-quark coupling constant, e.g. Z M = 1 Π (m 2 M) = 0 where Π(p 2 ) is the meson mass operator.

11 Z=0 : how it works + + free vertex mass operator mass renormalization ( 1 Π 2 ( m ) ) = Z = 0 + dressed vertex

12 Semileptonic B D transition b k + p1 O µ = γ µ γ µ γ 5 k + p2 B(p 1 ) D(p 2 ) c ū k ū φ B ( (k + w bu p 1 ) 2) φ D ( (k + w cu p 2 ) 2) w 12 = m 2 m 1 +m 2

13 Semileptonic B D transition b k + p1 O µ = γ µ γ µ γ 5 k + p2 B(p 1 ) D(p 2 ) c ū k ū φ B ( (k + w bu p 1 ) 2) φ D ( (k + w cu p 2 ) 2) w 12 = m 2 m 1 +m 2 Heavy quark limit: m H = m Q + E, m Q 1 1+ vi 1 m i k p i 2 kv i + E, v = p m

14 Isgur-Wise function M µ BD (p1, p2) = f+(q2 )(p 1 + p 2) µ + f (q 2 )(p 1 p 2) µ, f ± = ξ(w) = 1 Z 1 I HH Mb ± Mc 2 M bm c ξ(w), 0 dτ W Z 0» du φ 2 H(z) a(z) + p u/w b(z), W = 1 + 2τ(1 τ)(w 1), z = u 2E p u/w. Here the functions a(z), b(z) are the scalar and vector parts of the light quark propagator. The integral I HH comes from the compositeness condition Z H = 0 and provides the correct normalization ξ(w = 1) = 1.

15 The upper bound for the Isgur-Wise function is obtained if put E = 0. where ξ(w) ξ(w) = ξ(w) E=0 = 1 j ln[w + w2 1] + 2R ff 1 + R w w R = R 0 du φ 2 H(u) u b(u). R du φ 2 H (u) a(u) 0 As a consequence the slope parameter has the lower bound ρ 2 = ξ (1) 1 3.

16 B c -meson lowest ( bc)-bound state with open flavor B c can decay only weakly Mass and lifetime, update 2008: M(B c) = 6300 ± 14(stat) ± 5(syst)MeV Fermilab, 1998 τ(b c) = (stat) ± 0.032(syst) ps

17 B c -meson lowest ( bc)-bound state with open flavor B c can decay only weakly Mass and lifetime, update 2008: M(B c) = 6300 ± 14(stat) ± 5(syst)MeV Fermilab, 1998 τ(b c) = (stat) ± 0.032(syst) ps Semileptonic B c-decays Ivanov, Körner, Santorelli the CKM-enhanced b c-decays the CKM-suppressed b u-decays the CKM-enhanced c s-decays the CKM-suppressed c u-decays B + c (η c, J/ψ) l + ν B + c (D 0, D 0 ) l + ν B + c ( B 0 s, B 0 s ) l + ν B + c ( B 0, B 0 ) l + ν

18 Branching ratios (in %) of exclusive semileptonic B c decays into ground state charmonium states, and into ground state charm and bottom meson states. For the lifetime of the B c we take τ(b c) = 0.45 ps Mode This work Kiselev Chang Ebert Nobes B c η ceν B c η cτν B c J/ψeν B c J/ψτν B c D 0 eν B c D 0 τν B c D 0 eν B c D 0 τν B c B 0 seν B c B 0 s eν B c B 0 eν B B 0 eν

19 Charmonium states 2S+1 L J title quark current mass (GeV) J PC = S 0 = η c q iγ 5 q J PC = 1 3 S 1 = J/ψ q γ µ q J PC = P 0 = χ c0 q q J PC = P 1 = χ c1 q γ µ γ 5 q J PC = P 1 = h c q µ γ 5 q J PC = P 2 = χ c2 (i/2) q γ µ ν +γ ν µ q J PC = 2 3 D 2 = ψ(3836) (i/2) q γ µ γ 5 ν +γ ν γ 5 µ q 3.836

20 The branching ratios (in %) of exclusive semileptonic B c decays into p wave charmonium states, and into the 3 D 2 orbital excitation of the charmonium state ψ(3836). Mode This work Chang B c χ c0 e ν B c χ c0 τ ν B c χ c1 e ν B c χ c1 τ ν B c h c e ν B c h c τ ν B c χ c2 e ν B c χ c2 τ ν B c ψ(3836) e ν B c ψ(3836) τ ν

