Revisiting the Heavy Vector Quarkonium Leptonic Widths
aa r X i v : . [ h e p - ph ] J un Revisiting the Heavy Vector Quarkonium Leptonic Widths
Guo-Li Wang ∗ and Xing-Gang Wu †
1, Department of Physics, Hebei University, Baoding 071002, China2, Department of Physics, Chongqing University, Chongqing 401331, China
Abstract
We revisit the heavy quarkonium leptonic decays ψ ( nS ) → ℓ + ℓ − and Υ( nS ) → ℓ + ℓ − using theBethe-Salpeter method. The emphasis is on the relativistic correction. For the ψ (1 S − S ) decays,the relativistic effects are 22 +3 − %, 34 +5 − %, 41 +6 − %, 52 +11 − % and 62 +14 − %, respectively. For the Υ(1 S − S ) decays, the relativistic effects are 14 +1 − %, 23 +0 − %, 20 +8 − %, 21 +6 − % and 28 +2 − %, respectively.Thus, the relativistic corrections are large and important in heavy quarkonium leptonic decays,especially for the highly excited charmonium. Our results for Υ( nS ) → ℓ + ℓ − are consistent withthe experimental data. ∗ Electronic address: gl [email protected] † Electronic address: [email protected] . INTRODUCTION As it gives a clean experimental signal, the dilepton annihilation decay of the heavy vectorquarkonium plays an important role in determining the fundamental parameters such asthe strong coupling constant [1, 2], heavy quark masses [1, 3–5], heavy quarkonium decayconstants [2, 6–8], etc. Its decay amplitude is a function of the quarkonium wave function,and this process can be used to test various theories such as the quark potential model, non-relativistic Quantum Chromodynamics (NRQCD), QCD sums rules, lattice QCD, etc. TheStandard Model prediction of the universality of lepton flavor is questioned by the measuredratios R ( D ( ∗ ) ) and R ( K ( ∗ ) ) [9–14], and the quarkonium leptonic decay is another way totest the lepton flavor universality.The vector quarkonium leptonic decays have been studied since a long time [15–21]. Withthe progress in computer science and experimental technology, many advances have beenreported in literature. For example, on can find lattice QCD predictions of the leptonicdecays of the ground-state Υ and its first radial excitation Υ ′ in [22]; Ref. [23] reported thenext-to-leading non-perturbative prediction and Ref. [24] the next-to-leading-log perturba-tive QCD (pQCD) prediction; in Ref. [25], the two-loop QCD correction was computed;Ref. [26] studied the inclusive leptonic decay of Υ up to the next-to-next-to-leading order(NNLO) by including the re-summation of the logarithms (partly) up to the next-to-next-to-leading logarithmic (NNLL) accuracy; the NNNLO corrections have been discussed byvarious groups [27–31]. A pQCD analysis of the Υ(1 S ) leptonic decay up to NNNLO usingthe principle of maximum conformality (PMC) [32–35] was presented in Refs.[36, 37], wherethe renormalization scale ambiguity of the decay width is eliminated with the help of therenormalization group equation.Even though considerable improvements have been made, there are still deviations be-tween the theoretical predictions and the experimental data for the heavy vector quarkoniumleptonic decays. There are two sources which may cause such deviations. The first are theunknown higher order perturbative QCD corrections. By using PMC, the conventionalpQCD convergence of the series can be greatly improved by the elimination of the diver-gent renormalon terms, and a more accurate decay width can be obtained. However, thereare still large errors due to unknown high-order terms [36, 37]. The second source is therelativistic correction, which could be large. However, almost all pQCD predictions are cal-culated using NRQCD, in which the decay constant of quarkonium, or its wave function atthe origin is treated simply in the non-relativistic approximation.2ne may argue that the relativistic correction is small for a heavy quarkonium, sincethe relative velocity among the heavy constituent quarks is small, e.g. v ∼ . − . et al. computed the coefficients of the decay operators for the S heavyquarkonium decay into a leptonic pair and found large relativistic correction [39]; Gonzalez et al. pointed out that large relativistic and QCD corrections of the quarkonium leptonicdecays are necessary to fit the experimental data [7]; Geng et al. studied the B c mesonsemileptonic decays into charmonium and also found that the relativistic corrections arelarge [40], especially for highly excited charmonium states. Moreover, from the experimentalstandpoint, Ref. [41] showed that a careful study of leptonic decays is still needed for highlyexcited charmonium states.In this paper, we focus on the leptonic decays of charmonium and bottomonium, includ-ing their excited states, using the relativistic method. In a previous short letter [42], wepresented a relativistic calculation of the quarkonium decays into e + e − , where the resultsdisagree with the experimental data. As a step forward, we revisit this topic in more de-tail, and include the decays into µ + µ − and τ + τ − as well as the ratios R ττ . We present therelativistic effects in these quarkonium decays, and discuss the universality of lepton flavor.The paper is organized as follows. The general equation of the quarkonium leptonic decaywidth is given in Sec.2. In Sec.3, we give a brief review of the Bethe-Salpeter equation, andits instantaneous version, the Salpeter equation. We then show in Sec.4 in detail how tosolve the full Salpeter equation and obtain the relativistic wave function for a vector meson.The calculation of the decay constant in the relativistic method is given in Sec.5. Finally,in Sec.6, we give the numerical results and a discussion. A summary is presented in Sec.7. II. THE QUARKONIUM LEPTONIC DECAY WIDTH
The leptonic partial decay rate of a vector charmonium or bottomonium nS state V isgiven by Γ V → ℓ + ℓ − = 4 πα em e Q F V M nS × (cid:18) m ℓ M nS (cid:19) s − m ℓ M nS , (1)where α em is the fine structure constant, e Q is the electric charge of the heavy quark Q in units of the electron charge, e Q = +2 / e Q = − / M nS is the mass of the nS state quarkonium, m ℓ is the lepton mass, F V isthe decay constant of the vector meson that is defined by the following matrix element of3he electromagnetic current < | ¯ Qγ µ Q | V ( P, ǫ ) > = F V M nS ǫ µ , (2)where P is the quarkonium momentum, and ǫ is the polarization vector.In the non-relativistic method, the well-known formula for the decay constant is F NRV = r M nS | Ψ V (0) | , (3)where N R means the non-relativistic (NR), and Ψ V (0) is the non-relativistic wave functionevaluated at the origin. In the NR method, there is one radial wave function, and the vectormeson and its corresponding pseudoscalar have the same radial wave function and the samedecay constant. However, in the relativistic method, they have different wave functions anddifferent decay constants, and more than one radial wave function gives a contribution tothe vector meson decay constant.In the relativistic method, the decay constant F V = F ReV is not related to the wavefunction at the origin, but in the full region. In the following, we focus on the calculationof F ReV in the relativistic method.
