Tensions and correlations in |V_{cb}| determinations
Florian U. Bernlochner, Zoltan Ligeti, Michele Papucci, Dean J. Robinson
TTensions and correlations in | V cb | determinations Florian U. Bernlochner,
1, 2
Zoltan Ligeti, Michele Papucci, and Dean J. Robinson Physikalisches Institut der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn, 53115 Bonn, Germany Karlsruher Institute of Technology, 76131 Karlsruhe, Germany Ernest Orlando Lawrence Berkeley National Laboratory,University of California, Berkeley, CA 94720, USA Physics Department, University of Cincinnati, Cincinnati OH 45221, USA
Recently several papers extracted | V cb | using the Belle measurement [1] of the exclusive ¯ B → D ∗ (cid:96) ¯ ν unfolded differential decay rates, available for the first time. Depending on the theoretical inputs,some of the fits yield higher | V cb | values, compatible with those from inclusive semileptonic B decays.Since these four fits use mostly the same data, if their correlations were close to 100%, the tensionbetween them would be over 5 σ . We determine the correlations, find that the tension between theresults is less than 3 σ , and explore what might lead to improving the consistency of the fits. We findthat fits that yield the higher values of | V cb | , also suggest large violations of heavy quark symmetry.These fits are also in tension with preliminary lattice QCD data on the form factors. Withoutadditional experimental data or lattice QCD input, there are no set of assumptions under which thetension between exclusive and inclusive determinations of | V cb | can be considered resolved. I. INTRODUCTION
Using the unfolded ¯ B → D ∗ (cid:96) ¯ ν spectra from Belle [1],several theory papers [2–4] could perform fits to the datafor the first time, using different theoretical approaches.Using the BGL parametrization [5, 6] for the ¯ B → D ∗ (cid:96) ¯ ν form factors, a substantial shift in the extracted value of | V cb | was found [3, 4], compared to the Belle [1] analysisusing the CLN [7] parametrization, | V cb | CLN = (38 . ± . × − , [1] , (1a) | V cb | BGL = (41 . +2 . − . ) × − , [3] , (1b) | V cb | BGL = (41 . +2 . − . ) × − , [4] . (1c)The main result in Ref. [1] was | V cb | CLN = (37 . ± . × − , obtained from a fit inside the Belle framework, be-fore unfolding. Only Eq. (1a) quoted in the Appendixof [1] can be directly compared with Eqs. (1b) and (1c).These papers, as well as this work, use the same fixedvalue of F (1) [8] (see Eq. (4) below), so the differencesin the extracted values of | V cb | are due to the extrapola-tions to zero recoil, where heavy quark symmetry givesthe strongest constraint on the rate [9–13]. Intriguingly,the BGL fit results for | V cb | are compatible with thosefrom inclusive B → X c (cid:96) ¯ ν measurements [14]. If one as-sumed, naively, a 100% correlation between the fits yield-ing Eqs. (1a), (1b), and (1c), then the tension betweenEqs. (1a) and (1b) or between Eqs. (1a) and (1c) wouldbe above 5 σ .The BGL [5, 6] fit implements constraints on the B → D ∗ (cid:96) ¯ ν form factors based on analyticity and uni-tarity [15–17]. The CLN [7] fit imposes, in addition, con-straints on the form factors from heavy quark symme-try, and relies on QCD sum rule calculations [18–20] ofthe subleading Isgur-Wise functions [13, 21], without ac-counting for their uncertainties. Ref. [2] performed com-bined fits to ¯ B → D ∗ (cid:96) ¯ ν and ¯ B → D(cid:96) ¯ ν , using predictionsof the heavy quark effective theory (HQET) [22, 23], in-cluding all O (Λ QCD /m c,b ) uncertainties and their corre- lations for the first time. The effect of relaxing the QCDsum rule inputs in the CLN fit was found to be smallcompared to the difference of the CLN and BGL results.The recent papers using the BGL parametrization [3, 4]assert that the higher values obtained for | V cb | are dueto the too restrictive functional forms used in the CLNfits. It was previously also noticed that the CLN givesa poorer fit to the B → D(cid:96) ¯ ν data than BGL [24]. Theeffects on | V cb | due to additional theoretical inputs werealso explored in Refs. [25, 26].Based on our work in Ref. [2], we explore which differ-ences between the BGL and CLN fits are responsible forthe different extracted | V cb | values, study the consistencyand compatibility of the fits, and the significance of theshift in the extracted value of | V cb | . II. DEFINITIONS