21 Nonleptonic and rare B and B c -decays Plenty of nonleptonic decays by using the factorizing approximation Rare decays B Kl + l and B c D(D )l + l Effective b sl + l Hamiltonian H eff(b sl + l ) = GF 2 α λ t 2π 2mb q 2 C eff 7 n C eff 9 ( sb) V A ` ll V + C10 ( sb) ` ll V A A ff ` ll s iσ µν (1 + γ 5 ) q ν b V Account for the vector resonances o C eff 9 = C 9 + C 0 nh loop(m c, q 2 ) + h res(q 2 ) h res = 3π κ α 2 X V i =ψ(1s),ψ(2s) C eff 7 = C 7 1 C5 C6 3 Γ(V i l + l ) m Vi m Vi2 q 2 im Vi Γ Vi

22 B K form factors s F+ FT F s s Solid line: our result Dash line: A. Ali et al. PRD 61 (2000)

23 B meson branching ratios Ref. Br(B K µ + µ ) Br(B Kτ + τ ) Br(B K ν ν) Ali Ali 2002 (3.5 ± 1.2) 10 7 Melikhov Geng Our BaBar (3.9 ± 0.7 ± 0.2) 10 7 < arxiv: [hep-ex]

24 Heavy baryons Faessler, Ivanov,.Lyubovitskij, Körner Λ b Λ c ξ(ω) = ( 2 1+ω )1.7+1/ω ρ ξ = ξ (1) = 1.05 ± 0.3 Br(Λ b Λ ceν e) = (7.8 ± 1.1)% QCD SR: ρ ξ = Lattice QCD: ρ ξ = Skyrme model: ρ ξ = 1.3 Grozin,Yakovlev, 92/99, Dai et al, 1996 Bowler et al, 1998 Jenkins et al, 1993

25 Heavy baryons Faessler, Ivanov,.Lyubovitskij, Körner Λ b Λ c ξ(ω) = ( 2 1+ω )1.7+1/ω ρ ξ = ξ (1) = 1.05 ± 0.3 Br(Λ b Λ ceν e) = (7.8 ± 1.1)% QCD SR: ρ ξ = Lattice QCD: ρ ξ = Skyrme model: ρ ξ = 1.3 Grozin,Yakovlev, 92/99, Dai et al, 1996 Bowler et al, 1998 Jenkins et al, 1993 DELPHI Coll., PLB585 (2004) 63 ρ ξ = 2.03 ± 0.46(stat) Br(Λ b Λ ceν e) = ( (stat))%

26 Nonleptonic decays in factorizing approximation Process Our Experiment Λ + c Λπ ± 0.28 Λ + c Σ 0 π ± 0.32 Λ + c Σ + π ± 0.34 Λ + c p K ± 0.6 Λ + c Ξ 0 K ± 0.14 Λ + c pφ ± Ξ 0 c Ξ 0 π Ξ 0 c Σ + K 0.27 Ω 0 c Ξ 0 K Λ 0 b Λπ Λ 0 b pk < Λ 0 b J/ψΛ ± 0.028

27 Nonleptonic decays in factorizing approximation Process Our Experiment Λ + c Λπ ± 0.28 Λ + c Σ 0 π ± 0.32 Λ + c Σ + π ± 0.34 Λ + c p K ± 0.6 Λ + c Ξ 0 K ± 0.14 Λ + c pφ ± Ξ 0 c Ξ 0 π Ξ 0 c Σ + K 0.27 Ω 0 c Ξ 0 K Λ 0 b Λπ Λ 0 b pk < Λ 0 b J/ψΛ ± One-photon and one-pion decays

28 Semileptonic decays of double heavy baryons (heavy quark limit near zero recoil) Λ µ 2 n Ξ bc (v) Ξ cc(v ) = (3 + w)o µ + v µ + v µo η(w) 6 r Λ µ 2 = Ξ bc (v) Ξcc(v ) 3 r Λ µ 2 n 1 Ξ bc (v) Ξ cc (v,ν 1 ) = 3 2 Oµ v ν 1 g 1o µν η(w) Λ µ = n 1 2 Ξ bc (v) Ξ cc (v,ν 1 ) 2 (1 + w) gµν vν 1 v µ n 1 2 (1 + w) Oµ 1 2 (vµ + v µ )o η(w) Λ µ Ξ bc (v,ν) Ξcc(v ) = vν 1 γ µ γ 5o η(w) r 2 n o O µ v ν g µν η(w) 3 Λ µ Ξ bc (v,ν) Ξ cc (v,ν 1 ) = 2 nh 1 2 (vµ + v µ ) γ µ γ 5i g νν h g µν v ν 1 + g µν 1 v ν io η(w) where η(w) is universal function with η(1) = 1.

