B -> D* at zero recoil revisited
aa r X i v : . [ h e p - ph ] A p r SI-HEP-2010-07UND-HEP-10-BIG 04 B → D ∗ at Zero Recoil Revisited Paolo Gambino
Dipartimento di Fisica Teorica, Universit`a di Torino and
INFN Torino, I-10125 Torino, Italy
Thomas Mannel, Nikolai Uraltsev a ∗ Theoretische Physik 1, Fachbereich Physik, Universit¨at Siegen, D-57068 Siegen, Germany a also Department of Physics, University of Notre Dame du Lac, Notre Dame, IN 46556 U.S.A.
Abstract
We examine the B → D ∗ form factor at zero recoil using a continuum QCD approach rootedin the heavy quark sum rules framework. A refined evaluation of the radiative correctionsas well as the most recent estimates of higher order power terms together with more carefulcontinuum calculation are included. An upper bound on the form factor of F (1) < ∼ . F (1) ≈ .
86 with aboutthree percent uncertainty in the central value. ∗ On leave of absence from Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300, Russia
Introduction
The determination of the CKM matrix element V cb from exclusive decays has to rely on calcu-lations of the relevant form factors, which are usually defined as d Γ dω ( B → D ∗ ℓ ¯ ν ℓ ) = G F π | V cb | M D ∗ ( ω − / P ( ω )( F ( ω )) d Γ dω ( B → D ℓ ¯ ν ℓ ) = G F π | V cb | ( M B + M D ) M D ( ω − / ( G ( ω )) (1.1)with ω = v · v ′ = E D ( ∗ ) /M D ( ∗ ) (in the B rest frame), and where P ( ω ) is a known phase spacefactor. Based on the normalization of the form factors at ω = 1, V cb is extracted from anextrapolation of the data to the non-recoil point.In the heavy quark limit, the normalization of the form factors F (1) = G (1) = 1 is givenby heavy quark symmetry, and the main issue in the V cb determination becomes a reliablecalculation of the deviation from the heavy quark limit. The published extractions of V cb alongthis route rely solely on the lattice calculations currently cited as [1] F (1) = 0 . ± . G (1) = 1 . ± . ± .
016 (1.2)However, based on the dynamic heavy quark expansion in Minkowski space, it has been arguedthat larger deviations from the symmetry limit for F (1) are natural in continuum QCD [2];these have been further supported by the arguments [3] exploiting the relatively small kineticexpectation value µ π extracted from the fits to inclusive B decays. The same line of reasoningled to a rather precise estimate of G (1) in B → D [4], showing significantly smaller deviationsfrom unity compared to Eq. (1.2).Recent years were of primary importance for heavy flavor physics. Along with advancesin theory, many nontrivial nonperturbative predictions were verified with high precision, andheavy quark parameters experimentally extracted in accord with prior theoretical expectations;certain predictions were indirectly confirmed in dedicated lattice calculations. All this raisedthe credibility of OPE-based methods and favored an early onset of the short-distance expan-sion instrumental for high-precision predictions; confidence rose in the assumptions underlyingdynamic treatment of the nonperturbative physics in heavy quarks. This progress warrants acritical re-examination of the form factors. The goal is to incorporate the accumulated knowl-edge and to shift the focus from merely establishing the scale of the deviations from the heavyquark symmetry limit towards obtaining a refined estimate with a motivated error assessment.In the present note we discuss the B → D ∗ transition at zero recoil using a dynamic QCDapproach inspired by the original treatment of the zero-recoil sum rules for heavy flavor tran-sitions. The details of the analysis will be presented in the extended publication [5]. We consider the zero-recoil ( ~q = 0) forward scattering amplitude T zr ( ε ) of the flavor-changingaxial current ¯ c~γγ b off a B meson at rest: T zr ( ε ) = Z d x Z d x e − ix ( M B − M D ∗ − ε ) M B h B | iT ¯ cγ k γ b ( x ) ¯ bγ k γ c (0) | B i , (2.1)1igure 1: The analytic structure of T zr ( ε ) and the integration contour yielding the sum rule.Distant cuts are shown along with the physical cut. The radius of the circle is ε M .where ε is the excitation energy above M D ∗ in the B → X c transition (the point ε = 0 corre-sponds to the elastic B → D ∗ transition). The amplitude T zr ( ε ) is an analytic function of ε and has a physical decay cut at ε ≥
