Constrained second-order power corrections in HQET: $R(D^{(*)})$, $|V_{cb}|$, and new physics

We postulate a supplemental power counting within the heavy quark effective theory, that results in a small, highly-constrained set of second-order power corrections, compared to the standard approach. We determine all $\bar{B} \to D^{(*)}$ form factors, both within and beyond the standard model to $\mathcal{O}(\alpha_s/m_{c,b}, 1/m_{c,b}^2)$, under truncation by this power counting. We show that the second-order power corrections to the zero-recoil normalization of the $\bar{B} \to D^{(*)} l \nu$ matrix elements ($l = e$, $\mu$, $\tau$) are fully determined by hadron mass parameters, and are in good agreement with lattice QCD (LQCD) predictions. We develop a parametrization of these form factors under the postulated truncation, that achieves excellent fits to the available LQCD predictions and experimental data, and we provide precise updated predictions for the $\bar{B} \to D^{(*)} \tau \bar\nu$ decay rates, lepton flavor universality violation ratios $R(D^{(*)})$, and the CKM matrix element $|V_{cb}|$. We point out some apparent errors in prior literature concerning the $\mathcal{O}(1/m_cm_b)$ corrections, and note a tension between commonly-used simplified dispersive bounds and current data.


I. INTRODUCTION
The Heavy Quark Effective Theory (HQET) [1][2][3][4] underpins key foundations in our theoretical understanding of exclusive semileptonic b → clν decays (l = e, µ, τ ). HQET allows for a hadronic model-independent and high-precision determination of the CKM matrix element (see Ref. [10] for a review), similarly precise SM predictions are warranted. Moreover, |V cb | recovered from exclusive B → D * ν measurements currently exhibits a 3σ tension compared to the measured value from inclusive B → X c lν decays [5], with the magnitude of the deviation near the O(10%) level (also see [11]). Because the extraction of |V cb | (currently) to the B → D ( * ) matrix elements in the SM have been known for three decades [12][13][14][15], and explicit O(1/m c,b , α s ) results for all NP matrix elements were recently derived [7]. At first order in the heavy quark (HQ) expansion, six first-order wavefunctions are described by three subleading Isgur-Wise functions, while the one-loop O(α s ) perturbative corrections are calculable (see, e.g., Ref. [7] for their closed-form expressions). The O(1/m 2 c,b ) corrections have also been known for three decades [16].  Table I. Finally, the O(α s × 1/m c,b ) corrections are also long known (see, e.g., Ref. [14] for a review). and 10%, respectively (as the approximate small parameter is ∼ Λ QCD /m c,b or ∼ α s /π).
The second-order corrections at O(1/m 2 c ) can also be expected to contribute at the ∼ 5% level. Moreover, in the zero-recoil limit the B → D * form factor, F(1) (defined below), has vanishing first-order corrections, but its resulting value at O(α s , 1/m c,b ) differs at the 5% level from lattice QCD (LQCD) predictions [17]. These observations lead to the following possibilities: (a) given that second-order or higher corrections must fix the F(1) tension, it is possible the HQ expansion of the B → D ( * ) matrix elements could be 'badly behaved', such that 1/m 2 c terms may be unexpectedly large; or (b), while second-order power corrections must be important at zero recoil because of the vanishing first-order corrections at the phase space point, they are otherwise subdominant and the data beyond zero recoil will be predominantly described by first-order corrections. The latter is the approach used in Ref. [7], that performed the first combined and self-consistent analysis of B → D ( * ) ν decays at O(α s , 1/m c,b ). In this approach, only the shape of the differential distributions was used to constrain the subleading Isgur-Wise functions.
Recent analyses that attempt to quantify the effect of second-order power corrections [9,18,19] treat the six O(1/m 2 c ) wavefunctions (or a subset of them) as nuisance parameters in phenomenological fits. These analyses further make use of theoretical inputs from modeldependent calculations, such as QCD sum rules (QCDSR) or light cone sum rules (LCSR).
Such studies typically estimated that the HQ expansion appears well-behaved at O(1/m 2 c ). In addition, the constrained structure of the HQET for Λ b → Λ c lν decay has permitted its far simpler O(1/m 2 c ) contributions [20] to be extracted from combined fits to data and LQCD results [21,22]. These were found to be compatible with a well-behaved HQ expansion, too.
The goal of this work is to derive the set of O(α s × 1/m c,b , 1/m 2 c,b ) corrections to the B → D ( * ) form factors, in both the SM and beyond, under truncation at second order in the RC expansion. As a point of comparison to the RC expansion, we also consider the vanishing chromomagnetic (VC) interaction limit [16,[23][24][25], which also dramatically simplifies the number of subleading and subsubleading Isgur-Wise functions. We then confront the RC expansion and VC limit results with available experimental measurements and LQCD predictions, and obtain precise results for |V cb | and SM predictions for R(D ( * ) ).
These objectives require the assembly of a wide range of theoretical and phenomenological components. First, in order to ensure self-consistent conventions we carefully (re)develop the formal elements of the general HQ expansion that are required when working at second order, and then develop the RC conjecture, showing how it constrains and simplifies the structure of the power corrections (Sec. II and Apps A, B, and C). Second, we proceed to apply this to the B → D ( * ) system, deriving the corrections up to and including O(α s × 1/m c,b , 1/m 2 c,b ) under truncation by the RC expansion, incorporating zero-recoil and normalization constraints and removing redundant higher-order terms (Sec. III and Apps. D and E). We note apparent sign errors or inconsistencies for several O(1/m c m b ) wavefunctions derived in Ref. [16]. Third, we construct a parametrization of these corrections, implementing the 1S short-distance mass scheme for heavy quark masses and an analytic structure for the leading Isgur-Wise function that respects the HQ expansion at second order (Sec. IV). These results are encoded in the Hammer library [26,27]. We show that the tremendous simplification of the secondorder power corrections under the RC expansion constrains most zero-recoil corrections to be a combination of the hadron mass parameters, λ 1 and λ 2 . We investigate the zero-recoil predictions for various form-factors and their ratios, and find that the value of F(1) is in good agreement with LQCD results (Sec. V). Finally, in Sec. VI, the parametrizations of both the RC expansion and the VC limit are fitted against all available experimental measurements and LQCD data, examining various fit scenarios that consider different combinations of experimental and LQCD inputs and different assumptions. The latter includes fits that truncate at lower order in HQET, and fits that constrain only the shape of the distributions, as done in Ref. [7]. To properly identify optimal parameter subsets that describe the data and avoid potential overfitting, we employ a nested hypothesis test (NHT) prescription.

1.2)
These results can be compared to other recent results [17,18,[28][29][30][31]. (ii) While the inclusion of zero-recoil second-order power corrections in the RC expansion is crucial to good fits, the inclusion of second-order power corrections beyond zero recoil is not. This supports the approach used in Ref. [7]; (iii) The slope-curvature relation developed by Ref. [32] is in tension with the data, and leads to large upward biases in R(D); (iv) The VC limit, in contrast to the RC expansion, produces poor fits because of its structure at zero recoil, but using only shape information yields good fits.

