Revisiting the Heavy Vector Quarkonium Leptonic Widths *

Chinese Physics C Vol. 44, No. 6 (2020) 063104

Revisiting the heavy vector quarkonium leptonic widths *

1;1) 2;2) Guo-Li Wang(王国利) Xing-Gang Wu(吴兴刚) 1Department of Physics, Hebei University, Baoding 071002, China 2Department 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 the Bethe-Sal- peter method. The emphasis is on the relativistic corrections. For the ψ(1S − 5S ) decays, the relativistic effects are +3 +5 +6 +11 +14 Υ − 22−2%, 34−5%, 41−6%, 52−13% and 62−12%, respectively. For the (1S 5S ) decays, the relativistic effects are +1 +0 +8 +6 +2 14−2%, 23−3%, 20−2%, 21−7% and 28−7%, 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 with the experimental data. Keywords: quarkonium, leptonic decay, Bethe-Salpeter equation DOI: 10.1088/1674-1137/44/6/063104

1 Introduction 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-sum- As it gives a clean experimental signal, the dilepton mation of the logarithms (partly) up to the next-to-next- annihilation decay of the heavy vector quarkonium plays to-leading logarithmic (NNLL) accuracy; the NNNLO an important role in determining the fundamental para- corrections have been discussed by various groups meters such as the strong coupling constant [1, 2], heavy [27–31]. A pQCD analysis of the Υ(1S ) leptonic decay up quark masses [1, 3–5], heavy quarkonium decay con- to NNNLO using the principle of maximum conformal- stants [2, 6–8], etc. Its decay amplitude is a function of ity (PMC) [32–35] was presented in Refs. [36, 37], where the quarkonium wave function, and this process can be the renormalization scale ambiguity of the decay width is used to test various theories such as the quark potential eliminated with the help of the renormalization group model, non-relativistic Quantum Chromodynamics (NR- equation. QCD), QCD sums rules, lattice QCD, etc. The Standard Even though considerable improvements have been Model prediction of the universality of lepton flavor is made, there are still deviations between the theoretical ∗ ∗ questioned by the measured ratios R(D( )) and R(K( )) predictions and the experimental data for the heavy vec- [9–14], and the quarkonium leptonic decay is another tor quarkonium leptonic decays. There are two sources way to test the lepton flavor universality. which may cause such deviations. The first are the un- The vector quarkonium leptonic decays have been known higher order perturbative QCD corrections. By us- studied since a long time [15–21]. With the progress in ing PMC, the conventional pQCD convergence of the computer science and experimental technology, many ad- series can be greatly improved by elimination of the di- vances have been reported in literature. For example, one vergent renormalon terms, and a more accurate decay can find the lattice QCD predictions of the leptonic de- width can be obtained. However, there are still large er- cays of the ground-state Υ and its first radial excitation Υ′ rors due to unknown high-order terms [36, 37]. The in [22]; Ref. [23] reported the next-to-leading non-per- second source is the relativistic correction, which could turbative prediction and Ref. [24] the next-to-leading-log be large. However, almost all pQCD predictions are cal- perturbative QCD (pQCD) prediction; in Ref. [25], the culated using NRQCD, in which the decay constant of

Received 5 February 2020 Published online 23 April 2020 * Supported in part by the National Natural Science Foundation of China (NSFC) (11575048, 11625520, 11847301) 1) E-mail: [email protected] 2) E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd

