R(D ∗), |V cb |, and the Heavy Quark Symmetry relations between form factors

Journal of High Energy Physics, Nov 2017

Stringent relations between the B (∗) → D (∗) form factors exist in the heavy quark limit and the leading symmetry breaking corrections are known. We reconsider their uncertainty and role in the analysis of recent Belle data for B → D (∗)ℓν with model-independent parametrizations and in the related prediction of R(D (∗)). We find |V cb | = 41.5(1.3) 10−3 and |V cb | = 40.6( − 1.3 + 1.2 ) 10−3 using input from Light Cone Sum Rules, and R(D ∗) = 0.260(8).

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R(D ∗), |V cb |, and the Heavy Quark Symmetry relations between form factors

HJE ), jVcbj, and the Heavy Quark Symmetry Dante Bigi 0 1 Paolo Gambino 0 1 Stefan Schacht 0 1 0 I-10125 Torino , Italy 1 Dipartimento di Fisica, Universita di Torino & INFN , Sezione di Torino Stringent relations between the B( ) ! D( ) form factors exist in the heavy quark limit and the leading symmetry breaking corrections are known. We reconsider their uncertainty and role in the analysis of recent Belle data for B model-independent parametrizations and in the related prediction of R(D( )). We nd jVcbj = 41:5(1:3) 10 3 and jVcbj = 40:6(+11::23) 10 3 using input from Light Cone Sum Rules, Heavy Quark Physics; Quark Masses and SM Parameters - R(D ! D( )` with and R(D ) = 0:260(8). 1 Introduction 2 3 4 5 Uncertainty of the relations between form factors Strong unitarity bounds for B ! D form factors Fits to B ! D ` data Calculation of R(D ) 5.1 5.2 5.3 The standard way: normalizing P1 to A1 Normalizing P1 to V1 Enforcing a constraint at q2 = 0 6 Conclusions 1 Introduction Among the various avour observables showing a signi cant deviation from their Standard Model (SM) predictions ( avour anomalies ), those related to tree-level semileptonic B decays have received remarkable attention, as they potentially represent clean signals of New Physics. In addition to the long-standing discrepancy between the determination of the CKM element jVcbj from inclusive and exclusive semileptonic B decays, there is a anomaly [1] in the ratios of exclusive semileptonic decays to tau and to light leptons, R(D( )) = B(B ! D( ) ) ; B(B ! D( )` ) which could indicate a violation of lepton universality, and hence a clear departure from the SM. A full understanding of these relatively simple B decays is a necessary condition to pro t from the potential of the Belle-II and LHCb experiments in the search for New Physics, independently of these anomalies. For what concerns the B ! D channel, recent progress in the determination of the relevant form factors in lattice QCD [2, 3] and a new analysis of the q2 spectrum in B ! D` by the Belle Collaboration [4] have resulted [5, 6] in a more precise value of jVcbj in reasonable agreement with the inclusive determination [7, 8] and in the precise prediction R(D) = 0:299(3). The situation is not yet so favourable in the B ! D channel, which has so far provided the most accurate exclusive determination of jVcbj. First, unquenched lattice calculations of the relevant form factor [9, 10] are still limited to the zero-recoil point, where the jVcbj = 39:05(75) 10 3 [1], the parametrization [13] prefer a much higher value [14{16], ts performed with the Boyd-Grinstein-Lebed (BGL) jVcbj = (38:2 1:5) 10 3; jVcbj = (41:7 parametrization [11], and have published results in terms of the few parameters of this parametrization. Only recently, Belle has published a preliminary analysis [12] which, for the rst time, includes deconvoluted kinematic and angular distributions, without relying on a particular parametrization of the form factors. The new Belle results have allowed for ts of the experimental spectra with di erent CLN t leads to parameterizations, with surprising consequences on the resulting value of jVcbj: while a HJEP1(207)6 well compatible with the most recent inclusive result, jVcbj = 42:00(63) 10 3 [8]. As we have emphasized in ref. [14], this strong dependence of jVcbj on the parameterization should be interpreted with great care because i) it refers to a speci c set of data and the large discrepancy between eqs. (1.2) and (1.3) may not carry on to other sets of data; ii) the physical information encoded in the CLN and BGL parametrizations are not equivalent. Although they are grounded in the same foundations (analyticity, crossing symmetry, operator product expansion), the CLN parametrization makes use of Heavy Quark E ective Theory (HQET) relations between the B( ) ! D( ) form factors in various ways in order to reduce the number of independent parameters. Indeed, Heavy Quark Symmetry requires all of these form factors to be proportional to the Isgur-Wise function, and the leading symmetry breaking corrections of O( s; =mc;b) are known [17{19]. However, the residual uncertainty is not negligible and should be taken into account in the analysis of experimental data. There are also a few precise