Continuity and Semileptonic B (s) → D (s) Form Factors
CContinuity and Semileptonic B ( s ) → D ( s ) Form Factors
Andrew Kobach
Physics Department, University of California, San Diego, La Jolla, CA 92093, USA (Dated: August 26, 2020)Small changes in the masses of massive external scattering states should correspond tosmall changes in the non-perturbative parameterization of form factors in quantum fieldtheory, as long as the relevant energy range is not near strong deformations. Here, thedefinition of “small” is investigated and applied to SU (3) breaking in semileptonic B ( s ) → D ( s ) transitions. When unitarity and analyticity are imposed, the differences in the formfactors for semileptonic B → D versus B s → D s decays are found to be within O (1%)over the entire kinematic range, not just at zero recoil, which is consistent with results fromlattice calculations and differs from the expectation using HQET alone. I. INTRODUCTION
Matrix elements of local operators can be decomposed in to a set of non-perturbative scalarfunctions, called form factors. These form factors have served as a way to directly compare exper-imental measurements and non-perturbative theoretical predictions. If the masses of the externalscattering states are varied by an infinitesimal amount, the form factors should also vary by an in-finitesimal amount. This assumption of continuity is common, though not always explicitly stated.For example, the form factors for ¯ B → D +( ∗ ) (cid:96)ν are, to a very good approximation, the sameas those for ¯ B − → D ∗ ) (cid:96)ν , since ( M B − M B ± ) / Λ QCD (cid:28)
1, ( M D − M D ± ) / Λ QCD (cid:28)
1, and α QED / π (cid:28)
1. Generally put, the interactions of a theory have a nominal energy scale Λ, and theform factors for scattering processes in the theory will undergo small changes if the masses of theexternal states change by a small amount compared to Λ.If the masses of any external scattering state change by an amount ε , i.e., M (cid:55)→ M + ε , how largein value can ε be until the form factors begin to change significantly? B ( s ) and D ( s ) mesons aresuitable to this analysis, since ε (cid:39) M B s − M B (cid:39) M D s − M D . For example, consider purely leptonicdecays B ( s ) → (cid:96)ν and D ( s ) → (cid:96)ν , with associated decay constants f B ( s ) and f D ( s ) , respectively. Recent lattice measurements are f B s /f B (cid:39) .
22 and f D s /f D (cid:39) .
18 [1], which is consistent with theestimate that these ratios scale like ε/ Λ QCD , as in chiral perturbation theory [2, 3], and not ε/M ,since the masses of the B and D are not similar, i.e., M D /M B (cid:39) .
35. Turning to the differences inthe form factors for B → D ( ∗ ) (cid:96)ν versus B s → D ( ∗ ) s (cid:96)ν , one might again expect that they do differ byorder ε/ Λ QCD ∼ − M (cid:29) Λ QCD ,heavy particle effective field theory (HQET) predicts that the form factors would differ by the scale m s /M at zero recoil, and m s / Λ QCD away from zero recoil in q space. Recent results from thelattice observe that only the former is true, i.e., that the effect of the valence quark scales like m s /M ∼ O (1%) at zero recoil. In fact, the lattice observes that the form factors associated withthe semileptonic decays B → D versus the analogous ones for B s → D s differ from each other atthe level of O (1%) over the entire kinematic range [4–8]. Currently, there is no explanation as towhy this is in the literature. A quantitative estimate for how form factors change as the masses ofthe external states are varied is the purpose of this work.The discussion will focus on matrix elements between single-particle momentum eigenstates. Non-perturbative decay constants can be thought of as a form factor sampled at a singular point in momentum. a r X i v : . [ h e p - ph ] A ug The transition between single-particle states is particularly simple, since there is only a singlekinematic factor on which the form factors depend. More general scenarios are straightforward toconsider. The expectation of continuity is discussed in Section II, and its effects, combined with theconstraints from analyticity and unitarity, are illustrated for semileptonic B decays in Section III.It is estimated that the form factors for B → D versus B s → D s have, to a good approximation, thesame shape over the entire kinematic range, not just at zero recoil, and differ only in normalizationby a few percent, in agreement with lattice calculations. II.