063104-1 Chinese Physics C Vol. 44, No. 6 (2020) 063104 quarkonium, or its wave function at the origin, is treated matrix element of the electromagnetic current simply in the non-relativistic approximation. < 0|Q¯γµQ|V(P,ϵ) >= FV MnS ϵµ, (2) One may argue that the relativistic correction is small for a heavy quarkonium, since the relative velocity where P is the quarkonium momentum, and ϵ is the polar- among the heavy constituent quarks is small, e.g. ization vector. v2 ∼ 0.1 − 0.3 [38]. However, many analyses in literature In the non-relativistic method, the well-known for- mula for the decay constant is have found that the relativistic effect could be large. For √ example, Bodwin et al. computed the coefficients of the NR = 12 |Ψ |, 3 FV V (0) (3) decay operators for the S 1 heavy quarkonium decay into MnS a leptonic pair and found large relativistic correction [39]; where NR means the non-relativistic (NR), and ΨV (0) is Gonzalez et al. pointed out that large relativistic and the non-relativistic wave function evaluated at the origin. QCD corrections of the quarkonium leptonic decays are In the NR method, there is only one radial wave function, necessary to fit the experimental data [7]; Geng et al. and the vector meson and its corresponding pseudoscalar studied the Bc meson semileptonic decays into charmoni- have the same radial wave function and the same decay um and also found that the relativistic corrections are constant. However, in the relativistic method, they have large [40], especially for highly excited charmonium different wave functions and different decay constants, states. Moreover, from the experimental standpoint, Ref. and more than one radial wave function gives a contribu- [41] showed that a careful study of leptonic decays is still tion to the vector meson decay constant. needed for highly excited charmonium states. In the relativistic method, the decay constant In this paper, we focus on the leptonic decays of char- = Re FV FV is not related to the wave function at the origin, monium and bottomonium, including their excited states, but in the full region. In the following, we focus on the using the relativistic method. In a previous short letter Re calculation of FV in the relativistic method. [42], we presented a relativistic calculation of the quarkonium decays into e+e−, where the results disagree with the experimental data. As a step forward, we revisit 3 The Bethe-Salpeter equation and the Sal- this topic in more detail, and include the decays into µ+µ− peter equation + − and τ τ as well as the ratios Rττ. We present the relativistic effects in these quarkonium decays, and discuss the In this section, we briefly review the Bethe-Salpeter universality of lepton flavor. (BS) equation [43], which is a relativistic dynamic equa- The paper is organized as follows. The general equation describing the two-body bound state, and its instant- tion of the quarkonium leptonic decay width is given in aneous version, the Salpeter equation [44]. The BS equa- Sec. 2. In Sec. 3, we give a brief review of the Bethe-Sal- tion for a meson, which is a bound state of a quark, la- peter equation, and its instantaneous version, the Salpeter belled as 1, and anti-quark, labelled as 2, can be written equation. We then show in Sec. 4 in detail how to solve as [43] ∫ the full Salpeter equation and obtain the relativistic wave d4k (̸p − m )χ (q)(̸p + m ) = i V(P,k,q)χ (k) , (4) function for a vector meson. The calculation of the decay 1 1 P 2 2 (2π)4 P constant in the relativistic method is given in Sec. 5. Fi- χ nally, in Sec. 6, we give the numerical results and a dis- where P(q) is the relativistic wave function of the , , cussion. A summary is presented in Sec. 7. meson, V(P k q) is the interaction kernel between the quark and anti-quark, p1, p2,m1,m2 are the momenta and masses of the quark and anti-quark, P is the momentum 2 The quarkonium leptonic decay width of the meson, q is the relative momentum between quark and anti-quark. The momenta p1 and p2 satisfy the rela- m1 The leptonic partial decay rate of a vector charmoni- tions, p1 = α1P + q and p2 = α2P − q, where α1 = and m1+m2 m2 um or bottomonium nS state V is given by α2 = + . In the case of quarkonium, where m1 = m2, we √ m1 m2   α = α = . 4πα2 e2 F2  m2  m2 have 1 2 0 5. em Q V  ℓ  ℓ ΓV→ℓ+ℓ− = × 1 + 2  1 − 4 , (1) In the general case, the BS equation is hard to solve 3M M2 M2 nS nS nS due to the complex interaction kernel between the con- where αem is the fine structure constant, eQ is the electric stituent quarks. For the doubly heavy quarkonium con- charge of the heavy quark Q in units of the electron sidered here, the interaction kernel between the two charge, eQ = +2/3 for the charm quark and eQ = −1/3 for heavy constituent quarks can be treated as instantaneous, the bottom quark, MnS is the mass of the nS state leading to a simpler version of the BS equation. In this quarkonium, mℓ is the lepton mass, FV is the decay con- case, it is convenient to divide the relative momentum q µ = µ + µ µ ≡ · / 2 µ stant of the vector meson that is defined by the following into two parts, q q∥ q⊥, where q∥ (P q M )P and