lattice QCD calculations which test and complement these relations and should be taken into account. The main purpose of the present paper is to investigate to which extent the Heavy Quark Symmetry relations between the form factors a ect the results of our previous analysis, once their uncertainty is properly accounted for. The methodology developed to this end will then be applied to the calculation of R(D ), where the only available information on the scalar form factor comes from the form factor relations. In determining the uncertainty of the HQET relations between form factors we will use the form factor ratios and their uncertainties in deriving strong unitarity bounds on the coe cients of the BGL parametrization. We will then use these bounds directly in the t to experimental data, without deriving a simpli ed parametrization like in ref. [11]. Finally, we will apply the results of our ts to the calculation of R(D ). Our paper is organized as follows: in the next section we discuss the uncertainties due to higher order e ects in the HQET relations between form factors. In section 3 we compute the strong unitarity bounds on the form factors that enter the B ! D ` decay rate taking into account their uncertainties. In section 4 we discuss our new ts to the preliminary Belle data which incorporate the strong unitarity conditions. In section 5 we Uncertainty of the relations between form factors As explained in refs. [11, 13] the unitarity constraints on the parameters of the z-expansion can be made stronger by adding other hadronic channels which couple to c b currents. While in general this would require non-perturbative information on each form factor, in the case of the B( ) ! D( ) transitions the form factors are all related by Heavy Quark Symmetry, which can be used to simplify the task. These transitions are described by a total of 20 helicity amplitudes, which provide an appropriate basis of form factors. In the following we will adopt the notation of ref. [11] | see in particular eqs. (A.3){(A.6), in which all form factors reduce to the Isgur-Wise function (w) in the heavy quark limit. In this notation S1 3 couple with a scalar charm-bottom current, P1 3 with a pseudoscalar current, V1 7(A1 7) with a vector (axial-vector) current. For later convenience, we also provide in table 1 the relation with the notation of ref. [13]. The HQET ensures that the form factor Fi(w) (here w = v v0, with v and v0 the four-velocities of the incoming and outgoing mesons) can be expanded in inverse powers of the heavy quark masses and in s, which at the Next-to-Leading order (NLO) results in Fi(w) = (w) h1 + ci s s + cib b + cic c + : : : ; i where b;c = s(pmcmb) =2mb;c, the strong coupling is typically evaluated at 0:26, and the dots represent higher order corrections. We recall that the pmcmb with Isgur-Wise function (w) is normalized to 1 at zero recoil, (1) = 1. Some of the ratios are known at O( 0 s2) but the extra corrections are very small [24]. We will follow here the calculation of ref. [20] which updates those employed in the CLN paper. In particular, we will adopt the values of quark masses and of the subleading parameters given there, f+ = 1 + r 2pr V1 | | B ! D g^ = 1 mB pr V5 f^ = mB (1 + w)pr A2 F^1 = m2B (1 + w) (1 r) prA6 | F^2 = 1 + r p r P2 g = B ! D 1 mBpr V4 f = mBpr(1 + w)A1 ; the more common notation f0 = f0BGL=(m2B m2D), used, e.g., in ref. [5]. As the Isgur-Wise function cancels out in the ratios of form factors, the latter can be computed more accurately in the heavy quark expansion. The central values of the ratios of form factors Fj to V1 computed in this way and expanded in w1 = w 1, Fj (w) V1(w) =Aj 1 + Bj w1 + Cj w12 + Dj w13 + : : : ; are given in table 2, which updates table A.1 of ref. [11] and has very similar results. There is no obvious way to estimate the size of the higher order corrections to the NLO HQET expressions. Parametrically they are O( s2), O( s c), and O( c2), where roughly (2.3) (2.4) 2 s s c 2 c 6%; { 4 { HJEP1(207)6 which introduce a link between form factors protected by Luke's theorem and others which are not protected. As an e ect, the 1=m corrections tend to be smaller in V1;2;3 and P1;2;3 as well, as a sort of indirect Luke's protection. Finally, there are the following exact relations between form factors at zero-recoil (w = 1): S1;2;3(wmax) = V1;2;3(wmax); P1;2;3(wmax) = A5;6;7(wmax) S2(1) = S3(1); A1(1) = A5(1); A2(1) = A6(1); A3(1) = A4(1) = A7(1): but the choice of mc or of the scale s can easily change this estimate. Most importantly, the coe cients in front of these parameters can enhance or suppress signi cantly their contribution. For instance, the perturbative expansion is actually an expansion in s=4 and the two-loop is generally enhanced by 0 9. In the following we will mostly worry about power corrections. It is useful to recall that several of the form factors do not receive NLO power corrections at zero recoil because of Luke's theorem [18]. In