III. THE BETHE-SALPETER EQUATION AND THE SALPETER EQUATION
In this section, we briefly review the Bethe-Salpeter (BS) equation [43], which is a rela-tivistic dynamic equation describing the two-body bound state, and its instantaneous ver-sion, the Salpeter equation [44]. The BS equation for a meson, which is a bound state of aquark, labelled as 1, and anti-quark, labelled as 2, can be written as [43]( /p − m ) χ P ( q )( /p + m ) = i Z d k (2 π ) V ( P, k, q ) χ P ( k ) , (4)where χ P ( q ) is the relativistic wave function of the meson, V ( P, k, q ) is the interaction kernelbetween the quark and anti-quark, p , p , m , m are the momenta and masses of the quarkand anti-quark, P is the momentum of the meson, q is the relative momentum betweenquark and anti-quark. The momenta p and p satisfy the relations, p = α P + q and p = α P − q , where α = m m + m and α = m m + m . In the case of quarkonium, where m = m , we have α = α = 0 . q into two parts, q µ = q µ k + q µ ⊥ , where q µ k ≡ ( P · q/M ) P µ and q µ ⊥ ≡ q µ − q µ k , M is the mass of the bound state, and we have P = M . Then we have twoLorentz invariant variables, q P = ( P · q ) M and q T = p q P − q = p − q ⊥ . When ~P = 0, thatis in the meson center-of-mass frame, they reduce to the usual components q and | ~q | , and q ⊥ = (0 , ~q ).With this notation, the volume element of the relativistic momentum k can be writtenin an invariant form d k = dk P k T dk T dsdφ , where ds = ( k P q P − k · q ) / ( k T q T ) and φ is theazimuthal angle. Taking the instantaneous approximation in the center-of-mass frame ofthe bound state, the kernel V ( P, k, q ) changes to V ( k ⊥ , q ⊥ , s ). We introduce the three-dimensional wave function Ψ P ( q µ ⊥ ) ≡ i Z dq P π χ ( q µ k , q µ ⊥ ) , (5)and the notation η ( q µ ⊥ ) ≡ Z k T dk T dsdφ (2 π ) V ( k ⊥ , q ⊥ , s )Ψ P ( k µ ⊥ ) . (6)The BS equation Eq. (4) is then rewritten as χ ( q k , q ⊥ ) = S ( p ) η ( q ⊥ ) S ( p ) , (7)where S ( p ) and S ( p ) are propagators of quark 1 and anti-quark 2, respectively, whichcan be decomposed as S i ( p i ) = Λ + i ( q ⊥ )( − i +1 q P + α i M − ω i + iε + Λ − i ( q ⊥ )( − i +1 q P + α i M + ω i − iε . (8)Here, we have defined the constituent quark energy ω i = q m i + q T and the projectionoperators Λ ± i ( q ⊥ ) = ω i h /PM ω i ± ( − i +1 ( m i + /q ⊥ ) i , where i = 1 and 2 for quark and anti-quark, respectively.Using the projection operators, we can divide the wave function into four partsΨ P ( q ⊥ ) = Ψ ++ P ( q ⊥ ) + Ψ + − P ( q ⊥ ) + Ψ − + P ( q ⊥ ) + Ψ −− P ( q ⊥ ) , (9)with the definition Ψ ±± P ( q ⊥ ) ≡ Λ ± ( q ⊥ ) /PM Ψ P ( q ⊥ ) /PM Λ ± ( q ⊥ ). Here Ψ ++ P ( q ⊥ ) and Ψ −− P ( q ⊥ ) arecalled the positive and negative energy wave functions of the quarkonium.After integrating over q P on both sides of Eq. (7) using contour integration, we obtainthe famous Salpeter equation [44]:Ψ P ( q ⊥ ) = Λ +1 ( q ⊥ ) η ( q ⊥ )Λ +2 ( q ⊥ )( M − ω − ω ) − Λ − ( q ⊥ ) η ( q ⊥ )Λ − ( q ⊥ )( M + ω + ω ) . (10)5quivalently, the Salpeter equation can be written as four independent equations using theprojection operators: ( M − ω − ω )Ψ ++ P ( q ⊥ ) = Λ +1 ( q ⊥ ) η ( q ⊥ )Λ +2 ( q ⊥ ) , (11)( M + ω + ω )Ψ −− P ( q ⊥ ) = − Λ − ( q ⊥ ) η ( q ⊥ )Λ − ( q ⊥ ) , (12)Ψ + − P ( q ⊥ ) = 0 , (13)Ψ − + P ( q ⊥ ) = 0 . (14)The normalization condition for the BS wave function reads Z q T dq T π T r (cid:20) Ψ ++ P /PM Ψ ++ P /PM − Ψ −− P /PM Ψ −− P /PM (cid:21) = 2 M . (15)Note that usually in literature, it is not the full Salpeter equation Eq. (10) that issolved (or equivalently, the four Eqs. (11-14)), but only Eq. (11), which involves only thepositive wave function. There is a good reason why such an approximation is made: it isits effective range. The numerical value of M − ω − ω in Eq. (11) is much smaller than of M + ω + ω in Eq. (12), which means that the positive wave function Ψ ++ P ( q ⊥ ) is dominant,and that the contribution of the negative wave function Ψ −− P ( q ⊥ ) can be safely neglected.However, we point out that if only Eq. (11) for Ψ ++ P ( q ⊥ ) is considered, then not only isthe contribution of the negative wave function neglected, but so are the relativistic effectsof these wave functions. The reason is that the number of eigenvalue equations limits thenumber of radial wave functions, and as is shown below, only the four coupled equationsEqs. (11-14) can provide sufficient information to derive a relativistic wave function. IV. RELATIVISTIC WAVE FUNCTION AND THE KERNEL