FORM FACTORS AND CONTINUITY
The matrix element between single particle momentum eigenstates can be decomposed as follows: (cid:104) X f ( p (cid:48) , s (cid:48) ) | O { µ n } ( q ) | X i ( p, s ) (cid:105) = (cid:88) k f k ( q ) T { µ n } k ( p, p (cid:48) , s, s (cid:48) ) , (1)where p − p (cid:48) = q , the sum over k is finite, s and s (cid:48) signify spin degrees of freedom, and O { µ n } is someoperator with a set of Lorentz indices { µ n } inserting momentum q into the system. The functions f k ( q ) are a set of k dimensionless, scalar functions called form factors. The known functions T { µ n } k ( p, p (cid:48) , s, s (cid:48) ) inherit the Lorentz structure of the matrix element and, in general, depend on thespin degrees of freedom. Here, we will be considering the region of parameter space where M i and M f are both always nonzero. The states X i and X f are taken to be well-approximated as beingon-shell, with masses M i and M f , respectively.Considering the expansion of the form factor as a function of q around the point q : f k ( q ) = ∞ (cid:88) n =0 b n ( q − q ) n , (2)the coefficients b n can carry dimensions of mass. As the masses of the external states, M i and M f ,are changed, i.e., M i (cid:55)→ M (cid:48) i and M f (cid:55)→ M (cid:48) f , leaving all other quantum numbers of X i and X f thesame, one will have a new form factor, f (cid:48) k ( q ): f (cid:48) k ( q ) = ∞ (cid:88) n =0 d n ( q − q ) n . (3)A way to relate b n and d n is via dimensionless scalar functions F n that depend on M ( (cid:48) ) i , M ( (cid:48) ) f , andΛ: d n b n = 1 + F n . (4)As defined, F n is equal to zero when M (cid:48) i = M i and M (cid:48) f = M f . Here, we are interested in how F scales with small variances in the masses: M (cid:48) i = M i + ε and M (cid:48) f = M f + δ , while the remainingquantum numbers remain unchanged.There are a few limiting cases to consider. First, if the masses of the external states are much Their values will be bounded due to unitarity, and the exact details regarding such a unitarity bound are discussedfor a specific example in Section. III lower than the nominal scale of the underlying interactions, M (cid:28) Λ, then, in general, F n then canbe approximated as the following, to first order in ε and δ : F n (cid:39) A n ε Λ + B n δ Λ + O (cid:0) ε , δ , εδ (cid:1) . (5)This would be the expected behavior, for example, in chiral perturbation theory, containing only upand down quarks. The assumption of continuity is that this Taylor expansion in Eq. (5) is not onlypossible, but it is also useful, i.e., A n and B n , which can depend on M i , M f , and Λ, are constantsthat are not (cid:29) O (1). This is likely to occur when the range of physical q is not near any strongdeformations in the theory, where the form factor can vary significantly over a small range of q ,e.g., poles, branch cuts, or regions in the theory not near any “states,” but still exhibit significantvariations. On the other hand, if M ∼ Λ, there is only one energy scale, and F n (cid:39) A n ε Λ + B n δ Λ + C n εM + D n δM + O (cid:0) ε , δ , εδ (cid:1) . (6)Again, the assumption of continuity can be made, where none of the above coefficients are partic-ularly large. If M i and M f are greater than Λ, then a different expectation can be made about thescaling of F . The rest of this section will explore this statement.We will proceed without utilizing a Lagrangian, and instead relying solely on scaling argumentsas a function of the masses of the external states. This is similar in spirit to the methods usedin Ref. [9–11], before the discovery of the HQET Lagrangian. The final results in this sectionare, unsurprisingly, consistent with the full effective theory [12]. However, to our knowledge, theparticular argument presented here has not yet been illustrated in the literature, so a detailedderivation is discussed in Appendix A in order to remain self-contained.Making the small differences in masses explicit, let M (cid:48) i = M i + ε , and M (cid:48) f = M f + δ , and theratio the form factors in Eq. (4) scales as follows, where F n (cid:39) a εM i + a δM f + a (cid:0) χ ( v · v (cid:48) ) (cid:1) (cid:18) Λ M i εM i + a Λ M f δM f (cid:19) + χ ( v · v (cid:48) ) (cid:18) ε Λ + a δ Λ (cid:19) (cid:18) a Λ M i + a Λ M f (cid:19) + O (cid:18) Λ M , ε M , δ M (cid:19) . (7) The functions χ ( v · v (cid:48) ) and χ ( v · v (cid:48) ) are zero when v · v (cid:48) = 1, and a − are dimensionless constants,all of which are assumed not to be (cid:29) O (1), according to continuity. Of course, the number andlocation of functions such as χ ( v · v (cid:48) ) and χ ( v · v (cid:48) ) are not unique. Note that Eq. (7) describes thebehavior of the scaling; one cannot further take M → ∞ . The introduction of the factors a and a account for the possibility of radiative corrections to the ratio on the left-hand size of Eq. (7).Importantly, such radiative corrections can only scale like ε/M or δ/M , and not ε/ Λ or δ/ Λ, since,generically speaking, such corrections have a perturbative origin, coming from matching the theeffective theory to the full theory, and do not depend on Λ.Eq. (7) makes the prediction that at zero recoil, the ratio scales like ε/M i and δ/M f , not like ε/ Λ or δ/ Λ. Away from zero recoil, corrections to the ratio can scale like ε/ Λ or δ/ Λ. Because thetransition matrix elements are expected to scale as in Eq. (7) when
M >
Λ, then the form factorsassociated with those matrix elements, as defined in Eq. (1), are expected to obey the same scalingbehavior, i.e., the F in Eq. (4) scales similarly to the left-hand side to Eq. (7) when M > Λ. Non-analytic features are kinds of strong deformations, but one need not make the assertion that the only sourceof high variation in quantum field theories are only due to non-analytic features.