29 B- to light-meson transition form factors Dyson-Schwinger equations in QCD M.A. Ivanov, J.G. Körner, S.G. Kovalenko, C.D. Roberts 2007 C.D. Roberts and A.G. Williams 1994 One of the feature of the DSE study is that the dressed-quark propagator is confined S(p) = iγ p σ V(p 2 ) + σ S(p 2 ) The functions σ S,V(p 2 ) are algebraic combinations of the entire functions F(x) = 1 e x, x = p 2 /λ 2. x The entire range of physical momenta is directly accessible

30 Technical remark Calculation involves the numerical evaluation of 4-dimensional integral whose integrand is the convolution of entire functions and functions with a simple pole The straightforward use of spherical coordinates in the Euclidean loop integral works well only for p 2 B < M 2 b When p 2 B M 2 b one needs to shift the integration contour into complex plane Since that is not easily done numerically, we employ an alternate representation that can be used for any p 2 B

31 An example with the only Euclidean time integral: I[E B] = = = Z dk 4 π Z Z dα 0 Z 0 M 2 b e k (k4 + ieb)2 dk 4 π e k2 4 α[m2 b +(k 4+iE B ) 2 ] dα 1 + α e αm2 b + α 1+α E2 B Heavy quark limit E B = M b I[E B] = π 2 1 M b

32 Leptonic decays and normalization φh(k 2 ) k + w1p k w2p Γ(k + w1p, k + w1p) k + w1p k + w1p φh(k 2 ) φh(k 2 ) k + w1p w1 w2 k w2p φh(k 2 ) φh(k 2 ) k w2p k w2p Γ(k w2p, k w2p)

33 This work Other Reference f ρ (2) PDG f K (5) PDG f D (16.7) CLEO (25) 227 RCQM f Ds (12)(6) CLEO 283(17)(4)(14) BaBar 255 RCQM f Ds f D (11)(3) CLEO 1.12 RCQM f D (20) +3 2 LAT 249 RCQM f D s (16) LAT 266 RCQM f B (stat) (syst) BELLE 216(9)(19)(4)(6) HPQCD LAT 177 (17) UKQCD LAT 179 (18) LAT 210.5(11.4)(5.7) Chiral Lat 187 RCQM

34 This work Other Reference f Bs (32) HPQCD LAT 260 (7) (26) (8) (5) LAT 204 (12) UKQCD LAT 204 (16) LAT 218 RCQM f Bs f B (0.03) (0.01) HPQCD LAT 1.15 (0.02) UKQCD LAT 1.14 (0.03) LAT 1.16 RCQM f B (24) LAT 196 RCQM f B s (20) LAT 229 RCQM

35 Heavy to light transitions b k + p1 J = γ µ, γ µ γ 5, iσ µν qν, iσ µν qνγ 5 k + p2 d(s) B P(V ) ū k ū φ B (k 2 ) φ P(V ) ((k + w 2 p 2 ) 2 )

36 B-pi form factors 4 2 F + F- F T q 2, GeV B-K form factors F + F- F T q 2, GeV 2

37 B-rho form factors 4 A 0 A A- V q 2, GeV B-Kst form factors A 0 A + A- V q 2, GeV 2

38 4 B-rho+gamma form factors 3 2 a 0 a + g q 2, GeV B-Kst+gamma form factors a 0 a + g q 2, GeV 2

39 This work LCSR LCSR LCQM DQM RQM RCQM [1] [2] [3] [4] [5] [6] f + Bπ (0) ± ± f + BK (0) ± ± fbπ(0) T ± ± f T BK(0) ± ± V Bρ (0) ± V BK (0) ± A Bρ 1 (0) ± A1 BK (0) ± (0) ± A Bρ 2 A BK 2 (0) ± (0) ± T Bρ 1 T BK 1 (0) ± [1] A. Khodjamirian, T. Mannel, N. Offen, Phys. Rev. D 75, (2007). [2] P. Ball and R. Zwicky, Phys. Rev. D 71, (2005). [3] C. D. Lü, W. Wang, Z. T. Wei, Phys. Rev. D 76, (2007) [4] D. Melikhov, N. Nikitin, S. Simula, Phys. Rev. D 57, 6814 (1998). [5] D. Ebert, R. N. Faustov, V. O. Galkin, Phys. Rev. D 75, (2007). [6] A. Faessler, T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij, Eur. Phys. J. direct C 4, 18 (2002).

40 Conclusion A wide-ranging analysis of B, B c meson and single and double heavy baryon exclusive semileptonic and rare decays are presented by using relativistic phenomenological framework based on the quark structure of heavy and light hadrons. This framework provides the relativistic description of the physical amplitudes. All transition form factors are directly calculable on the entire physical region of accessible momentum transfer. In the heavy quark limit m b, m c the leptonic decay constants evolve as m 1/2 Q and the matrix elements describing semileptonic heavy-heavy decays can be expressed in terms of a single Isgur-Wise function according to heavy-quark effective theory. The calculated transition form factors are used to evaluate the numerous physical observables.