Form-factor-independent test of lepton universality in semileptonic heavy meson and baryon decays
Stefan Groote, Mikhail A. Ivanov, Jürgen G. Körner, Valery E. Lyubovitskij, Pietro Santorelli, Chien-Thang Tran
aa r X i v : . [ h e p - ph ] F e b MITP/21-005
Form factor independent test of lepton universalityin semileptonic heavy meson and baryon decays
Stefan Groote, ∗ Mikhail A. Ivanov, † J¨urgen. G. K¨orner, ‡ Valery E. Lyubovitskij,
4, 5, 6, 7, § Pietro Santorelli,
8, 9, ¶ and Chien-Thang Tran
8, 9, ∗∗ F¨u¨usika Instituut, Tartu ¨Ulikool, W. Ostwaldi 1, EE-50411 Tartu, Estonia Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia PRISMA+ Cluster of Excellence, Institut f¨ur Physik,Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany Departamento de F´ısica y Centro Cient´ıfico Tecnol´ogico de Valpara´ıso-CCTVal,Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile Department of Physics, Tomsk State University, 634050 Tomsk, Russia Tomsk Polytechnic University, 634050 Tomsk, Russia Dipartimento de Fisica “E. Pancini”, Universit`a di Napoli Federico II,Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio 6, 80126 Napoli, Italy Instituto Nazionale di Fisica Nucleare, Sezione di Napoli, 90126 Napoli, Italy (Dated: February 26, 2021)In the semileptonic decays of heavy mesons and baryons the lepton mass dependence factors outin the quadratic cos θ coefficient of the differential cos θ distribution. We call the correspondingnormalized coefficient the convexity parameter. This observation opens the path to a test of leptonuniversality in semileptonic heavy meson and baryon decays that is independent of form factoreffects. By projecting out the quadratic rate coefficient, dividing out the lepton mass dependentfactor and restricting the phase space integration to the τ lepton phase space, one can defineoptimized partial rates which, in the Standard Model, are the same for all three ( e, µ, τ ) modes ina given semileptonic decay process. We discuss how the identity is spoiled by New Physics effects.We discuss semileptonic heavy meson decays such as ¯ B → D ( ∗ )+ ℓ − ¯ ν ℓ and B − c → J/ψ ( η c ) ℓ − ¯ ν ℓ , andsemileptonic heavy baryon decays such as Λ b → Λ c ℓ − ¯ ν ℓ for each ℓ = e, µ, τ . I. Introduction
Recently there has been an extraordinary amount of experimental and theoretical activity on the analysis of semilep-tonic heavy meson and baryon decays. Starting with the
BABAR papers [1, 2], this upsurge of activity has beenfuelled by possible observations of the violation of lepton flavor universality which, if true, would signal possibleNew Physics (NP) contributions in these decays. The many papers on this subject can be traced back to the abovetwo experimental papers [1, 2]. The present situation concerning the so-called flavor anomalies is summarized inRefs. [3, 4].The present tests of lepton flavor universality suffer from their dependence on the assumed form of the q behaviorof the transition form factors. In the Standard Model (SM) the three semileptonic ( ℓ = e, µ, τ ) modes of a given decayare governed by the same set of form factors. However, due to the kinematical constraint m ℓ ≤ q ≤ ( m − m ) theform factors are probed in different regions of q . Furthermore, the helicity flip factor δ ℓ = m ℓ / q multiplying thehelicity flip contributions provides an additional weight factor depending on q and the lepton mass which differ forthe three modes. All in all, the tests of lepton universality based on rate measurements alone suffer from a complexinterplay of the above two effects which is difficult to control. Ultimately, such tests require the exact knowledge ofthe q behavior of the various transition form factors which is difficult to obtain with certainty (see, e.g., Ref. [5]). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected],[email protected]
Instead, one would prefer tests of lepton universality which are independent of form factor effects such as we areproposing in this paper.It turns out that the above two obstacles to a clean test of lepton universality can be overcome by (i) restricting theanalysis to the phase space of the τ mode, and (ii) choosing angular observables for which the helicity flip contributionscan be factored out. Fortunately, such an observable is provided by the coefficient of the cos θ contribution in thedifferential cos θ distribution.The restriction to a reduced phase space will lead to a loss in rate for the ℓ = µ, e modes which will hopefully becompensated by the 40-fold increase in luminosity provided by the SuperKEKb accelerator at the Belle II detector.For example, the loss in rate through the phase space reduction (Γ tot − Γ red ) / Γ tot is given by O (50)% and O (30)% forthe decays ¯ B → D + ℓ − ¯ ν ℓ and ¯ B → D ∗ + ℓ − ¯ ν ℓ ( ℓ = e, µ ), respectively. Much