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µ ≡ µ − µ Ψ−+ = . q⊥ q q∥ , M is the mass of the bound state, and we P (q⊥) 0 (14) have P2 = M2. Then, we have two Lorentz invariant √ √ vari- The normalization condition for the BS wave func- · ables, q = (P q) and q = q2 − q2 = −q2 . When P⃗ = 0, tion reads P M T P ⊥ ∫ [ ] that is in the meson center-of-mass frame, they reduce to q2 dq ++ ̸ ̸ −− ̸ ̸ T T Ψ P Ψ++ P − Ψ P Ψ−− P = . |⃗| = ,⃗ Tr P P 2M (15) the usual components q0 and q , and q⊥ (0 q). 2π2 M P M M P M With this notation, the volume element of the relativ- Note that usually in literature, it is not the full Sal- istic momentum k can be written in an invariant form peter equation Eq. (10) that is solved (or equivalently, the d4k = dk k2 dk dsdϕ, where ds = (k q − k · q)/(k q ) and ϕ P T T P P T T four Eqs. (11)-(14)), but only Eq. (11), which involves is the azimuthal angle. Taking the instantaneous approx- only the positive wave function. There is a good reason imation in the center-of-mass frame of the bound state, why such an approximation is made: it is its effective V P,k,q V k⊥,q⊥, s the kernel ( ) changes to ( ) . We introduce range. The numerical value of M − ω − ω in Eq. (11) is the three-dimensional wave function 1 2 ∫ much smaller than of M + ω1 + ω2 in Eq. (12), which µ q µ µ ++ Ψ ≡ d P χ , , means that the positive wave function Ψ (q⊥)is domin- P(q⊥) i (q∥ q⊥) (5) P 2π ant, and that the contribution of the negative wave func- Ψ−− and the notation tion P (q⊥) can be safely neglected. However, we point ∫ Ψ++ 2 ϕ out that if only Eq. (11) for P (q⊥)is considered, then µ kT dkT dsd µ η(q⊥) ≡ V(k⊥,q⊥, s)Ψ (k⊥). (6) not only is the contribution of the negative wave function (2π)3 P neglected, but so are the relativistic effects of these wave The BS eqution Eq. (4) is then rewritten as functions. The reason is that the number of eigenvalue χ(q∥,q⊥) = S 1(p1)η(q⊥)S 2(p2), (7) equations limits the number of radial wave functions, and as is shown below, only the four coupled equations Eqs. where S 1(p1) and S 2(p2) are propagators of quark 1 and anti-quark 2, respectively, which can be decomposed as (11)-(14) can provide sufficient information to derive a + relativistic wave function. Λ (q⊥) S p = i i( i) i+1 (−1) qP + αi M − ωi + iε − 4 Relativistic wave function and the kernel Λ (q⊥) + i . (8) − i+1 + α + ω − ε ( 1) qP i M i i Although BS or the Salpeter equation is the relativist- Here,√ we have defined the constituent quark energy ic dynamic equation describing the two-body bound state, ω = m2 + q2 Λ q⊥ = the equation cannot by itself provide the information i [ i T and the projection] operators i ( ) 1 ̸P ω − i+1 + ̸ = about the wave function. This means that we need to ω M i ( 1) (mi q⊥) , where i 1 and 2 for quark 2 i provide an explicit form of the relativistic kinematic and anti-quark, respectively. wave function as input, which can be constructed using Using the projection operators, we can divide the all allowable Lorentz and γ structures. wave function into four parts From literature, we have the familiar form of the non- Ψ = Ψ++ + Ψ+− + Ψ−+ + Ψ−− , − P(q⊥) P (q⊥) P (q⊥) P (q⊥) P (q⊥) (9) relativistic wave function for the 1 vector meson, e.g. Ψ ≡ Λ ̸P Ψ ̸P Λ Ψ ⃗q = ̸P + M ̸ϵψ ⃗q , with the definition P (q⊥) 1 (q⊥) M P(q⊥)M 2 (q⊥). P( ) ( ) ( ) (16) Ψ++ Ψ−− Here, P (q⊥) and P (q⊥) are called the positive and where M, P and ϵ are the mass, momentum and polariza- negative energy wave functions of the quarkonium. tion of the vector meson, ⃗q is the relative momentum After integrating over qP on both sides of Eq. (7) us- between the quark and anti-quark. There is only one un- ing contour integration, we obtain the famous Salpeter known wave function ψ(⃗q) in Eq. (16), which can be ob- equation [44]: tained numerically by solving Eq. (11) or the non-relativ- + + − − Λ (q⊥)η(q⊥)Λ (q⊥) Λ (q⊥)η(q⊥)Λ (q⊥) istic Schrödinger equation. The relative momentum ⃗q is Ψ = 1 2 − 1 2 . P(q⊥) related to the relative velocity ⃗v between the quark and (M − ω1 − ω2) (M + ω1 + ω2) ⃗ = m1m2 ⃗ (10) anti-quark in the meson, q m +m v . A relativistic wave 1 2 ⃗ Equivalently, the Salpeter equation can be written as function should depend on the relative velocity v or mo- ⃗ ψ ⃗ four independent equations using the projection operat- mentum q separately, not merely on the radial part (q), ψ |⃗| ψ ⃗2 ors: because the radial part is in fact ( q ) or equally (q ). ++ + + To obtain the form of the relativistic wave function, (M − ω − ω )Ψ (q⊥) = Λ (q⊥)η(q⊥)Λ (q⊥) , (11) 1 2 P 1 2 we start from J pc of a meson, because J p or J pc are in any + ω + ω Ψ−− = −Λ− η Λ− , (M 1 2) P (q⊥) 1 (q⊥) (q⊥) 2 (q⊥) (12) case good quantum numbers, where J is the total angular momentum, and p and c are the parity and the charge Ψ+− = , P (q⊥) 0 (13) conjugate parity of the meson. The parity transform