While the prediction that at zero recoil the form factor scales like 1 /M is fairly robust, andunsurprising to those familiar with HQET, one may be tempted to further conclude, based onthese argument alone, that away from zero recoil that the differences in the slope of the formfactors in q at zero recoil would scale like 1 / Λ. However, this is is not what is seen in somephysical systems. For example, calculations on the lattice consistently claim that the B ( s ) → D ( s ) are only O (1%) different over the entire kinematic range [4–8]. What is missing are the constraintsfrom unitarity and analyticity, which are discussed in the following section. III.
SEMILEPTONIC B → D AND B s → D s DECAYS
The matrix element for semileptonic B ( s ) → D ( s ) transitions in the standard model, decomposedinto a finite set of form factors, is: (cid:104) D ( s ) ( p (cid:48) ) | ¯ cγ µ b | B ( s ) ( p ) (cid:105) = ( p + p (cid:48) ) µ f ( s )+ ( q ) + ( p − p (cid:48) ) µ f ( s ) − ( q ) , (8)where q = p − p (cid:48) . For simplicity, we consider that the weak current is conserved, so f ( s ) − does notcontribute to the decay. The constraints on the behavior of these form factors due to analyticityand unitarity were developed by the authors of Refs. [13–17]. Such methods were utilized inRefs. [18–21] to not only constrain the behavior of the form factors that describe the semileptonictransitions B → D ( ∗ ) , but also provide a parameterization of the form factors, relying on theremarkable fact that the entire kinematic range over which these particular transitions occur canbe conformally mapped to a small analytic region within the unit disc. This parameterization hasbeen quite successful in extrapolating experimental data in order to determine the exclusive valueof | V cb | at zero recoil [22, 23]. The parameterization of the form factors developed by the authorsof Refs. [18–21] is, very generically, f k ( z ) = 1 P k ( z ) φ k ( z ) ∞ (cid:88) n =0 a n z n , (9)and ∞ (cid:88) n =0 | a n | ≤ , z ≡ √ w + 1 − √ a √ w + 1 + √ a , w ≡ M i + M f − q M i M f . (10)In Eq. (9), the function φ k ( z ) is known and depends on the specificities of the matrix element, thedetails of the chosen dispersion relation, and the unitarity bound, and P k ( z ) is a Blaschke factorwhere | P k ( z ) | = 1 when | z | = 1, and chosen to be zero at the known location of any poles in therange 0 ≤ q < ( M i + M f ) . The factor a in the definition of z in Eq. (10) is a free parameterassociated with what value of z corresponds to what value of q . Typically, a = 1 is chosen, asdone here, which corresponds to z = 0 being the point of zero recoil, i.e., when q = ( M i − M f ) .Different choices of the value of a can provide nominal improvement of the convergence of the Taylorexpansion in Eq. (9) when fitting to data, but these this choice does not affect the discussion inthis work. Importantly, the left-hand side of Eq. (9) is analytic for | z | <
1, which justifies theTaylor expansion in z on the right-hand side. If considering the process where X i spontaneouslytransitions, (due to a local operator) into X f , then M i > M f , q ≥
0, 0 ≤ z ≤ z max , and this Provided contributions from the B c π in the continuum are negligible [18–21]. expansion converges rapidly if z max ≡ (cid:0) √ M i − (cid:112) M f (cid:1) ( M i − M f ) (cid:28) . (11)To illustrate, if z max (cid:39) .