The B → D ∗ (cid:96) ¯ ν form factors which occur in the stan-dard model are defined as (cid:104) D ∗ | ¯ cγ µ b | B (cid:105) = i √ m B m D ∗ h V ε µναβ (cid:15) ∗ ν v (cid:48) α v β , (cid:104) D ∗ | ¯ cγ µ γ b | B (cid:105) = √ m B m D ∗ (cid:2) h A ( w + 1) (cid:15) ∗ µ (2) − h A ( (cid:15) ∗ · v ) v µ − h A ( (cid:15) ∗ · v ) v (cid:48) µ (cid:3) , where v is the four-velocity of the B and v (cid:48) is that of the D ∗ . The form factors h V,A , , depend on w = v · v (cid:48) =( m B + m D ∗ − q ) / (2 m B m D ∗ ). Neglecting lepton masses,only one linear combination of h A and h A is measur-able. In the heavy quark limit, h A = h A = h V = ξ and h A = 0, where ξ is the Isgur-Wise function [9, 10].Each of these form factors can be expanded in powers ofΛ QCD /m c,b and α s . It is convenient to parametrize de-viations from the heavy quark limit via the form factorratios R ( w ) = h V h A , R ( w ) = h A + r D ∗ h A h A , (3) a r X i v : . [ h e p - ph ] S e p form factors BGL CLN CLNnoR noHQSaxial ∝ (cid:15) ∗ µ b , b h A (1) , ρ D ∗ h A (1) , ρ D ∗ h A (1) , ρ D ∗ , c D ∗ vector a , a (cid:26) R (1) , R (1) (cid:26) R (1) , R (cid:48) (1) R (1) , R (cid:48) (1) (cid:26) R (1) , R (cid:48) (1) R (1) , R (cid:48) (1) F c , c TABLE I. The fit parameters in the BGL, CLN, CLNnoR, and noHQS fits, and their relationships with the form factors. which satisfy R , ( w ) = 1 + O (Λ QCD /m c,b , α s ) in the m c,b (cid:29) Λ QCD limit, and r D ∗ = m D ∗ /m B .The B → D ∗ (cid:96) ¯ ν decay rate is given bydΓd w = G F | V cb | m B π ( w − / ( w + 1) r D ∗ (1 − r D ∗ ) × (cid:20) ww + 1 1 − wr D ∗ + r D ∗ (1 − r D ∗ ) (cid:21) F ( w ) , (4)and the expression of F ( w ) in terms of the form factorsdefined in Eq. (2) is standard in the literature [27]. Inthe heavy quark limit, F ( w ) = ξ ( w ). We further denote ρ D ∗ = − h A (1) d h A ( w )d w (cid:12)(cid:12)(cid:12)(cid:12) w =1 , (5)which is a physical fit parameter in the CLN approach,and is a derived quantity in the other fits. III. NEW FITS, LATTICE QCD, AND THEIRTENSIONS
The constraints built into the CLN fit can be relaxedby ignoring the QCD sum rule inputs and the condition R , ( w ) = 1 + O (Λ QCD /m c,b , α s ) following from heavyquark symmetry. (Ref. [2] showed that only ignoring theQCD sum rule inputs, and using only w = 1 lattice QCDdata, leaves | V cb | = (38 . ± . × − .) Thus, we write R ( w ) = R (1) + ( w − R (cid:48) (1) ,R ( w ) = R (1) + ( w − R (cid:48) (1) , (6)and treat R , (1) and R (cid:48) , (1) as fit parameters. We referto this fit as “CLNnoR”. It has the same number of fitparameters as BGL, and allows O (1) heavy quark sym-metry violation, but the constraints on the form factorsare nevertheless somewhat different than in BGL.While this CLNnoR fit is a simple modification of theCLN fit widely used by BaBar and Belle, it still re-lies on heavy quark symmetry and model-dependent in-put on subleading Isgur-Wise functions. The reason isthat both CLN and CLNnoR use a cubic polynomial in z = ( √ w + 1 − √ / ( √ w + 1 + √
2) to parametrize theform factor h A , with its four coefficients determined bytwo parameters, h A (1) and ρ D ∗ , derived from unitarityconstraints on the B → D form factor. Therefore, wealso consider a “noHQS” scenario, parametrizing h A bya quadratic polynomial in z , with unconstrained coeffi-cients, h A ( w ) = h A (1) (cid:2) − ρ D ∗ z + (53 . c D ∗ − . ) z (cid:3) , (7) CLN CLNnoR noHQS BGL | V cb |× . ± . . ± . . ± . . ± . ρ D ∗ . ± .
15 1 . ± . . ± . . ± . c D ∗ ρ D ∗ ρ D ∗ . ± . R (1) 1 . ± .
09 0 . ± .
35 0 . ± .