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′ changes the momentum q = (q0,⃗q) into q = (q0,−⃗q), so for scalar interaction λr suggested by the lattice QCD calcu- a meson, after applying the parity transform, the four-di- lations [47]. In the Coulomb gauge and in the leading or- ′ mensional wave function χP(q) changes to p · γ0χP′ (q )γ0, der, the kernel is the famous Cornell potential where p is the eigenvalue of parity. The charge conjugate α = λ + − γ ⊗ γ0 4 s , χ ·CχT − C−1 V(r) r V0 0 (19) transform changes P(q) to c P( q) , where c is the 3 r eigenvalue of charge conjugate parity, C = γ γ is the 2 0 where λ is the string tension, V0 is a free constant appear- charge conjugate transform operator, and T is the trans- ing in the potential to fit the data, and αs is the running pose transform. Since the Salpeter equation is instantan- coupling constant. In order to avoid infrared divergence Ψ eous, the input wave function P(q⊥) is also instantan- and incorporate the screening effects, an exponential − eous, and the general form of the wave function for the 1 factor e−αr is added to the potential [48], i.e. vector meson can be written as [45, 46] λ α [ −αr 0 4 s −αr ̸ ̸ V(r) = (1 − e ) + V0 − γ0 ⊗ γ e . (20) 1− P q⊥ α Ψ (q⊥) =q⊥ · ϵ⊥ ψ (q⊥) + ψ (q⊥) + ψ (q⊥) 3 r P 1 M 2 M 3 ] It is easy to check that when αr ≪ 1, Eq. (20) reduces to ̸P̸q⊥ Eq. (19). In the momentum space and in the rest frame of + ψ (q⊥) + M̸ϵ⊥ψ (q⊥) M2 4 5 the bound state, the potential takes the form: + ̸ϵ ̸ ψ ⊥ + ̸ ̸ϵ − ⊥ · ϵ⊥ ψ ⊥ 0 ⊥P 6(q ) (q⊥ ⊥ q ) 7(q ) V(⃗q) = Vs(⃗q) + γ0 ⊗ γ Vv(⃗q), (21) 1 + (̸P̸ϵ⊥q̸ ⊥ − ̸Pq⊥ · ϵ⊥)ψ8(q⊥). (17) where M ( λ ) λ 3 1 There are in total 8 radial wave functions ψ (q⊥) = ψ (|⃗q|) Vs(⃗q) = − + V0 δ (⃗q) + , i i α π2 (⃗q2 + α2)2 with i = 1 ∼ 8, which obviously cannot be obtained by 2 αs(⃗q) solving only one equation, e.g. Eq. (11), but can be ob- Vv(⃗q) = − , tained by solving the full Salpeter Eqs. (11)-(14). The 3π2 (⃗q2 + α2) above expression does not include the terms with P · q, 12π 1 αs(⃗q) =  . −  2  since the condition of instantaneous interaction is 33 2N f  ⃗q  · = · = loge +  P q P q⊥ 0. There are also no higher order q⊥ terms Λ2 2 3 4 QCD like q⊥, q⊥, q⊥, etc., because the even powers of q⊥ can be ⃗ absorbed into the radial part of ψi(q⊥), while the odd Here, αs(q) is the running coupling of the one loop QCD powers of q⊥ can be changed to lower power, for ex- correction, and e = 2.71828. The constants λ, α, V0 and 3 ′ Λ ample, ̸q⊥ψ (q⊥) = ̸q⊥ψi(q⊥). By