1, then M f /M i (cid:39) .
27, and this corresponds to ( M i − M f ) / ( M i + M f ) (cid:46) .
33, which is not a small number compared to unity (one may note that this latter ratio of massesis known as the Shifman-Voloshin parameter [9]), which indicates that in this system, the smallnessof the Shifman-Voloshin is not a necessary ingredient, and instead the constraints of unitarity andanalyticity introduce a new small parameter: z max .The form factors f ( s )+ for semileptonic B ( s ) → D ( s ) decays can be parameterized as follows: f + ( z ) = 1 P ( z ) φ ( z ) ∞ (cid:88) n =0 a n z n , ∞ (cid:88) n =0 | a n | ≤ , (12)for B → D , and f s + ( y ) = 1 P ( y ) φ ( y ) ∞ (cid:88) n =0 b n y n , ∞ (cid:88) n =0 | b n | ≤ , (13)for B s → D s . The only differences between Eq. (12) and Eq. (13) is that a n (cid:54) = b n , in general, andone uses the masses of the B and D in Eq. (12), and the masses of the B s and D s in Eq. (13),i.e., y has the same definition as z , but using the B s and D s masses in instead of the ones for B and D . The masses of the poles and the unitarity bound are the same in both cases [18–21].Here, M B (cid:39) .
28 GeV, M D (cid:39) .
87 GeV, M B s (cid:39) .
37 GeV, M D s (cid:39) .
97 GeV, and these Taylorseries converge very quickly, because Eq. (11) is satisfied, where z max (cid:39) .
4% and y max (cid:39) . f + ( z ) (cid:39) P ( z ) φ ( z ) ( a + a z ) , f s + ( y ) (cid:39) P ( y ) φ ( y ) ( b + b y ) , (14)where a , a , b , and b are unknown constants, whose values can be determined by fitting toexperimental or lattice data (the a ’s here have no relation to those in Eq. (7)).The quantum numbers for the B and B s are the same, and likewise for the D and D s , exceptthat their masses differ by O (1%). Furthermore, the values of z or y over which the transitions B → D and B s → D s occur is known to be much smaller than the values of z or y at whichthe theory begins experiencing strong deformations, e.g., the locations of the B c poles or the ¯ BD threshold. If so, one may expect that the result in Eq. (7) is applicable, since M B > M D > Λ QCD .To compare the two form factors, the differences in the definitions of z and y can be bounded fromabove by this ratio: z max y max (cid:39) m s √ M B M D + O (cid:0) m s (cid:1) (15)where m s (cid:39) M B s − M B (cid:39) M D s − M D . Therefore, one can let z (cid:39) y for the sake of the scalingarguments that follow, and the error associated with this is subdominant, being at the sub-percentlevel. Using the scaling argument at zero recoil (where z = 0) in Eq. (7): a b (cid:39) ± O (cid:16) m s M (cid:17) (16)= 1 ± O (1%) , (17)and away from zero recoil: a b (cid:39) ± O (cid:18) m s Λ QCD (cid:19) , (18)= 1 ± O (10% − . (19)This means that the analogous form factors for B → D versus B s → D s should have approximatelythe same shape over the entire physical range of q , and differ in normalization by order of a fewpercent, because the relevant small parameter is z max m s / Λ QCD (cid:39) O (1%).These results can then be used to produce a parameterization of the ratio R s ≡ f s ( q ) /f ( q ),which can be directly used B → D and B s → D s data, either from experiment or the lattice. Aderivation can be found in Appendix B, for which the final result is: R s ≡ f s ( q ) f ( q ) (cid:39) c + c y + O ( y ) , (20)where c − ∼ O ( m s /M ) and c − ∼ O ( m s / Λ QCD ). Because R s encapsulates the appropriatelevel of detail regarding the scaling arguments presented in this work, the linear expansion in theconformal variable on the left-hand side of Eq. (20) is completely generic - it applies equally to all B ( s ) → D ( ∗ )( s ) form factors. The result in Eq. (20) is to be distinguished from the result using HQETalone (which neglects analyticity and unitarity), where, as discussed at the end of Section II, onewould expect this behavior around zero recoil: R s (cid:39) c + ∞ (cid:88) n =1 c n ( w ( q ) − n , (21)where w is defined in Eq. (10), which is essentially an expansion in q . With HQET alone, it isalso expected that c − ∼ O ( m s /M ), however it is naively expected that c n − ∼ O ( m s / Λ QCD )for all n ≥ B → D calculated on thelattice, multiply by the parameterization of R s in Eq. (20) or Eq. (21), and the result can be directlycompared to a lattice calculation for the corresponding form factors for B s → D s . For example, wecan use the interpolated value of f + ( q ) calculated on the lattice in Ref. [4], multiply by R s eitherin Eq. (20) or Eq. (21), vary the c n ’s by their appropriate amounts, and compare with the valuesof f s + ( q ) calculated in the same work. Fig. 1 shows the drastic differences in expectation betweenthe scaling expectation in Eqs. (20) and