Model independent analysis of semileptonic $B$ decays to $D^{**}$ for arbitrary new physics

We explore semileptonic $B$ decays to the four lightest excited charm mesons, $D^{**} = \{D_0^*,\, D_1^*,$ $D_1,\, D_2^*\}$, for nonzero charged lepton mass and for all $b\to c \ell\bar\nu$ four-Fermi interactions, including calculation of the ${\cal O}(\Lambda_\text{QCD}/m_{c,b})$ and ${\cal O}(\alpha_s)$ corrections to the heavy quark limit for all form factors. In the heavy quark limit some form factors are suppressed at zero recoil, therefore, the ${\cal O}(\Lambda_\text{QCD}/m_{c,b})$ corrections can be very important. The $D^{**}$ rates exhibit sensitivities to new physics in $b\to c\tau\bar\nu$ mediated decays complementary to the $D$ and $D^*$ modes. Since they are also important backgrounds to $B\to D^{(*)}\tau\bar\nu$, the correct interpretation of future semitauonic $B\to D^{(*)}$ rate measurements requires consistent treatment of both the $D^{**}$ backgrounds and the signals. Our results allow more precise and more reliable calculations of these $B\to D^{**}\ell\bar\nu$ decays, and are systematically improvable by better data on the $e$ and $\mu$ modes. As an example, we show that the $D^{**}$ rates are more sensitive to a new $\bar c\, \sigma_{\mu\nu} b$ tensor interaction than the $D^{(*)}$ rates.


I. INTRODUCTION
The measurements of the ratio of semitauonic B decays compared to the light-lepton final states, show a 4σ tension with the standard model (SM) expectations [1], when the X = D and D * results are combined. Improving our understanding of the B → D ( * ) form factors, required for precision calculations of R(D ( * ) ), has received renewed attention recently [2][3][4][5][6][7][8]. To maximize future sensitivity to new physics (NP) contributions, measuring and understanding contributions for additional semileptonic decay modes mediated by the same parton-level transition is important and necessary, not only as they can give complementary information on the new physics, but also as they constitute backgrounds to the R(D ( * ) ) measurements. In this paper we study B → D * * ν decays, where denotes the four lightest excited charmed mesons, above the {D, D * } ground-state doublet of heavy quark symmetry (HQS) [9][10][11]. (The D * * notation is common in the experimental literature; these are the 1P orbitally excited states in the quark model.) In Ref. [12], SM predictions for R(D * * ) were derived, extending results for massless leptons [13,14], but a comprehensive study of NP effects has not been carried out yet. We include contributions from all possible four-fermion operators (assuming no right-handed neutrinos), and derive the O(Λ QCD /m c,b ) and O(α s ) terms in the expansions of the form factors, going beyond the leading order in the heavy quark expansion. The O(Λ QCD /m c,b ) corrections to the SM matrix elements were calculated a long time ago [13,14], and can be substantial, due to the suppressions of certain leading order matrix elements near zero recoil, imposed by heavy quark symmetry. We show that the available B → D * 2 lν and B → D 1 lν data are in severe tension with the heavy quark limit, that is alleviated by including O(Λ QCD /m c,b ) corrections. Similarly, O(Λ QCD /m c,b ) terms can generate numerically dominant contributions to NP matrix elements as well, and must be included.
Understanding the B → D * * ν decays as precisely as possible, both theoretically and experimentally, is important for several reasons. First, as in B → D ( * ) ν decays, certain form factor combinations are suppressed by the light lepton mass, and thus cannot be constrained by B → D * * lν measurements, but enter unsuppressed in the semitauonic rates. The use of heavy quark effective theory (HQET) [15,16] allows more precise future measurements of B → D * * lν to systematically improve the predictions for B → D * * τν [12], which will provide complementary sensitivity to new physics compared to B → D ( * ) τν. Second, B → D * * ν decays also constitute a significant background to the measurements of R(D ( * ) ), contributing significantly to its uncertainty at present. As certain B → D * * τν modes may exhibit high sensitivity to NP, good theoretical control of these backgrounds is required in order to understand which NP operators may best fit the data. Third, better theoretical control of these modes will help to improve the determinations of the CKM elements |V cb | and |V ub |, both from exclusive and inclusive B decays. The study of these decay modes [17] and their contributions to the Bjorken sum rule [18] will help understanding the composition of the inclusive B → X c ν decay in terms of exclusive modes.