the way, if we delete all QCD are the parameters which characterize the potential, i ¯ q⊥ terms except those inside the radial wave functions, and N f = 3 for the cc¯ system, N f = 4 for the bb system. then the wave function Eq. (17) reduces to The reader may wonder why we have chosen a simple M̸ϵ⊥ψ5(q⊥) + ̸ϵ⊥̸ Pψ6(q⊥). If we further set basic kernel, and not a relativistic one [47, 49] which in- ψ5(q⊥) = −ψ6(q⊥) = ψ(q⊥), the wave function reduces to cludes details of the spin-independent potential and the the non-relativistic case, e.g. Eq. (16). Thus, the terms spin-dependent potential, like the spin-spin interaction, with ψ1, ψ2, ψ3, ψ4, ψ7 and ψ8 in Eq. (17) are all relativist- spin-orbital interaction, tensor interaction, etc. The reas- ic corrections. on is that in our relativistic method, with a relativistic When the charge conjugate parity is taken into ac- wave function for the bound state, we only need the ba- count, the terms with ψ2(q⊥) and ψ7(q⊥) vanish because of sic potential and not a relativistic one, otherwise we the positive charge conjugate parity c = +, and the gener- would have double counting. To explain this, let us show al instantaneous wave function for the 1−− quarkonium how the relativistic potential is obtained: the potential becomes between a quark and anti-quark is constructed from the [ ] ̸ ̸ ̸ on-shell qq¯ scattering amplitude in the center-of-mass 1−− q⊥ Pq⊥ Ψ (q⊥) =q⊥ · ϵ⊥ ψ (q⊥) + ψ (q⊥) + ψ (q⊥) frame motivated by single gluon exchange, where the P 1 M 3 M2 4 gluon propagator is given in the Coulomb gauge. The ba- + M̸ϵ⊥ψ (q⊥) + ̸ϵ⊥̸Pψ (q⊥) α 5 6 sic non-relativistic vector potential − 4 s is obtained at 1 3r + (̸P̸ϵ⊥̸q⊥ − ̸Pq⊥ · ϵ⊥)ψ8(q⊥). (18) leading-order from the amplitude (usually in the mo- M mentum space). To obtain the relativistic corrections of Before moving on, we would like to discuss the inter- the potential, the on-shell Dirac spinors of the quark and action kernel V(r). We know from Quantum Chromody- anti-quark are expanded in quantities like the mass, mo- namics that the strong interaction between a quark and mentum, etc. The relativistic potential is then obtained, antiquark is given by the exchange of gluon(s), and that and the relativistic corrections from the free spinors µ the basic kernel contains a short-range γµ ⊗ γ vector in- (wave functions for a bound state) are moved to the po- α − 4 s ⊗ teraction 3r plus a long-range 1 1 linear confining tential. The corresponding wave function becomes non-

