arXiv:hep-ph/0510262v1 20 Oct 2005
Light-cone sum rules: A SCET-based formulation
F. De Fazioa? , Th. Feldmannb and T. Hurthc?
a b c
Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy Fachbereich Physik, Universit¨t Siegen, D-57068 Siegen, Germany a
CERN, Dept. of Physics, Theory Division, CH-1211 Geneva 23, Switzerland and SLAC, Stanford University, Stanford, CA 94309, USA.
We describe the construction of light-cone sum rules (LCSRs) for exclusive B-meson decays into light energetic hadrons from correlation functions within soft-collinear e?ective theory (SCET). As an example, we consider the SCET sum rule for the B → π transition form factor at large recoil, including radiative corrections from hard-collinear loop diagrams at ?rst order in the strong coupling constant.
1. INTRODUCTION Form factors parameterizing hadronic matrix elements de?ning B decays to a light pseudoscalar (P ) or vector meson (V ) play an important role in several respects. For example, they enter in the determination of |Vub | from exclusive modes. However, since they are non-perturbative objects, their determination is a di?cult task. Let us consider for de?niteness the decay of a B meson in its rest frame into a highly energetic pion. Several energy scales are involved: i) Λ = few × ΛQCD , the soft scale set by the typical energies and momenta of the light degrees of freedom in the hadronic bound states; ii) mb , the hard scale set by the b-quark mass; iii) the hard√ collinear scale ?hc = mb Λ appearing via interactions between soft and energetic modes. The dynamics of hard and hard-collinear modes can be described perturbatively in the heavy-quark limit. The separation of the two perturbative scales from the non-perturbative hadron dynamics is formalized within the framework of softcollinear e?ective theory (SCET) [1,2]. The small expansion parameter in SCET is given by λ = Λ/mb , such that λ2 mb ? ?hc ? λmb ? mb . SCET describes B decays to light hadrons with energies much larger than their masses, assum? Speaker ? Heisenberg
ing that their constituents have momenta almost collinear to the hadron momentum p? . Introducing two light-like vectors n? = (1, 0, 0, ?1), + n? = (1, 0, 0, 1) one can generically write: p? = ? p? + p? + p? , with p? = (n? p)/2n? , p? = + ? + + ? ⊥ (n+ p)/2n? ; momenta are then classi?ed accord? ing to the scaling of their light-cone coordinates (p+ , p? , p⊥ ). In order to see how SCET can be exploited in the case of heavy-to-light B decays, we have to recall some general features of the form factors relevant to these kind of transitions. In the large energy limit of the ?nal state, B → P, V form factors obey spin symmetry relations , broken by hard gluon corrections to the weak vertex and hard spectator interactions. In the heavy-quark limit one can write  (see also [5,6]): ? π|ψ Γi b|B = Ci (E, ?I ) ξπ (?I , E) + Ti (E, u, ω, ?II ) ? φB (ω, ?II ) + ? φπ (u, ?II ) + (1) ...,
where Γ is a generic Dirac structure and the dots stand for sub-leading terms in Λ/mb . The matrix elements in (1) get therefore two contributions. The ?rst one contains the short-distance functions Ci , arising from integrating out hard modes: ?I < mb , and a “soft” form factor ξπ which does not depend on the Dirac structure of the decay current. In this contribution, the hard-collinear interactions are not factorizable, so 1
2 that the “soft” form factor is in general a nonperturbative object of order (αs )0 . The second term in (1) factorizes into a hard-scattering kernel Ti and the light-cone distribution amplitudes φB and φπ . Ti contains the e?ect of both hard and hard-collinear dynamics: ?II < ?hc . Both Ci and Ti can be computed as perturbative series in αs , and potentially large logarithms ln mb /?I and ln ?hc /?II can be resummed by renormalizationgroup techniques (the e?ective theories for the two short-distance regimes are known as SCETI and SCETII , respectively). A still controversial question is to what extent the ?rst contribution is numerically suppressed by Sudakov e?ects. Let us consider eq. (1) for Γi = γ? , i.e. for the QCD vector current. This can be matched onto SCETI currents as follows : ? q γ? b → C4 n? + C5 v ? ξhc Whc Ys? hv + . . . ? ? (2)
