We present OPE predictions for the hadronic mass and q2 moments in inclusive semileptonic .... therefore we can express the moments of M2. X in term o...

0 downloads 0 Views 104KB Size

Hadronic mass and q 2 moments in B → Xulν Giovanni Ossola Dipartimento di Fisica Teorica, Universit`a di Torino, and INFN, Sez. Torino, I-10125 Torino, Italy [email protected]

Abstract We present OPE predictions for the hadronic mass and q 2 moments in inclusive semileptonic charmless B decays, with a lower cut on the charged lepton energy and an upper cut on the hadronic invariant mass. We include non-perturbative corrections through O(1/m3b ) and perturbative contributions through O(α2s β0 ). We also investigate the range of the cut on the hadronic mass for which the local OPE can be considered valid and give estimates of the residual theoretical uncertainty. The study of these moments is important to constrain the effect of the Weak Annihilation (WA) contributions and the shape of the distribution function, providing a more precise inclusive determination of |Vub |.

To appear in the proceedings of the International Europhysics Conference on High Energy Physics July 21st - 27th 2005, Lisboa, Portugal

1

Introduction and General Motivations

The determination of the Vub element of the CKM matrix, together with a sensible estimate of the theoretical uncertainties involved in its calculation, is an important but challenging task. The magnitude of Vub , combined with Vcb, determines the length of the left side of the Unitary Triangle, allowing for a stringent test of the Standard Model and a providing a powerful tool to detect hints of new physics [1]. While Vcb , thanks to recent developments, has reached a theoretical uncertainty below 2%, the situation of Vub is much worse, with uncertainties about 10% [2, 3]. Therefore improvements are needed here. The difficulties related with the determination of the Vub from inclusive semileptonic decays are well known. Inclusive decay rates can be calculated using a double expansion in αs (parton model) and ΛQCD /mb (Heavy Quark Expansion): " n # α i Λ 2 5 X G |V | m F ub QCD s b ¯ → Xu ℓ¯ Γ(B ν) = . (1) 1+ C(i,n) 192π 3 π mb i, n

¯ → Xu ℓ¯ ¯→ The measurement of B ν requires experimental cuts to suppress the large B Xc ℓ¯ ν background. Unfortunately, the introduction of kinematic cuts restricts the phase ¯ → Xu ℓ¯ space available for the B ν decay, spoiling the global properties of the OPE. Several possibilities have been explored in the literature [4], involving cuts in the hadronic invariant mass (MX ), lepton energy (El ), leptonic invariant mass (q 2 ), lightcone component of the hadronic four-momentum (P+ ), and combinations of them. Each of these cases requires different techniques and a careful estimates of the uncertainties involved. In this talk, I will present a calculation of the moments in the hadronic invariant mass ¯ → Xu ℓ¯ of the B ν distribution [5], and try to suggest the reason for its usefulness. Of course, as soon as the measurements of such moments will be available, one should verify their consistency with moments of B → Xc ℓ¯ ν and B → Xs γ in the OPE framework. The present combined fit [6] of all moments of inclusive B decays (hadronic mass, lepton energy) shows very good consistency of the available data and allows for the determination of the b quark mass and expectation values of the dominant power suppressed operators. However the main motivations for our analysis comes from recent experimental developments. The high statistics accumulated at the B-factories by BaBar and Belle allows for the measurement, based on fully reconstructed events, of hadronic invariant mass ¯ → Xu ℓ¯ distributions in inclusive B ν . This new generation of analysis can discriminate between charmless events and charmed background, even for relatively high values of hadronic invariant mass MX . For instance, BaBar has been able to measure invariant mass distribution and first two moments with promising accuracy [7]. However, the experimental error is much smaller if one applies a cut on the hadronic mass MXcut close to the

kinematic boundary for charm production. On the other hand, theoretical calculations of inclusive B decays rely on the OPE, whose convergence is spoiled by severe cuts. A possible solution could be to raise MXcut just enough to suppress non-perturbative effects that cannot be accounted for by the OPE (B meson distribution function effects). I will show a simple example in the Discussion (Sect. 4), but before let me briefly describe the content of our calculation (Sect. 2) and succinctly discuss the theoretical uncertainties involved (Sect. 3).