063104-4 Chinese Physics C Vol. 44, No. 6 (2020) 063104 relativistic. function Eq. (17) or Eq. (18), we are ready to solve the In our case, we have a relativistic wave function and coupled Salpeter equation Eqs. (11)-(14). Substituting the the potential is non-relativistic. If both of them are re- wave function Eq. (18) into Eq. (13) and Eq. (14), taking lativistic, then there is double counting. In general, a re- the trace on both sides, multiplying with the polarization ∗ ∗ lativistic method should have a relativistic wave function vector on both sides, e.g. q⊥ · ϵ or ̸ϵ⊥ · ̸P, and then using with a non-relativistic potential, or a non-relativistic wave the completeness of the polarization vector, we obtain the function with a relativistic potential. In principle, a half- relations relativistic wave function with a half-relativistic potential 2 2 q⊥ψ3(q⊥) + M ψ5(q⊥) ψ6(q⊥)M is also permitted, but one has to be careful to avoid ψ1(q⊥) = , ψ8(q⊥) = − , double counting. The method with a non-relativistic wave Mm1 m1 function and a relativistic potential is usually good for where we have used m1 = m2 for a quarkonium state. We calculating the mass spectrum of bound states, while the now have only four independent unknown radial wave method with a relativistic wave function and a non-re- functions, ψ3(q⊥), ψ4(q⊥), ψ5(q⊥), ψ6(q⊥), whose numeric- lativistic potential is not only good for calculating the al values can be obtained by solving Eq. (11) and Eq. mass spectrum as an eigenvalue problem, but is also good (12). Substituting the wave function Eq. (18) into Eq. for calculating the transition amplitude. (11) and Eq. (12) and taking the trace again, we finally With the kernel Eq. (21) and the relativistic wave obtain four coupled equations    {( ) ( ) } ∫  ⃗q2 ⃗q2 m d3⃗k 2   ⃗k2  M − ω ψ ⃗q − ψ ⃗q + ψ ⃗q + ψ ⃗q 1 =  V + V ψ ⃗k − ψ ⃗k  ⃗k ·⃗q ( 2 1) 3( ) 2 5( ) 4( ) 2 6( ) 3 2 ( s v) 3( ) 2 5( )( ) M M ω1 (2π) ω M     1    ⃗ ·⃗ 2   ⃗ ·⃗ 2   2  ⃗ (k q) ⃗   ⃗ (k q) ⃗  − (Vs − Vv)m ψ3(k) − ψ5(k) +m1ω1 ψ4(k) + ψ6(k) , (22) 1 M2⃗q2 M2⃗q2     {( ) ( ) } ∫  ⃗q2 ⃗q2 m d3⃗k 2   ⃗k2  M + ω ψ ⃗q − ψ ⃗q − ψ ⃗q + ψ ⃗q 1 = −  V + V ψ ⃗k − ψ ⃗k  ⃗k ·⃗q ( 2 1) 3( ) 2 5( ) 4( ) 2 6( ) 3 2 ( s v) 3( ) 2 5( )( ) M M ω1 (2π) ω M     1    ⃗ ·⃗ 2   ⃗ ·⃗ 2   2  ⃗ (k q) ⃗   ⃗ (k q) ⃗  − (Vs − Vv)m ψ3(k) − ψ5(k) −m1ω1 ψ4(k) + ψ6(k) , (23) 1 M2⃗q2 M2⃗q2  {( ) ( ) } m ⃗q2 ω ⃗q2 (M − 2ω ) ψ (⃗q) + ψ (⃗q) 1 − 3 ψ (⃗q) − ψ (⃗q) 1 − ψ (⃗q) = 1 3 4 ω M2 5 6 m 6 m ω  1  1 1 1 ∫  d3⃗k 1   2ω ⃗k2  −  V + V − 1 ψ ⃗k − ψ ⃗k + ψ ⃗k  ⃗k ·⃗q 3 2 ( s v) 6( ) 3( ) 2 5( )( ) (2π) ω m1 M  1         ⃗2   ⃗2   ⃗ ·⃗ 2   2  ⃗ k ⃗   ⃗ k ⃗   ⃗ (k q) ⃗ 2 + (Vs − Vv)ω ψ3(k) − 3ψ5(k) + m1ω1 ψ4(k) + 3ψ6(k) −ψ3(k) − ψ5(k)⃗q  , (24) 1 M2 M2 M2  {[ ] ( ) } m ⃗q2 ω ⃗q2 (M + 2ω ) ψ (⃗q) − ψ (⃗q) 1 − 3 ψ (⃗q) + ψ (⃗q) 1 + ψ (⃗q) = 1 3 4 ω M2 5 6 m 6 m ω  1  1  1 1 ∫  d3⃗k 1  2ω ⃗k2   V + V  1 ψ ⃗k − ψ ⃗k + ψ ⃗k  ⃗k ·⃗q 3 2 ( s v) 6( ) 3( ) 2 5( )( ) (2π) ω m1 M 1         ⃗2   ⃗2   ⃗ ·⃗ 2   2  ⃗ k ⃗   ⃗ k ⃗   ⃗ (k q) ⃗ 2 + (Vs − Vv)ω ψ3(k) − 3ψ5(k) − m1ω1 ψ4(k) + 3ψ6(k) −ψ3(k) − ψ5(k)⃗q  , (25) 1 M2 M2 M2 