F. De Fazio, Th. Feldmann and T. Hurth theoretical input. A theoretical approach for this purpose is represented by QCD/light-cone sum rules (see for instance [8,9,10]). In  we have shown that it is possible to formulate light-cone sum rules within SCET, in a di?erent way with respect to the traditional method. We summarize below the main features of this new formulation. 2. SUM RULES IN SCET: THE CASE OF B → π DECAY In contrast to the traditional approach where the B meson is represented by an interpolating current, we treat it as an external ?eld and not as a propagating particle in the correlation function (see also ). Actually, the heavy quark is nearly on-shell in the end-point region. In SCETI this is re?ected by the fact that hard sub-processes (virtualities of order m2 ) are already integrated out b and appear in coe?cient functions multiplying J0 . Instead, the short-distance (o?-shell) modes in SCETI are the hard-collinear quark and gluon ?elds. Hence, our starting point is the correlator Π(p′ ) = i d4 x eip x 0|T [Jπ (x)J0 (0)]|B(pB ) , (5)
where dots represent subleading terms and C4 = 1 + O(αs ), C5 = O(αs ). v ? is the heavy-quark velocity with n± v = 1. The direction of the momentum of the (massless) pion is given by p? = π n n (n+ pπ ) n? /2. Besides, ξhc (x) = /?4/+ ψhc (x) is ? a hard-collinear light-quark ?eld in SCETI and hv is the usual heavy quark ?eld in HQET. The hard-collinear and soft Wilson lines Whc and Ys appear to render the de?nition gauge-invariant. The soft form factor in (1) can be de?ned as  ? π(p′ )|(ξhc Whc )(0) (Ys? hv )(0)|B(mB v) = (n+ p′ ) ξπ (n+ p′ , ?I ) , (3) Neglecting O(αs ) e?ects the approximate symmetry relations mentioned above between the vector and tensor form factors for B → π transitions read [3,7]: f+ (q 2 ) ? ? mB mB f0 (q 2 ) ? fT (q 2 ) n+ pπ mB + mπ ξπ (q 2 ) . (4)
where p? = mB v ? , and B ? J0 (0) = ξhc (0)Whc (0)Y ? (0)hv (0) ,
? = ?i ξhc (x) n+ γ5 ξhc (x) / (7) ?hc Whc (x)n+ γ5 Y ? qs (x) + h.c. , ?i ξ / s where qs is the soft quark ?eld in SCET and 0|Jπ |π(p′ ) = (n+ p′ ) fπ . In the following we will consider a reference frame where p′ = v⊥ = 0 ⊥ and n+ v = n? v = 1. In this frame the two independent kinematic variables are (n+ p′ ) ? 2Eπ = O(mb ), 0 > (n? p′ ) = O(Λ), with |n? p′ | ? m2 /(n+ p′ ). The dispersive analysis will be perπ formed with respect to (n? p′ ) for ?xed values of (n+ p′ ). As with all QCD sum rule calculations, the procedure consists in writing the correlator (5) in two di?erent ways: we will refer to them as the hadronic side and the SCET side. On the hadronic side, one can write: ΠHAD (n? p′ ) = Π(n? p′ )
? ≡ ?i ψ(x) n+ γ5 ψ(x) /
SCET thus provides a ?eld-theoretical framework to achieve the factorization of short- and long-distance physics, and to calculate the former in renormalization-group-improved perturbation theory. However, non-perturbative quantities such as the soft form factors remain undetermined without further phenomenological or
+ Π(n? p′ )
Light-cone sum rules: A SCET-based formulation the ?rst term represents the contribution of the pion, while the second takes into account the role of higher states and continuum above an e?ective threshold ωs = O(Λ2 /n+ p′ ). One has Π(n? p′ )
3 in SCET. Finally, a Borel transformation with parameter ωM is applied to both sides, giving the following sum rule at tree level: ξπ (n+ p′ ) = f B mB fπ (n+ p′ )
0|Jπ |π(p ) π(p )|J0 |B(pB ) m2 ? p′2 π (n+ p′ )ξπ (n+ p′ )fπ = ? , (9) n? p′
dω e?ω/ωM φB (ω) .(12) ?
obtained in the chosen frame where p′ = 0 and ⊥ neglecting the pion mass. At tree level, the SCET
J0 s s
v @ @
The inclusion of radiative corrections to the correlation function (5) comes from hard-collinear loops, as shown in Fig. 2 for the leading order in αs . The explicit calculation shows that the scaledependence of the correlation function cancels with that of the Ci (?) at the considered leading logarithmic order (involving double logs). As for
J0 J0 s hc (a1) s Jπ (a2) s hc Jπ
Figure 1. Leading contribution to the correlation function for the SCET current J0 .
J0 s hc (a3) s Jπ (a4) s s
J0 hc Jπ
side stems from calculating the diagram in Fig. 1, with the result: ∞ φB (ω) ? , (10) Π(n? p′ ) = fB mB dω ω ? n? p′ ? iη 0 where ω = n? · k, k ? being the momentum of the soft light quark that ends up as spectator in the B. In (10) we used the momentum-space representation of LCDAs for B mesons as in [13,7], ifB mB MB = ? × βα 4 1+v / B . φ+ (ω)n+ + φB (ω)n? + . . . γ5 / / ? 2 βα Notice that (10) has already the form of a dispersion relation in the variable n? p′ : Π(n? p′ ) = with 1 π
J0 s hc (b1) s Lξq
J0 s Jπ (b2) s hc Lξq
Figure 2. Diagrams contributing to the sum rule for ξπ to order αs with hard-collinear loops and no external soft gluons.