II. THE RESIDUAL CHIRAL EXPANSION
A. General HQET preliminaries The standard construction of HQET follows from a reorganization of the QCD Lagrangian for a heavy quark Q with mass m Q , in terms of the mass-subtracted fields The parameter v is a heavy quark velocity-defined up to reparametrization freedom [33] via p Q = m Q v + k, in which k ∼ Λ QCD is a residual momentum-and the projectors Π ± = (1 ± / v)/2. This yields Here D µ is a gauge covariant derivative of QCD, and the transverse derivative D µ ⊥ = D µ − (v · D)v µ . Because of the mass subtraction in the phase of Q ∼ e −ip Q ·x , the derivative D ∼ k ∼ Λ QCD , so that in the heavy quark regime, m Q Λ QCD , one may integrate out the double heavy field Q v − yielding an effective theory for the light field Q v + with order-by-order corrections in 1/m Q . This HQET Lagrangian reads Writing L HQET = n=0 L n /(2m Q ) n to second order, Here the field strength ig G αβ = [D α , D β ], σ αβ ≡ i 2 [γ α , γ β ], and we have made use of the equation of motion, iv · D Q v + = 0 in the free effective theory. 1 The coefficient of the L 1 chromomagnetic operator a Q (µ) is renormalized by the strong interactions, where µ is an arbitrary matching scale of QCD onto HQET. Its deviation from unity is important when considering corrections at O(α s /m Q ) and higher. Therefore it can be consistently neglected everywhere except when discussing the O(α s /m Q ) radiative corrections, for which we use [35] (see also Refs. [14,36]), and except for explicit evaluation of the λ 2 parameter (see Eqs. (A9) and (A10)). The renormalization of the coefficients of the L 2 terms can be neglected at O(α s /m Q ).
At any order in 1/m Q , one may compute Lagrangian corrections to a particular HQET correlator via an operator product involving the L i . In addition, the quark source term JQ for a QCD correlator can be expressed with respect to mass-subtracted quantities via The time-ordered correlators of full QCD will then match onto time-ordered HQET correlators determined by functional derivatives with respect to J v , order-by-order in 1/m Q . Applying the equations of motion from Eq. (2.2), the source term becomes keeping terms to second order, and noting the second order term arises via Expanding the current factor 1 In the full effective theory, the equation of motion receives corrections, such that beyond leading order iv ·DQ v As usual in any perturbation theory, consistent power counting in 1/m Q mandates use of the free equation of motion at each order in 1/m Q (see e.g. [34]).
in which Z is the partition function of the free theory generated by L 0 . As is the usual practice, we use a notational convention that labels charm parameters with primes while In Appendix A we present this derivation from first principles, including precise definitions of the HQ mass parametersΛ, λ 1 and λ 2 and pertinent conventions used in this work, that are important to a self-consistent derivation of the second-order power corrections. In Eq. (2.10) we have explicitly restored the renormalization factor for the chromomagnetic operator. For a pseudoscalar (P ) and vector (V ) meson, furnishing a heavy quark doublet with brown muck spin-parity s π = 1 2 − , the factor d P = 3 and d V = −1, respectively.
The HQET eigenstates of L 0 , |H v , are normalized such that Note this normalization choice differs from that in Ref. [16], which normalized the HQET states with respect to an HQ mass scale, m Q +Λ. Similarly the matching (2.9) is defined with respect to normalized QCD states. To second-order, Eq. (2.9) then becomes The Π − projectors on the J 1 terms have been eliminated in some terms by the Q v ( ) + equation of motion. Here, the • operator denotes an operator product. For instance, 13) in which P 2 (x − z) is the (dressed) heavy quark two-point function.
Of particular utility when working at second order is the observation that the two-point function is a Green's function Appendix B we derive these relations from first principles (cf. Appendix C of Ref. [16]).
It is important to keep in mind that the notation Q = c or b here and throughout is merely a convenient reminder of which HQET operator acts on the ingoing and outgoing states. From the point of view of HQET, there is no distinction between b v + and c v + : The heavy quark flavor symmetry is only broken by the masses m b = m c . We will therefore switch as convenient between writing Q v

B. Interaction operator basis
Writing the QCD current J Γ =c Γ b, then a full operator basis entering the QCD matrix The pseudotensor contribution is determined by the identity σ µν γ 5 ≡ ±(i/2) µνρσ σ ρσ , in which the sign is subject to a convention choice. For B → D ( * ) , the sign convention most often chosen is such that σ µν γ 5 ≡ −(i/2) µνρσ σ ρσ , which implies Tr[γ µ γ ν γ ρ γ σ γ 5 ] = +4i µνρσ . This is the opposite of the sign convention often used c . Perturbative corrections to the currents (2.15) may be computed by matching QCD onto HQET local operators [37][38][39] at a suitable matching scale µ. We present the general derivation of these corrections in Appendix C.

C. Modified power counting
We are interested in exploring whether it is possible to develop a supplemental power counting, on top of the heavy quark expansion, that may systematically reorganize the second order power corrections into a small set of dominant terms, plus a larger set of subdominant contributions that can be truncated.
The heavy quark expansion arises from a reorganization of the QCD Lagrangian into the L 0 term that obeys heavy quark spin-flavor symmetry, plus symmetry-breaking corrections suppressed by powers of 1/m Q (and by α s ). The order of any given correction in the 1/m Q expansion is effectively determined by the number of insertions of Q v − Q v − into a QCD correlator of interest, which are then integrated out to form the corresponding HQET matrix element. This expansion does not assign any relative importance to the local current corrections versus non-local Lagrangian insertions, that enter at each fixed order. However, with respect to the structure of 1/m c,b corrections in B → D ( * ) decays, it has been hypothesized (albeit based on model-dependent calculations, such as QCDSR), that corrections from the chromomagnetic operator in the L 1 Lagrangian may be numerically small compared to the current corrections from J 1 [23][24][25]. These expectations are also supported somewhat by fits to B → D ( * ) data at O(1/m c,b ) [7], and compatible with fits to B → D * * data, which find that first-order chromomagnetic contributions are consistent with zero [40,41].
With this in mind, a distinguishing feature between a current and a Lagrangian correction is the number of / D ⊥ insertions involved: there is one for the former, and two for the latter, as follows immediately from Eqs. (2.6) and (2.3). Thus, one may contemplate an additional expansion that resembles counting in / D ⊥ /Λ QCD , in which each Lagrangian insertion amounts to two powers of / D ⊥ /Λ QCD , while a current insertion involves just a single power of / D ⊥ /Λ QCD . As we will see in this section and the next, this counting can be related to an expansion in the number of operator products inserted along the heavy quark line, with the additional counting rule that a current insertion counts for half that of a Lagrangian one.
Before discussing further such an expansion, which can be fully defined within HQET (the low energy effective field theory), it is useful to consider first the origin of the difference in the number of / D ⊥ 's entering the current and Lagrangian insertions. This is better understood by looking at the matching between QCD and HQET. In particular, apart from counting the number of insertions of Q v − Q v − , one may additionally count the number of insertions of the cross term Q v + i / D ⊥ Q v − into a QCD correlator (after which Q v − is integrated out to form an HQET correlator). This counting is not the same as for 1/m Q , because of the equation of motion for Q v − . The cross terms break the accidental U (1) 2 chiral symmetry, respected by the Q v ± kinetic terms, to a diagonal U (1). Therefore, although there is no small parameter in the QCD Lagrangian (2.2) that parametrizes this chiral symmetry breaking, one may nonetheless systematically organize the contributions to any matrix element by power counting in the number of insertions of the chiral symmetry breaking cross-term that enter into each correlator.
Referring to Eq. (2.2), one may implement this power counting by introducing a chiral symmetry breaking parameter θ, such that  respect to the heavy quark expansion of the Lagrangian, it follows from Eq. (2.3) that the leading term L 0 ∼ θ 0 , while all the L n≥1 ∼ θ 2 . In the heavy quark expansion of the source term (2.6), however, all the current correction terms J n≥1 ∼ θ. Moreover, all product current correction terms from J m J n -i.e., terms at order 1/(m m Q m n Q )-are then ∼ θ 2 . The θ power counting is summarized in Table II.