2

Hadronic mass moments

We determine the normalized integer moments of the squared invariant mass R dM 2 M 2n dΓ/dMX2 2n hMX i = R X 2 X dMX dΓ/dMX2

(2)

and in particular the central moments defined as

U1 = hMX2 i , U2,3 = h MX2 − hMX2 i

2,3

i.

The physical hadronic invariant mass is related to parton level quantities by ¯ 2 + 2mb ΛE ¯ 0 + m2 s0 , MX2 = Λ b therefore we can express the moments of MX2 in term of combinations of moments of parton energy E0 and invariant mass of the parton s0 . Using the same approach, we can easily obtain also the moments of the distribution in q 2 . The building blocks that allow to construct both MX2 and q 2 moments are: Z d3 Γ 1 i j dE0 ds0 dEℓ s0 E0 (3) M(i,j) = Γ0 dE0 ds0 dEℓ µ2π αs2 β0 (2) µ2G ρ3D ρ3LS αs (1) = T(i,j) + 2 B(i,j) + 2 C(i,j) + 3 D(i,j) + 3 E(i,j) + A(i,j) + A(i,j) . mb mb mb mb π π2 This expression does not include non-perturbative terms of O(1/m4b ), O(αs /m2b ), and perturbative corrections of O(αs2 ). The moments of E0 and s0 and of their product are obtained in the local OPE and are expressed in terms of the heavy quark parameters. We employ the “kinetic” scheme [8] (Wilsonian scheme with a hard factorization scale µ ≃ 1 GeV). We start from on-shell expressions, and express the on-shell parameters in terms of the µ-dependent “kinetic” parameters: kinetic mass of the b quark mb (µ), kinetic

expectation value µ2π (µ), and Darwin expectation value ρ3D (µ). At µ → 0 one recovers the results of the on-shell scheme. In the case of charmless decay, we introduce an additional parameter Xµ ≡ 8 ln m2b /µ24q . It is well known [9, 10] that the coefficient function D(0,0) (see Eq. 3) has a logarithmic divergence as mq → 0, i.e. contains a term proportional to ln (m2b /m2q ). This problem is solved when we include the contribution of the Weak Annihilation (WA) operator. In fact, a one-loop penguin diagram that mixes the four-quark operator into ρ3D replaces ln m2u /m2b by ln µ24q /m2b , where µ4q is the normalization point of the WA operator. Varying the parameter Xµ allows us to estimate the effects of WA. The associated variation in the results for the moments accounts for the non-valence piece in the expectation value of the WA operator (flavor singlet WA). This contribution does not distinguish between B + and B 0 (non-singlet WA effect). The calculation is implemented in a FORTRAN code1 and includes non-perturbative corrections O(1/m2b ) [11] and O(1/m3b ) [9], and perturbative corrections O(αs ) and O(αs2 β0 ) [12]. Partial checks have been performed to reproduce previous results [13].

3

Theoretical Uncertainties

There are several sources of theoretical uncertainty in our predictions for the moments. Let me briefly list them and describe the methods used to estimate their size. We should consider: (i) Uncalculated O(αs2 ) and perturbative corrections to the Wilson coefficients; (ii) Missing O(1/m4b ) non-perturbative effects; (iii) Error from the scale in Xµ ; (iv) WA contributions; (v) Effects of Fermi Motion. We estimate the effect of missing higher order corrections (i) and (ii), by varying the parameters µ2π , µ2G , αs , ρ3D , ρ3LS , and mb in an uncorrelated way. Alternatively, we can estimate the size of missing higher orders comparing the results obtained for the moments using different rearrangements of non-perturbative corrections. As I mentioned previously, the effects of (iii) and (iv) are closely related [5]: varying µ4q over a reasonable interval, we estimate the effect of the flavor singlet WA. Those contributions are concentrated at maximal q 2 (small hadronic invariant mass) and are therefore suppressed in the moments of MX2 . Finally, we estimate the effect of Fermi motion (v), introduced by the MX cut, by smearing the tree-level differential rate with an exponential distribution function. This estimate depends on the functional form adopted, for instance an exponential form leads to more conservative estimates than a Gaussian ansatz. While find relatively small effects for the first moment, the second and the third one are more sensitive. A more complete and satisfactory treatment of these effects is required, and it is currently under investigation. 1

Available upon request from the authors of Ref. [5].