Although BS or the Salpeter equation is the relativistic dynamic equation describingthe two-body bound state, the equation cannot by itself provide the information about thewave function. This means that we need to provide an explicit form of the relativistickinematic wave function as input, which can be constructed using all allowable Lorentz and γ structures. 6rom literature, we have the familiar form of the non-relativistic wave function for the1 − vector meson, e.g. Ψ P ( ~q ) = ( /P + M ) /ǫψ ( ~q ) , (16)where M , P and ǫ are the mass, momentum and polarization of the vector meson, ~q is therelative momentum between the quark and anti-quark. There is only one unknown wavefunction ψ ( ~q ) in Eq.(16), which can be obtained numerically by solving Eq. (11) or thenon-relativistic Schr¨odinger equation. The relative momentum ~q is related to the relativevelocity ~v between the quark and anti-quark in the meson, ~q = m m m + m ~v . A relativistic wavefunction should depend on the relative velocity ~v or momentum ~q separately, not merely onthe radial part ψ ( ~q ), because the radial part is in fact ψ ( | ~q | ) or equally ψ ( ~q ).To ontain the form of the relativistic wave function, we start from J pc of a meson, because J p or J pc are in any case good quantum numbers, where J is the total angular momentum,and p and c are the parity and the charge conjugate parity of the meson. The paritytransform changes the momentum q = ( q , ~q ) into q ′ = ( q , − ~q ), so for a meson, afterapplying the parity transform, the four-dimensional wave function χ P ( q ) changes to p · γ χ P ′ ( q ′ ) γ , where p is the eigenvalue of parity. The charge conjugate transform changes χ P ( q ) to c · C χ T P ( − q ) C − , where c is the eigenvalue of charge conjugate parity, C = γ γ isthe charge conjugate transform operator, T is the transpose transform. Since the Salpeterequation is instantaneous, the input wave function Ψ P ( q ⊥ ) is also instantaneous, and thegeneral form of the wave function for the 1 − vector meson can be written as [45, 46]Ψ − P ( q ⊥ ) = q ⊥ · ǫ ⊥ (cid:20) ψ ( q ⊥ ) + PM ψ ( q ⊥ ) + q ⊥ M ψ ( q ⊥ ) + P q ⊥ M ψ ( q ⊥ ) (cid:21) + M ǫ ⊥ ψ ( q ⊥ )+ ǫ ⊥ P ψ ( q ⊥ ) + ( q ⊥ ǫ ⊥ − q ⊥ · ǫ ⊥ ) ψ ( q ⊥ ) + 1 M ( P ǫ ⊥ q ⊥ − 6 P q ⊥ · ǫ ⊥ ) ψ ( q ⊥ ) . (17)There are in total 8 radial wave functions ψ i ( q ⊥ ) = ψ i ( | ~q | ) with i = 1 ∼
8, which obviouslycan not be obtained by solving only one equation, e.g. Eq. (11), but can be obtained bysolving the full Salpeter Eqs. (11-14). The above expression does not include the termswith P · q , since the condition of instantaneous interaction is P · q = P · q ⊥ = 0. Thereare also no higher order q ⊥ terms like q ⊥ , q ⊥ , q ⊥ , etc., because the even powers of q ⊥ canbe absorbed into the radial part of ψ i ( q ⊥ ), while the odd powers of q ⊥ can be changed tolower power, for example, q ⊥ ψ ′ i ( q ⊥ ) = q ⊥ ψ i ( q ⊥ ). By the way, if we delete all q ⊥ termsexcept those inside the radial wave functions, then the wave function Eq. (17) reduces to M ǫ ⊥ ψ ( q ⊥ ) + ǫ ⊥ P ψ ( q ⊥ ). If we further set ψ ( q ⊥ ) = − ψ ( q ⊥ ) = ψ ( q ⊥ ), the wave function7educes to the non-relativistic case, e.g. Eq. (16). Thus, the terms with ψ , ψ , ψ , ψ , ψ and ψ in Eq. (17) are all relativistic corrections.When the charge conjugate parity is taken into account, the terms with ψ ( q ⊥ ) and ψ ( q ⊥ )vanish because of the positive charge conjugate parity c = +, and the general instantaneouswave function for the 1 −− quarkonium becomesΨ −− P ( q ⊥ ) = q ⊥ · ǫ ⊥ (cid:20) ψ ( q ⊥ ) + q ⊥ M ψ ( q ⊥ ) + P q ⊥ M ψ ( q ⊥ ) (cid:21) + M ǫ ⊥ ψ ( q ⊥ )+ ǫ ⊥ P ψ ( q ⊥ ) + 1 M ( P ǫ ⊥ q ⊥ − 6 P q ⊥ · ǫ ⊥ ) ψ ( q ⊥ ) . (18)Before moving on, we would like to discuss the interaction kernel V ( r ). We know fromQuantum Chromodynamics that the strong interaction between a quark and antiquark isgiven by the exchange of gluon(s), and that the basic kernel contains a short-range γ µ ⊗ γ µ vector interaction − α s r plus a long-range 1 ⊗ λr suggestedby the lattice QCD calculations [47]. In the Coulomb gauge and in the leading order, thekernel is the famous Cornell potential V ( r ) = λr + V − γ ⊗ γ α s r , (19)where λ is the string tension, V is a free constant appearing in the potential to fit thedata, and α s is the running coupling constant. In order to avoid infrared divergence andincorporate the screening effects, an exponential factor e − αr is added to the potential [48],i.e. V ( r ) = λα (1 − e − αr ) + V − γ ⊗ γ α s r e − αr . (20)It is easy to check that when αr ≪