0, and other distant singularities. The analytic structure of T zr ( ε ) is shown in Fig. 1.The contour integral I ( ε M ) = − πi I | ε | = ε M T zr ( ε ) d ε (2.2)with the contour running counterclockwise from the upper side of the cut, see Fig. 1, leads tothe sum rule involving F (1). Using the analytic properties of T zr ( ε ) the integration contourcan be shrunk onto the decay cut; the discontinuity there is related to the weak transitionamplitude squared of the axial current into the final charm state with mass M X = M D ∗ + ε .Separating out explicitly the elastic transition contribution B → D ∗ at ε = 0 we have I ( ε M ) = F (1) + w inel ( ε M ) , w inel ( ε M ) ≡ πi ε M Z ε> disc T zr ( ε ) d ε , (2.3)where w inel ( ε M ) is related to the sum of the differential decay probabilities into the excitedstates with mass up to M D ∗ + ε M in the zero recoil kinematics.The OPE allows us to calculate the amplitude in (2.1) – and hence I ( ε M ) – in the short-distance expansion provided | ε | is sufficiently large compared to the ordinary hadronic massscale. It should be noted that strong interaction corrections are driven not only by | ε | , but alsoby the proximity to distant singularities. Therefore, ε M cannot be taken too large either, andthe hierarchy ε M ≪ m c has to be observed.The sum rule Eq. (2.3) can be cast in the form F (1) = p I ( ε M ) − w inel ( ε M ) (2.4)which is the master identity for the considerations to follow. Since w inel ( ε M ) is strictly positive,we get an upper bound on the form factor F (1) ≤ p I ( ε M ) (2.5)which relies only on the OPE calculation of I . Note that this bound depends on the parameter ε M , while (2.4) is independent of ε M since the dependence in I and w inel cancel. Furthermore,including an estimate of w inel ( ε M ) we obtain an evaluation of F (1).The correlator in (2.1) can be computed using the OPE, resulting in an expansion of T zr ( ε )in inverse powers of the masses m c and m b . This results in the corresponding expansion of2 ( ε M ). This OPE takes the following general form I ( ε M ) = ξ pert A ( ε M , µ ) + X k C k ( ε M , µ ) M B h B | O k | B i µ m d k − Q (2.6)= ξ pert A ( ε M , µ ) − ∆ /m Q ( ε M , µ ) − ∆ /m Q ( ε M , µ ) − ∆ /m Q ( ε M , µ ) − ... where O k are local b -quark operators ¯ b...b of increasing dimension d k ≥ C k ( µ ) are Wilsoncoefficients for power-suppressed terms, and ξ pert A is the short-distance renormalization (corre-sponding to the Wilson coefficient of the unit operator), which is unity at tree level. We havealso introduced a Wilsonian cutoff µ used to separate long and short distances. The completeresult, of course, does not depend on µ since the µ -dependence cancels between the Wilsoncoefficients and the matrix elements of the operators. At tree level ∆ does not depend on ε M .The choice of µ is subject to the same general constraints as that of ε M , and it is thereforeconvenient to choose µ = ε M . The perturbative renormalization ξ pert A ( µ ) can be expanded in power series in α s . We use theWilsonian OPE and benefit from well-behaved perturbative series for ξ pert A ( µ ). The exact formof the perturbative coefficients depends on the definition chosen for higher-dimension operators;we adopt the often used kinetic scheme [6].In one-loop perturbative calculations there is a simple connection between the normalizationpoint of the heavy quark operators in the kinetic scheme and the hard cut-off on the gluonmomentum in the diagram. This allows to obtain the analytic expression for ξ pert A ( µ ) to thisorder even without explicit calculation of the Wilson coefficients C k in Eq. (2.6). The expressionis rather lengthy and will be presented in Ref. [5]. By the same trick one also obtains all higher-order BLM corrections by performing the one-loop calculations with massive gluon [7].A similar argument does not apply to non-BLM corrections starting α s where ε M -dependenceof ξ pert A has to be determined expanding in 1 /m Q ; for O ( α s ) corrections this was done in Ref. ([8])through order 1 /m Q . The corresponding coefficient was found to be small numerically, whichsuggests that omitted terms ∝ α s ε M /m Q and higher should not produce a significant change.Perturbative corrections to ξ pert A ( ε M ) appear to be small for practical values of ε M between0 . ε M = 0 .