where we have used the relation ω1 = ω2 for a quarkoni- function is ⃗ ⃗ ∫ { um, and Vs = Vs(⃗q − k), Vv = Vv(⃗q − k). Since we have four 3⃗ ω 2 d q 8 1 ψ ⃗ ψ ⃗ M coupled equations, the four independent radial wave 3 5(q) 6(q) (2π)3 3M 2m functions can be obtained numerically, and the mass [ (1 )]} ⃗q2 ⃗q2 spectrum obtained simultaneously as the eigenvalue prob- + ψ ⃗q ψ ⃗q − ψ ⃗q ψ ⃗q + ψ ⃗q = . 4( ) 5( ) 3( ) 4( ) 2 6( ) 1 lem. 2m1 M The normalization condition Eq. (15) for the 1−− wave (26)

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5 The decay constant in the Salpeter method now on, we use the symbols |⃗q| = q and |⃗v| = v for simpli- city. The relativistic decay constant FRe in Eq. (2) for a As described in Sec. 4, the terms with radial wave V functions ψ and ψ in the total wave function Eq. (18), vector quarkonium can be calculated in the BS method as 5 6 ∫ are non-relativistic, while all the others are relativistic √ 4 Re d q F Mϵµ = N Tr[χ (q)γµ] corrections. Figure 1 shows that the relativistic wave V c (2π)4 P ψ ψ ∫ functions 3 and 4 are small and could be safely neg- √ d3⃗q lected, but in fact this is not the case. Figure 1 only shows =i N Tr[Ψ (⃗q)γµ], (27) c (2π)3 P the relative importance of the wave functions in the region of small q, and to see the relative importance of the where N = 3 is the color number, and Tr is the trace oper- c wave functions in the whole q region, we plot the ratio ator. We note that when calculating the decay constant, 2 2 ψ5/(q ψ3/M ) in Fig. 2. It can be seen that in the large q the Salpeter wave function ΨP(⃗q), and not merely the pos- ++ region, the value of ψ is only a few times larger than of itive wave function Ψ (⃗q) gives a contribution. For a 5 P (q2ψ /M2). Thus, the terms which are proportional to ψ vector quarkonium with the relativistic wave function Eq. 3 3 (ψ and others) may have a sizable contribution in the (18), we obtain the relativistic decay constant 4 ∫ [ ] large q region, leading to possibly important relativistic √ d3⃗q ⃗q2 FRe = 4 3 ψ (⃗q) − ψ (⃗q) , (28) corrections. V (2π)3 5 3M2 3 6.2 Charmonium leptonic decay widths where we note that the ψ5 and ψ3 terms both contribute. Our results for ψ(nS ) → ℓ+ℓ− are shown in Table 2, 6 Results and discussion where in the second column, ‘NR ’, the non-relativistic decay rates are shown, meaning that in Eq. (28) the ψ3 6.1 Input parameters and the heavy quarkonium wave term is ignored, so that the only contribution is from the ψ functions 5 term. The third column, ‘Re ’ , show the relativistic results including the contributions of ψ5 and ψ3. One can The input parameters can be fixed by fitting the mass see that for charmonium the relativistic results are differ- spectra of charmonium and bottomonium. We choose ent from the non-relativistic ones. To see this clearly, we mb = 4.96 GeV, mc = 1.60 GeV, α = 0.06 GeV, and add the fourth column in Table 2 with the ratio (NR- 1) 2 ΛQCD = 0.21 GeV [42] . We also choose λ = 0.23 GeV Re)/Re, whose value can be called the ‘relativistic effect’. and V0 = −0.249 GeV for the charmonium system, and Table 2 indicates that the relativistic effect is about 2 λ = 0.2 GeV and V0 = −0.124 GeV for the bottomonium 22% for the J/ψ decay, which is consistent with the usu- system. al power relation for the relativistic terms, e.g. 2 ∼ . − . The mass spectra of vector charmonium and bot- vc 0 2 0 3. For the excited states, the relativistic ef- tomonium are shown in Table 1. The theoretical predic- fects are much larger than for the ground state. For the tions are consistent with the experimental data given by 2S, 3S, 4S and 5S states, the relativistic effects are about the Particle Data Group (PDG). As an example of the 34%, 41%, 52% and 62%, respectively. These results are wave functions, we present four J/ψ radial wave func- consistent with our previous study of the semi-leptonic ψ + → + ℓ+ + ν tions in Fig. 1: the dominant radial wave functions 5 and decays Bc cc¯ ℓ, where higher excited charmoni- 2 2 2 22) ψ6 and the two minor ones ⃗q ψ3/M and ⃗q ψ4/M . From um states were shown to have larger relativistic effects