1, Eq. (20) reduces to Eq. (19). In the momentumspace and in the rest frame of the bound state, the potential takes the form: V ( ~q ) = V s ( ~q ) + γ ⊗ γ V v ( ~q ) , (21)where V s ( ~q ) = − ( λα + V ) δ ( ~q ) + λπ ~q + α ) , V v ( ~q ) = − π α s ( ~q )( ~q + α ) ,α s ( ~q ) = 12 π − N f e + ~q Λ QCD ) . Here, α s ( ~q ) is the running coupling of the one loop QCD correction, and e = 2 . λ , α , V and Λ QCD are the parameters which characterize the potential, and N f = 3 for the c ¯ c system, N f = 4 for the b ¯ b system.8he reader may wonder why we have chosen a simple basic kernel, and not a relativisticone [47, 49] which includes details of the spin-independent potential and the spin-dependentpotential, like the spin-spin interaction, spin-orbital interaction, tensor interaction, etc. Thereason is that in our relativistic method, with a relativistic wave function for the boundstate, we only need the basic potential and not a relativistic one, otherwise we would havedouble counting. To explain this, let us show how the relativistic potential is obtained:the potential between a quark and anti-quark is constructed from the on-shell q ¯ q scatteringamplitude in the center-of-mass frame motivated by single gluon exchange, where the gluonpropagator is given in the Coulomb gauge. The basic non-relativistic vector potential − α s r is obtained at leading-order from the amplitude (usually in the momentum space). To obtainthe relativistic corrections of the potential, the on-shell Dirac spinors of the quark and anti-quark are expanded in quantities like the mass, momentum, etc. The relativistic potentialis then obtained, and the relativistic corrections from the free spinors (wave functions for abound state) are moved to the potential. The corresponding wave function becomes non-relativistic.In our case, we have a relativistic wave function and the potential is non-relativistic. Ifboth of them are relativistic, then there is double counting. In general, a relativistic methodshould have a relativistic wave function with a non-relativistic potential, or a non-relativisticwave function with a relativistic potential. In principle, a half-relativistic wave functionwith a half-relativistic potential is also permitted, but one has to be careful to avoid doublecounting. The method with a non-relativistic wave function and a relativistic potential isusually good for calculating the mass spectrum of bound state, while the method with arelativistic wave function and a non-relativistic potential is not only good for calculatingthe mass spectrum as an eigenvalue problem, but is also good for calculating the transitionamplitude.With the kernel Eq. (21) and the relativistic wave function Eq. (17) or Eq. (18), we areready to solve the coupled Salpeter equation Eqs. (11-14). Substituting the wave functionEq. (18) into Eq. (13) and Eq. (14), taking the trace on both sides, multiplying with thepolarization vector on both sides, e.g. q ⊥ · ǫ ∗ or /ǫ ∗⊥ · /P , and then using the completeness ofthe polarization vector, we obtain the relations ψ ( q ⊥ ) = q ⊥ ψ ( q ⊥ ) + M ψ ( q ⊥ ) M m , ψ ( q ⊥ ) = − ψ ( q ⊥ ) Mm , where we have used m = m for a quarkonium state. We now have only four independentunknown radial wave functions, ψ ( q ⊥ ), ψ ( q ⊥ ), ψ ( q ⊥ ), ψ ( q ⊥ ), whose numerical values can9e obtained by solving Eq. (11) and Eq. (12). Substituting the wave function Eq. (18) intoEq. (11) and Eq. (12), and taking the trace again, we finally obtain four coupled equations( M − ω ) (cid:26)(cid:18) ψ ( ~q ) ~q M − ψ ( ~q ) (cid:19) + (cid:18) ψ ( ~q ) ~q M + ψ ( ~q ) (cid:19) m