D. Operator product conjecture
Because pure current corrections act on one of the external states, the single / D ⊥ term that is inserted by these corrections amounts to inserting a factor / k ⊥ (. 2 We similarly expect a current-current product correction to involve a fac- . By contrast, a pure Lagrangian correction involves an operator product with two / D ⊥ insertions, producing a factor of the form Q v . This, in turn, entails an integral of the form which can be thought of as a second moment of the (dressed) two-point function, with respect to the transverse residual momentum, plus higherorder HQET corrections.
2 With some abuse of notation we track here only the transverse momentum contributions originating from the Fourier transform of ∂ ⊥ . The same power counting would also apply to the soft gluon interactions contained in the covariant derivative D ⊥ .
Recalling, as mentioned in the previous section, that corrections in B → D ( * ) decays from the first-order chromomagnetic operator are hypothesized to be numerically small compared to the current corrections [23][24][25], and that chromomagnetic contributions are consistent with zero in fits to B → D * * data [40,41], one might hypothesize that, generally, A mixed current and Lagrangian correction involves three / D ⊥ 's, yielding a factor ∼ Given Eq. (2.17), this leads us to the conjecture regarding the magnitudes of integrals of the form with l = 0 or 1 and m ≥ 1, and treating as a small parameter. I.e., the greater the number of operator products in a correlator, the smaller its value.
The conjectured expansion in Eq. (2.18) requires at least one operator product, and is formally different from that of the θ expansion, because at O(θ 2 ) the product current corrections enter, that do not involve an operator product. Further, radiative corrections in HQET may induce mixing under the renormalization group evolution (RGE), such that the Wilson coefficient of an operator containing n time-ordered operator products may induce a (α s /π-suppressed) correction to one containing m ≤ n [42]. 3 In the context of the power counting, this amounts to higher-order operators generating contributions to lower order ones. This, however, is not a problem (and not dissimilar to what happens with conventional perturbative expansions) as long as is small, which is the basic assumption motivating this expansion: it is based on empirical evidence at O(1/m Q ) and ultimately involves a question about non-perturbative QCD dynamics, that can only be determined by comparing this constrained expansion to experimental or lattice data.
The first occurrence of this phenomenon-higher-order operators generating contributions to lower order ones-is at second order in the heavy quark expansion, at which for example the operator product L 1 •L 1 induces an O(α s /m 2 Q ) correction to the L 2 Wilson coefficient [43,44]. However, the L 2 Wilson coefficient or that of any other term up to and including O(θ 2 ) cannot radiatively generate contributions to O( 3 ) or higher-order operators. Moreover, at O(θ 3 ) and beyond, the θ and tree-level power countings coincide, so that the θ expansion becomes a convenient tool for tracking, within HQET, the conjectured suppressions. That is, from the conjectured expansion in HQET, one may deduce that O(θ 2 ) and lower-order terms dominate those at O(θ 3 ) and higher, while any RGE-induced counterterms from the latter will be captured by O(θ 2 ) terms that are already present. Thus we may truncate the expansion at O(θ 2 ). We refer to this as the residual chiral (RC) expansion.
writing the HQET current operator J Γ+ (z) =c v + (z)Γb v + (z). These allow us to relate the IW functions associated with second order current corrections to λ 1,2 times the leading IW function.
One additional point of importance is that in this expansion the O(θ 3 ) terms at second order-the terms corresponding to mixed current and Lagrangian corrections ∼ J 1 L 1vanish at zero recoil [16]. Thus, if the residual chiral expansion is a good approximation, we may expect it to be particularly useful at zero recoil, because only O(θ 4 ) corrections enter. The D and D * (or B and B * ) mesons belong to a HQ spin-symmetry doublet, formed by the tensor product of a spin-1/2 heavy quark with brown muck in the s π = 1 2 − spin-parity state. This doublet, containing the pseudoscalar (P ) and vector (V ) mesons with a single heavy quark, can be represented as [37,45,46] H in which µ denotes the polarization vector of the spin-1 state with velocity v. Here and hereafter X = γ 0 X † γ 0 for any Dirac object X.
With reference to the reduced terms in Eq. (2.19) at O(1/m 2 c,b , θ 2 ) the matching of HQET to the QCD matrix elements becomes defining Here we have included in H (1,1) bc only those terms relevant for matching at O(1/m 2 c,b , θ 2 ). The full expressions are given in Appendix G. The recoil parameter is defined as 4) and the HQ expansion parameters, implies that ξ(1) = 1. We discuss further zero-recoil constraints below in Sec. III D. As done in, e.g., Refs. [7,14,47] (but not in Refs. [12,15]) we have further normalized terms in the expansion with respect toΛ, such that all Isgur-Wise functions are dimensionless.
We use the notation that hatted functions of w are normalized to the leading Isgur-Wise for any Isgur-Wise function or form factor. In particular, theL (n) i andM i denote linear combinations of higher-order Isgur-Wise functions 4 normalized by ξ. 4 In the notation of Ref. [16],ΛL We have added the superscript index to theL's in order to make clearer at which order they enter into the power expansion. We use the standard numbering for the subscripts of theL (n) i , while our numbering for theM i is the same as those used for the m i in Ref. [16].