48 0 . ± . R (1) 0 . ± .
08 1 . ± .
19 0 . ± .
36 1 . ± . R (cid:48) (1) fixed: − .
12 5 . ± . . ± . . ± . R (cid:48) (1) fixed: 0.11 − . ± .
61 0 . ± . − . ± . χ / ndf 35.2 / 36 27.9 / 34 27.6 / 33 27.7 / 34 TABLE II. Summary of CLN, CLNnoR, noHQS, and BGL fitresults. | V cb | CLN | V cb | CLNnoR | V cb | noHQS | V cb | BGL | V cb | CLN
1. 0.75 0.69 0.76 | V cb | CLNnoR
1. 0.95 0.97 | V cb | noHQS
1. 0.97 | V cb | BGL | V cb | val-ues. For BGL the outer functions of Ref. [4] were used. Allresults are derived by bootstrapping [28] the unfolded distri-butions of Ref. [1] using the published covariance. keeping the same prefactors as in CLN, to permit com-parison between ρ D ∗ and c D ∗ (in the CLN fit c D ∗ = ρ D ∗ ).The fit parameters in the BGL, CLN, CLNnoR, andnoHQS fits are summarized in Table I. The results ofthese fits for | V cb | , ρ D ∗ , c D ∗ , R , (1), and R (cid:48) , (1) areshown in Table II. The BGL, CLNnoR, and noHQS re-sults are consistent with each other, including the un-certainties, and the fit quality. The correlations of thesefour fit results for | V cb | are shown in Table III and havebeen derived by creating a bootstrapped [28] ensemble ofthe unfolded distributions of Ref. [1], using the publishedcovariance. Each set of generated decay distributions inthe ensemble is fitted with the BGL, CLN, CLNnoR, andnoHQS parametrizations, and the produced ensemble of | V cb | values is used to estimate the covariance betweenthem. The correlation of the CLN fit with either BGL,CLNnoR, or noHQS is substantially below 100%. Thisreduces the tension between these fits to below 3 σ .As soon as R (cid:48) , (1) are not constrained to their valuesimposed in the CLN framework, large deviations fromthose constraints are observed. The BGL, CLNnoR, andnoHQS results favor a large value for R (cid:48) (1), in tensionwith the heavy quark symmetry prediction, R (cid:48) (1) = ���������������������� ���� / ���� � * ���� / ���� � + ���� ��� ��� ��� ��� ��� �������� � � � ( � ) ���������������������� ���� / ���� � * ��� ��� ��� ��� ��� ��������������� � � � ( � ) FIG. 1. The form factor ratios R ( w ) (left) and R ( w ) (right) for the BGL (red long dashed), CLN (gray dashed), CLNnoR(orange dotted) fits, and noHQS (purple dot-dot-dashed). The BGL, CLNnoR, and noHQS fits for R suggest a possibly largeviolation of heavy quark symmetry, in conflict with lattice QCD predictions. The blue lines show our estimated bounds, basedon preliminary FNAL/MILC lattice results [29]. The black data point for R (1) follows from the FNAL/MILC B → D (cid:96) ¯ ν result and heavy quark symmetry (see details in the text). O (Λ QCD /m c,b , α s ).These aspects of the BGL, CLNnoR, and noHQS fitsare also in tension with lattice QCD results. Recently thefirst preliminary lattice results were made public on the B → D ∗ (cid:96) ¯ ν form factors away from zero recoil, at finitelattice spacing [29]. The results are fairly stable over arange of lattice spacings. Assuming that the continuumextrapolation will not introduce a sizable shift (the chirallogs are not large [30, 31]) we can estimate the projec-tions for the R , ( w ) form factor ratios. We approximatethe predicted form factors in a narrow range of w us-ing a linear form, with a normalization and slope chosensuch that they encompass all reported lattice points anduncertainties in Ref. [29]. At zero recoil we obtain the es-timates R (1) (cid:39) . ± . R (1) (cid:39) . ± .
45, whichshould be viewed as bounds on these values, as the actuallattice QCD results will likely have smaller uncertainties.Figure 1 shows R , ( w ) derived from the results of ourfit scenarios, as well as these lattice QCD constraints.We can obtain another independent prediction for R (1) based on lattice QCD and heavy quark symmetry,using the result for the B → D (cid:96) ¯ ν form factor [32]. Usingthe O (Λ QCD /m c,b , α s ) expressions [2], the f + form factor(see Eq. (2.1) in Ref. [32]) and the subleading Isgur-Wisefunction η are related at zero recoil via2 √ r D r D f + (1) = 1 + ˆ α s (cid:18) C V + C V r D r D + C V
21 + r D (cid:19) − ( ε c − ε b ) 1 − r D r D [2 η (1) −
1] + . . . , (8)since other subleading Isgur-Wise functions enter sup-pressed by w −
1. Here r D = m D /m B , ε c,b = ¯Λ /m c,b is treated as in Ref. [2], and hereafter the ellipsis de-notes O ( ε c,b , α s ε c,b , α s ) higher order corrections. Us- ing f + ( w = 1) = 1 . ± .