ω (cid:27) = Z d ~k (2 π ) ω ( ( V s + V v ) ψ ( ~k ) ~k M − ψ ( ~k ) ! ( ~k · ~q ) − ( V s − V v ) " m ψ ( ~k ) ( ~k · ~q ) M ~q − ψ ( ~k ) ! + m ω ψ ( ~k ) ( ~k · ~q ) M ~q + ψ ( ~k ) ! , (22)( M + 2 ω ) (cid:26)(cid:18) ψ ( ~q ) ~q M − ψ ( ~q ) (cid:19) − (cid:18) ψ ( ~q ) ~q M + ψ ( ~q ) (cid:19) m ω (cid:27) = − Z d ~k (2 π ) ω ( ( V s + V v ) " ψ ( ~k ) ~k M − ψ ( ~k ) ! ( ~k · ~q ) − ( V s − V v ) " m ψ ( ~k ) ( ~k · ~q ) M ~q − ψ ( ~k ) ! − m ω ψ ( ~k ) ( ~k · ~q ) M ~q + ψ ( ~k ) ! , (23)( M − ω ) (cid:26)(cid:18) ψ ( ~q ) + ψ ( ~q ) m ω (cid:19) ~q M − (cid:18) ψ ( ~q ) − ψ ( ~q ) ω m (cid:19) − ψ ( ~q ) ~q m ω (cid:27) = − Z d ~k (2 π ) ω ( ( V s + V v ) " − ω m ψ ( ~k ) − ψ ( ~k ) ~k M + ψ ( ~k ) ( ~k · ~q )+( V s − V v ) " ω ψ ( ~k ) ~k M − ψ ( ~k ) ! + m ω ψ ( ~k ) ~k M + 3 ψ ( ~k ) ! − ψ ( ~k ) ( ~k · ~q ) M − ψ ( ~k ) ~q ! , (24)( M + 2 ω ) (cid:26)(cid:20) ψ ( ~q ) − ψ ( ~q ) m ω (cid:21) ~q M − (cid:18) ψ ( ~q ) + ψ ( ~q ) ω m (cid:19) + ψ ( ~q ) ~q m ω (cid:27) = Z d ~k (2 π ) ω ( ( V s + V v ) " ω m ψ ( ~k ) − ψ ( ~k ) ~k M + ψ ( ~k ) ( ~k · ~q )+( V s − V v ) " ω ψ ( ~k ) ~k M − ψ ( ~k ) ! − m ω ψ ( ~k ) ~k M + 3 ψ ( ~k ) ! − ψ ( ~k ) ( ~k · ~q ) M − ψ ( ~k ) ~q ! , (25)where we have used the relation ω = ω for a quarkonium, and V s = V s ( ~q − ~k ), V v = V v ( ~q − ~k ). Since we have four coupled equations, the four independent radial wave functions10an be obtained numerically, and the mass spectrum obtained simultaneously as the as theeigenvalue problem.The normalization condition Eq. (15) for the 1 −− wave function is Z d ~q (2 π ) ω M (cid:26) ψ ( ~q ) ψ ( ~q ) M m + ~q m (cid:20) ψ ( ~q ) ψ ( ~q ) − ψ ( ~q ) (cid:18) ψ ( ~q ) ~q M + ψ ( ~q ) (cid:19)(cid:21)(cid:27) = 1 . (26) V. THE DECAY CONSTANT IN THE SALPETER METHOD
The relativistic decay constant F ReV in Eq. (2) for a vector quarkonium can be calculatedin the BS method as F ReV
M ǫ µ = p N c Z d q (2 π ) Tr[ χ P ( q ) γ µ ] = i p N c Z d ~q (2 π ) Tr[Ψ P ( ~q ) γ µ ] , (27)where N c = 3 is the color number, and Tr is the trace operator. We note that whencalculating the decay constant, the Salpeter wave function Ψ P ( ~q ), and not merely the positivewave function Ψ ++ P ( ~q ) gives a contribution. For a vector quarkonium with the relativisticwave function Eq. (18), we obtain the relativistic decay constant F ReV = 4 √ Z d ~q (2 π ) (cid:20) ψ ( ~q ) − ~q M ψ ( ~q ) (cid:21) , (28)where we note that the ψ and ψ term both contribute. VI. RESULTS AND DISCUSSIONA. Input parameters and the heavy quarkonium wave functions
The input parameters can be fixed by fitting the mass spectra of charmonium and bot-tomonium. We choose m b = 4 .
96 GeV, m c = 1 .
60 GeV, α = 0 .
06 GeV, and Λ
QCD = 0 . . We also choose λ = 0 .
23 GeV and V = − .
249 GeV for the charmonium system,and λ = 0 . and V = − .
124 GeV for the bottomonium system. In previous Letter [42], we have chosen different Λ
QCD for charmonium and bottomonium. Since thisparameter appears only in α s , which depends on ~q , and for more convenience of fitting the data, wechoose the same Λ QCD for the two systems. ABLE I: Mass spectra of the S wave c ¯ c and b ¯ b vectors in units of MeV. ‘Th’ is the theoreticalprediction, ‘Exp’ are the experimental data from PDG [50]. nS Th( c ¯ c ) Exp( c ¯ c ) Th( b ¯ b ) Exp( b ¯ b )1 S S S S S -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.50.00.51.01.52.02.53.03.54.0 R ad i a l w a v e f un c t i on s o f J / ( G e V - ) q (GeV) q -q - FIG. 1: Four typical radial wave functions of
J/ψ . The mass spectra of vector charmonium and bottomonium are shown in Table I. Thetheoretical predictions are consistent with the experimental data given by the Particle DataGroup (PDG). An an example of the wave functions, we present four
J/ψ radial wavefunctions in Figure 1: the dominant radial wave functions ψ and ψ and the two minorones ~q ψ /M and ~q ψ /M . From now on, we use the symbols | ~q | = q and | ~v | = v for Here we show the curves of ~q ψ /M and ~q ψ /M other than ψ and ψ , because they always appear insuch a combined form in the applications. R a t i o o f t he r ad i a l w a v e f un c t i on s o f J / q (GeV) (q ) FIG. 2: Ratio ψ q ψ /M of the J/ψ radial wave functions.