In Sec. II we establish notations and calculate all B → D * * form factors, including the complete set of order O(Λ QCD /m c,b ) and O(α s ) effects. Section III contains expressions for the differential decay rates for arbitrary currents and charged lepton mass. In Sec. IV we study observables that are particularly sensitive to the O(Λ QCD /m c,b ) corrections, and plot effects of a NP ten-  sor interaction which could not be evaluated previously with comparable accuracy. Section V concludes.

A. Spectroscopy
The spectroscopy of the D * * states is important, because in addition to the impact on the kinematics, it also affects the HQET expansion of the form factors [13,14]. The isospin averaged masses and widths for the six lightest charm meson states are shown in Table I. (The level of agreement between the measurements of the masses and widths of the D * * mesons, especially those of D * 0 in the top row of Table I, is presently unsatisfactory [12].) In the heavy quark limit, the spin-parity of the light degrees of freedom, s π l l , is a conserved quantum number, yielding doublets of heavy quark symmetry, as the spin s l is combined with the heavy quark spin [11]. In the quark model, the four D * * states correspond to combining the heavy quark and light quark spins with L = 1 orbital angular momentum. The masses of each heavy quark spin symmetry doublet of hadrons, H ± , with total spin J ± = s l ± 1 2 can be expressed in HQET as  [12].
where n ± = 2J ± + 1 is the number of spin states of each hadron, and the ellipsis denote terms suppressed by more powers of Λ QCD /m Q . The parameterΛ H is the energy of the light degrees of freedom in the m Q → ∞ limit, and plays an important role, as it is related to the semileptonic form factors [13,14]. The λ H 1 and λ H 2 parameters are related to the heavy quark kinetic energy and chromomagnetic energy in the hadron H. We use the notationΛ,Λ , andΛ * for the 1 2 − , 3 2 + , and 1 2 + doublets, respectively, and for the states in each doublet The current data suggest that the m D * 1 − m D * 0 mass splitting is substantially larger than m D * 2 − m D1 . This possibility was not considered in Refs. [13,14], since at that time both of these mass splittings were about 40 MeV. The smallness of MeV was taken as an indication that the chromomagnetic operator matrix elements are suppressed for the four D * * states, in agreement with quark model predictions. We relax this constraint, as in Ref. [12].
While the measured masses of the broad D * 0 and D * 1 states changed substantially over the last twenty years, their 2J + 1 weighted average is essentially unchanged compared to Ref. [14]. We useΛ −Λ = 0.40 GeV and Λ −Λ * 0.04 GeV, and summarize the parameters used in Table II. The uncertainty ofΛ is substantially greater than that ofΛ −Λ andΛ −Λ * ; as we see below, the form factors are less sensitive toΛ than to these splittings.

B. Matrix elements to order ΛQCD/mc,b and αs
It is simplest to calculate theB → D * * matrix elements in HQET using the trace formalism [20][21][22]. It allows a straightforward evaluation of the matrix elements of the five operators in Eq. (3), as well as those of additional operators generated by perturbative corrections (for a review, see Ref. [23]). The O(α s ) corrections are given explicitly in Appendix A of Ref. [2], extracted from Refs. [20,24,25]. The O(α s Λ QCD /m c,b ) corrections are known for the SM currents and would be straightforward to calculate for any new physics, but are neglected below.
The three heavy quark spin symmetry doublets relevant for this paper can be represented by the (super)fields, which have the correct transformation prop-erties under Lorentz and heavy quark symmetries [22], In this paper each representation occurs for only one heavy quark flavor, so for simplicity we denote the components of H v by B v and B * v . The αβ denote a normalized traceless symmetric spin-2 polarization tensor.