compared to the HQET-only expectation in Eq. (21). Notshown is the positive consistency with other lattice results [4–8]. Of course, similar arguments holdsfor B ( s ) → D ∗ ( s ) , and only one form factor comparison is shown here for simplicity. It is clear fromlattice data that R s ( q ) scales like the parameterization Eq. (20).The first version of this present work appeared in Oct. 2019, and since then the form factors for B s → D ( ∗ ) s (cid:96)ν have been measured by the LHCb experiment [24], though the data is still dominatedby statistical uncertainties, so comparing the form factors for B → D ∗ compared to B s → D ∗ s is FIG. 1: The value of f s + ( q ) (cid:39) R s ( q ) f + ( q ), where f + is a form factor associated with B → D , for theparameterizations of R s using HQET only in Eq. (21) and those of this work in Eq. (20). The interpolatedfunction of f + ( q ) is used from Bailey et al. (Ref. [4]), its uncertainties are ignored, and the results for R s ( q ) f + ( q ) are shown in light and darker blue, with the parameterizations for R s in Eq. (21) and Eq. (20),varied by their nominal scales, respectively. In this plot specifically, the HQET-only parameterization inEq. (21) is truncated at linear order in the expansion, and c and c are varied between 1 ± .
01 and 1 ± . R s in Eq. (21) and Eq. (20). The results from Ref. [4] for f s + ( q )are also shown with dark blue points with corresponding uncertainties. While the differences between f + ( q )and f s + ( q ) are consistent with HQET alone, it is clear that there is additional relevant information in thesystem, i.e., that from analyticity and unitarity, which constrain the further differences between f + ( q ) and f s + ( q ), and are the primary result of this work. not yet possible at the percent level. Also recently, a state-of-the-art result from the lattice wasreported in Ref. [25], which shows clearly that the form factors for B → D differ from those for B s → D s at the percent level, over the entire kinematic range of the semileptonic decay. Theseresults from experiment and the lattice provide a robust confirmation of the scaling arguments innonperturbative QCD presented here. IV.
SUMMARY
An assumption of continuity is that when the masses of massive external states are varied bya small amount ε , the form factors that parametrize the scattering process should also change bysmall amounts, as long as the process is far away from any strong deformations in the theory.Because the form factors are dimensionless functions, the changes can either scale like ε/M or ε/ Λ,where M is a typical mass scale in the process, and Λ is the energy scale of the interactions.When the masses of the external scattering states are larger than Λ, and both the initial andfinal states have one heavy particle, then corrections away from the M → ∞ limit are constrainedto take on a very particular form, i.e., corrections at zero recoil scale like ε/M and corrections awayfrom zero recoil scale like ε/ Λ in q space. A discussion of this result is presented in Section II andthe derivation is presented in Appendix A, which is consistent with conclusions drawn from explicitcalculations studying chiral symmetry breaking in HQET [26, 27], despite the fact that the resultsin Section II are make no mention of chiral symmetry or a Lagrangian. When combined with thestringent constraints from unitarity and analyticity, as developed by the authors of Refs. [18–21],the result is that away from zero recoil, corrections do indeed scale like ε/ Λ, but in z space, notin q space . When applied to SU (3) breaking in semileptonic B ( s ) → D ( s ) transitions, there isan accidentally small number, z max m s / Λ ∼ O (1%), and this means that the semileptonic B → D and B s → D s should differ by more than O (1%) over the entire kinematic range, not just at zerorecoil. This conclusion is consistent with recent lattice calculations and distinguishable from theHQET-only expectation that the slope in q of the form factors for semileptonic B decays at zerorecoil scale like ε/ Λ QCD , as can be seen in Fig. 1.These results can provide valuable motivation for experimental analyses that combine resultsfrom B → D ∗ and B s → D ∗ s for a measurement of | V cb | , or for lattice calculations to further confirmthis non-trivial expectation in nonperturbative QCD. Acknowledgments
This work is funded in part by the US DOE Office of Nuclear Physics and by the LDRD programat Los Alamos National Laboratory. AK is grateful for useful feedback from Tanmoy Bhattacharya,Wouter Dekens, Benjam´ın Grinstein, Andrew Lytle, John McGreevy, Emanuele Mereghetti, andVarun Vaidya, and for the hospitality of the physics department at UC San Diego.