1 Im[Π(ω ′ )] = fB mB φB (ω ′ ). The ?nal sum ? π rule is obtained by writing also Π(n? p′ ) accont. cording to a dispersion relation in which the spectral function is identi?ed with the one computed
Im[Π(ω )] , ? n? p′ ? iη
the numerical analysis, we ?x the sum rule parameters ωS and ωM from the sum rule for fπ , which provides us with the default values ωM ? ωS ? 0.2 GeV. For φB (ω), we use the parametriza? tion proposed in : φB (ω) = e?ω/ω0 /ω0 with ? 1/ω0 = φB (0) = 2.15 GeV?1 . Fixing one of the ? two parameters to its default value and varying the other, we may investigate the dependence on such quantities. It turns out that going from LO
4 to NLO such a dependence becomes moderate (see ref.  for details). Taking into account the various uncertainties, we obtain: Ci (?) · ξπ (mB , ?) = 0.27+0.09 , ?0.11 Ci (mb ) (13)
F. De Fazio, Th. Feldmann and T. Hurth thus con?rming the power-counting adopted in the QCD-factorization approach. The improvement of the SCET sum rule for the B → π form factor and the extension to other decays requires a better understanding of both, the size and the renormalization-group behaviour, of the light-cone wave functions for higher Fock states in the B meson. These issues are left for future investigations. REFERENCES 1. C. W. Bauer et al., Phys. Rev. D 63 (2001) 114020; C. W. Bauer and I. W. Stewart, Phys. Lett. B 516 (2001) 134. 2. M. Beneke et al., Nucl. Phys. B 643 (2002) 431; M. Beneke and T. Feldmann, Phys. Lett. B 553 (2003) 267. 3. J. Charles et al., Phys. Rev. D 60 (1999) 014001. 4. M. Beneke and T. Feldmann, Nucl. Phys. B 685 (2004) 249. 5. R. J. Hill and M. Neubert, Nucl. Phys. B 657 (2003) 229; B. O. Lange and M. Neubert, Nucl. Phys. B 690 (2004) 249. 6. C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 67 (2003) 071502. 7. M. Beneke and T. Feldmann, Nucl. Phys. B 592 (2001) 3. 8. A. Khodjamirian and R. R¨ ckl, Adv. Ser. Diu rect. High Energy Phys. 15 (1998) 345. 9. P. Ball and V. M. Braun, Phys. Rev. D 58 (1998) 094016. P. Ball, JHEP 9809 (1998) 005. 10. P. Colangelo and A. Khodjamirian, hep-ph/0010175. 11. F. De Fazio, T. Feldmann and T. Hurth, arXiv:hep-ph/0504088. 12. A. Khodjamirian, T. Mannel and N. O?en, Phys. Lett. B 620 (2005) 52. 13. A. G. Grozin and M. Neubert, Phys. Rev. D 55 (1997) 272. 14. M. Beneke et al., Phys. Rev. Lett. 83 (1999) 1914; Nucl. Phys. B 606 (2001) 245. 15. C. H. Chen, Y. Y. Keum and H. n. Li, Phys. Rev. D 64 (2001) 112002. 16. C. W. Bauer et al., Phys. Rev. D 70 (2004) 054015.
which compares well with other estimates for the B → π form factor in full QCD. Our approach can also be applied to calculate the factorizable form factor contribution, which comes from spectator scattering terms. This can be obtained starting from the correlator: Π1 (p′ ) = i d4 x eip x 0|T [Jπ (x)J1 (0)]|B(pB ) ,
? where J1 = ξhc g A⊥ hv in the light-cone gauge. /hc The remarkable result of the SCET-sum-rule for the B → π form factor is that the ratio of factorizable and non-factorizable contributions is independent of the B-meson wave function to ?rst approximation and amounts numerically to about ≈ 6% , which is in line with the power counting used in QCD factorization [14,7], but contradicts the assumptions of the pQCD approach  and the results of a recent study in . 3. CONCLUSIONS We have described the approach derived in  consisting in the derivation of light-cone sum rules for exclusive B-decay amplitudes at large recoil within soft-collinear e?ective theory (SCET). This formalism de?nes a consistent scheme to calculate both factorizable and non-factorizable contributions to exclusive B decays as a power expansion in Λ/mb . The non-perturbative information is encoded in the light-cone wave functions of the B meson, and in the sum-rule parameters. An explicit example is provided by the study of the factorizable and non-factorizable contributions to the B → π form factor at leading power in Λ/mb . The result for the central value of the “soft”/non-factorizable B → π form factor is consistent with corresponding estimates in full QCD. In particular, to ?rst approximation, the ratio of factorizable and non-factorizable contributions is independent of the B-meson wave function and small (formally of order αs at the hardcollinear scale, numerically of the order of 5-10%),