particular, all of the scalar and axial-vector form factors, S1 3 and A1 7, do not receive 1=m corrections at zero recoil. There are also exact kinematic relations between the (pseudo)scalar and (axial)vector form factors at maximal recoil w = wmax (corresponding to q2 = 0) In some cases, such as V1, the leading power corrections at zero-recoil are known to be suppressed [25]. The actual pattern of the NLO HQET corrections re ects these two qualitative suppressions: the form factors protected directly or indirectly by Luke's theorem receive small or moderate power corrections over the whole w range (1 while the others (V4;5;6;7) are a ected by leading power corrections as large as 50%. The magnitude of the coe cients of b;c reaches 2.1. The total NLO correction is almost 60% in V6=V1, see table 2. As Luke's theorem does not protect the form factors from 1=m2 corrections, it is therefore natural to expect 1=m2 corrections of order 10{20%, and one cannot exclude that occasionally they can be even larger. The comparison with recent lattice QCD results is instructive, even though it is limited to the few cases for which the form factors have been computed at zero or small recoil. Considering only unquenched lattice results, we average those by the Fermilab/MILC and HPQCD collaborations [2, 3, 9, 10], neglecting correlations between their results. We also mention that there is some tension between the preliminary value of A1(1) = 0:857(41) by HPQCD and the result of Fermilab/MILC, A1(1) = 0:906(13). Incidentally we note that the rst value agrees well with the heavy quark sum rule estimate of ref. [26]. The results w < 1:59(1:51) for B ! D( )), { 5 { (2.5) (2.6) S1 S2 S3 P1 P2 P3 1.0208 1.0208 0:0436 0:0749 0:0710 0:2164 0:0949 0:2490 0 0:2251 0:2651 0:1492 0:0440 0:1835 0:1821 0:0704 0:0280 0:0629 0:0009 0:3488 0:2548 0:0528 0:0201 0:0846 0:1903 0:0026 0:0034 0:0030 0 0:0000 0:0000 0:0012 0:0014 0:0009 0:0011 0:0580 0:0074 0:0969 0:1475 0:2944 0:0978 0:0942 0:0105 0:0418 0:0947 0:0007 0:0009 0:0008 0 at or near zero recoil are from which it follows that S1(w) = 1:027(8) V1(w) = 1:053(8) A1(1) = 0:902(12) ; 1:154(32)(w 1:236(33)(w Notice that in the case of S1=V1 both numerator and denominator have been computed at small recoil by the Fermilab/MILC and HPQCD collaborations, and we therefore have also a lattice determination of the slope of the ratio. { 6 { On the other hand, the HQET calculation at NLO of ref. [20] gives S1(w) V1(w) HQET A1(1) V1(1) HQET S1(1) where the errors represent only the parametric uncertainty on mb, s and the QCD sum HJEP1(207)6 rules parameters. Comparing the zero-recoil values of the ratios in eqs. (2.8) to those in eqs. (2.9) one observes deviations between 5% and 13%, which are obviously due to higher order corrections unaccounted for in eq. (2.9). In all cases the deviation is larger than the NLO correction. While it is quite possible that lattice uncertainties are somewhat underestimated, here we are not interested in a precision determination. What matters here is that the size of these deviations is consistent with our discussion above. The slope of the ratio S1=V1 computed on the lattice has a di erent sign from the one in (2.9) and their di erence induces a 6% shift at maximal w. However, since S1=V1 = 1 at maximal recoil, it is not surprising that higher order corrections are moderate in this case. In conclusion, higher order corrections to the form factor ratios computed in HQET at NLO are generally sizeable and can naturally be of the order of 10{20%.1 3 Strong unitarity bounds for B D form factors ! In the following we refer to the setup based on [13] which we have employed in [14] to perform a t to the recent Belle B ! D ` di erential distributions. In this framework the generic form factor Fi (already in CLN notation) can be expressed as Fi(w) = pi(w) Bi(z) i(z) N X a(ni)zn n=0 1The CLN form factors Fi we consider are helicity amplitudes which are linear combinations of the form factors in terms of which the matrix elements are decomposed. This sometimes leads to a correlation between numerator and denominator in the ratios of helicity amplitudes, which could a ect our error estimates. There are only a few such cases among the ratios we employ in this paper: S1=V1, P1=A1, A5=A1, P1=A5. The correlation is maximal at zero recoil where A1 = A5 and one has V1 S1 = 1 + 0:48. This suggests that higher order corrections are somewhat suppressed, as in fact we found by comparing with LQCD above. In the second line the NLO corrections are sizeable despite the suppression factor, and also here one can naturally expect NNLO corrections between 10% and 20%. (2.9) (3.1) B D B D Type Mass (GeV) Method Decay const.