As described in Sec 4, the terms with radial wave functions ψ and ψ in the total wavefunction Eq.(18), are non-relativistic, while all the others are relativistic corrections. Figure1 shows that the relativistic wave functions ψ and ψ are small and could be safely neglected,but in fact this is the case. Figure 1 only shows the relative importance of the wave functionsin the region of small, and to see the relative importance of the wave functions in the whole q region, we plot the ratio ψ / ( q ψ /M ) in Figure 2. It can be seen that in the large q region, the value of ψ is only a few times larger than of ( q ψ /M ). Thus, the terms whichare proportional to ψ ( ψ and others) may have a sizable contribution in the large q region,leading to possibly important relativistic corrections. B. Charmonium leptonic decay widths
Our results for ψ ( nS ) → ℓ + ℓ − are shown in Table II, where in the second column, ‘NR’,the non-relativistic decay rates are shown, meaning that in Eq. (28) the ψ term is ignored, sothat the only contribution is from the ψ tern. The third column, ‘Re’, show the relativisticresults including the contributions of ψ and ψ . One can see that for charmonium therelativistic results are different from the non-relativistic ones. To see this clearly, we add13 ABLE II: Decay rates of ψ ( nS ) → ℓ + ℓ − in units of keV. ‘NR’ is the non-relativistic result, ‘Re’is the relativistic result, ‘Exp’ are the experimental data from PDG [50].modes NR Re NR − ReRe
Exp
J/ψ → e + e − . +2 . − . . +1 . − . . +2 . − . % 5.55 ± ψ (2 S ) → e + e − . +1 . − . . +0 . − . . +5 . − . % 2 . ± ψ (2 S ) → τ + τ − . +1 . − . . +1 . − . . +3 . − . % 0.91 ± ψ (3 S ) → e + e − . +0 . − . . +0 . − . . +5 . − . % 0.86 ± ψ (3 S ) → τ + τ − . +0 . − . . +0 . − . . +6 . − . % / ψ (4 S ) → e + e − . +0 . − . . +0 . − . . +11 . − . % 0.48 ± ψ (4 S ) → τ + τ − . +0 . − . . +0 . − . . +15 . − . % / ψ (5 S ) → e + e − . +0 . − . . +0 . − . . +14 . − . % 0.58 ± ψ (5 S ) → τ + τ − . +0 . − . . +0 . − . . +16 . − . % / the fourth column in Table II with the ratio (NR-Re)/Re, whose value can be called the‘relativistic effect’.Table II indicates that the relativistic effect is about 22% for the J/ψ decay, which isconsistent with the usual power relation for the relativistic terms, e.g. v c ∼ . − . S , 3 S , 4 S and 5 S states, the relativistic effects are about 34%, 41%, 52% and62%, respectively. These results are consistent with our previous study of the semi-leptonicdecays B + c → c ¯ c + ℓ + + ν ℓ , where higher excited charmonium states were shown to havelarger relativistic effects [40]. This conclusion can also be obtained qualitatively from theplots of radial wave functions. We mentioned that the relative momentum q concerns therelative velocity v Q between the quark and antiquark in quarkonium, q = 0 . m Q v Q . Asshown in Figure 1, two non-relativistic J/ψ radial wave functions always dominate over therelativistic wave functions in the whole q region, leading to a small relativistic correction.For the excited states, see Figure 3 as an example of the radial wave functions of ψ (2 S ),the non-relativistic wave functions still dominate in the small q region, but there is a nodestructure in each curve where the wave function changes sign. The contributions in the low q region may cancel each other, and the wave functions for large q ( v Q ) may give sizablecontributions, resulting in large relativistic correction.14 R ad i a l w a v e f un c t i on s o f ( S )( G e V - ) q (GeV) q -q - FIG. 3: Radial wave functions of ψ (2 S ). There are other methods for considering the relativistic effects in heavy quarkoniumdecays. For example, Bodwin et al. [39] and Brambilla et al. [51] computed the v Q andthe v Q corrections of the decay rate of Q ¯ Q quarkonium in the framework of NRQCD. Inthe case of J/ψ [39], the predicted relativistic effect is 34 .
1% for v c ∼ .
3, and is 23 .
0% for v c ∼ .
18. These values are consistent with our prediction of 22 . ±
10% of their central values,and take the largest variation as the uncertainty. With the errors, most predictions aremuch larger than the experimental data. The only exception is the channel ψ (2 S ) → τ + τ − ,which has large uncertainties . We note that a calculation of the J/ψ leptonic decayin lattice QCD with fully relativistic charm quarks was reported in Ref. [52], and giveΓ(
J/ψ → e + e − ) = 5 . et al. [53] calculated the quarkonium leptonic decayusing the Cornell potential in a non-relativistic version and with the pQCD correction up to The reason is that the ψ (2 S ) mass is only a slightly heavier than that of two τ , so the phase space of thischannel is very sensitive to the variation of parameters. et al. [54] calculated the decay rates of the ψ (1 S − S ) leptonic decays withQCD correction at NLO using the Cornell potential and the semi-Salpeter equation, andobtained 5.47 keV, 2.68 keV, 1.97 keV, and 1 .