B. Form factor matching
We use the standard HQET definitions for the B → D ( * ) form factors. The B → D matrix elements are and for B → D * , Here the h Γ i are functions of w. While typically we are not interested in B * → D ( * ) decays, because the B * decays to Bγ, the vector current matrix element for B * → D * is important for mass normalization constraints at second order: a generalization of Luke's theorem. In particular, we need also consider which is the only form factor that contributes in the zero recoil, equal mass limit.
For the sake of writing the form factors in terms ofL (n) i andM i in a compact manner, it is convenient to defineL i , Q = c, b . (3.11) In this notation, the hatted B → D form factors (see Eq. (3.7)) at O(α s , 1/m 2 c,b , θ 2 ) arê The hatted B → D * form factors at this order arê We have included here the leading perturbative corrections inα s = α s /π, that are given in Eq. (C2). The higher-orderα s /m c,b corrections are discussed in Appendix E. Finally, the B * → D * vector form factor (3.14) In Eqs.
which must be regular. By definition W (1) = W (1), the gradient at zero recoil.

C. Chiral corrections
A full assessment of potential percent-level corrections to the form factors requires consideration of chiral corrections, which originate from strong dynamics of the brown muck at momentum scales below that of chiral symmetry breaking, sensitive to the light mesons spectrum and the heavy mesons mass differences. This dynamics may therefore be represented using Heavy Hadron Chiral Perturbation Theory (HHχPT) [48,49], under which the dominant chiral corrections to the HQET matrix elements are generated by loops containing a light pseudoscalar, P = π, K, η. Schematically the structure of the chiral corrections can be expanded in powers of the heavy mass scale M H (e.g., a heavy hadron mass) as where log(m 2 P /µ 2 ) denotes chiral logarithms of the light meson masses, and B n contain finite and counterterm contributions. The scale µ ∼ O(1 GeV) is where HQET is matched onto HHχPT. The expressions for B → D ( * ) decays have been known for a long time [50][51][52].
The terms A n , B n are in general different for B 0 , B + and B s decays.
At zero recoil, the leading and subleading corrections vanish, and the leading nonzero contribution is proportional to the hyperfine mass splitting ∆m 2 H ∼ λ 2 2 [51]. Because λ 2 2 ∼ O(θ 4 ) in the RC expansion, the chiral corrections can be neglected at zero recoil. Similarly, in the VC limit they vanish because λ 2 → 0.
Away from zero recoil, both leading and subleading corrections in powers of 1/M H are present. Parametrically the size of A n , B n is controlled by the chiral loop factor (g P m P /4πf P ) 2 1%, where g P is the coupling of the light meson P to the heavy hadrons and m P and f P are its mass and decay constant. At the order of precision we are interested in, only the leading corrections are important, because the subleading ones contribute Importantly, the leading chiral corrections are universal for any HQ current. As a result, they can be reabsorbed via a redefinition of the leading order Isgur-Wise function, up to induced corrections of order ∼ 1% × Λ QCD /m Q or ∼ 1% × α s /π that can be neglected. s decays are not considered in this paper, we do not discuss them further.

D. Zero-recoil normalization constraints and redefinitions
The mass normalization condition (3.6) implies that in the equal mass limit the vector current matrix elements satisfy, to all orders, (3.17) In this limit, both the perturbative corrections and the power corrections to h +,1 (1) vanish order-by-order. At first order, Eqs. (3.12) and (3.14) then imply L 1 (1) = L 2 (1) = 0, which is a part of Luke's theorem [12]χ At second order, the mass normalization constraints for pseudoscalar and vector mesons respectively require which results in the zero recoil constraintŝ Just as in Eq. (D15), we therefore writê in which the quotient functions β 1,3 are regular near zero recoil.
The three Isgur-Wise functions β 1 , χ 1 and ξ arise from the same leading-order trace, as can be seen by comparing Eqs.
The constraint χ 1 (1) = 0 ensures that the normalization condition ξ(1) = 1 is preserved. Since the second terms in Eq. (3.21) vanish at zero recoil by construction, then at while preserving ξ(1) = 1. Note, however, one cannot generally redefine χ 2,3 to absorb β 2,3 , because although β 2 and χ 2 enter via an identical trace, as do β 3 and χ 3 , each trace violates heavy quark spin symmetry. As a result χ 2,3 + ε c β 2,3 enters into the B → D * form factors with a different prefactor than corrections, then one can absorb β 2,3 via the redefinitions   Table I. In this table, moving from leading order to first order to second order, we only count the new, independent Isgur-Wise functions entering at each order (though 1/m 2 c,b counts functions also counted at 1/m 2 c ). The counting of Isgur-Wise functions is performed after redefini- Ref. [7] Form factors

F. Vanishing chromomagnetic limit
Besides the RC expansion truncated at O(θ 2 ), we also consider another Ansatz, in which the field strength G αβ is set to zero. 5 This vanishing chromomagnetic (VC) limit, already considered in Ref. [16], also significantly reduces the number of Isgur-Wise functions at . It is motivated by the smallness ofχ 2,3 calculated using QCD sum rules [24,25], and is also consistent with O(1/m c,b ) fits (see, e.g., Ref. [7]). For the sake of completeness and consistency of notation, we revisit the derivation of this limit in Appendix F, noting a few differences with respect to Ref. [16].
The expressions for the nonvanishingL 1 =L Concerning the perturbative corrections, in the vanishing chromomagnetic limit the O(α s ) expressions remain the same, while at O(α s /m c,b ) the terms proportional to C c,b g vanish (see Appendix E), as they correspond to insertions of the chromomagnetic operator. Finally, the effects of the chiral corrections in the VC limit are the same as for the RC expansion, as the zero recoil effects proportional to λ 2 2 vanish altogether instead of being higher order.