Appendix A: Derivation of scaling behavior in M → ∞ limit In the limit that M → ∞ , the momentum of an on-shell degree of freedom with mass M is p µ = M v µ , where v µ is the 4-velocity, and v = 1, in order to match with the classical expectation.Such states of heavy degrees of freedom can be prepared with a well-defined velocity and position,which can be understood heuristically as [26, 27]:∆ x · ∆ p ≥ (cid:126) / → ∆ x · ∆ v ≥ O (cid:18) M (cid:19) . (A1)In the limit that Λ /M (cid:28)
1, one can define a state with precise velocity, so the support of thespatial degrees of freedom of the heavy state obey the equation of motion: dv µ dt = 0 . (A2)Transformations that preserve this equation of motion elucidate some of the symmetries of thesystem in question. Eq. (A2) is related to heavy-quark symmetry. Such heavy particles X arenon-relativistic, so these states can be factorized between their spatial degrees of freedom andeverything else, which can be stated in momentum space as: | X (cid:105) = | p (cid:105) ⊗ | other (cid:105) , (A3)where “other” means anything other than the momentum degrees of freedom, which includes spin.Because of this factorization, the momentum degrees of freedom can be treated as those in a freetheory, so, at this point in the discussion, we can speak of creation and annihilation operators actingon the vacuum corresponding to single-particle states. In a relativistic theory, the momentumeigenstates | p (cid:105) are typically normalized as | p (cid:105) = (cid:112) E p a † p | (cid:105) , but in the M → ∞ limit, thisbecomes: | p (cid:105) = √ M a † p | (cid:105) , (cid:104) | a p (cid:48) a † p | (cid:105) = (2 π ) δ ( p (cid:48) − p ) . (A4)A heavy particle travels in a straight line through spacetime in the M → ∞ limit; there are nointeractions in the Hilbert space that can change the particle’s mass or velocity. So, there are onlytransition amplitudes in the forward direction: (cid:104) X f | O | X i (cid:105) = 2 M (cid:104) other f | O | other i (cid:105) , (A5)which follows from Eqs. (A3) and (A4). Because this is a matrix element, the momentum-conservingdelta function has been stripped off, and can be reintroduced when performing the phase spaceintegral. The M appearing in this equation is the physical mass of the particle. Now considerintroducing a second heavy particle in the spectrum with mass M (cid:48) , where the underlying inter-actions do not turn one heavy particle into another. Taking the ratio of single-particle transitionamplitudes for the same external momentum: (cid:104) X f | O | X i (cid:105)(cid:104) X (cid:48) f | O | X (cid:48) i (cid:105) = MM (cid:48) (cid:104) other f | O | other i (cid:105)(cid:104) other (cid:48) f | O | other (cid:48) i (cid:105) , (A6)Note here that the quantum numbers of X i may be different than X f , and likewise with X (cid:48) i and X (cid:48) f , but the support of the momentum degrees of freedom are the same. In the special case wherethe quantum numbers of X i are equal to X (cid:48) i and those of X f are equal to X (cid:48) f (which is the casebeing considered henceforth), then: M (cid:48) M (cid:104) X f | O | X i (cid:105)(cid:104) X (cid:48) f | O | X (cid:48) i (cid:105) = 1 . (A7)This ratio is trivially 1 if M = M (cid:48) . Eq. (A7) is the same expectation as in HQET [9–12, 27, 28].Note that in the limit that both M → ∞ and M (cid:48) → ∞ , that there is no additional hierarchyinduced, e.g., neither M/M (cid:48) nor M (cid:48) /M become large in those limits.Moving away from the M, M (cid:48) → ∞ limit, the interactions can change the velocity of the heavyparticle, so neither the equation of motion in Eq. (A2) nor the factorization in Eq. (A3) hold. Againkeeping the quantum numbers identical between initial and final states, the ratio in Eq. (A7) canonly scale like the following: M (cid:48) M (cid:104) X f | O | X i (cid:105)(cid:104) X (cid:48) f | O | X (cid:48) i (cid:105) ∼ A + B δv ) (cid:18) Λ M − Λ M (cid:48) (cid:19) + O (cid:18) M (cid:19) , (A8)since the ratio on the right-hand side must be 1 in the limit when M, M (cid:48) → ∞ or when M = M (cid:48) .Here, A and B are some constants and δv is a number directly proportional to the change inthe velocity