(GeV) resonance as it is very close to threshold and its value is very uncertain. Predictions for the decay constant of the second 0 resonance are also very uncertain and we do not include them here. where z = ( w + 1 p p2)=(pw + 1 + p2) and the prefactors pi(w) are the ratios between helicity amplitudes in the CLN and BGL notations which can be read o series in z in (3.1) is truncated at power N and we will set N = 2 from the outset, which is su cient at the present of level accuracy as 0 < z < 0:056 in the physical region for semileptonic B ! D decays to massless leptons. The Blaschke factors, Bi(z), take into account the subthreshold Bc resonances with the same quantum numbers as the current involved in the de nition of Fi. As the exact location of the threshold (mB( ) + mD( ) )2 depends on the particular B( ) ! D( ) channel, Bi(z) may di er even between form factors with the same quantum numbers. We will employ the resonances given in table 3. Finally, the outer functions i(z) can be read from eq. (4.23) and tables I and IV of ref. [13] and are given explicitly in a few cases in [5, 14]. We will use nI = 2:6 for the number of spectator quarks (three), decreased by a large and conservative SU(3) breaking factor. The other inputs we use are given in table 4 (all uncertainties are small and can be neglected), where the ~ are the constants after taking into account the one-particle exchanges, see [5, 11]. Analyticity ensures that the coe cients of the z-expansion (3.1) for the form factor Fi satisfy the weak unitarity condition but there are a number of two body channels (BD; BD ; B D; B D ; b c; : : : ) with the right quantum numbers, as well as higher multiplicity channels, that give positive contriN n=0 X(a(ni))2 < 1; { 8 { [27{29] L 1+ are needed for the scalar and pseudoscalar formfactors [13]; they are related to 0+ and 0 , respectively. numbers 1+, butions to the absorptive part to the two-point function and can strengthen the unitarity bound on the coe cients of each form factor. For instance, the form factors A1;5 which appear in the B ! D ` decays both contribute to the same unitarity sum with quantum aA1 2 n + aA5 2 n < 1: However, the (strong ) unitarity sums including all the B( ) numbers 0+, 0 , 1 , 1+ are ! D( ) channels with quantum Input Now we can use the relations between the 20 form factors we have presented in the previous section to obtain constraints on the coe cients of any speci c form factor Fi [5, 11, 13]. It is su cient to replace Fj by (Fj =Fi) Fi and expand the product in powers of z to re-express each coe cient anFj in terms of a linear combination of the anFi . In the case N = 2 which is relevant here, each unitarity sum can then be reduced to a quadratic form in a0Fi , a1Fi , a2Fi , and each unitarity condition in eqs. (3.4) represents an ellipsoid in the (a0Fi , a1Fi , a2Fi ) space. To take into account the uncertainties in the relations between form factors we generate replicas of the set of ratios Fj =Fi which satisfy the kinematic relations of eqs. (2.5), (2.6) and incorporate the lattice QCD results of eqs. (2.8) within their uncertainties. Each replica must also have all of the ratios contained within a band around their central values computed at NLO in HQET as presented in the previous section, improved whenever { 9 { 3 X N X possible with existing lattice data. At zero-recoil the band has a width corresponding to the maximum between 25% and 15% + 2 HQET. At the endpoint, corresponding to q 2 = 0, the width is slightly larger and corresponds to the maximum between 30% and 20% + 2 HQET. Here HQET is the total parametric relative uncertainty of the NLO HQET calculation, obtained combining in quadrature the uncertainty from the QCD sum rule parameters, mb, and s . Another condition that the replicas (Fj =Fi) must comply with is that the coe cients a(nFj) of the form factor Fj , computed by expanding in powers of z the expression Fj (w) = (Fj =Fi) (Fi=V1) V exp(w) 1 (3.5) satisfy weak unitarity, i.e. eq. (3.2). Here V exp(w) is the result of the t to B ! D` 1 experimental data and lattice results performed in [5]. The replicas (Fj =Fi) are acceptable if unitarity is satis ed for values of coe cients of V1exp(w) within 3 s from their central values. Each replica therefore represents a viable model of the form factors. An example of a few replicas of the ratios S1;2=A1 passing all tests is shown in gure 1. We will be primarily interested in the four form factors which enter the B ! D ` decays, namely A1; A5, V4, and P1. The latter contributes only for massive nal leptons. Each set of replicas of Fi=A1 gives rise to four ellipsoids in the (a0A1 ; a1A1 ; a2A1 ) space, corresponding to the four possible conditions in eq. (3.4). As A1(1) is known relatively well from lattice QCD calculations, we can x a0A1 and obtain 4 ellipses in the (a1A1 ; a2A1 ) plane. Samples of such ellipses from the S and V sectors are shown in gure 2: there is very little sign of correlation between a1A1 , a2A1 , but the regions identi ed in the two cases are similar. We have repeated the same procedure with a large number of replicas. The envelope formed by all the ellipses represents the allowed region in the (a1A1 ; a2A1 ) plane and is shown in gure 3. It is quite remarkable that the