A. 1S scheme and numerical inputs
Cancellation of the leading renormalon ambiguities [54,55] from the mass parameterΛ against those in the factorially growing coefficients in the α s perturbative power series can be achieved by use of a short distance mass scheme. We use the 1S scheme [56][57][58], which defines m 1S b as half of the perturbatively computed Υ(1S) mass. It is related to the pole . This may be inverted to express the pole mass The splitting of the bottom and charm quark pole mass δm bc ≡ m b − m c is subject to a renormalon ambiguity only at third order when one computes just the leading n f -dependence at high orders [59][60][61], so we fix m c = m b − δm bc . Thus, when working at second order in the HQ expansion, we may parametrize observables in terms of m 1S b and δm bc . In practice, however, because m 1S b and δm bc are extracted from fits to inclusive spectra at O(1/m 3 Q ) [62][63][64], third-order terms must be retained numerically in the expansion of the hadron mass, even though we formally work to second order in the expressions for the form factors. In particular, the spin-averaged mass of the HQ pseudoscalar-vector doublet, noting from Eq. (A10) that the λ 2 dependence cancels. Here we have included only the O(θ 2 ) contribution to the hadron mass from ∆m H 3 , proportional to the parameter ρ 1 , as defined in Ref. [65]. This leads tō .
That is, truncating the hadron masses at third order in the HQ power expansion and second order in the residual chiral expansion,Λ and λ 1 are parametrized in terms of m 1S b , δm bc , and ρ 1 .
The fits to inclusive B → X c lν spectra and other determinations of m 1S b , find that [64] which we use as inputs to our fits. For ρ 1 we use corresponding numerically to the range λ 1 = (−0.3 ± 0.1) GeV 2 , commensurate with the ranges quoted in Ref. [64]. 7 We choose the HQET to QCD matching scale One may apply these inputs to Eq. (4.1), combined with the renormalization group evolution [66,67]. One finds up to small uncertainties that are negligible when working at O(1/m 2 c,b , α s /m c,b ). We shall therefore treat α s (µ bc ) as a fixed external parameter.
0.11 ± 0.02 GeV 2 , (4.8) using the 1S prescription and Eq. (2.5), in which we have assigned an inflated ∼ 20% uncertainty to λ 2 to absorb possible higher-order renormalon effects. The fits to inclusive B → X c ν decays [62][63][64], from which we obtain the 1S inputs, use the leading log approximation for a Q (µ). This leads to an enhancement of the extracted λ 2 by approximately a factor (1 + 13α s /6π) 1.2, which is formally higher order, O(α s /m 2 c,b ). This difference is also covered by the assigned uncertainties in λ 2 .
7 These somewhat arbitrary uncertainties, to be used in fit inputs, should not be confused with the recovered uncertainties from fit results, that determine the uncertainties in predictions of, e.g., R(D ( * ) ). 8 In Ref. [7] we used α s = 0.26 due to the slightly higher scale choice. Both formally and numerically, the difference between these choices only enters at higher order compared to O(1/m 2 c , α s /m c ). 9 Order α 2 s corrections will be included at zero recoil only for F(1) in Sec. V B because 1/m c,b corrections vanish there. While at that order one should consider also second-order corrections in the 1S expansion, these would multiply ε c,b and therefore vanish in F(1).
Differentiating Eq. (4.11) at w = 1, one may relate ρ 2 and c to ρ 2 * and c * . In the fits below, we keep ρ 2 * and c * as free parameters. This differs from the approach of Ref. [32] (see also Ref. [7]), in which one expands the B → D form factor G(w) with respect to z * (using the same w + = −1 branch point), and then applies dispersive bounds to constrain the curvaturec * and the slopeρ 2 * (and higher coefficients) to lie in an elongated ellipsoidal region. Further fixingc * to the central value-the major axis-of the allowed region yields [32] with V 21 57. and V 20 7.5. As we discuss further in Sec. VI F below, this approach leads to fit biases when applied to current data. Because G(w) = ξ(w) ĥ + (w) − ρ Dĥ− (w) , one may relate the coefficients of Eqs. (4.11) and (4.13) directly, ± (w 0 ) can be expanded to arbitrary order in HQET as desired. These relations allow the fitted ρ 2 * and c * parameters to be compared to the dispersive bounds forρ 2 * andc * (see Sec. VI F).

C. Sub(sub)leading Isgur-Wise functions
We approximate the subleading Isgur-Wise functions (as in Ref. [7] and elsewhere), aŝ parameters, given the precision of the available data, we treat these functions as constantŝ The relevant parameters for the residual chiral expansion are shown in Table III. Applying the 1S scheme, the full set of Isgur-Wise parameters in our parametrization of the form factors are The SM differential rates for B → D ( * ) lν with respect to w have the well-known expressions in the massless lepton limit where r D ( * ) = m D ( * ) /m B and η EW 1.0066 [68] is the electroweak correction (see e.g. Ref. [10] for full expressions including the lepton mass). The form factors in which the form factor ratios are In addition, in which all functions are evaluated at w = 1 and we use the notationL (2) i as in Eq. (3.11).
in which all C X functions are evaluated at w = 1. In R 2 (1), two additional subleading  from the LQCD values. Fits at first-order must therefore consider nuisance parameters for higher-order terms, or consider shape-only fits as in Ref. [7]; see Sec. VI C.  [7], at which order λ 1,2 also formally vanish. However, note Ref. [7] used the nonzero value λ 1 = −0.3 GeV 2 , which amounts to a correction of about 5% in the value ofΛ used here.
In the G αβ → 0 limit, the expressions for the F(1) and G(1) form factors and the ratios R 1 (1) and R 2 (1) become, in which all C X functions are evaluated at w = 1. Here, one finds a downward shift in F(1) towards the LQCD prediction by requiringĉ 0 (1) < 0. However, unlike in the RC expansion, the same downward shift from the second-order power correction enters into both F(1) and G(1)both have the same 2(ε c − ε b ) 2ĉ 0 (1) term-resulting in a large downward shift in G(1). It is again useful to examine the numerical forms for the truncated expressions, In most of our fits, we only fit to the w spectra of Refs. [73] and [74], as there is little information constraining the form factors encoded in the projections of the angular distributions. We combine and unfold the reported results for electrons and muons of Ref. [74] using the provided migration matrices and efficiency corrections. Systematic uncertainties are incorporated into the unfolding procedure using nuisance parameters that act upon the resulting yields to avoid the d'Agostini bias (cf. Eq. (3) in Ref. [75]).   [76], and for the B → D * form factors at w = 1.03, 1.10, 1.17 [31]. The correlations can be found in Table VII of Ref. [76], and in the ancillary files of Ref. [31], respectively.

B. Lattice QCD inputs
For B → D decay, LQCD predictions for the SM form factors f + and f 0 have long been available at and beyond zero recoil [76,77]. Their relationship to h + and h − is Ref. [76] is currently the most precise, and conveniently provides a synthetic dataset at three values of w = 1.0, 1.08 and 1.16, including statistical and systematic correlations. These may be incorporated directly into the combined fits; the values are shown in Table IV ( Table IV. Results for the B → D * form factors beyond zero recoil are also expected soon from HPQCD.