between the initial and final states, i.e., δv need not be small, and when δv is zero,this corresponds to there being no change in the velocity of the initial and final state particles.Continuity assumes that neither A nor B are (cid:29) O (1). Interestingly, if M (cid:48) = M + ε , and B = 0,0then the right-hand side of Eq. (A8) scales like ε/M .Now consider adding two more heavy particles, where there is another interaction, in additionto the one at scale Λ, which can turn heavy particles into other heavy particles, i.e., it allows thetransition X i → X f and X (cid:48) i → X (cid:48) f , as before, but now the masses of the states X i , X f , X (cid:48) i and X (cid:48) f are M i , M f , M (cid:48) i and M (cid:48) f , respectively. Even in the limit that all of these masses are infinite,this new interaction can insert momentum q µ , which can give rise to a change in the velocity of theheavy-particle trajectory, so the equation of motion in Eq. (A2) is not always conserved, i.e., oneis moving away from a region in the phase space protected by heavy-quark symmetry. However,at the special point where q ( (cid:48) )max = v ( (cid:48) ) i ( M ( (cid:48) ) i − M ( (cid:48) ) f ), then v ( (cid:48) ) i = v ( (cid:48) ) f , and the equation of motionin Eq. (A2) maintains its form through the X ( (cid:48) ) i → X ( (cid:48) ) f transitions, and heavy-quark symmetryholds. This point is called zero recoil, at which the ratio of amplitudes should have the same formas Eq. (A7), since the spatial equation of motion is preserved: (cid:113) M (cid:48) i M (cid:48) f (cid:112) M i M f (cid:104) X f | O ( q max ) | X i (cid:105)(cid:104) X (cid:48) f | O ( q (cid:48) max ) | X (cid:48) i (cid:105) = 1 . (A9)The value of this ratio is the one calculated in the HQET literature [9–12]. Moving away from thezero recoil point by a small amount, the ratio scales like the following: (cid:113) M (cid:48) i M (cid:48) f (cid:112) M i M f (cid:104) X f | O ( q ) | X i (cid:105)(cid:104) X (cid:48) f | O ( q ) | X (cid:48) i (cid:105) ∼ C δv (cid:32) M i − M (cid:48) i Λ + M f − M (cid:48) f Λ (cid:33) + O (cid:18) M (cid:19) , (A10)where C is a constant. Now δv can depend non-trivially on q . The right-hand side of Eq. (A10)must be equal to 1 when δv = 0 since it must reduce to Eq. (A9) in that limit, or when M i = M (cid:48) i and M f = M (cid:48) f since the left-hand of Eq. (A10) side becomes 1. Including 1 /M corrections, theratio takes the form: (cid:113) M (cid:48) i M (cid:48) f (cid:112) M i M f (cid:104) X f | O ( q ) | X i (cid:105)(cid:104) X (cid:48) f | O ( q ) | X (cid:48) i (cid:105) ∼ A + B δv ) (cid:18) Λ M i − Λ M (cid:48) i + Λ M f − Λ M (cid:48) f (cid:19) + C δv (cid:18) M i − M (cid:48) i Λ + M f − M (cid:48) f Λ (cid:19) (cid:32) D Λ M i + E Λ M (cid:48) i + F Λ M f + G Λ M (cid:48) f (cid:33) + O (cid:18) M (cid:19) , (A11) since it must reduce to Eq. (A10) as M → ∞ , and the right-hand side of Eq. (A11) must be equalto 1 when M i = M (cid:48) i and M f = M (cid:48) f , since in that case the right-hand side of Eq. (A11) becomes 1.This scaling argument does not rely on the fact that the δv is small; its purpose is to showthat there is a difference in scaling away from zero recoil. One could replace any individual termproportional to δv in the above equation with its own function, which depends on v · v (cid:48) , which goesto zero when v · v (cid:48) = 1. Making the small differences in masses explicit, let M (cid:48) i = M i + ε , and The scaling in Eq. (A10) is consistent with the scaling in the calculation in Refs. [12, 27, 28], where the SU (3) V flavor breaking effects were calculated in HQET for the ratio of the leading-order semileptonic Isgur-Wise functionsfor B → D versus B s → D s . M (cid:48) f = M f + δ , and the ratio of matrix elements in Eq. (A11) can be written as: (cid:104) X f | O ( q ) | X i (cid:105)(cid:104) X (cid:48) f | O ( q ) | X (cid:48) i (cid:105) ∼ a εM i + a δM f + a (cid:0) χ ( v · v (cid:48) ) (cid:1) (cid:18) Λ M i εM i + a Λ M f δM f (cid:19) + χ ( v · v (cid:48) ) (cid:18) ε Λ + a δ Λ (cid:19) (cid:18) a Λ M i + a Λ M f (cid:19) + O (cid:18) Λ M , ε M , δ M (cid:19) , (A12) This is the result quoted in Eq. (7).