more demanding in terms of experimentalaccuracy is the fact that the proposed test requires an angular analysis which is not required for the analysis of therate alone.The proposed test of lepton universality will lead to the SM equality of certain optimized (“optd”) rates Γ optd U − L inthe three ( e, µ, τ ) modes, i.e. one has Γ optd U − L ( e ) = Γ optd U − L ( µ ) = Γ optd U − L ( τ ) . (1)While the actual values of the optimized rates in Eq. (1) are form factor dependent, the unit ratio of any of thetwo optimized rates in Eq. (1) or, equivalently, the ratio of the corresponding branching fractions are form factorindependent, i.e. one has R optd ( ℓ, ℓ ′ ) = Γ optd U − L ( ℓ )Γ optd U − L ( ℓ ′ ) = B optd U − L ( ℓ ) B optd U − L ( ℓ ′ ) = 1 . (2)In this way one can test µ/e , τ /µ and τ /e lepton flavor universality independent of form factor effects. NP contributionsdesigned to weaken the τ rate will clearly lead to a violation of the equalities (1) or to the unit ratio of optimizedrates (2). The size of the NP violations can be used to constrain the parameter space of the NP contributions in amodel dependent way. II. Generic differential cos θ distribution We discuss three kinds of semileptonic heavy hadron decays involving the b → c current transition, namely thedecays P (0 − ) → P ′ (0 − ) ℓ ¯ ν , P (0 − ) → V (1 − ) ℓ ¯ ν and B (1 / + ) → B ′ (1 / + ) ℓ ¯ ν . We expand the generic differential( q , cos θ ) distribution for these decays in terms of their helicity structure functions [6–13] d Γ dq d cos θ = 22 S + 1 38 Γ | ~q | q v m (cid:16) A + A cos θ + A cos θ (cid:17) , (3)where S is the spin of the initial hadron, Γ = G F | V cb | m π . (4)is the fundamental rate occurring in the weak three-body decay transitions of particle with mass m and governed bythe weak coupling G F | V cb | , the coefficients A , A , and A are given by A = H U + 2 H L + 2 δ ℓ ( H U + 2 H S ) (5) A = − (cid:16) H P + 4 δ ℓ H SL (cid:17) (6) A = v ( H U − H L ) . (7)In (7) we have introduced the velocity-type parameter v = 1 − m ℓ /q which, when expressed in terms of the helicityflip factor δ ℓ = m ℓ / q , reads v = 1 − δ ℓ . The helicity structure functions H X ( X = U, L... ) are bilinear combinationsof the helicity amplitudes which will be specified later on. Note that the coefficient A factors into the q and leptonmass dependent factor v = 1 − m ℓ /q , and the q dependent helicity structure function combination H U ( q ) − H L ( q ).We mention that instead of expanding the ( q , cos θ ) distribution in terms of helicity structure functions as in (3) onecan also expand the decay distribution in terms of invariant structure functions [14–17].The momentum transfer is denoted by q = p − p , and | ~q | = | ~p | = p Q + Q − / m is the momentum of thedaughter particle in the rest system of the parent particle with Q ± = ( m ± m ) − q . The polar angle of the chargedlepton in the ( l, ν l ) c.m. system relative to the momentum direction of the W off − shell is denoted by cos θ . The cosineof the polar angle θ can be related to the energy E ℓ of the lepton measured in the rest system of the parent particle.The relation reads (see e.g. [6, 9]) cos θ = 2 E ℓ − q (1 + 2 δ ℓ ) | ~q | v (8)with − ≤ cos θ ≤
1. The energy of the off-shell W boson in the rest system of the parent particle is given by q = ( m − m + q ) / (2 m ) . (9)For our purposes, it is more convenient to rewrite the cos θ distribution in terms of the Legendre polynomials. Oneof the advantages of the Legendre representation is that one can project out the coefficient A in a straightforwardway. One has d Γ dq d cos θ = 12 S + 1 Γ | ~q | q v m (cid:26) H tot ( q , m ℓ ) P (cos θ ) + H ( q , m ℓ ) P (cos θ ) + v H ( q ) P (cos θ ) (cid:27) . (10)The coefficient functions H tot , H , and H are given by H tot ( q , m ℓ ) = (1 + δ ℓ )( H U + H L ) + 3 δ ℓ H S , H ( q , m ℓ ) = − (cid:16) H P + 4 δ ℓ H SL (cid:17) , H ( q ) = 12 ( H U − H L ) = 12 H U − L . (11)For the convenience of the reader we list some properties of the Legendre polynomials, P (cos θ ) = 1 , P (cos θ ) = cos θ, P (cos θ ) = 12 (3 cos θ − . (12)The Legendre polynomials satisfy the orthonormality relation Z +1 − dxP m ( x ) P n ( x ) = 22 n + 1 δ mn . (13)It is now straightforward to extract the observables H tot , H and H from Eq. (10) by folding the angular distributionwith the relevant Legendre polynomial. For instance, the differential decay rate is obtained by folding in P (cos θ ), d Γ dq = Z − d cos θ d Γ dq d cos θ P (cos θ ) = 22 S + 1 Γ | ~q | q v m H tot ( q , m ℓ ) . (14)The partial differential rate d Γ U − L /dq can be projected out by folding in P (cos θ ) according to d Γ U − L dq = 10 Z − d cos θ d Γ dq d cos θ P (cos θ ) = 22 S + 1 Γ | ~q | q v m H U − L ( q ) , (15)where the helicity structure