allowed regions are very similar for the S, P, V channels, while the A channel is less constraining. The intersection of the S, P, V and A channels is the allowed region we will consider in the following. In the same way we have derived bounds in the (a1A5 ; a2A5 ) plane. Indeed, A5(1) is xed by the same lattice calculations which x A1(1). The nal results are also shown in 0.6 0.4 0.2 1 A2 gure 3. The case of V4 is slightly di erent because there is no lattice calculation that xes a0V4 . In principle one should keep the three-dimensional envelope of all ellipsoids. However, in line with the previous discussion, we can assume that V4(1) is within about 30% from the values it takes when one uses A1(1) or V1(1) together with HQET form factor ratios. This leads to 0:0209 < aV4 < 0:0440. The bounds in the (a1V4 ; a2V4 ) plane depend little on 0 the exact value of a0V4 in that range, besides being anyway much weaker than those on the coe cients of A1;5. Therefore, also in this case we obtain a two-dimensional allowed region, shown in gure 3. The case of P1 is very similar to that of V4 and one similarly nds 0:041 < aP1 < 0:089 and then a two-dimensional allowed region in the (a1P1 ; a2P1 ).2 0 Two comments are in order at this point. First, the weak or absent correlation between a1 and a2 that we observe in most cases does not imply the absence of a strong correlation between slope and curvature of the form factors when they are expressed in terms of the variable w. The latter was observed long ago in refs. [11, 13] and is a simple consequence of the change of variable from z to w and of the outer functions structure, combined with the weak unitarity bounds on a1;2. Indeed, if we proceed as in ref. [11], we con rm their bounds on slope and curvature of V1. The only exception are the constraints from the vector channel, which we nd more constraining than in [11].3 Second, we see no point in modifying the parametrization to include these stronger unitarity bounds. The bounds we have found should be used directly in ts to experimental and lattice data based on the BGL parametrization. In the future, when new lattice information on the slopes of these form factors will become available, the bounds can be simpli ed; they will become one-dimensional bounds on a2Fi only. 2The two-dimensional numerical regions are available from the authors upon request. 3We traced the origin of the discrepancy to the exponent of ( j2 (w + 1)=2) in the denominator of the third row in their eq. (5) (sum over j = 4 7) which should be 4 instead of 5. The main results of [11] are una ected. 1.0 0.5 (S), pseudoscalar (P), vector (V) and axial-vector (A) channels. 4 Fits to B D ` data ! We will now employ the results of the previous section in a t to the available experimental data for B ! D ` , in order to illustrate the relevance of strong unitarity bounds in the present situation. To this end, we repeat here the analysis of ref. [14], based on the preliminary Belle data of [12], and refer to [14] for all the details. The only additional piece of data we will include in the t is the HFLAV average for the branching ratio of B0 ! D + where we added the errors in quadrature. Combining it with the total lifetime B0 = (152:0 0:4) 10 14s [ 27 ], we get a rather precise value for the total width of this decay. The above branching ratio can be compared with B(B0 ! D + l l) = 0:0495 0:0025 reported in [12]. One would expect the lower value in (4.1) to drive the t towards slightly lower values of jVcbj but we will see that the precision of the new input changes the t in an unexpected way. We stress that the branching ratio is, to good approximation, independent of the parametrization of the form factors used in the experimental analyses and it is therefore the only piece of data that we can use from older experimental results. We will also neglect all correlations of the total width with the binned angular and kinematic distributions included in the t. For what concerns the lattice determination of the form BGL Fit: Data + lattice Data + lattice + LCSR Data + lattice Data + lattice + LCSR unitarity 2=dof jVcbj aA1 0 aA1 1 aA1 2 aA5 1 aA5 2 aV4 0 aV4 1 aV4 2 weak 28:2=33 unitarity constraints. In the BGL ts a0A5 is related to the value of a0A1 , a0A5 = 0:1675 a0A1 . factor at zero recoil, A1(1), we will use the average given in eq. (2.7), which di ers slightly from the value employed in ref. [14]. The results of the constrained t are shown in table 5, where we consider ts in the BGL parametrization with weak and strong unitarity bounds, with and without the inclusion of the constraints computed with Light Cone Sum Rules at w = wmax in [36]: A1(wmax) = 0:65(18); R1(wmax) = 1:32(4); R2(wmax) = 0:91(17); (4.2) where V4(w) A1(w) ; R1(w) = R2(w) = w w r 1 1 1 w r A5(w) r A1(w) : We have also performed ts with the CLN parametrization (with free parameters A1(1); 2; R1(1); R2(1)) in the same way as in [14]. We obtain jVcbj = 0:0393(12) ( 2=dof = 35:4=37) without the