Similar to Ref. [26], including Λ QCD /m c,b corrections, theB → D * * matrix elements can be written as where ε c,b = 1/(2m c,b ), Γ is an arbitrary Dirac matrix, and At leading order, in the heavy quark limit, allB → D 1/2 + form factors are determined by one Isgur-Wise function, ζ(w), while allB → D 3/2 + form factors are determined by another, τ (w). (In the notation of Ref. [17], ζ(w) is twice the function τ 1/2 and τ (w) is √ 3 times the function τ 3/2 .) All form factors are viewed as functions of the dimensionless kinematic variable w, instead of Here v = p B /m B and v = p D * * /m D * * are the fourvelocities of the initial and final states. The coefficientsM i andN i contain order Λ QCD /m c,b corrections. These are expressed in terms of subleading Isgur-Wise functions, which arise either from corrections to the HQET Lagrangian or from matching the current operators onto HQET [27][28][29]. Specifically: (i) matrix elements of the O(Λ QCD /m c,b ) current operators give rise to the subleading (dimensionful) Isgur-Wise functions ζ 1 and τ 1,2 ; (ii) matrix elements involving the Λ QCD /m c,b suppressed kinetic energy operator, h v (iD) 2 h v /(2m Q ), are spin symmetry conserving, and generate the functions χ c,b ke and η c,b ke ; (iii) matrix elements involving the chromomagnetic operator in the HQET Lagrangian, (g s /2)h v σ µν G µν h v /(2m Q ), which violates spin symmetry, generate the functions χ c,b 1,2 and η c,b 1,2,3 . The notation for these subleading Isgur-Wise functions are summarized in Table III.
As we consider only B (and not B * ) decays, the Λ QCD /m b corrections from the chromomagnetic operator in the Lagrangian -the terms involving χ b 1,2 (η b 1,2,3 ) -enter in just one linear combination for all D 1/2 + (D 3/2 + ) form factors, as do the heavy quark spin symmetry conserving subleading Isgur-Wise functions, χ c,b ke (η c,b ke ). These can therefore be absorbed into the leading order Isgur-Wise functions via the replacements, These replacements formally introduce O(Λ 2 QCD /m 2 c,b ) errors. Because the Λ QCD /m c,b terms themselves can be dominant near zero recoil, these O(Λ 2 QCD /m 2 c,b ) corrections may in practice be sizable. doublet leading order 1/m current 1/m Lagrangian Hereafter it is understood that the replacements in Eq. (10) are made, unless explicitly noted otherwise. As in Table III, we omit the c superscript from the remaining (unabsorbed) subleading Isgur-Wise functions. We further defineχ 1,2 = χ 1,2 /ζ,ζ 1 = ζ 1 /ζ,η 1,2,3 = η 1,2,3 /τ , andτ 1,2 = τ 1,2 /τ . With these conventions, theM i and N i coefficient functions in Eq. (8) arê The Λ QCD /m b corrections not absorbed into the leading order Isgur-Wise functions occur exclusively in theF b andĜ b linear combinations. (The sign difference between these terms in Eqs. (7a) and (7b) is simply due to defining their knownΛ ( , * ) parts to be positive.) We define form factors in agreement with those in Refs. [13,14] for the SM currents.
and forB → D * 1 , These form factors are dimensionless functions of w, and hereafter we often suppress displaying the w variable. In the heavy quark limit, each of these form factors either vanishes or is determined by the Isgur-Wise function, Unlike the leading order Isgur-Wise function inB → D ( * ) decays, the function ζ(w) is not subject to any symmetry imposed normalization condition. Near zero recoil, only g P , g V1 , and the linear combination g T1 + g T2 contribute to the decay rates without a (w − 1) suppression (in addition to the √ w 2 − 1 phase space factor; see Eqs. (30) and (31) below), so heavy quark symmetry implies that these form factors in the heavy quark limit are suppressed near zero recoil as (w − 1) ζ. This is why the order Λ QCD /m c,b terms have enhanced significance for these decays [13,14]. At order Λ 2 QCD /m 2 c,b and higher, the expansion is expected to behave as suggested by the power counting.