function H U − L ( q ) defined in Eq. (11) is a function of q only [see also Refs. [15–17]].The overall factor 10 in Eq. (15) has been chosen such to have the same normalization of Eq. (14) and Eq. (15).In Refs. [10, 11] we have defined a convexity parameter C F ( q , ℓ ) as a measure of the curvature of the cos θ distribution by taking the second derivative of the cos θ distribution. The relation of the convexity parameter to theratio of the two differential rates (14) and (15) is given by C F ( q , ℓ ) = 34 d Γ U − L ( q , ℓ ) /dq d Γ( q , ℓ ) /dq . (16)Also we introduce the average values of the convexity parameter h C ℓF i where the average is taken in the interval m τ ≤ q ≤ ( m − m ) for both µ and τ modes: h C ℓF i = 34 R ( M − M ) m τ dq d Γ U − L ( q , ℓ ) /dq R ( M − M ) m τ dq d Γ( q , ℓ ) /dq , ℓ = µ, τ (17) III. Optimized observables
The possible breaking of lepton flavor universality is usually studied by analyzing the ratios of rates or, equivalently,the ratio of branching ratios for the tau and muon modes. As discussed in the introduction one can remove thedependence on lepton mass effects by introducing two improvements. First, we propose to analyze observables in thecommon phase space region m τ ≤ q ≤ ( m − m ) as has been suggested before in Refs. [18–20]. As an example,in Fig. 1 we show the ( q , cos θ ) phase space for the decay ¯ B → D + + ℓ − + ¯ ν ℓ where the hatched area shows thecommon phase space region m τ ≤ q ≤ ( m − m ) . Second, we reweight suitable observables in which the leptonmass dependence factors out by dropping the overall lepton mass dependent factor. As Eqs. (3) and (10) show,such an observable is available through the coefficient of the quadratic cos θ term in the angular decay distributionproportional to the helicity structure function v H U − L . FIG. 1: ( q , cos θ ) phase space for ¯ B → D + + ℓ − + ¯ ν ℓ . The hatched region shows the ℓ = τ phase space. Based on Eq. (15) we define an optimized differential partial rate by dividing out the factor v . One has d Γ optd U − L ( q , ℓ ) dq = v − d Γ U − L ( q , ℓ ) dq = 2Γ S + 1 | ~q | q m H U − L ( q ) , (18)which by construction does not depend on the lepton mass. In terms of the ratios of branching fractions B optd U − L ( q , ℓ ) = τ d Γ optd U − L ( q , ℓ ) dq , (19)where τ is the lifetime of the respective hadron, Eq. (18) leads to R optd U − L ( q ; ℓ, ℓ ′ ) = B optd U − L ( q , ℓ ) B optd U − L ( q , ℓ ′ ) = 1 . (20)Eq. (20) can be used to test lepton universality on the differential q level by analyzing the ratios of the optimizedbranching fractions R optd U − L ( q ; τ, µ ) = R optd U − L ( q ; τ, e ) = R optd U − L ( q ; µ, e ) in the reduced phase space region m τ ≤ q ≤ q . In practice one would lump the light lepton modes together and concentrate on the ratio of branching fractions R optd U − L ( q ; τ, ( µ + e )) = 1 / q integration over the reduced phase space region one hasΓ optd U − L ( ℓ ) = Z ( m − m ) m τ dq d Γ optd U − L ( q , ℓ ) dq . (21)The proposed test of lepton universality will lead to the equality of the optimized partial rates Γ optd U − L ( ℓ ) in the three( e, µ, τ ) modes, Γ optd U − L ( e ) = Γ optd U − L ( µ ) = Γ optd U − L ( τ ) (22)or, equivalently, to the equality of the three corresponding optimized branching ratios B optd U − L ( e ) = B optd U − L ( µ ) = B optd U − L ( τ ) . (23)The equality of the three optimized rates or optimized branching ratios is independent of form factor effects, whilethe actual value of the optimized rates or optimized branching ratios is form-factor dependent and is thus modeldependent. However, the ratio of the ( e, µ, τ ) branching fractions are predicted to be equal to one independent ofform factor effects, i.e. one has R optd U − L ( ℓ, ℓ ′ ) = B optd U − L ( ℓ ) B optd U − L ( ℓ ′ ) = 1 . (24)Since the ( q , cos θ ) phase space is rectangular the q and cos θ integrations can be interchanged. One can thereforefirst integrate over q and then do the U − L projection rather than first projecting out H U − L and then doing the q integration. This may be of advantage in the experimental analysis.Note that our definition of the optimized rates or branching ratios differs from the one used in Ref. [20]. In orderto differentiate between the two definitions we denote our optimized rates by the label “optd” instead of the label“opt” used in Ref. [20]. The authors of Ref. [20] define an optimized rate ratio R opt = R ( m − m ) m τ d Γ opt ( τ ) /dq R ( m − m ) m τ (1 − δ τ ) (1 + δ τ ) d Γ opt ( µ ) /dq > . (25)The numerator exceeds the denominator because of the additional positive definite scalar contribution in the numer-ator.The idea behind the definition (25) is to define an R-measure R opt which minimizes the propagation of form factorerrors to the optimized R-measure R opt . This goal is, in fact, achieved by the R-measure R opt (25) as shown inRef. [20]. IV. Three classes of semileptonic decays