C. Fitting setup and scenarios
To determine the leading and subleading Isgur-Wise functions and |V cb |, we carry out a simultaneous χ 2 fit of the experimental and lattice data (and in some scenarios include constraints from QCDSR). To take into account the uncertainties in m 1S b and δm bc , we introduce both as nuisance parameters into the fit, assuming Gaussian constraints (see Eq. (4.4) for their value and uncertainties). The constraints from LQCD are incorporated into the fit assuming multivariate Gaussian errors. The χ 2 function is numerically minimized and uncertainties are evaluated using the asymptotic approximation by scanning the χ 2 contour to find the ∆χ 2 = 1 crossing point, providing the 68% confidence level.
As mentioned in Sec. V, in the B → D ( * ) transitions many first-order corrections vanish at zero recoil, such that the HQ expansion is more constrained at w = 1 than beyond zero recoil, prospectively leading to higher sensitivity to second-order contributions. For this reason, when working at O(α s , 1/m c,b ) as in Ref. [7], it is a well-motivated approach to consider information from 'shape-only' fits. In these fits, information concerning the overall one in which no lattice information was used (denoted 'NoL' in Ref. [7]); and one in which beyond zero recoil LQCD predictions for B → D were included (denoted 'L w≥1 ' in Ref. [7]).
At zero recoil, the O(1/m 2 c,b ) corrections are also more constrained. As can be seen in Eqs. (3.12), (3.13) and (3.25), at O(1/m 2 c,b , θ 2 ) all corrections to the matrix elements at zero recoil are determined just by λ 1,2 , while beyond zero recoil effects fromφ 1 can become important. Along similar lines, in Fig. 1 we see that the O(α 2 s ) corrections are relevant at zero recoil. It therefore remains interesting to consider similar 'shape-only' fits, that probe the structure of second-order power corrections at O(θ 2 ) beyond zero recoil, and we therefore also include the correction in Eq. (5.16) in all our fits.
The various fit scenarios and their inputs considered in this work are summarized in Table V. They comprise the following: (i) Our baseline fit scenario uses all published LQCD data-i.e., except for the not-yetpublished B → D * form factors beyond zero recoil [31]-plus all available experimental data from Belle. This fit is denoted 'L D;D * w≥1;=1 ', adapting from the notation in Ref. [7]. (ii) We also perform fit (i), with the relative normalization between the D and D * -in effect the relation between G(1) and F(1)-allowed to float, so that only shape information is used. This fit is denoted as 'L D;D * w≥1;=1 Shape'. (iii) We further consider a fit using all available LQCD data, denoted by 'L D;D * w≥1;≥1 '. (iv) A fit that includes only experimental data, but no LQCD inputs, labelled 'NoL'.
In addition, we consider the same L D;D * w≥1;=1 fit, with the following variations: (v) Using only either 2017 or 2019 B → D * data from Belle, denoted with a '17' or '19' suffix, respectively (vi) Including QCDSR as discussed above, denoted with a '+SR' . This fit provides an interesting contrast to the L D;D * w≥1;≥1 fit. To further characterize the role of the second order power corrections, finally we consider: (viii) A fit at first-order in the HQ expansion, similar to the abovementioned 'L w≥1 +SR' fit of Ref. [7], which we denote here with a 'NLO' suffix.

D. Nested hypothesis tests
Before proceeding to obtain results for our various fit scenarios, we employ a nested hypothesis test (NHT)-based prescription to determine the optimal number of parameters for the L D;D * w≥1;=1 fit scenario. Such a prescription not only allows systematic determination of those parameters to which the current data has sensitivity, but also prevents overfitting. The optimal parameter set obtained through this prescription depends on the precision of the available experimental data, such that the prescription permits systematic improvements as future data becomes available.
We use here a variation of the prescription developed in Ref. [79]. The core idea of an NHT is to test a N -parameter fit hypothesis versus alternative fit hypotheses that use one additional parameter. The difference in χ 2 , provides a convenient test statistic, because it is distributed as a χ 2 in the large N limit [80] with a single degree of freedom. We choose ∆χ 2 = 1 as the decision boundary: the (N + 1)parameter hypothesis is then rejected in favor of the N -parameter fit at 68% CL.
As in Ref. [79], we apply the NHT starting from a suitably small initial number of parameters. In this case, based on the parameters entering at zero-recoil (see Sec. V B) we pick all the HQ mass parameters, the leading Isgur-Wise parameters, andη(1). Thus, the initial parameters are |V cb | ; m 1S b , δm bc , ρ 1 , λ 2 ; ρ 2 * , c * ;η(1) . (6.5) We then incrementally add all combinations of the remaining seven candidate parameterŝ one by one. This generates a 'graph' of fit hypotheses, with each node of the graph representing a possible set of fit parameters, and each edge denoting the addition of one of the candidate parameters. Over the graph, we identify a 'terminating node'-a parameter set-as a fit hypothesis that is preferred over all hypotheses that nest it. In order to avoid runaways in fit parameters, we constrainφ 1 (1),β 2 (1), andβ 3 (1) to be at most O(1) (in practice less than approximately 9.) in a terminating node. We further require that no two parameters are more than approximately 95% correlated, in order to avoid flat directions and consequent overfitting and/or non-Gaussian uncertainties. The terminating node with the fewest parameters (and hence the largest number of degrees of freedom) and lowest χ 2 is then selected as the optimal fit.
To characterize the behavior of the selected fit hypothesis S1, in Fig. 2(a) we show the various form factor ratios R 1,2,0 (w), along with the leading-order Isgur-Wise function ξ(w), and the form factors F(w) and G(w). The uncertainties in all the form factor ratios are well controlled. The small uncertainties in F(w) and G(w) are directly determined by the precision of the LQCD and experimental data. For comparison, we also show in Fig. 2(b) the same ratios and form factors for the S3 hypothesis, which has the lowest χ 2 of those fits with 30 degrees of freedom. The S3 fit results exhibit slightly larger uncertainties, while F(w) and G(w) remain almost entirely unchanged, as expected, and ξ(w) deviates from S1 only very slightly at high recoil. Most notable is an overall downward shift in R 2 , and a small disagreement in R 0 at high recoil, both at the 1σ level or so (depending on correlations).  the corresponding small variations in ξ(w) and R 0 (w), such that other physical observables hardly change.
In Fig. 3(a) we show the R(D ( * ) ) predictions for the S1 and S3 scenarios, finding good agreement. In Fig. 3(b) we show the R(D ( * ) ) predictions for the other six terminating nodes: The R(D ( * ) ) predictions are very similar between all hypotheses, providing good evidence that the truncation of the fit parameters determined by our NHT prescription has not introduced a model dependence associated with the choice of parameters into the fit.

E. Fit results
Using the selected S1 fit hypothesis for the L D;D * w≥1;=1 scenario as our baseline, in Table VII we present the fit results for the various scenarios discussed in Sec. VI C and summarized in Table V. In Table VIII we show for each scenario the corresponding recovered parameters: R(D), R(D * ), and their correlation; F(1) and G(1) and their correlation; and the zerorecoil values for the form-factor ratios R 1 (1), R 2 (1), and R 0 (1). Figure 4 shows the fitted experimental and LQCD data, and the predicted differential spectra for B → D ( * ) τ ν as a function of w for the baseline fit. We note the fitted value forη(1) in the L D;D * w≥1;=1 scenario is in excellent agreement with Eq.  Table VIII) are F(1) = 0.938 (22) and G(1) = 1.055 (11), whose uncertainties are similarly larger but remain compatible with the LQCD data. We see from this that the differential shape information has mild constraining power on parameters entering the zero recoil predictions, such that the tension of the first-order prediction F(1) NLO HQET 0.966 (2) (see Eq. (5.12)) with the LQCD prediction is relaxed.
The NoL scenario fit also results in larger uncertainties, as expected, but is compatible with the baseline fit. Put in other words, the included LQCD data is in agreement with the experimental data, in the context of an RC expansion-based parametrization. By contrast, the L D;D * w≥1;≥1 scenario, which uses LQCD predictions for B → D * beyond zero recoil, produces a fit of notably poorer quality, due to known tensions between the LQCD beyond zero-recoil B → D * lν predictions [31] and experimental measurements. However, including just the beyond zero-recoil LQCD data for h    (2) 3.40 (2) 3.40 (2) 3.40 (2) 3.41 (2) 3.41 (2) 3.41 (2) 3.41(2)  by about 1σ but with notably larger uncertainties. We note in Fig. 5 scenario is in excellent agreement with the L D;D * w≥1;=1 . While the parameters of the L D;D * w≥1;=1 NLO fit scenario appear naively compatible with the baseline results, the zero-recoil slope and curvature are significantly different. We see in    Between L D;D * w≥1;=1 17 and L D;D * w≥1;=1 19 we further note significant differences in the slope and curvature, suggesting a mild tension between these two datasets. We explore the implications of this further in Sec. VI F below.