To write the O(Λ QCD /m c,b ) corrections in a compact form, we defineĝ Denotingα s = α s /π we obtain The C i 's encode the α s corrections (and are given in Appendix A in Ref. [2]), and correspond to integrating out the b and c quarks at a common scale, chosen as µ = √ m c m b . For the order Λ QCD /m c,b and α s contributions to thē B → D * 1 form factors we obtain In Eqs. (17) and (18) the expressions for the SM terms (g + , g − , g V1 , g V2 , g V3 , and g A ) agree with Refs. [13,14].
Only four functions of w are needed to parametrize all twelveB → D 1/2 + form factors in Eqs. (17) and (18)

D.B → D 3/2 + form factors
We define the form factors forB → D 3/2 + such that they agree with those in Refs. [13,14] for the SM terms, The form factors f i and k i are again dimensionless functions of w. In the heavy quark limit, each of these form factors either vanishes or is determined by the Isgur-Wise function, τ (w). (This Isgur-Wise function is different from ζ(w), although model calculations can relate the two). The simple parametrizations in Eqs. (19) and (20) yield the slightly complicated relations At zero recoil where w = 1 and v = v , only the f V1 form factor can contribute (as well as the linear combination f T1 + f T2 ), since µ v µ and µν v ν vanish. Heavy quark symmetry implies that f V1 (1) is either of order Λ QCD /m c,b , or its dependence on the leading Isgur-Wise function, τ (w), is suppressed by (w − 1) [17]. This is why, as explained above, the O(Λ QCD /m c,b ) terms are so significant for semileptonic decays to excited charmed mesons.
We define in analogy with Eq. (16), For the order Λ QCD /m c,b and α s contributions to thē B → D 1 form factors, we obtain For the order Λ QCD /m c,b and α s contributions to thē B → D * 2 form factors we obtain In Eqs. (23) and (24) the relations for the SM terms Refs. [13,14]. In this case six functions of w are needed to parametrize all sixteenB → D 3/2 + form factors in Eqs. (23) and (24), including all O(Λ QCD /m c,b ) corrections: τ ,τ 1,2 , andη 1,2,3 .

E. Equations of motion
As in Ref. [2], we can verify the relations stemming from the QCD equations of motion between the (pseudo)scalar and the (axial)vector matrix elements, imply that all first order Λ QCD /m c,b and α s corrections agree in Eq. (25) as they must. Note that the left-hand sides of Eq. (25) contain the running quark masses at the common scale µ. At order α s the results are sensitive to this, and the O(α s ) terms from the expansion of are required to be present for Eq. (25) to be satisfied. As emphasized in Ref. [2], it is probably better to evaluate the scalar and pseudoscalar matrix elements using Eqs. (17), (18), (23), and (24) instead of Eq. (25), because the natural choice for µ is below m b . In the MS scheme fermions do not decouple for µ < m, introducing artificially large corrections in the running, compensated by corresponding spurious terms in the β-function computed without integrating out heavy quarks [30].

III. B → D * * ν RATES FOR GENERIC NP
It is straightforward to calculate the B → D * * ν rates including lepton mass effects and all possible four-fermion operators. The double differential distributions in the SM were written down in Ref. [12], the integrals of which agree with the expressions below. Here we give the single differential distributions as the SM plus generic NP. These expressions can be used with any form factor input to study the SM predictions and possible patterns of NP in R(D * * ). In Appendix A we provide explicit results for theB → D * * ν amplitudes, which may be used in combination with D * * and τ decay amplitudes to derive fully differential distributions of the visible decay products in B → D * * ν, including all interference effects [31].
We consider the following complete basis for the four-Fermi operators mediating b → c ν decay The NP couplings to the quark and lepton currents are denoted byα i j andβ i j , respectively. The lower index ofβ denotes the ν helicity and the lower index ofα is that of the b quark. (This notation is a variation of the conventions chosen in Ref. [31], whence the seemingly superfluous tildes. See App. A for details.) The NP couplingsα i j andβ k l may be complex, andα i jβ i l products are normalized with respect to the SM couplings, such that settingα V Lβ V L = 1 would amount to doubling the coefficient of the (cγ µ P L b)(¯ γ µ P L ν) operator compared to its SM value. The CP conjugate operators forb →c¯ ν are obtained by Hermitian conjugation. The operators involving right-handed neutrinos are included for completeness, but do not interfere with the SM (neglecting m ν suppressed terms).