We now discuss three classes of prominent b → c induced semileptonic decays in turn. We begin with the decay P (0 − ) → P ′ (0 − ) ℓ ¯ ν ℓ . A. P (0 − ) → P ′ (0 − ) ℓ ¯ ν ℓ decay The decays ¯ B → D + ℓ − ¯ ν ℓ and B + c → η c ℓ + ν ℓ belong to this class of decays. The two form factors describing the B → D transition are defined by (see, e.g., Refs. [6, 11]) h P | J Vµ | P i = F + ( q )( p + p ) µ + F − ( q ) q µ (26)The corresponding helicity amplitudes H λ W read H = 2 m | ~q | p q F + ( q ) , H ± = 0 , H t = 1 p q ( m + m − F + ( q ) + q F − ( q )) , (27)where m ± = m ± m .The longitudinal and scalar helicity structure functions are given in terms of the bilinear combinations H L = | H | , H U = | H + | + | H − | = 0 , H S = | H t | . (28)Note that the longitudinal structure function H L is proportional to | ~q | . Since the unpolarized transverse structurefunction H U is zero, one has H U − L ∼ | ~q | . B. P (0 − ) → V (1 − ) ℓ ¯ ν ℓ decay Interesting decays in this class are ¯ B → D ∗ + ℓ − ¯ ν ℓ and B − c → J/ Ψ ℓ − ¯ ν ℓ . We define invariant form factors accordingto the expansion (see, e.g., Refs. [6, 11]) h V | J V − Aµ | P i = ǫ † α m + (cid:18) − g µα P qA ( q ) + P µ P α A + ( q ) + q µ P α A − ( q ) + iε µαP q V ( q ) (cid:19) . (29)One has to specify the helicity amplitudes H λ W λ V by the two helicities λ W and λ V of the off-shell W boson andthe daughter vector meson. The helicity amplitudes are given by H t = m m − | ~q | m p q (cid:18) − A + A + + q m + m − A − (cid:19) ,H ± ± = m − (cid:18) − A ± m m + m − | ~q | V (cid:19) ,H = m − m p q (cid:18) − ( m + m − − q ) A + 4 m m + m − | ~q | A + (cid:19) . (30)The helicity structure functions read H U = | H +1+1 | + | H − − | , H L = | H | , H S = | H t | . (31)Note that H S , H U − L ∼ | ~q | . This is obvious for H S and requires a little algebra for H U − L based on the use ofidentity: | ~q | = ( m + m − − q ) m − m m q . (32) C. B (
12 + ) → B ′ (
12 + ) ℓ ¯ ν ℓ decay One defines invariant form factors by writing (see, e.g., Refs. [10, 12]) h B | J V/Aµ | B i = ¯ u p ( p ) (cid:20) F V/A ( q ) γ µ − i F V/A ( q ) m σ µν q ν + F V/A ( q ) m q µ (cid:21) ( I/γ ) u n ( p ) . (33)The corresponding helicity amplitudes H V/Aλ λ W read H V/A t = p Q ± p q (cid:18) m ∓ F V/A ( q ) ± q m F V/A ( q ) (cid:19) ,H V/A = p Q ∓ p q (cid:18) m ± F V/A ( q ) ± q m F V/A ( q ) (cid:19) ,H V/A = p Q ∓ (cid:18) F V/A ( q ) ± m ± m F V/A ( q ) (cid:19) . (34)From parity or from an explicit calculation one has H V − λ − λ W = + H Vλ λ W and H A − λ − λ W = − H Aλ λ W . The relevanthelicity structure functions read H U = 2 (cid:16) | H V + +1 | + | H A + +1 | (cid:17) , H L = 2 (cid:16) | H V + | + | H A + | (cid:17) , H S = 2 (cid:16) | H V + t | + | H A + t | (cid:17) . (35)With a little algebra one finds H U − L ∼ | ~q | .In all three classes of decays one finds that the helicity structure function combination H U − L = H U − H L isproportional to | ~q | . This leads to a depletion of the partial rate d Γ opt U − L /dq close to the zero recoil q = ( m − m ) where | ~q | = 0. In this paper we do not study the parity odd helicity structure functions H P and H SL , which scale as H P , H SL ∼ | ~q | [9–12]. V. Numerical results
We are now in the position to discuss the numerical values for the optimized observables introduced in our paper.Their values are calculated by using the form factors obtained in the framework of the covariant confined quarkmodel (CCQM). The behavior of all CCQM form factors were found to be quite smooth in the full kinematical rangeof the semileptonic transitions. In fact, they are well represented by a two-parameter representation in terms of adouble-pole parametrization F ( q ) = F (0)1 − as + bs , s = q m . (36)The values of the fit parameters a and b and the q = 0 values of the form factors F (0) are listed in Eq. (34) ofRef. [11] for the B → D ( ∗ ) transition, in Table I of Ref. [21] for the B c → η c and B c → J/ψ transitions, and inEq. (59) of Ref. [10] for the Λ b → Λ c transition. The values of the lepton and hadron masses, their lifetimes as wellas the value of the CKM matrix element V cb are taken from the PDG [22].In Table I we list the average values of the convexity h C ℓF i . For the two transitions B → D and B c → η c we get h C µF i = − . ≃ − / µ -mode. The reasons are that there are no transverse contribution in the P → P ′ transitions and the muon mass is strongly suppressed in comparison with the τ lepton mass ( m µ /m τ ≪ m µ /m τ ≡ h C µF i ≡ − /
2. In case of the τ -mode for the two P → P ′ transitions the average convexityparameter is quite small: − .
26 for the B → D transition and − .