75 GeV, m c = 1 . m b = 4 . α s ( m b ) =0 .
22 we get the numeric estimates at different orders q ξ pert A = 1 − .
019 + (0 . − . . ... (2.7)Here the first term is the tree value, second is O ( α s ) evaluated with α s = 0 .
3, the next pair ofvalues show the shift upon passing to the O ( α s ) order (positive for the BLM part and negativefrom the non-BLM contribution); the last term shows β α s term as an estimate of even higher-order perturbative corrections. Fig. 2 shows the dependence on ε M of these predictions for q ξ pert A . In particular, taking the full two-loop result as the central estimate we find q ξ pert A (0.75 GeV) = 0 . ± .
01 ; (2.8)we will use ε M = 0 . .4 0.5 0.6 0.7 0.8 0.90.9650.9700.9750.9800.9850.9900.9951.000 ε M , GeV q ξ pert A ε M , GeV
Figure 2:
Left: q ξ pert A to order α s (blue), including β α s (green), full O ( α s ) (red) and including β α s (magenta), assuming α MS s ( m b ) = 0 . m c = 1 . m b = 4 . Right:
Upper bound (2.5) on F (1) depending on ε M , with or without β α s term. Afixed ∆ = 0 .
11 is used; assuming the perturbative evolution of ∆ with ε M wouldflatten the dependence. The leading power corrections to I were calculated in Refs. [9, 10] to order 1 /m Q and to order1 /m Q in Ref. [11] and read∆ /m = µ G m c + µ π − µ G (cid:18) m c + 23 m c m b + 1 m b (cid:19) , ∆ /m = ρ D − ρ LS m c + 112 m b (cid:18) m c + 1 m c m b + 3 m b (cid:19) ( ρ D + ρ LS ) . (2.9)The nonperturbative parameters µ π , µ G , ρ D and ρ LS all depend on the hard Wilsonian cutoff.The ε M -dependence of ξ pert A is linked to the power-like scale dependence of the nonperturbativematrix elements through their mixing with lower-dimension operators.In the numerics we use the values µ π (0 .
75 GeV) = 0 . , ρ D (0 .
75 GeV) = 0 .
15 GeV , whilefor the quark masses m c = 1 . m b = 4 . µ G and on ρ LS is minimal and their precise values do not matter; we use for them 0 . and − .
12 GeV , respectively. We then get∆ /m = 0 . , ∆ /m = 0 .
028 ; (2.10)If we employ the values of the OPE parameters extracted from a fit to inclusive semileptonicand radiative decay distributions [12, 13], we find a consistent result∆ /m + ∆ /m = 0 . ± . . (2.11)An important question is how well the power expansion for the sum rule converges. Recently,the OPE for the semileptonic B -meson structure functions has been extended to order 1 /m Q and 1 /m Q [14, 15]. Combined with the estimates [16] of the corresponding expectation valuesdiscussed in Ref. [15], this leads to∆ /m ≃ − . , ∆ /m Q ≃ − . . (2.12)We then observe that the power series for I appears well-behaved at the required level ofprecision. For what concerns the loop corrections to ∆, the O ( α s ) correction to the Wilson4oefficient for the kinetic operator in Eq. (2.9) was calculated in Ref. [8] and turned out nu-merically insignificant. At O (1 /m Q ), even if radiative corrections change the coefficient for theDarwin term by 30% the effect on the sum rule would still be small.Taking into account all the available information, our estimate for the total power correctionat ε M = 0 .
75 GeV is ∆ = 0 .
105 (2.13)with a 0 .
015 uncertainty due to higher orders. On theoretical grounds, larger values of µ π and/or ρ D are actually favored; they tend to increase ∆. Combining the above with the perturbativecorrections we arrive at an estimate for I and, according to Eq. (2.5) at a bound for the formfactor, which in terms of central values at ε M = 0 .