We define the dimensionless ratios as well aŝ In the rest of this section, for brevity, we suppress indication of absolute value squared for the NP 2 For the narrow D 1 state (s π = 3 2 + ) we find The result for the broad D * 1 state (s π = 1 2 + ) can be obtained from Eqs. (31a) and (31b) via the replacements In the heavy quark limit, for the SM and the tensor coupling contributions, using Eqs. (15) and (21) [32] (see also Ref. [33]), which studied the R(D * * ) predictions for a NP tensor interaction using QCD sum rule predictions for the leading order Isgur-Wise functions. In Appendix B, for completeness we include the analogous expressions for the B → D ( * ) rates. In the SM, the only form factors that enter these rates without additional (w−1) suppressions are g + for D * 0 , g V1 for D * 1 , and f V1 for D 1 . In the infinite mass limit, these form factors must vanish at w = 1 due to heavy quark symmetry. The model independent result derived in the SM is that the O(Λ QCD /m c,b ) leading terms for these form factors at w = 1 are determined by hadron mass splittings and the leading order Isgur-Wise functions [14], where the ellipses denote O[α s (w − 1), ε c,b (w − 1), ε 2 c,b ] and higher order terms. All terms in dΓ D * 2 /dw have an overall (w 2 − 1) 3/2 suppression, so there is no similar constraint for that channel.
In the presence of new physics, the g P (for D * 0 ), g T1,2 (for D * 1 ), and f T1,2 (for D 1 ) form factors also have unsuppressed contributions at w = 1. Of these, g P (1) vanishes in the heavy quark limit, and is determined at order Λ QCD /m c,b by hadron mass splittings and the leading order Isgur-Wise function similar to Eq. (33) abovê The f T1,2 and g T1,2 form factors are proportional to the respective Isgur-Wise functions, ζ and τ , in the heavy quark limit, without any (w − 1) factors, and their contributions to dΓ/dw neither include a (w − 1) suppression. However, in the heavy quark limit, g T1 = −g T2 and f T1 = −f T2 , so only f T1 + f T2 and g T1 + g T2 terms appear in the rates. One sees in Eq. (31b) that the sum of the terms proportional to f 2 T1 , f 2 T2 , and f T1 f T2 vanishes at w = 1 in the heavy quark limit, as it must (and similarly for the g 2 T1 , g 2 T2 , and g T1 g T2 terms). The contributions to the rate proportional to the linear combinations (f T1 −f T2 ) and (g T1 −g T2 ) are suppressed by (w−1), while (35) are again determined at order Λ QCD /m c,b by hadron mass splittings and the leading order Isgur-Wise function, similar to Eqs. (33) and (34).

IV. SOME PREDICTIONS
The data which can be used to constrain the leading and subleading Isgur-Wise functions are the fourB → D * * lν branching ratios, theB → D * 2 lν andB → D * 0 lν spectra measured [34] in four and five bins of q 2 , respectively, and the B → D 3/2 + π rates which are related to the semileptonic rates at q 2 = m 2 π using factorization. The available data, used in our fits, is identical to that collected in Tables V-VII in Ref [12], and are not repeated here. In all fits we perform, we expand the leading-order Isgur-Wise functions in (w − 1) to linear order, In our predictions for theB → D * * τ ν rates and R(D * * ), for simplicity we assume that the l = e, µ rates are given by the SM, and that NP may only enter the τ mode.

A. Fits in the heavy quark limit
To understand the importance of the O(Λ QCD /m c,b ) corrections, it is instructive to attempt fitting the data using the form factor parametrizations in the heavy quark limit. Fitting the dΓ/dw data for D * 2 and D * 0 and the branching fractions of all four states in the heavy quark limit, using Eqs. (15) and (21), yields an unacceptably poor fit (χ 2 /dof = 80./4). The fit is not improved with the addition of quadratic terms to Eqs. (36). Given the branching ratios and some gain in efficiencies in Belle II over Belle, one may expect sensitivity to such quadratic terms with about (5-10)/ab of Belle II data.