24 for the B c → η c transition. Note that theentries in Table I are form factor dependent. In case of the P → V transitions one can see that the average convexityparameter is again suppressed for the τ modes. Also we notice that h C ℓF i is more suppressed for the P → V transitionsin comparison with the P → P ′ transitions. Finally, for the Λ b → Λ c transition we get the h C ℓF i parameters, whichlie in between the ones for the P → V and P → P ′ transitions. TABLE I: q averages of the convexity parameters h C µF i and h C τF i in the range m τ ≤ q ≤ ( m − m ) .Obs. B → D B c → η c B → D ∗ B c → J/ψ Λ b → Λ c h C µF i − . − . − . − . − . h C τF i − . − . − . − . − . In Fig. 2 we show the behavior of d Γ optd U − L /dq and d Γ U − L /dq = v d Γ optd U − L /dq ( τ -mode) in the region m τ ≤ q ≤ ( m − m ) . In the case of ℓ, µ modes the two above rates coincide each other with high accuracy.The differential rates are largest at threshold q = m τ and go to zero at the zero recoil point q = ( m − m ) withthe characteristic | ~q | dependence. The (form factor dependent) numerical values of the integrated observables aregiven in Table II. We also list their average values for the range 4 GeV ≤ q ≤ ( m − m ) to highlight the fact thatthe differential rates are largest close to threshold where, in the τ -mode, the division by v is potentially problematicfrom the experimental point of view.Next we address the question of how to compare the numerical values calculated in Table II with the outcome ofthe corresponding experimental measurements. We first assume that the number of the produced parent particles isknown which, in the case of produced ¯ B ’s, we will refer to as N ( ¯ B − tags). For example, in e + e − annihilations onthe Υ(4 S ) resonance the bottom mesons are produced in pairs and the identification of a B on one side can be usedas a tag for the ¯ B on the opposite side. In an experimental analysis one counts the number of events of a given decayand relates these to the known number of produced particles given by N ( ¯ B − tags). TABLE II: The optimized partial rate Γ optd U − L in units of 10 − GeV. q B → D B c → η c B → D ∗ B c → J/ψ Λ b − Λ c m τ − . − . − . − . − .
904 GeV − . − . − . − . − . One can then define an experimental branching fraction by writing B ( ¯ B → D + ℓ − ¯ ν ℓ ) = N ( ¯ B → D + ℓ − ¯ ν ℓ ) N ( ¯ B − tags) (37)which can be compared to the theoretical branching fraction B ( ¯ B → D + ℓ − ¯ ν ℓ ) = τ ( ¯ B )Γ tot ( ¯ B → D + ℓ − ¯ ν ℓ ) . (38)In the same way one can define an experimental optimized branching fraction by writing B optd U − L ( ¯ B → D + ℓ − ¯ ν ℓ ) = N optd U − L ( ¯ B → D + ℓ − ¯ ν ℓ ) N ( ¯ B − tags) (39)which, again, can be compared to the corresponding theoretical branching fraction B optd U − L ( ¯ B → D + ℓ − ¯ ν ℓ ) = τ ( ¯ B )Γ optd U − L ( ¯ B → D + ℓ − ¯ ν ℓ ) . (40) q (GeV ) -4-3-2-101 U-2L-tauU-2L-opt B - D q (GeV ) -3-2-101 U-2L-optU-2L-tau B - D * q (GeV ) -4-3-2-101 U-2L-optU-2L-tau B c - η c q (GeV ) -2-101 U-2L-optU-2L-tau B c - J/ ψ q (GeV ) -3-2-101 U-2L-optU-2L-tau Λ b - Λ c FIG. 2: q dependence of the optimized partial rate d Γ optd U − L /dq (solid curve) and d Γ U − L /dq = v d Γ optd U − L /dq ( τ -mode,dashed curve) in units of 10 − GeV − . One then defines optimized rate ratios R optd U − L ( ℓ, ℓ ′ ) by R optd U − L ( ℓ, ℓ ′ ) = B optd U − L ( ℓ ) B optd U − L ( ℓ ′ ) = N optd U − L ( ℓ ) N optd U − L ( ℓ ′ ) = Γ optd U − L ( ℓ )Γ optd U − L ( ℓ ′ ) = 1 , (41)which are predicted to be equal to one.As the ratios (41) show, tagging is not really required when measuring the optimized rate ratio R optd U − L ( ℓ, ℓ ′ ), since thedenominators N ( ¯ B − tags) drop out when taking the ratio (41). This shows that the optimized rate ratio R optd U − L ( ℓ, ℓ ′ )can be experimentally determined even for untagged decays as in the B − c and Λ b decays. VI. New Physics Contributions