LCSR and jVcbj = 0:0392(12) ( 2=dof = 35:9=40) with the LCSR. As expected, the di erence between the values of jVcbj obtained with the BGL and CLN parametrization is reduced by the use of strong unitarity bounds, but it remains as large as 3.5{5%, depending on whether LCSR results are included or not. Comparing the ts in table 5 with those in ref. [14] we note that the inclusion of the world average for the branching ratio has a signi cant impact on jVcbj: the central value increases by 1.2 to 1.7% and the error is reduced by 10{20%. Using the average of eq. (2.7) instead of the Fermilab/MILC result alone also leads to a minor increase of the jVcbj central value. Comparing the ts in table 5 with weak and strong unitarity bounds we observe that the strong constraints decrease jVcbj by 1.5{2.2% and tighten its uncertainty quite a bit, especially in the less constrained t without LCSR input. HJEP1(207)6 ,121.5 R combined in quadrature with a 15% theoretical uncertainty. It is also interesting to compare the e ects of the strong unitarity bounds we have derived with the help of heavy quark symmetry relations with a naive rescaling of the weak unitarity conditions of eq. (3.3). This gives an idea of how strong the strong unitarity bounds really are and helps us understanding their usefulness. The e ects of the strong unitarity bounds is roughly similar to that of using 2 n=0 X(aVn4 )2 < 2 n=0 X[(anA1 )2 + (anA5 )2] < V ; A with A;V < 0:2, depending on the inputs. In e ect, the strong unitarity bounds introduce little correlations among the aiFj coe cients: they mostly bound their individual size. This is unsurprising, as the unitarity sum rules cannot be saturated by one or two amplitudes only. We now want to verify a posteriori that the results of our ts are compatible with heavy quark symmetry within reasonable uncertainties. Indeed, the form factor ratios R1;2(w) de ned above after eq. (4.2) can be determined from the results of our ts. A deviation from the NLO HQET predictions signi cantly larger than 20% would signal an unexpected and unnatural breakdown of the heavy mass expansion. This point has been emphasised in refs. [14] and [37]. The two plots in gure 4 show that the ts without/with LCSR lead to R2 in good agreement with HQET (with input from QCD sum rules) and the same holds for R1 when LCSR are included. On the other hand, without LCSR R1 is well compatible with HQET only at small or moderate recoil: at large w there is a clear tension with both HQET and LCSR predictions. Lattice calculations will compute A1 and R1;2 at small recoil in the near future4 and are likely to settle the whole jVcbj determination. In the meantime, the t without LCSR appears somewhat disfavoured. Finally, we comment on the di erences of our ts with those performed in [20]. The main di erences are that the authors of ref. [20] employ the CLN parametrization for 4Preliminary and incomplete results have been presented recently [38]. They seem to exclude large deviations from HQET at small recoil. 2q2 (q2)3 d dw ; 3 m2(m2 q2)2r3(1+r)2(w2 1) 2 P1(w)2 : Here d =dw represents the di erential width for the decay to massless leptons, see e.g. [14], and depends on the form factors A1;5 and V4. In the second term k = E2WjVcbj2G2F m5B=32 3, r = mD =mB, with EW ' 1:0066 the leading QED correction. The second term depends on a new form factor, P1(w), whose z-expansion is the reference form factor A1 and assume the NLO HQET calculation for the form factor ratios R1;2 without accounting for a theoretical uncertainty due to unknown higher order corrections. They also perform a combined t to B ! D and B ! D Belle data. In our ts we do not employ directly the HQET relations because we believe their present uncertainty does not make them useful and therefore a combined t to B ! D and B ! D data would give the same results of the two separate ts presented here and in [5]. Indeed, the coupling between the two sets of data through the unitarity bounds would be extremely small. 5 In the case of a massive ( ) lepton the di erential width for B ! D can be written as the sum of two terms HJEP1(207)6 where where where the outer function is given by P1(w) = p r (1 + r)B0 (z) P1 (z) n=0 1 X anP1 zn ; P1 = s nI ~1L+ (0) p1 p 8 2 r2(1 + z)2 z ((1 + r)(1 z) + 2pr(1 + z))4 : The Blaschke factor B0 (z) takes into account the rst three 0 resonances, see table 3. The ratio R(D ), de ned in eq. (1), can be split into two parts R(D ) = R ;1(D ) + R ;2(D ) ; R ;1(D ) = R ;2(D ) = 1 1 R w ;max dw d ;1=dw 1R wmax dw d =dw R w ;max dw d ;2=dw 1R wmax dw d =dw ; ; w ;max = (m2B + m2D m2)=(2mBmD ) 1:355: (5.1) (5.2) (5.3) (5.4) Unfortunately, the experimental q2-spectrum of B ! D cannot be reliably used to constrain the form of P1 and there is no lattice calculation of this form factor. We therefore consider three options: to use P1 = (P1=A1)A1 where A1(w) is taken from the t and the ratio from HQET; to use P1 = (P1=V1)V1 where V1(w) is taken from the t of [5] and the ratio from HQET; to use the HQET expression for P1(1) and the constraint P1(wmax) = A5(wmax) HJEP1(207)6 together with unitarity. Having three alternative derivations will give us an additional handle to estimate the overall uncertainty. For a reference, we recall that the experimental world average for R(D ) is [1] (see update online) R(D )exp = 0:304 0:013 The standard way: normalizing P1 to A1 The rst option corresponds to the usual way of computing R(D ), see e.g. [20, 39]. The relevant ratio is traditionally denoted by R0: R0(w) = P1(w) A1(w) ; and of course R0(w) ! 