To better quantify this tension, we instead fit τ (1) and τ (ζ(1) and ζ ) using the branching ratio and dΓ/dw for the D * 2 (D * 0 ) alone, and theB → D * 2 π rate. This yields good fits, with the results shown in Table IV. From these fits one can predict the ratios, 1.67 ± 0.09 , to be compared to the current experimental values [12,34], R 3/2 0.45 ± 0.07 and R 1/2 2.2 ± 0.7. (The   Table II in Ref. [14].) The severe tension for R 3/2 , about 10σ, is evidence for the presence of large deviations from the heavy quark limit. Adding quadratic terms to Eqs. (36) does not resolve this tension. However, including the O(Λ QCD /m c,b ) corrections yields good fits [12], in alignment with the expectation that these corrections can be large because of the zero recoil suppression of the heavy quark limit terms. Therefore, hereafter we consider only fits including both Λ QCD /m c,b and α s terms.

B. Fits including the subleading terms
To deal with the several unknown subleading Isgur-Wise functions and the limited amount of experimental data, some approximations need to be made. In what is called "Approximation A" in Ref. [14], (w − 1) is treated as a small parameter of order Λ QCD /m c,b , up to second order terms are kept, and chromomagnetic terms are neglected. This reduces the number of subleading Isgur-Wise functions that enter, and allows parametrization of the rates with a few numbers (rather than functions). In "Approximation B" one does not expand in (w − 1), but still neglects chromomagnetic terms. Finally, Ref. [12] introduced an "Approximation C", which treatsτ 1,2 and ζ 1 as constant fit parameters. It also does not neglect the subleading Isgur-Wise functions parametrizing matrix elements of the chromomagnetic term in the Lagrangian, motivated by the fact that the m D * 1 −m D * 0 mass splitting no longer appears much smaller than m D * − m D .
We have updated the Approximation C fit to include the α s corrections neglected in Ref. [12]. This changes the fit parameters shown in Table V only slightly compared to Ref. [12]. For R(D * * ) defined in Eq. (1) we obtain R(D * 0 ) = 0.08 ± 0.03 , R(D * 0 ) = 0.24 ± 0.05 , R(D * 1 ) = 0.05 ± 0.02 , R(D * 1 ) = 0.18 ± 0.02 , R(D 1 ) = 0.10 ± 0.02 , R(D 1 ) = 0.20 ± 0.02 , R(D * 2 ) = 0.07 ± 0.01 , R(D * 2 ) = 0.17 ± 0.01 , (38) and the phase-space constrained ratio is defined as Our results in Eq. (38) are nearly identical with Eq. (38) in Ref. [12]. The ratio for the sum of the four D * * states is (O S ) and pseudoscalar (O P ) matrix elements could be estimated in the past using Eq. (25) [12]. A right-handed vector current, O V + O A , does not help to fit the current data, and O V − O A is the SM operator. Hence, we study here in some detail NP generating the tensor operator, O T , which has best fit for the present data atα T Lβ T L 0.35, assuming it is real (corresponding to C T 0.48 in the conventions of Ref. [35]). This is for primarily for illustration: While exclusively O T cannot be generated by a dimension-6 new physics operator, it can arise from Fierzing interactions generated in viable scenarios.  Figure 2 shows the predicted dΓ/dw spectra both for the SM and forα T Lβ T L 0.35. UnderstandingB → D * * lν is also important because they give some of the largest experimental backgrounds to the measurements of R(D ( * ) ). For example, for Belle, Table IV in Ref. [36] showed that theB → D * * ν shapes and composition are significant backgrounds to the R(D ( * ) ) measurements. For BaBar, Table V in Ref. [37] lists separately the uncertainties due toB → D * * lν andB → D * * τν, which are both significant. The sensitive dependence of theB → D * * ν rates onα T Lβ T L shown in Figs. 1 and 2 illustrate the importance of treating these D * * backgrounds toB → D ( * ) ν consistently, when fitting NP to more precise future R(D ( * ) ) data. Figure 3 shows the possible ranges of R(D 1 ) and

R(D *
2 ), for the two narrow states, allowing any one of the four non-SM interactions, with arbitrary relative phases compared to the SM. The Approximation C best fit result is used, including Λ QCD /m c,b and α s corrections.