Possible NP contributions to the semileptonic decays ¯ B → D ( D ∗ ) τ − ¯ ν τ and ¯ B c → η c ( J/ψ ) τ − ¯ ν τ have been studiedin our papers [21, 23, 24]. The NP transition form factors have been calculated in the full kinematic q range employingagain the covariant confined quark model (CCQM). The modifications of the partial differential rates d Γ U − L ( τ ) /dq from the differential ( q , cos θ ) distributions of the decays ¯ B → Dτ − ¯ ν τ and ¯ B → D ∗ τ − ¯ ν τ are presented in Eqs. (14)and (C1), respectively, in Ref. [23]. One has d Γ U − L (NP) dq = 2Γ S + 1 | ~q | q m (1 − δ τ ) H U − L (NP) (42)where H U − L (NP) = − | V L + V R | | H | + 32 | T L | | H T | ( P − P ′ )-transition , ( | V L | + | V R | )( | H ++ | + | H −− | − | H | ) − V R ( H ++ H −− − | H | ) − | T L | ( | H + T | + | H − T | − | H T | ) ( P − V )-transition . If we recall the relations of helicities with the Lorentz form factors then one gets H P − P ′ U − L + NP = 4 m | ~q | q × n − | V L + V R | F + 32 | T L | q m F T o | H ++ | + | H −− | − | H | = 2 m | ~q | m m q × n − ( P q ) A + 2 h m q V + P q ( P q − q ) A A + i − m | ~q | A o ,H ++ H −− − | H | = m | ~q | m m q × n − ( P q ) A − h m q V − P q ( P q − q ) A A + i − m | ~q | A o , | H + T | + | H − T | − | H T | = 2 m | ~q | m n m q G − ( G + G ) + 2 m h ( m + 3 m − q ) G + ( P q − q ) G i G − m | ~q | m G o . (43)One can see that the differential rate d Γ U − L vanishes as | ~q | at zero recoil. Here V L/R and T L are the complexWilson coefficients governing the NP contributions. It is assumed that NP only affects leptons of the third generation,i.e. the τ lepton mode. Note that the lepton mass dependent factor v also factors out in the NP contributions to the( U − L ) helicity structure function. The parameters of the dipole approximation for the calculated NP form factorsare listed in Eqs. (10) and (11) of Ref. [23] for B − D and B − D ∗ transitions, and in Table I of the Ref. [21] for0 B c − η c and B c − J/ψ transitions. The allowed regions for the NP Wilson coefficients have been found by fitting theexperimental data for the ratios R ( D ( ∗ ) ) by switching on only one of the NP operators at a time.In each allowed region at 2 σ the best-fit value for each NP coupling was found. The best-fit couplings read V L = − . − . i, V R = 0 .
03 + 0 . i,S L = − . − . i, T L = 0 .
38 + 0 . i. (44)We define optimized rates for the NP contributions in the same way as has been done for the SM in Eq. (18). InFig. 3 we plot the SM differential q distributions of the optimized rates d Γ optd U − L /dq together with the corresponding(SM+NP) distributions for the τ -mode. The P → P ′ optimized differential rates are enhanced by the NP V L and V R contributions, and reduced by the NP tensor contribution T L . For the P → V transitions the enhancement due tothe NP tensor contribution T L is quite pronounced over the whole q range. q (GeV ) -5-4-3-2-101 B → D + τ + ν τ U-2L+T L U-2L+V R U-2L (GeV )-15-10-50 B → D * + τ + ν τ U-2L U-2L+T L q (GeV ) -5-4-3-2-101 B c → η c + τ + ν τ U-2L+T L U-2L+V R U-2L (GeV )-10-50 B c → J/ ψ + τ + ν τ U-2L U-2L+T L FIG. 3: P → P ′ ( V ) semileptonic transitions taking into account NP effects for the τ mode. The q dependence of the optimizedpartial rates are shown in units of 10 − GeV − . In the figures we make use of the short hand notation U − L = d Γ optd U − L /dq . The enormous size of the NP tensor contribution to the P → V transitions also shows up in Table III where we listthe integrated optimized rates and the the τ /µ ratio of optimized branching fractions R optd U − L ( τ, µ ) = Γ optd U − L (SM + NP)Γ optd U − L (SM) . (45)The deviations of the ratio of optimized branching fractions from the SM value of 1 is substantial and huge for the P → V transitions. One should be remindful of the fact that the NP optimized τ rates and thereby the ratio ofbranching fractions B optd U − L ( τ, µ ) are form factor dependent.1 TABLE III: Optimized ( U − L ) rates in units of 10 − GeV and rate ratios. NP effects are included in the τ -mode only.Obs. NP-coupling B → Dℓν ℓ B c → η c ℓν ℓ B → D ∗ ℓν ℓ B c → J/ψℓν ℓ Γ optd U − L (SM) − . − . − . − . optd U − L (SM + NP) V L − . − . − . − . V R − . − . − . − . T L − . − . − . − . R optd U − L ( τ, µ ) V L .
32 1 .
31 1 .
32 0 . V R .
42 1 .
42 1 .
29 0 . T L .
75 0 .
77 6 .