1 in the heavy quark limit. We use the updated NLO HQET calculation for R0 [20], include gaussian uncertainties from the QCD sum rules parameters and mb in the same way as [20], and in addition we assign to the NLO HQET result a 15% uncertainty from higher order corrections. This corresponds to over 30% uncertainty on R ;2(D ). Here and in the remainder of this section all the errors are meant to be gaussian errors and are combined in quadrature whenever appropriate: this di ers from what we did in section 3 where we were looking for absolute bounds. In addition, we also impose strong unitarity constraints on the parameters of the z-expansion for P1, see gure 3: this moves the central value slightly o the one computed using the central values in table V and reduces somewhat the uncertainty. Using our t with LCSR and strong unitarity bounds, we nd R ;1(D ) = 0:232 R ;2(D ) = 0:026 ; R(D ) = 0:258(5)(+87) ; (5.6) where the rst error refers to the B ! D ` t parameters and the full parametric uncertainty of R0, while the second one is related to the 15% uncertainty due to higher order corrections to R0. The contribution of R ;2(D ) to the nal result is about 10%. The uncertainty on P1, which a ects only R ;2(D ), has therefore a comparably small impact on the total uncertainty of the SM prediction of R(D ). It turns out, however, that this is the largest single source of uncertainty. The results obtained with the t without LCSR and with strong unitarity bounds are very similar, where the two errors have the same meaning as in (5.6). Combining all errors in quadrature we end up with R(D ) = 0:258(+190) and 0:257(+180), respectively. These results agree well with those obtained in ref. [20] using the same normalization to A1, except for the uncertainty due to higher order corrections which is not considered there. They are also 0:003 [39] which has been used so far as reference SM prediction in most papers on the subject. Let us now proceed to compute R(D ) in the second way. Only the calculation of R ;2(D ) is di erent from the above derivation. Here we use the precise determination of V1(w) from experimental B ! D` data and lattice QCD calculations of [5]. In particular, with the BGL N = 2 parametrization of V1 and our t with LCSR and strong unitarity bounds we get where the rst error comes from parametric and t uncertainties, and the second one from the 15% higher orders error. Using instead our t without LCSR input one gets R ;1(D ) = 0:232 ; R ;2(D ) = 0:036 ; R(D ) = 0:268(+98)(+1120) ; R ;1(D ) = 0:232 ; R ;2(D ) = 0:038 ; R(D ) = 0:270(+98)(+1120) : The values of R(D ) in eqs. (5.8), (5.9) are substantially higher and have a larger uncertainty than those obtained with the rst method, although they are compatible within errors. The higher value of R(D ) is mostly due to the large di erence, already noticed in section 2, between the NLO HQET and the lattice QCD predictions for A1(1)=V1(1), see eqs. (2.8), (2.9). A lattice QCD determination of the form factor P1, even only at zero-recoil, would drastically decrease the uncertainty in R(D ). 5.3 Enforcing a constraint at q2 = 0 The ts presented in section 3 allow for a 5% determination of A5 at the endpoint w = wmax. This is outside the physical range for the semileptonic decay to taus, see (5.4), but the relation P1(wmax) = A5(wmax) still constrains P1(w) signi cantly. We will now use only the t with strong unitarity bounds and LCSR, which gives A5(wmax) = 0:545 0:025: (5.10) (5.7) (5.8) (5.9) This is signi cantly lower than P1(wmax) ' 0:69 obtained using the normalization to V1 considered in the previous subsection, and also lower than the P1(wmax) ' 0:62 obtained normalizing P1 to A1. For what concerns the value at zero recoil, w = 1, we can again use P1(1) = (P1=V1)HQETV1(1)lat or P1(1) = (P1=A1)HQETA1(1)lat, where the lattice values V1(1)lat and A1(1)lat are taken from eq. (2.7), leading to P1(1) = 1:27(21) and P1(1) = 1:12(18). Here we have combined in quadrature the parametric uncertainty with a 15% theoretical uncertainty. An intermediate choice consists in using the HQET relation between P1 and the Isgur-Wise function, which is 1 at zero recoil. At the NLO we nd where the rst error is parametric, and the second corresponds to the 15% theoretical uncertainty considered above. Using eq. (5.1) this amounts to a determination of a0P1 , P1(1) = 1:21 0:06 0:18 which can be combined with (5.10) to