4

Discussion and Conclusions

As an example of the possibilities offered by the analysis based on the moments, we present an estimate of the error on a the determination of mb based on U1 only. We consider a mild MX2 <5.6 GeV2 cut, that has the advantage of removing the experimentally most poorly known region of phase-space for charmless decays, reducing the experimental error by almost a factor two, respect to the case without the cut [14]. The experimental uncertainty, considering the statistics that can be accumulated in the near future, can be estimated as δU1exp = 11% [14]. On the theoretical side, considering the sources of uncertainty listed above, we obtain δU1th = 9% at MXcut =2.5 GeV[5]. The combined effect of theoretical and experimental uncertainties, together with the use of the linearized formulas of Ref. [5], roughly leads to δmb ≃ 80 MeV. To summarize, we presented predictions for the first three moments of the hadronic ¯ → Xu ℓ¯ invariant mass distribution in B ν , that includes corrections through O(αs2β0 ) and O(1/m3b ) and cuts on the lepton energy and on the invariant hadronic mass. The calculation is implemented in a FORTRAN code, available upon request. The work is in progress: we are currently improving our code, allowing all possible combined cuts (including q 2 and light-cone variables). A lot is still to be done, in order to constrain WA (singlet/non-singlet) and better understand Fermi motion effects. An analysis based on moments with high MXcut is challenging for experiments, but reduces the theoretical uncertainties related with OPE predictions. The preliminary results are promising. Acknowledgments Work done in collaboration with Paolo Gambino and Nikolai Uraltsev, supported in part by the EU grant MERG-CT-2004-511156 and by MIUR under contract 2004021808-009.

References [1] M. Bona et al. [UTfit Collaboration], JHEP 0507 (2005) 028 [hep-ph/0501199] and hep-ph/0509219. [2] Heavy Flavor Averaging Group (HFAG), http://www.slac.stanford.edu/xorg/hfag/ [3] P. Gambino, hep-ph/0510085; M. Morii, Mod. Phys. Lett. A 20 (2005) 2183 [hep-ex/0505070]. [4] B. O. Lange, M. Neubert and G. Paz, Phys. Rev. D 72 (2005) 073006 [hep-ph/0504071]; A. H. Hoang, Z. Ligeti and M. Luke, Phys. Rev. D 71 (2005) 093007 [hep-ph/0502134]; C. W. Bauer, hep-ph/0408100. [5] P. Gambino, G. Ossola and N. Uraltsev, JHEP 0509 (2005) 010 [hep-ph/0505091].

[6] O. Buchmuller and H. Flacher, hep-ph/0507253. [7] B. Aubert et al. [BABAR Collaboration], hep-ex/0408068. [8] I. I. Y. Bigi, M. A. Shifman, N. Uraltsev and A. I. Vainshtein, Phys. Rev. D 56 (1997) 4017 [hep-ph/9704245] and Phys. Rev. D 52 (1995) 196 [hep-ph/9405410]. [9] M. Gremm and A. Kapustin, Phys. Rev. D 55 (1997) 6924 [hep-ph/9603448]. [10] B. Blok, R. D. Dikeman and M. A. Shifman, Phys. Rev. D 51 (1995) 6167 [hep-ph/9410293]. [11] I. I. Y. Bigi, N. G. Uraltsev and A. I. Vainshtein, Phys. Lett. B 293 (1992) 430 [hep-ph/9207214]; I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein, Phys. Rev. Lett. 71 (1993) 496 [hep-ph/9304225]; B. Blok, L. Koyrakh, M. A. Shifman and A. I. Vainshtein, Phys. Rev. D 49 (1994) 3356 [Erratum-ibid. D 50 (1994) 3572] [hep-ph/9307247]; A. V. Manohar and M. B. Wise, Phys. Rev. D 49 (1994) 1310 [hep-ph/9308246]. [12] V. Aquila, P. Gambino, G. Ridolfi and N. Uraltsev, Nucl. Phys. B 719 (2005) 77 [hep-ph/0503083]. [13] F. De Fazio and M. Neubert, JHEP 9906 (1999) 017 [hep-ph/9905351]. A. F. Falk and M. E. Luke, Phys. Rev. D 57 (1998) 424 [hep-ph/9708327]. [14] K. Tackmann, talk presented at CKM 2005, San Diego, http://ckm2005.ucsd.edu/WG/WG2/fri2/tackman1-WG2-S4.pdf.

03/2005,