11 3 . VII. Some concluding remarks
As the authors of Ref. [15] have emphasized, it is important to also have a look at the ( q , E ℓ ) distribution insemileptonic decays when testing lepton universality. We briefly discuss the merits of using the ( q , E ℓ ) distributionfor form-factor independent tests of lepton universality. One merit of ( q , E ℓ ) distribution is is obviously that cos θ isa derived quantity whereas the lepton energy can be directly measured.The ( q , cos θ ) distribution (3) can be transformed to the ( q , E ℓ ) distribution by making use of the the relation (8)between cos θ and E ℓ . One obtains d Γ dq dE ℓ = 12 S + 1 3 q | ~q | Γ m (cid:18) B ( q , m ℓ ) + B ( q , m ℓ ) (cid:16) E ℓ m (cid:17) + B ( q ) (cid:16) E ℓ m (cid:17)(cid:19) . (46)where the coefficients B ( q , m ℓ ) , B ( q , m ℓ ) and B ( q ) are given by B ( q , m ℓ ) = 14 (cid:16) q (1 + 2 δ ℓ ) ( H U − H L ) + v | ~q | ( H U + 2 H L + 2 δ ℓ ( H U + 2 H S ))+2 q | ~q | (1 + 2 δ ℓ )( H P + 4 δ ℓ H SL ) (cid:17) , (47) B ( q , m ℓ ) = − m (cid:16) q (1 + 2 δ ℓ ) ( H U − H L ) + | ~q | ( H P + 4 δ ℓ H SL ) (cid:17) , (48) B ( q ) = m (cid:16) H U − H L (cid:17) . (49)The ( q , cos θ ) distribution (46) can be seen to be well defined in the limit | ~q | → H P , H SL ∼ | ~q | and H U − H L ∼ | ~q | in all three classes of decays as discussed in Sec. IV.In Fig. 4 we show the ( q , E ℓ ) phase space boundaries of the three ( e, µ, τ ) modes of the semileptonic decay¯ B → D + + ℓ − ¯ ν ℓ . The phase space boundaries are determined by the curves [7, 9] q ± = 1 a (cid:16) b ± p b − ac (cid:17) , (50)where a = m + m ℓ − m E ℓ ,b = m E ℓ ( m − m + m ℓ − m E ℓ ) + m ℓ m ,c = m ℓ (cid:16) ( m − m ) + m ℓ m − ( m − m )2 m E ℓ (cid:17) . FIG. 4: ( q , E ℓ ) phase space for ¯ B → D + + ℓ − + ¯ ν ℓ for the three ( e, µ, τ ) modes. From the relation (8) linking cos θ and E ℓ it is not difficult to see that the coefficients of the cos θ and E ℓ termsare simply related. In particular, as Eq. (49) shows, the coefficient B ( q ) of the quadratic E ℓ term is proportionalto H U − L and, differing from the corresponding coefficient H of the ( q , cos θ ) distribution, does not depend on thelepton mass. A gratifying feature of the ( q , E ℓ ) analysis is the fact that the (model dependent) ratio A ( q ) /A ( q , m ℓ )is quite large over the whole q range [15].Similar to Eq. (15), the second order coefficient B ( q ) = m H U − L ( q ) can be projected from the distribution (46)by folding the distribution with the second order Legendre polynomial expressed in terms of the lepton energy, i.e. P (cos θ ( E ℓ )) = 32 1 | ~q | v (cid:16) E ℓ − E ℓ q (1 + 2 δ ℓ ) + q (1 + 2 δ ℓ ) − | ~q | v (cid:17) . (51)The folding has to be done within the limits ( E + ℓ , E − ℓ ) where (see, e.g., Refs. [7, 9]) E ± ℓ = 12 (cid:16) q (1 + 2 δ ℓ ) ± | ~q | v (cid:17) . (52)The zero and first order coefficients B and B in Eq. (46) are removed by the folding process since Z E + ℓ E ℓ − dE ℓ P (cos θ ( E ℓ )) = Z E + ℓ E ℓ − E ℓ dE ℓ P (cos θ ( E ℓ )) = 0 (53)as can be seen by direct calculation or by considering the orthogonality relations Z E + ℓ E ℓ − dE ℓ P , (cos θ ( E ℓ )) P (cos θ ( E ℓ )) = 0 . (54)Similar to Eq. (15) one finds d Γ U − L dq = 10 E + ℓ Z E − ℓ dE ℓ d Γ dq dE ℓ P (cos θ ( E ℓ )) = 22 S + 1 Γ | ~q | q v m H U − L ( q ) . (55)To be sure, we have done the somewhat lengthy E ℓ integration in Eq. (55) and confirmed the expected result on ther.h.s. of Eq. (55). From here on one would proceed as in Sec. III, i.e. one would proceed by defining an optimized rateby dividing out the lepton mass dependent factor v = (1 − m ℓ /q ) . Differing from the ( q , cos θ ) analysis discussedin the main text the ( q , E ℓ ) phase space is not recctangular which means that the q and E ℓ integrations are notinterchangeable. The projection of the relevant B coefficient Eq.