derive a0P1 = 0:0595 0:0093; a1P1 = 0:318 0:170 0:056a2P1 where the last term must satisfy ja2P1 j < 1 and a1P1 is consistent with strong unitarity for almost any a2P1 . Using the last two relations, scanning in the relevant range of a2P1 , and combining the errors in quadrature we get R ;2(D ) = 0:028 ; R(D ) = 0:260(5)(6) ; where the rst error refers to the parametric uncertainty in R ;1(D ) and the second one is related to P1 and parametric uncertainty in R ;2(D ) only. The correlation between the two errors is small. We observe that the uncertainty is slightly smaller than those of the other methods. The three methods we have employed to compute R(D ) lead to results which are consistent within uncertainties. The third method has a slightly smaller error and bene ts from an important constraint at q2 = 0 which is not taken into account with the rst two methods. In particular, eqs. (5.8), (5.9) are likely to somewhat overestimate R(D ). We therefore adopt as our nal result the one obtained with the third method, R(D ) = 0:260 0:008; which still di ers 2:6 from the experimental world average, but it is higher and has an uncertainty almost three times larger than existing estimates.5 We apply the same methodology also to the prediction of the longitudinal lepton polarization [39{42] 5Only ref. [16], which appeared together with the rst version of this paper, has a larger uncertainty, nding R(D ) = 0:257(5). (5.11) (5.12) where in our notation and H00; H P = 0:38 in the ts, is H0t = mB p pr(1 + r)pw2 1 + r2 2wr 1 P1 ; P = 0:47 are the integrated decay rates for de nite lepton helicity. One has [39] d dq2 = d + dq2 = G2F jVcbj2jp~jq2 Recently, Belle reported the measurement 0:51+00::2116 [43, 44]. Our SM prediction, independently of the use of LCSR 6 Unitarity bounds are an essential part of the model independent form factor parametrization in semileptonic B decays. They can be made stronger using Heavy Quark Symmetry relations between the B( ) ! D( ) form factors, and are solid and reliable constraints, provided one takes into account conservative uncertainties and recent input from lattice calculations and experiment. In this paper we have obtained bounds on the z-expansion parameters of the form factors relevant in the calculation of B ! D ` decays. Since we keep only terms up to z2 in the expansion, and we generally have lattice QCD information on the rst coe cient a0, the bounds are expressed as allowed regions in the (a1; a2) planes for each of the form factors, see gure 3. As lattice QCD calculations extend beyond the zero recoil point, they will soon provide a relatively precise determination of the slopes of some of the form factors. Our bounds will then become rather strict one-dimensional bounds on the curvature, or on the a2 parameters. In practice, we have revisited the CLN methodology 20 years later, and used experimental and lattice data to estimate the uncertainties in the HQET relations and to reduce the errors. Unlike CLN, however, we do not provide a simpli ed parametrization. On the contrary, our results on unitarity bounds applied to the BGL parametrization should form the basis of a new generation of model independent analyses of B ! D ` data at both Belle-II and LHCb. For what concerns the determination of jVcbj, we con rm and reinforce the conclusions of our recent analysis [14]. The present world average of the exclusive determination of jVcbj [1] relies on the CLN parametrization, but does not include a reliable estimate of the related theoretical uncertainties and is likely to be biased. Although the strong unitarity bounds have important consequences on the determination of jVcbj and reduce its value by about 2%, our ts to recent Belle's and lattice data (complemented by the world average for the B0 ! D +` branching ratio) show a large persisting di erence (3.5{5%) in the value of jVcbj extracted using the BGL and CLN parametrizations. As already observed in [14], it is possible that such a large di erence is accidentally related to the only Belle data we could analyse for the B ! D ` di erential distributions, and that future global averages of Babar and Belle data will lead to a smaller di erence between the CLN and BGL ts. However, our approach now includes HQET constraints with realistic uncertainties and improves on the CLN parametrization in several ways. Our nal results for jVcbj are consistent with the inclusive determination but the error is signi cantly larger, about 3% instead of 1.5%. We have also reconsidered the SM prediction of R(D ) in the light of the above results. Our analysis points to a higher central value and a signi cantly larger theoretical error than found in previous analyses [20, 39]. Our nal result is reported in eq. 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Dante Bigi, Paolo Gambino, Stefan Schacht. R(D ∗), |V cb |, and the Heavy Quark Symmetry relations between form factors, Journal of High Energy Physics, 2017, 61, DOI: 10.1007/JHEP11(2017)061