58 keV, respectively, which are smaller thanour charmonium results. These studies indicate that the relativistic corrections and QCDcorrections are large for the charmonium system.
C. Bottomonium leptonic decay widths
TABLE III: Decay rates of Υ( nS ) → ℓ + ℓ − in units of keV. ‘NR’ is the non-relativistic result, ‘Re’is the relativistic result, ‘Exp’ are the experimental data [50].modes NR Re NR − ReRe
ExpΥ(1 S ) → e + e − +0 . − . +0 . − . +0 . − . % 1.340 ± ± S ) → τ + τ − +0 . − . +0 . − . +1 . − . % 1.40 ± S ) → e + e − +0 . − . +0 . − . +0 . − . % 0.612 ± S ) → τ + τ − +0 . − . +0 . − . +0 . − . % 0.64 ± S ) → e + e − +0 . − . +0 . − . +7 . − . % 0.443 ± S ) → τ + τ − +0 . − . +0 . − . +7 . − . % 0.47 ± S ) → e + e − +0 . − . +0 . − . +6 . − . % 0.272 ± ± S ) → τ + τ − +0 . − . +0 . − . +6 . − . % /Υ(5 S ) → e + e − +0 . − . +0 . − . +1 . − . % 0.31 ± S ) → τ + τ − +0 . − . +0 . − . +1 . − . % / We present the non-relativistic and relativistic results of the bottomonium leptonic decaywidths in Table III. Similarly to charmonium, the relativistic corrections are also sizable.For the ground state Υ(1 S ), the relativistic effect is about 14%, and for the excited statesΥ(2 S − S ), they vary from 20% to 28%. These predictions agree with those in literature.For example, Bodwin et al. [39] predicted the relativistic effect of 13 .
2% for v b ∼ .
10 usingNRQCD up to the v b accuracy, and a lattice QCD prediction indicated that the relativisticeffects are about (15 − S ) and Υ(2 S ) up to the v b accuracy.16e should point out that the above large relativistic effects are specific for bottomo-nium leptonic decays Υ( nS ) → ℓ + ℓ − , and are not universal for processes involving a bot-tomonium. In the di-lepton decays, the amplitude is proportional to the wave function as R d ~q h ψ ( ~q ) − ~q M ψ ( ~q ) i , i.e. the wave function is to the power of one. For other processes,such as meson A to meson B semileptonic decays, the amplitude is proportional to theoverlapping integral of the wave functions for the initial and final states R d ~q ψ A · ψ B . Be-cause the wave functions are large in the small q region, the product of two wave functionsis suppressed in the large q region compared to the case with one wave function, and thecontributions from the relativistic terms are greatly suppressed.In Table III, we also give the theoretical uncertainties, which are obtained by varying allparameters simultaneously within ±
10% of the central values, and the largest variations aretaken as the errors. Our relativistic results agree well with the experimental data. We alsonote that, for Υ(1 S ) → e + e − , PDG gives two different results: directly listed is Γ ee = 1 . Br = 2 .
38% is also given, leading to Γ
Υ(1 S ) → e + e − = 1 .
29 keVusing the full width Γ
Υ(1 S ) = 54 .
02 keV [50]. The second value is the same as our relativisticresult. Similarly, in the case of Υ(4 S ) → e + e − , PDG directly lists Γ ee = 0 .
272 keV [50], butfrom the branching ratio also given in PDG, we get Γ ee = 0 .
322 keV. We hope PDG willupdate the data in the near future.Table III shows that all relativistic results for Γ ℓℓ (1 S − S ) are consistent with theexperimental data. Our predictions also agree with the lattice QCD prediction [22],Γ(Υ(1 S ) → e + e − ) = 1 . S ) → e + e − ) = 0 . S ) → e + e − ) = 1 .
25 keV, and with the NRQCD prediction withNNNLO pQCD corrections [30], Γ(Υ(1 S ) → e + e − ) = 1 . ± . α s ) +0 . − . ( µ ) keV. D. Lepton flavor university
To test the lepton flavor university, we give the ratios R ψ nS ττ and R Υ nS ττ in Table IV. Theirdefinitions are similar, for example, R ψ nS ττ = Γ( ψ nS → τ + τ − )Γ( ψ nS → µ + µ − ) . The deviation of the ratio R ψ nS ττ from the lepton flavor universality indicates the presence ofnew physics beyond the Standard Model.Table IV shows the ratios calculated with the ‘Re’ values. The uncertainties of the ratioare from the variation of the input parameters. In the case of charmonium, the ratios R ψ nS ττ ABLE IV: Ratios R ψ nS ττ = Γ( ψ nS → τ + τ − )Γ( ψ nS → µ + µ − ) and R Υ nS ττ . The experimental data are from PDG [50],with the statistical and systematic uncertainties added together. R ψ S ττ R ψ S ττ R ψ S ττ R ψ S ττ Ours 0.391 +0 . − . +0 . − . +0 . − . +0 . − . R Υ S ττ R Υ S ττ R Υ S ττ R Υ S ττ R Υ S ττ Ours 0.992 +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . CLEO[12] 1 . ± .
07 1 . ± .
09 1 . ± .