010 [32] one finds η (1) =0 . ± .
10. The uncertainty in this relation and theextracted value of η (1) is dominated by O (Λ /m c )corrections parametrized by several unknown matrix el-ements [33], which we estimate with ε c ∼ .
05. Thus, R (1) = 1 . − . η (1) + . . . = 1 . ± . . (9)(Recall that both the α s terms and a ¯Λ / (2 m c ) correctionenhance R (1).) This estimate is shown with the blackdot and error bar in the left plot in Fig. 1. It shows goodconsistency with our estimate from the preliminary directcalculation of the B → D ∗ (cid:96) ¯ ν form factors, as shown inthe region bounded by the blue curves.Another clear way to see that the central values of theBGL, CLNnoR, and noHQS fit results cannot be accom-modated in HQET, without a breakdown of the expan-sion, is by recalling [2] that besides Eq. (9), also R (1) = 0 . − . η (1) − .
54 ˆ χ (1) + . . . ,R (cid:48) (1) = − .
15 + 0 . η (1) − . η (cid:48) (1) + . . . , (10) R (cid:48) (1) = 0 . − .
54 ˆ χ (cid:48) (1) + 0 . η (1) − . η (cid:48) (1) + . . . . Here η and ˆ χ are subleading Isgur-Wise functions.Eqs. (9) and (10) have no solutions close to the BGL,CLNnoR, or noHQS fit results in Table II with O (1) val-ues for η (1), η (cid:48) (1), ˆ χ (1), and ˆ χ (cid:48) (1).Figure 2 shows dΓ / d w in the four fit scenarios, as wellas the Belle data [1]. The shaded bands show the uncer-tainties of the CLN and noHQS fits, which are compara-ble to the uncertainties of the other two fits. The BGL,CLNnoR, and noHQS fits show larger rates near zeroand maximal recoil, in comparison to CLN. The CLN fitshows a larger rate at intermediate values of w . ���������������������� ���� ��� ��� ��� ��� ��� ���������������� � � Γ ( � → � * � ν � ) / � � [ �� - �� � � � ] FIG. 2. dΓ / d w for the fit scenarios shown in Fig. 1. IV. CONCLUSIONS
Our results show that the tensions concerning the ex-clusive and inclusive determinations of | V cb | cannot beconsidered resolved. The central values of the BGL,CLNnoR, and noHQS fits, which all give good descrip-tions of the data, suggest possibly large deviations fromheavy quark symmetry. These results are also in ten-sion with preliminary lattice QCD predictions for theform factor ratio R , which use the same techniques as for the determination of F (1) used to extract | V cb | from B → D ∗ (cid:96) ¯ ν . If the resolution of the tension betweenlattice QCD and the fits for R is a fluctuation in thedata, then we would expect the extracted value of | V cb | to change in the future. If the resolution of the tension ison the lattice QCD side, then it may also affect the cal-culation of F (1) used to extract | V cb | . We look forwardto higher statistics measurements in the future, and abetter understanding of the composition of the inclusivesemileptonic rate as a sum of exclusive channels [34, 35],which should ultimately allow unambiguous resolution ofthese questions. ACKNOWLEDGMENTS
FB and ZL thank Prof. Toru Iijima for organizing the“Mini-workshop on D ( ∗ ) τ ν and related topics”, and thekind hospitality in Nagoya, where this work started. Wealso thank the Aspen Center of Physics, supported bythe NSF grant PHY-1066293, where this paper was com-pleted. We thank Ben Grinstein and Bob Kowalewskifor helpful conversations, not only over sushi and sake.FB was supported by the DFG Emmy-Noether GrantNo. BE 6075/1-1. ZL and MP were supported in partby the U.S. Department of Energy under contract DE-AC02-05CH11231. DR acknowledges support from theUniversity of Cincinnati. [1] A. Abdesselam et al. (Belle Collaboration), (2017),arXiv:1702.01521 [hep-ex].[2] F. U. Bernlochner, Z. Ligeti, M. Papucci, andD. J. Robinson, Phys. Rev. D95 , 115008 (2017),arXiv:1703.05330 [hep-ph].[3] D. Bigi, P. Gambino, and S. Schacht, Phys. Lett.