Appendix B: Derivation for R s Starting with the expressions for the form factors in Eq. (14) and taking their ratio: R s ≡ f s ( q ) f ( q ) = P ( z ) φ ( z ) ( b + b y ) P ( y ) φ ( y ) ( a + a z ) . (B1)The presence of the P ( z ) φ ( z ) and P ( y ) φ ( y ) terms are due to the non-trivial constraints from uni-tarity, though the justification for truncating the Taylor expansion P ( z ) φ ( z ) f ( z ) and P ( y ) φ ( y ) f ( y )at linear order also applies to the ratio R s : R s = d + d z + d y + O (cid:0) z , y , z max y max (cid:1) , (B2)where d , d , and d are constants, known functions of a , a , b , b , M B , M D , M B s , M D s , andthe M B c pole masses contained in the definition of the Blaschke factor P . One can check via astraightforward calculation that if Eqs. (16) and (18) are true, and if a /a (or equivalently b /b )is not (cid:29)
1, then the Taylor series in Eq. (B2) is justified for the physical values of the mesons(meaning the values of d , d , and d are not (cid:29) d = 1 + δ, where δ ∼ O (cid:16) m s M (cid:17) , (B3)and d , d = 1 + ∆ , where ∆ ∼ O (cid:18) m s Λ QCD (cid:19) , (B4)These results are derived using Eqs. (16) and (18), respectively.To illustrate the result in Eq. (B3) is correct with an explicit choice of the form factors associatedwith B ( s ) → D ( ∗ )( s ) , we can choose the form factor f ( s ) for B ( s ) → D ∗ ( s ) , as defined in Refs. [21, 22],from which one can calculate d explicitly: d = M B s M B b a r (cid:0) √ r s + r s (cid:1) r s (cid:0) √ r (cid:1) (cid:89) i =1 z P i y P i , (B5)where r ≡ M D /M B , r s ≡ M B s /M D s , and z P i and y P i are the locations of the B c poles in z and y space, respectively. Using the results in Eqs. (15) and (16), it is then clear that Eq. (B5) is consistentwith Eq. (B3). The same expectation is also true for all the other form factors in B ( s ) → D ( ∗ )( s ) .The same can be done with d and d , though the calculation is considerably more complicatedand will not be illustrated here for the sake of brevity.2Continuing with the parameterization for R s , using the fact that Eq. (15) is an upper bound onthe differences between z and y across the entire kinematic region, R s (cid:39) c + c y . (B6)Here, y , and not z , is chosen, because the there are values of q in B → D that are beyond thekinematic limits of B s → D s , allowing for a comparison between the form factors in q . Again, c − ∼ O ( m s /M ) and c − ∼ O ( m s / Λ QCD ). This is the result stated in Eq. (20). [1] A. Bazavov et al., B - and D -meson leptonic decay constants from four-flavor lattice QCD , arXiv:1712.09262 .[2] B. Grinstein, E. E. Jenkins, A. V. Manohar, M. J. Savage, and M. B. Wise, Chiral perturbation theoryfor f D(s) / f D and B B(s) / B B , Nucl. Phys.
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