13 / /BABAR[14] 1 . ± .
035 / / / /PDG [50] 1 . ± .
06 1 . ± .
20 1 . ± .
24 / / are quite different from each other, since the charmonium mass is a bit higher than of two τ . For the same reason, we get a large uncertainty. For bottomonium since the τ massis much smaller than the bottomonium mass, we get almost the same values for all ratios R Υ nS ττ . Their uncertainty is also very small due to the cancellation between the numeratorand denominator. Even though all central values of the ratios R Υ nS ττ are smaller than 1, theyare consistent with the existing experimental data within errors. TABLE V: Ratio Γ( ψ ( nS ) → e + e − ) / Γ( J/ψ → e + e − ). Γ( ψ (2 S ))Γ( J/ψ ) Γ( ψ (3 S ))Γ( J/ψ ) Γ( ψ (4 S ))Γ( J/ψ ) Γ( ψ (5 S ))Γ( J/ψ ) Ours 0.495 +0 . − . +0 . − . +0 . − . +0 . − . Exp [50] 0.42 ± ± ± ± To cancel the model dependence of the theoretical predictions, we give in Table V andTable VI the ratios Γ( ψ ( nS ) → e + e − ) / Γ( J/ψ → e + e − ) and Γ(Υ( nS ) → ℓ + ℓ − ) / Γ(Υ(1 S ) → ℓ + ℓ − ). For the Υ( nS ) decay, we obtain the same central values for the e and τ final states,so we only present the ratio Γ(Υ( nS ) → ℓ + ℓ − ) / Γ(Υ(1 S ) → ℓ + ℓ − ) in Table V and Table VI,which are calculated using the e + e − final states listed in Table III.Table V shows that the ratio Γ( ψ (2 S ))Γ( J/ψ ) is larger but close to the experimental data, while theratios for highly excited states are much larger than the experimental data. Table VI showsthe bottomonium leptonic decay ratios. In the row ‘Exp1’, the value of Γ ee (1 S ) = 1 . ± ABLE VI: Ratio Γ(Υ( nS ) → ℓ + ℓ − ) / Γ(Υ(1 S ) → ℓ + ℓ − ). ‘Exp1’ are the experimental data withΓ ee (1 S ) = 1 . ± .
018 keV, ‘Exp2’ are the experimental data with Γ ee (1 S ) = 1 . ± .
09 keV.For Υ(4 S ), Γ ee (4 S ) = 0 . ± .
029 (0 . ± . Γ(Υ(2 S ))Γ(Υ(1 S )) Γ(Υ(3 S ))Γ(Υ(1 S )) Γ(Υ(4 S ))Γ(Υ(1 S )) Γ(Υ(5 S ))Γ(Υ(1 S )) Ours 0.488 +0 . − . +0 . − . +0 . − . +0 . − . Exp1 [50] 0.457 ± ± ± ± ± ± ± ± ± ± .
018 keV is used, which is directly listed in PDG. In the row ‘Exp2’, Γ ee (1 S ) = 1 . ± . S ) → e + e − given in PDG.For Υ(4 S ), the results outside the brackets were obtained using Γ ee (4 S ) = 0 . ± . ee (4 S ) = 0 . ± .
056 keVobtained from the PDG branching ratio. It can be seen that all our theoretical predictionsare consistent with the experimental data.
VII. SUMMARY
In this paper, we studied the leptonic decays of heavy vector quarkonia. For the char-monium decays, not all states are consistent with the experimental data, while for thebottomonium decays, almost all S wave states are in good agreement with the data.Theoretical results of the ratios Γ( ψ ( nS ) → e + e − ) / Γ( J/ψ → e + e − ) and Γ(Υ( nS ) → ℓ + ℓ − ) / Γ(Υ(1 S ) → ℓ + ℓ − ) were given in Ref. [55], where the potential model was used in-cluding the v Q relativistic corrections and pQCD corrections at NLO. These results arecomparable with ours, i.e. the charmonium leptonic decay widths are not consistent withthe experimental data and the bottomonium leptonic widths are in good agreement withthe data. This situation was also observed in Ref. [56]. It seems that the same theoreticaltool cannot provide satisfactory results for both the charmonium and bottomonium systems[57]. There are several possible reasons for this difference in our study. It may be that theinstantaneous approximation works well for bottomonium, but is not good enough for char-monium. An improvement of the Cornell potential may be needed, and more importantly,the perturbative QCD corrections may have larger effect in charmonium decays than in bot-19omonium decays. Since the BS equation is an integral equation, the QCD corrections fromthe gluon ladder diagrams are already included, but other QCD corrections in the kernel orin the quark propagators may need to be improved in future calculations.The Bethe-Salpeter method provides a strict way to deal with the relativistic effects. Inthis framework, we found that the relativistic corrections are large and important for theleptonic decays ψ ( nS ) → ℓ + ℓ − and Υ( nS ) → ℓ + ℓ − . For the ψ (1 S − S ) leptonic decays, therelativistic effects are 22 +3 − %, 34 +5 − %, 41 +6 − %, 52 +11 − % and 62 +14 − %, respectively. Therefore,for the highly excited states ψ ( nS ), the relativistic corrections give dominant contributions.For the Υ(1 S − S ) decays, the relativistic effects are 14 +1 − %, 23 +0 − %, 20 +8 − %, 21 +6 − % and28 +2 − %, respectively. Thus, relativistic effects should be considered for a sound prediction ofthe heavy quarkonium decays. VIII. ACKNOWLEDGMENTS
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