B769 ,441 (2017), arXiv:1703.06124 [hep-ph].[4] B. Grinstein and A. Kobach, Phys. Lett.
B771 , 359(2017), arXiv:1703.08170 [hep-ph].[5] C. G. Boyd, B. Grinstein, and R. F. Lebed, Nucl. Phys.
B461 , 493 (1996), arXiv:hep-ph/9508211 [hep-ph].[6] C. G. Boyd, B. Grinstein, and R. F. Lebed, Phys. Rev.
D56 , 6895 (1997), arXiv:hep-ph/9705252 [hep-ph].[7] I. Caprini, L. Lellouch, and M. Neubert, Nucl. Phys.
B530 , 153 (1998), arXiv:hep-ph/9712417 [hep-ph].[8] J. A. Bailey et al. (Fermilab Lattice, MILC), Phys. Rev.
D89 , 114504 (2014), arXiv:1403.0635 [hep-lat].[9] N. Isgur and M. B. Wise, Phys. Lett.
B232 , 113 (1989).[10] N. Isgur and M. B. Wise, Phys. Lett.
B237 , 527 (1990).[11] M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. , 511 (1988), [Yad. Fiz. 47, 801 (1988)].[12] S. Nussinov and W. Wetzel, Phys. Rev. D36 , 130 (1987).[13] M. E. Luke, Phys. Lett.
B252 , 447 (1990).[14] Y. Amhis et al. (Heavy Flavor Averaging Group), (2016),and updates at , arXiv:1612.07233 [hep-ex].[15] C. Bourrely, B. Machet, and E. de Rafael, Nucl. Phys.
B189 , 157 (1981). [16] C. G. Boyd, B. Grinstein, and R. F. Lebed, Phys. Rev.Lett. , 4603 (1995), arXiv:hep-ph/9412324 [hep-ph].[17] C. G. Boyd, B. Grinstein, and R. F. Lebed, Phys. Lett. B353 , 306 (1995), arXiv:hep-ph/9504235 [hep-ph].[18] M. Neubert, Z. Ligeti, and Y. Nir, Phys. Lett.
B301 ,101 (1993), arXiv:hep-ph/9209271 [hep-ph].[19] M. Neubert, Z. Ligeti, and Y. Nir, Phys. Rev.
D47 , 5060(1993), arXiv:hep-ph/9212266 [hep-ph].[20] Z. Ligeti, Y. Nir, and M. Neubert, Phys. Rev.
D49 , 1302(1994), arXiv:hep-ph/9305304 [hep-ph].[21] A. F. Falk, B. Grinstein, and M. E. Luke, Nucl. Phys.
B357 , 185 (1991).[22] H. Georgi, Phys. Lett.
B240 , 447 (1990).[23] E. Eichten and B. R. Hill, Phys. Lett.
B234 , 511 (1990).[24] D. Bigi and P. Gambino, Phys. Rev.
D94 , 094008 (2016),arXiv:1606.08030 [hep-ph].[25] D. Bigi, P. Gambino, and S. Schacht, (2017),arXiv:1707.09509 [hep-ph].[26] S. Jaiswal, S. Nandi, and S. K. Patra, (2017),arXiv:1707.09977 [hep-ph].[27] A. V. Manohar and M. B. Wise, Camb. Monogr. Part.Phys. Nucl. Phys. Cosmol. , 1 (2000).[28] K. G. Hayes, M. L. Perl, and B. Efron, Phys. Rev. D39 ,274 (1989).[29] A. Vaquero et al. (Fermilab/MILC Collaboration),Talk at the Lattice 2017 Conference, (2017), https://makondo.ugr.es/event/0/session/92/contribution/120/material/slides/0.pdf . [30] C.-K. Chow and M. B. Wise, Phys. Rev. D48 , 5202(1993), arXiv:hep-ph/9305229 [hep-ph].[31] L. Randall and M. B. Wise, Phys. Lett.
B303 , 135(1993), arXiv:hep-ph/9212315 [hep-ph].[32] J. A. Bailey et al. (Fermilab Lattice and MILC Collabora-tions), Phys. Rev.
D92 , 034506 (2015), arXiv:1503.07237[hep-lat]. [33] A. F. Falk and M. Neubert, Phys. Rev.
D47 , 2965 (1993),arXiv:hep-ph/9209268 [hep-ph].[34] F. U. Bernlochner, Z. Ligeti, and S. Turczyk, Phys. Rev.
D85 , 094033 (2012), arXiv:1202.1834 [hep-ph].[35] F. U. Bernlochner, D. Biedermann, H. Lacker, andT. Luck, Eur. Phys. J.