(55) has to be done for each q value, or for each q bin, before q integration. In the τ -mode the range of E ℓ becomes very small close to threshold q = m τ and to thezero recoil point q = ( m − m ) In summary, we have proposed a form-factor independent test of lepton universality for semileptonic B meson, B c meson and Λ b baryon decays by analyzing the two-fold ( q , cos θ ) decay distribution. We have defined optimizedrates for the e, µ, τ modes the ratios of which take the value of 1 in the SM independent of form factor effects. Theform-factor independent test involves a reduced phase space for the light lepton modes which will somewhat reduce3the data sample for the light modes. The requisite angular analysis of the two-fold ( q , cos θ ) distribution will be quitechallenging from the experimental point of view. We have discussed New Physics effects for the τ -mode the inclusionof which will lead to large aberrations from the SM value of 1 for the ratio of the optimized rates. As a by-line wehave also included a discussion of the ( q , E ℓ ) decay distribution as a possible candidate for form factor independenttests of lepton universality.We conclude with two remarks. We have made a wide survey of polarization observables in semileptonic b hadrondecays to find an observable with the requisite property that the helicity flip dependence factors out of the observable.In fact, in semileptonic polarized Λ b decay one can identify the observable v ( H P − H L − ) which posesses the desiredproperty [9, 12]. We want to emphasize that the tests proposed in this paper are necessary but not sufficient tests oflepton universality. All in all, we are looking forward to experimental tests of lepton universality using the optimizedbranching ratios proposed in this paper. Acknowledgments
J.G.K. acknowledges discussions with H.G. Sander on the experimental aspects of the problem. M.A.I. and J.G.K.thank the Heisenberg-Landau Grant for providing support for their collaboration. The research of S.G. was supportedby the European Regional Development Fund under Grant No. TK133. The research of S.G. and M.A.I. was supportedby the PRISMA+ Clusters of Excellence (project No. 2118 and ID 39083149) at the University of Mainz. Bothacknowledge the hospitality of the Institute for Theoretical Physics at the University of Mainz. The research ofV.E.L. was funded by the BMBF (Germany) “Verbundprojekt 05P2018 - Ausbau von ALICE am LHC: Jets undpartonische Struktur von Kernen” (F¨orderkennzeichen: 05P18VTCA1), by ANID PIA/APOYO AFB180002 (Chile)and by FONDECYT (Chile) under Grant No. 1191103. [1] J. Lees et al. (BABAR Collaboration), Phys. Rev. Lett. , 101802 (2012) [arXiv:1205.5442 [hep-ex]].[2] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. D , 072012 (2013) [arXiv:1303.0571 [hep-ex]].[3] Y. S. Amhis et al. (HFAG Collaboration), arXiv:1909.12524 [hep-ex].[4] F. U. Bernlochner, M. F. Sevilla, D. J. Robinson, and G. Wormser, arXiv:2101.08326 [hep-ex].[5] T. D. Cohen, H. Lamm, and R. F. Lebed, Phys. Rev. D , 094503 (2019) [arXiv:1909.10691 [hep-ph]].[6] J. G. K¨orner and G. A. Schuler, Phys. Lett. B , 306 (1989).[7] J. G. K¨orner and G. A. Schuler, Z. Phys. C , 93 (1990).[8] P. Bialas, J. G. K¨orner, M. Kr¨amer, and K. Zalewski, Z. Phys. C , 115 (1993).[9] A. Kadeer, J. G. K¨orner, and U. Moosbrugger, Eur. Phys. J. C , 27 (2009) [hep-ph/0511019].[10] T. Gutsche, M. A. Ivanov, J. G. K¨orner, V. E. Lyubovitskij, P. Santorelli, and N. Habyl, Phys. Rev. D , 074001 (2015);D , 119907(E) (2015) [arXiv:1502.04864 [hep-ph]].[11] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Rev. D , 114022 (2015) [arXiv:1508.02678 [hep-ph]].[12] S. Groote, J. G. K¨orner, and B. Meli´c, Eur. Phys. J. C , 948 (2019) [arXiv:1910.00283 [hep-ph]].[13] E. Di Salvo, F. Fontanelli, and Z. J. Ajaltouni, Int. J. Mod. Phys. A , 1850169 (2018) [arXiv:1804.05592 [hep-ph]].[14] M. Fischer, S. Groote, J. G. K¨orner, and M. C. Mauser, Phys. Rev. D , 054036 (2002) [hep-ph/0101322].[15] N. Penalva, E. Hern´andez, and J. Nieves, Phys. Rev. D , 113007 (2019) [arXiv:1908.02328 [hep-ph]].[16] N. Penalva, E. Hern´andez, and J. Nieves, Phys. Rev. D (2020) 113004 [arXiv:2004.08253 [hep-ph]].[17] N. Penalva, E. Hern´andez, and J. Nieves, Phys. Rev. D , 096016 (2020) [arXiv:2007.12590 [hep-ph]].[18] M. Freytsis, Z. Ligeti, and J. T. Ruderman, Phys. Rev. D , 054018 (2015) [arXiv:1506.08896 [hep-ph]].[19] F. U. Bernlochner and Z. Ligeti, Phys. Rev. D , 014022 (2017) [arXiv:1606.09300 [hep-ph]].[20] G. Isidori and O. Sumensari, Eur. Phys. J. C , 1078 (2020) [arXiv:2007.08481 [hep-ph]].[21] C. T. Tran, M. A. Ivanov, J. G. K¨orner, and P. Santorelli, Phys. Rev. D , 054014 (2018) [arXiv:1801.06927 [hep-ph]].[22] P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. (2020) 083C01.[23] M. A. Ivanov, J. G. K¨orner, and C. T. Tran, Phys. Rev. D94