75 GeV is F (1) < . . (2.14)As stated above, the upper bound in Eq. (2.5) depends on ε M , see Fig. 2, becoming strongerfor smaller ε M . It is advantageous to choose the minimal value of ε M for which the OPE-basedshort-distance expansion of the integral (2.2) for I ( ε M ) sets in. This directly depends onhow low one can push the renormalization scale µ while still observing the expectation valuesactual µ -dependence in the kinetic scheme approximated by the perturbative one. Since in thisscheme µ π ( µ ) ≥ µ G ( µ ) holds for arbitrary µ , in essence this boils down to the question at whichscale µ min the spin sum rule and the one for µ G get approximately saturated, e.g. µ G ( µ min ) ≃ . . The only vital assumption in the analysis is that the onset of the short-distanceregime is not unexpectedly delayed in actual QCD and hence does not require ε M > B decays of the predicted 3 / w inel On general grounds w inel is expected to be comparable to the power correction ∆ consideredabove. To actually estimate it we consider another contour integral I ( ε M ) = − πi I | ε | = ε M T zr ( ε ) ε d ε (2.15)for which we can write w inel ( ε M ) = I ( ε M )˜ ε (2.16)where ˜ ε is the average excitation energy (it depends on ε M ). The integral is expected to bedominated by the lowest radial excitations of the ground state, with ˜ ε ≈ ǫ rad ≈
700 MeV. I ( ε M ) can also be calculated in the OPE [10]; the result including 1 /m Q terms reads I = − ( ρ πG + ρ A )3 m c + − ρ ππ − ρ πG m c m b + ρ ππ + ρ πG + ρ S + ρ A (cid:18) m c + 23 m c m b + 1 m b (cid:19) (2.17)where the non-local zero momentum transfer correlators ρ ππ , ρ πG , ρ S and ρ A are defined in [10].They can be estimated along the lines described in [15], based on saturation by the appropriateintermediate states. We shall defer the details of this estimate to [5]. Here we note that onlythe first term survives in the BPS limit [4], where it is positive; the second and third terms are5f first and second order in the deviation from the BPS limit, respectively. The last term ispositive being the correlator of two identical operators ¯ b ( ~σ~π ) b . Since the middle term comeswith a small coefficient 1 / m c m b , the expression has only a shallow minimum where the firstterm is decreased by less than 10%; the sum of the last two terms becomes larger than thatonly if it is positive. On the other hand, the combination − ( ρ πG + ρ A ) + ρ LS (2.18)determines the hyperfine splitting to order 1 /m Q and can be constrained from the observedmasses of B ( ∗ ) and D ( ∗ ) mesons. We finally obtain I ( ε M ) > ∼ .
48 GeV m c (2.19)with some uncertainty from perturbative corrections, implying for ˜ ε = ǫ rad ≃
700 MeV w inel ≈ I ǫ rad > ∼ . . (2.20)This estimate is derived at leading order in 1 /m Q and may be corrected by higher-orderterms by as much as 30%. We observe that w inel is similar in size to the power term in I andeven exceeds it. Using (2.20) at face value we arrive at our estimate for the expected value ofthe form factor F (1) < ∼ . . (2.21)The quasi-resonant states are expected to dominate w inel ( µ ) at intermediate µ ≈ w inel ( µ ) is parametrically 1 /N c -suppressed and usually smaller.The D ( ∗ ) π continuum can independently be evaluated in the soft-pion approximation. It turnsout that numerically the dominant effect originates from the heavy quark symmetry breakingdifference between the B ∗ Bπ , D ∗ Dπ and D ∗ D ∗ π -couplings which until recently has not beenaccounted for in this context, although is expected to be significant [19]. The result is shown inFig. 3; it depends on the upper cutoff in the pion momentum p π max marking the end of the softcontinuum domain for D ( ∗ ) π , presumably somewhat below ǫ rad . We expect about 5% combinedyield for Γ D ∗ + = 96 keV, i.e. about a third of the overall w inel in Eq. (2.20) in accord with the1 /N c arguments. This contribution alone would lower the upper bound in Eq. (2.14) by 0 . The direct OPE-based 1 /m Q expansion of the zero recoil form factor which we have analyzedthrough the sum rule for the correlator of the zero-recoil axial currents yields the unbiasedestimate F (1) ≈ .
86, and suggests a lower bound F (1) < .
93. All the values refer to pureQCD form factors and exclude possible electromagnetic effects, in particular the universalshort-distance semileptonic enhancement factor of 1 . µ π , ρ D and m c . Their determination from inclusive measure-ments will soon be improved thanks to refined theoretical calculations. On the other hand, The D ∗ D ∗ π channel has not been previously considered while generally required by the heavy quark sym-metry. .3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.000.010.020.030.040.050.060.07 p π max , GeV
D π p π max , GeV D ∗ π Figure 3:
Non-resonant Dπ and D ∗ π contribution to w inel at g D ∗ Dπ = 4 . − corresponding toΓ D ∗ + = 96 keV. The plots show, from bottom to top, g B ∗ Bπ g D ∗ Dπ = 1, 0 .