F. Biases and the major axis of doom
The astute reader will have noted that our correlated R(D ( * ) ) predictions from the L D;D * two sources of external biases that are mainly responsible for this shift.
The first of these is a so-called major-axis approximation introduced in Ref. [32], which is a core feature of the CLN parametrization. In Ref. [32], the application of dispersive bounds from unitarity constraints to the B → D form factor G(w) was shown to constrain the allowed region in theρ 2 * −c * plane (slope and curvature, defined in Eq. (4.13)) in the form of two elongated overlapping ellipses for the J P = 0 − and 0 + currents, respectively. QCDSR results were applied to the first-order HQET corrections, in order to relate bounds on the J P = 0 − current to G(w). These ellipses, which incorporated also estimates of theoretical uncertainties in the first-order corrections, are reproduced in Fig. 6 in blue. 11 Ref. [32] approximated these allowed regions simply by the major axis of the most constraining ellipse (perhaps because the size of the minor axes of these ellipses were far smaller than the experimental uncertainties inc * at the time), shown by the purple dashed line in Fig. 6.
This imposes the relationship betweenc * andρ 2 * in Eq. (4.14), leading to a polynomial form G(w)/G(w 0 ) = 1 − 8a 2ρ2 * z * + (57.ρ 2 * − 7.5)z 2 * + . . ., and, after application of HQET relations at O(α s , 1/m c,b ), to similar polynomial forms in z * for h A 1 (w)/h A 1 (w 0 ). The CLN parametrization, and all parametrizations derived from it, implicitly apply this constraint on theρ 2 * −c * plane. The experimental data and LQCD predictions have reached a level of precision, however, such that the size of theρ 2 * −c * allowed region recovered from fits is now comparable to the minor axes. To see this, we show in Fig. 6 the recovered 68% and 95% CLs for the L D;D * w≥1;=1 , L D;D * w≥1;=1 Shape, and L D;D * w≥1;=1 17 scenarios, by red, orange and green ellipses, respectively. Constrainingρ 2 * andc * to the major axis is barely compatible with these fits at 95% CL. Thus, imposing the CLN constraint in Eq. (4.14) introduces fit biases into the analysis of current data.
To demonstrate this explicitly, we show in Fig. 7(a) the recovered R(D) -R(D * ) CLs arising from applying the CLN constraint (4.14) to the L D;D * w≥1;=1 17 scenario (blue ellipse) versus the L D;D * w≥1;=1 17 scenario without such a constraint (gray ellipse). We do the same for the L D;D * w≥1;=1 scenario (red ellipse), i.e., using all Belle data, versus the L D;D * w≥1;=1 scenario without such a constraint (orange ellipse). One observes a significant shift in R(D), commensurate with a bias introduced into G(w): R(D * ) remains unaffected because the parameters entering the first-order power corrections may compensate for the bias when translated to h A i and h V . In Fig. 7 Because of the tension in the R(D * ) predictions from fits using either the Belle 2017 or 2019 dataset, we adopt a scale factor for its uncertainty, χ 2 for 2 experiments [81], in order to account for the differences between the two datasets. From the results in Table VIII, this leads to the R(D * ) prediction R(D * ) = 0.249 (3) , (6.8) in which the scale factor is 2.6.

G. Vanishing chromomagnetic limit fits
Applying the VC limit to the fit scenarios in Table V instead of the RC expansion, we find poor fits for the L D;D * w≥1;=1 scenario and for all its variations that include zero-recoil normalization constraints. Typically, the χ 2 /ndf corresponds to a p value of less than one percent. This is caused by the tensions in the predicted value of F(1) from the VC expansion versus LQCD data: to describe the experimental spectra at nonzero recoil, the fit parameterĉ 0 (1) is pushed to small values, which result in F(1) 0.96. This is far from the LQCD constraint F(1) LQCD = 0.906 (13) [17], yielding a large contribution to the fit χ 2 . This behavior also matches the approximate expectations discussed in Sec. V C: The zero-recoil structure of the first-and second-order power corrections in the VC limit appears inconsistent with the LQCD data (5.9) and the recovered ratios R 1,2 (1) from first-order fits. The recovered values for F(1) and G(1) for the L D;D * w≥1;=1 scenario are shown in Fig. 8(a). We next consider the L D;D * w≥1;=1 Shape scenario, that relaxes the zero-recoil normalization constraints. In this scenario, the VC limit parametrization achieves excellent fits. To avoid overfitting, we again apply our NHT prescription, considering all combinations of the candidate parameter setη(1),η (1),φ 0 (1),ê 3 (1),ê 3 (1) andĉ 0 (1). The prescription identifies three terminating nodes. Of these, two are the same as the third under the approximate replacements ∼ ε cê3 (1) →η(1) or ∼ ε cê 3 (1) →η (1), matching our expectation in Sec. III F thatη can reabsorb ε cê3 as in Eq.  Table IX, which has χ 2 = 27.1 for 29 degrees of freedom.
The fitted value for |V cb | is in good agreement with the L D;D * w≥1;=1 result for the RC expansion, which must be the case as F(1) and G(1) are constrained to the LQCD data. The corresponding zero recoil slope and curvature parameters are ρ 2 = 1.20(3) and c = 2.10(15), 38.98(68) 1.055 (7)    which are in moderate tension with those for the L D;D * w≥1;=1 Shape in the RC expansion. A similar difference arises inη (1), which leads to a larger R 2 (1) and a smaller R 0 (1) as in Eqs. (5.17). One sees respective up and down shifts in these form-factor ratios over the entire w range, as shown in Fig. 9. While R(D) is mainly determined by lattice data and is unchanged versus the RC expansion fits, these shifts in R 2,0 result in a significantly smaller R(D * ): One finds R(D) = 0.290(4) and R(D * ) = 0.246(1) with correlation 0.54. The corresponding CL is shown in Fig. 8(b) and compared to the RC baseline fit.
More concerning than the shift in R(D * ), however, is that the VC limit L D;D * w≥1;=1 Shape fits have no sensitivity toĉ 0 (1), which solely determines the second-order power correction to F(1) in the VC limit (see Eq. (5.17b)). This insensitivity arises by construction: because  (2) and G(1) = 1.050 (5). Therefore the optimal VC L D;D * w≥1;=1 Shape fit does not address the tension with LQCD predictions for F(1). While O(1/m 3 c ) corrections may be of percent size, it seems unlikely that the third-order VC limit corrections could resolve the remaining tension in F(1) at the 5% level. Therefore, while the VC limit parametrization can describe the shape of the B → D ( * ) ν spectra, it is unlikely to be able to provide a full description of the data.