Predictions forB s → D * * s ν can be made using the same formalism with the appropriate hadron masses, thē Λ ( , * ) parameters in Table II, and flavor SU (3) symmetry for the form factors. Three of the D * * s states have widths below a few MeV, and may become the first B (s) → D * * (s) τν decay modes measured by LHCb. The R(D * * s ) predictions are numerically close to those for B → D * * ν, but larger uncertainties arise from SU (3) violation and questions remain regarding the interpretation of D * * s as simply orbitally excitedsc states [12].

V. CONCLUSIONS
We derived theB → D * * ν decay rates for arbitrary beyond SM b → c currents and finite charged lepton mass, including all order Λ QCD /m c,b and α s terms in the heavy quark effective theory expansion of the form factors.
With the above results, we have now all ingredients in place to consistently study semileptonic B decays to the six lightest charm mesons,B → D ( * , * * ) ν, for arbitrary new physics and for arbitrary charged lepton masses. These results are being implemented in the Hammer [38] analysis software, which will allow reweighing fully simulated data for fully differential decays to arbitrary new physics, including arbitrary NP contributions for each of the three lepton flavors. This will lead to better control of both theoretical and experimental uncertainties in R(D ( * , * * ) ) measurements, as well as in the determinations of |V cb | and |V ub | from semileptonic B decays. Unlike calculations using model-dependent inputs on the form factors, our predictions are systematically improvable with more data on theB → D * * lν decays to light lepton final states and/or input from lattice QCD. The upcoming much larger data sets at LHCb and Belle II will answer many important questions. In this appendix we provide explicit results for the B →D * * ¯ ν amplitudes themselves. These B →D * * amplitudes correspond to those used in the hammer code [38].
As in Ref. [31], we consider theb →c amplitudes, defining the basis of NP operators to be The lower index of β denotes the ν helicity and the lower index if α is that of the c quark. The correspondence between these coefficients and those defined in Eq. (27) is (equivalent to) Operators for the CP conjugate b → c processes are obtained by Hermitian conjugation. The B →D * * ¯ ν process features only a single physical polar helicity angle, θ , defined in Fig. 4 below. (Helicity angles and momenta are defined with respect to theb →c process. Definitions for the conjugate process follow by replacing all particles with their antiparticles.) With respect to the D * * → DY decay products, one may define φ and φ D as twist angles of the -ν and D-Y decay planes, in accordance with Fig. 4: The combination φ − φ D becomes a physical phase in the presence of D * * decays. Anticipating the need to account for interference effects once D * * and τ decays are included, we therefore write results for the helicity amplitudes including this physical phase combination. We adopt conventions that match the spinor conventions of Ref. [31] for the τ decay amplitudes. For left-handed neutrino amplitudes, this is achieved by including an extra D * * phase factor e iλ D * * φ D for D * * spin λ D , and an additional spinor phase function, h s = 1, e −iφ for s = 1, 2 respectively. (We label the leptonic spin by 1 and 2, to distinguish it from the D * * spins as well as to match the conventions of Ref. [31] for massive spinors.) Under these conventions, the fully differential B → D * * ¯ ν rates may be written as In the following subsections, we write the A λ D * * s helicity amplitudes for theb →c processes, for left-handed neutrinos only. As in Ref. [31] , the amplitudes for righthanded neutrinos are straightforward to include. The CP conjugate b → c processes are obtained via where s is the set of external state quantum numbers, ands denotes their corresponding CP conjugates. For the D * 2 processes, the λ D * 2 = ±2 amplitudes are all zero, and are not included. We label the remaining amplitudes via λ D * 2 = ±, 0.