8, 0 . .
4, and g B ∗ Bπ g D ∗ D ∗ π = 1, 0 . .
6, respectively even if these QCD parameters were accurately known, sharpening the upper bound and theestimate of F (1) would require additional nonperturbative input. To some extent the uncer-tainties may be reduced by dedicated measurements in semileptonic decays or by direct latticecalculation of the relevant heavy quark parameters. In our opinion, the latter can be done in astraightforward way, which will be discussed elsewhere.In our numerical estimates we used the lowest values of µ π and ρ D consistent with thetheoretical bounds; larger values would typically further lower both the upper bound and thecentral expectation for F (1). We estimate the perturbative and non-perturbative uncertaintiesin our approach to be about ± . pt and ± . np ( ± . np for the upper bound), mostly relatedto the unavoidable truncation of the 1 /m c expansions. The uncertainty in the prediction isdominated by w inel , is not symmetric, and allows for larger deviations towards lower values of F (1).There is an alternative way of determining | V cb | , using inclusive semileptonic decays. Itrests on the heavy quark expansion in 1 /m b and on the OPE which, in this case, should beconsidered a mature tool. Recent estimates of higher orders in 1 /m Q as well as in α s did notshow signs of failure of the heavy quark expansion in this application. Hence one can employthe value of V cb = (41 . ± . ± . × − from the inclusive decays [12] and determine theform factors from the data on B → D ( ∗ ) ℓν decays; this yields F (1) = 0 . ± .
02 (3.1) G (1) = 1 . ± .
04 (3.2)The value for G (1) is in a perfect agreement with the HQE prediction [4]; F (1) hits the cen-tral value of our estimate. Both form factors in Eq. (3.1) are noticeably below the quotedlattice values Eqs. (1.2); this explains the observed tension between the inclusive and exclusivedeterminations of | V cb | .Our analysis suggests that the lattice determinations of both form factors are systematicallyhigh. The central values of the currently quoted lattice F (1) is very close to our upper bound.Lattice calculations – provided they accommodate the experimentally measured B -meson ex-pectation values – seem to imply an extremely small inelastic contribution, which is a prioriunnatural. Moreover, it appears in contradiction with the large non-local correlators encoun-tered to order 1 /m Q . The estimate of the non-resonant soft-pion D ( ∗ ) π rates also confirmsthis assessment. We conclude that values for F (1) in excess of 0 . F (1) larger than 0 .
93 should be viewed as violation of unitarity assuming that usualshort-distance expansion in QCD works. 7ith the unbiased estimate based on the lower-end µ π and ρ D yielding F (1) ≈ .
86 – oreven smaller for larger µ π , ρ D – we find a surprisingly good agreement (probably, somewhataccidental) between the exclusive and inclusive approaches. In view of the inherent 1 /m c expansion for F (1) and of the approximations used for proliferating hadronic expectation values,matching the theoretical precision already attained for V cb from the inclusive fits does not lookprobable. Nevertheless, additional experimental and/or lattice input would make the suggested3% uncertainty interval more robust.We estimate that the contribution of excited radial states with mass below 3 GeV constitutesabout 10% or more of the total yield. This refers only to the zero-recoil kinematics and onlyto the axial current; it does not include P -wave states. This fraction may even be larger whenapplied to the full phase space. This suggests that the observed ‘broad state’ yield in this massrange routinely attributed to the P -wave excitations is actually dominated by states withdifferent quantum numbers, thus resolving the ‘ > puzzle’. The suppression of the broad P -wave yield was predicted based on the spin sum rules and confirmed indirectly in nonleptonic B decays [18]; a recent discussion can be found in Ref. [20]. Acknowledgments
We gratefully acknowledge discussions with Ikaros Bigi and invaluable help from Sascha Turczykregarding higher-order power corrections. We thank the Galileo Galilei Institute for TheoreticalPhysics for the hospitality and the INFN for partial support during the completion of thiswork. The study enjoyed a partial support from the NSF grant PHY-0807959 and from theRSGSS-65751.2010.2 grant. PG is supported in part by a EU’s Marie-Curie Research TrainingNetwork under contract MRTN-CT-2006-035505 (HEPTOOLS). TM is partially supportedby the German research foundation DFG under contract MA1187/10-1 and by the GermanMinistry of Research (BMBF), contracts 05H09PSF.
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