H. Branching ratios, forward-backward asymmetries, and polarizations
With our fits we can produce precise predictions for several additional observables. We quote predictions here based on our L D;D * w≥1;=1 baseline scenario in the RC expansion, unless stated otherwise. First, for the D -D * ratio we find R e,µ D/D * = 0.417 (12) and R τ D/D * = 0.479 (8) . The light lepton value R e,µ D/D * can be compared to the world averages of Ref. [5]. Averaging the branching fractions from both B 0 and B + decays assuming isospin (including a correction for their relative lifetimes), one obtains The R τ D/D * predictions for the various fit scenarios in the RC expansion are compared to this value in Fig. 10(b), all showing good agreement with the world average. In addition, we can also predict the difference ∆A FB and sum A FB ,

The forward-backward asymmetry is defined by
To compare to the experimental value of ∆A FB from Ref. [74], one needs to also include a small phase-space cut q 2 > 0.08 GeV 2 , for which the quoted measurement is not corrected.

VII. SUMMARY
We developed a supplemental power counting for HQET, based on counting insertions of the transverse residual momentum, / D ⊥ , within HQET correlators: the residual chiral expansion. We conjectured that higher-order terms within this power counting may be suppressed, and showed how this leads to a dramatic simplification of the second-order power corrections in HQET, when truncating at O(θ 2 ) in the RC expansion. In doing so, we presented a review of the formal elements of the general HQ expansion, that are required when working at second order and beyond. Though these formal developments are not new per se, we are unaware of a self-contained and self-consistent presentation of these elements in the literature. Whether these errors also affect the additional terms that arise when including all secondorder corrections remains to be checked.
Based on the RC expansion results, we developed a form factor parametrization, applying the 1S short distance mass scheme that is self-consistent at second order in the HQ expansion. These results are encoded in the Hammer library [26,27]. We showed that the resulting zero-recoil predictions for B → D ( * ) form factors, G(1) and F(1), are in good agreement with zero recoil LQCD data, in particular resolving the prior tension of the LQCD data with the first-order prediction for F(1).
Confronting our parametrization of the form factors with experimental and LQCD data, we identified optimal parameter sets for the RC expansion under a nested hypothesis test prescription. We found that the RC expansion can achieve excellent agreement with the data, with relatively few parameters, and without using any QCDSR or LCSR model-dependent inputs. The VC limit parametrization produces poor fits due to its restricted structure at zero recoil, but using only shape information yields good fits.
We recovered for our best fit second-order power corrections in the RC expansion was crucial to good fits, but the inclusion of second-order power corrections beyond zero recoil was not. This supports the approach used in Ref. [7], which used only the shape of the differential distributions to constrain the subleading Isgur-Wise functions, under the premise that second-order corrections are important only at zero recoil. We found that the simplified linear CLN slope-curvature relation advocated in Ref. [32] is in tension with the data, and leads to large upward biases in R(D) predictions obtained in previous analyses. Our fitting prescription is systematically improvable with more precise future data, that will simultaneously allow further tests of The equation of motion for Q v + applied to Eq. (A3) implies that at leading order which can be thought of as the energy of the brown muck in the heavy quark limit. One may further deduce from Eq. (A2) that the individual matrix elements must take the form, These ensure that the equations of motion for Q v + and Q v + are satisfied at leading order, and the forward-scattering matrix element of the HQET operator is subleading, i.e., With reference to Eq. (2.9), at first order the left side of Eq. (A6) matches onto HQET as where we made use of the contact term (2.14) plus the Q v + equation of motion. The hadronic mass can then be expanded to second order as 13 In our normalization (2.11), It is conventional to define the parameters where d H is a spin combinatoric factor, specific to a given hadronic state. The mass correction then becomes in which we have explicitly restored the scale-dependent renormalization of the chromomagnetic operator. For a pseudoscalar (P ) and vector (V ) meson, which fill a HQ spin symmetry doublet with brown muck spin-parity s π = 1 2 − , the factor d P = 3 and d V = −1, respectively. (For the ground-state baryon, the brown muck is in a s π = 0 + state, and the chromomagnetic parameter λ 2 vanishes.) generated (using the notation of Ref. [15]) from which one may read off the Γ i basis for each of the currents in Eq. (2.15).
The O(α s ) corrections for all five currents were computed in Ref. [39]; explicit expressions are given in Ref. [7]. The vector and axial-vector currents in QCD are (partially) conserved and so not renormalized, but the corresponding HQET currents have nonzero anomalous dimensions, leading to µ-dependence for C V 1 and C A 1 for w = 1. The scalar, pseudoscalar, and tensor currents are renormalized in QCD, and thus C S , C P , and C T 1 are also µ-dependent.
In the MS scheme, the remaining C Γ j (j ≥ 2) are scale independent. reference to Eq. (2.9), these additional contributions read [14,86] δ cΓb →c v with C Γ = ∂C Γ /∂w. In the last line we have written the nonlocal contributions from L 1 alongside the explicit O(α s ) correction from the renormalization of the chromomagnetic operator that arises from the O(1/m c ) operator product in Eq. (2.12).
The first term on the second line may be determined by applying the relations (A5) after inserting the current into a correlator, and expanding to first order. Thus the second line may be rewritten as The proportionality toΛ of the first term is explicit although all the remaining terms in in terms of the standard definitions of subleading Isgur-Wise functionsχ i (w) andη(w) (see below). Note that we retained explicitly the term parametrizing the correction from the kinetic energy operator,χ 1 (cf. Ref. [7]).
If one includes higher-order terms in the RC expansion these relations become more complicated, but Eq. (D9) remains valid at zero recoil.
These results may be applied to the mixed current-Lagrangian matrix elements (F2), that generate corresponding second-order power corrections in Eq.
with two Isgur-Wise functions C 0 (v, v ) = 2c 0 (w) and D 0 (v, v ) = 2d 0 (w). These matrix elements trivially match ontoL to the form factors, according to the notation used in this paper. See footnote 6 for a summary of (apparent) typographical errors in the expressions in Appendix A of Ref. [16].