Table 1. Mass spectra of the S wave cc¯ and bb¯ vectors in units of MeV. 'Th' is the theoretical prediction, 'Exp' are the experimental data from PDG [50].

nS Th(cc¯) Exp(cc¯) Th(bb¯) Exp(bb¯) 1S 3097.3 3096.9 9460.7 9460.3 2S 3686.4 3686.1 10020.5 10023.3 3S 4059.3 4039 10362.6 10355.2 4S 4337.5 4421 10622.2 10579.4 5S 4559.4 − 10835.1 10889.9

1) In this previous Letter, we have chosen different ΛQCD for charmonium and bottomonium. Since this parameter appears only in αs, which depends on ⃗q , and for more convenience of fitting the data, we choose the same ΛQCD for the two systems. 2 2 2 2 2) Here we show the curves of ⃗q ψ3/M and ⃗q ψ4/M other than ψ3 and ψ4, because they always appear in such a combined form in the applications.

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ψ 5 /ψ Fig. 1. (color online) Four typical radial wave functions of Fig. 2. Ratio 2 2 of the J radial wave functions. q ψ3/M J/ψ.

Table 2. 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].

NR−Re modes NR Re Re Exp + − . +2.20 +1.57 + . J/ψ → e e 10 95− . 8.95− . . 2 7 5.550.16 1 86 1 38 22 3−2.2 % + . ψ → + − . +1.05 . 0 60 + . . (2S ) e e 5 92− . 4 43−0.54 . 5 0 2 33 0.04 0 89 33 6−4.3 % + − . +1.68 +1.29 + . ψ(2S ) → τ τ 2 31− . 1.73− . . 3 2 0.910.14 2 31 1 73 33 6−4.1 % ψ → + − . +0.69 . +0.35 + . (3S ) e e 4 30− . 3 04− . . 5 6 0.860.07 0 66 0 35 41 4−6.1 % + . + . ψ → τ+τ− . 0 61 . 0 48 + . (3S ) 2 87−1.23 2 03−0.87 . 6 9 − 41 4−6.3% ψ → + − . +0.65 . +0.24 + . (4S ) e e 3 53− . 2 32− . . 11 1 0.480.22 0 66 0 26 52 2−12.9% + . ψ → τ+τ− . 0 26 . +0.25 + . (4S ) 2 70−0.60 1 78− . . 15 4 − 0 43 52 2−13.0 % ψ → + − . +0.55 . +0.16 + . (5S ) e e 3 05− . 1 88− . . 14 3 0.580.07 0 52 0 19 62 2−12.5% + − . +0.30 . +0.21 + . ψ(5S ) → τ τ 2 49− . 1 54− . . 16 4 − 0 58 0 31 62 2−23.1 %

2 4 [40]. This conclusion can also be obtained qualitatively vQ and the vQ corrections of the decay rate of QQ¯ from the plots of the radial wave functions. We men- quarkonium in the framework of NRQCD. In the case of tioned that the relative momentum q concerns the relat- J/ψ [39], the predicted relativistic effect is 34.1% for 2 ∼ . 2 ∼ . ive velocity vQ between the quark and antiquark in vc 0 3, and is 23.0% for vc 0 18. These values are quarkonium, q = 0.5mQvQ. As shown in Fig. 1, two non- consistent with our prediction of 22.3%. relativistic J/ψ radial wave functions always dominate In Table 2, we also show the theoretical uncertainties over the relativistic wave functions in the whole q region, caused by the choice of input parameters. We vary all leading to a small relativistic correction. For the excited parameters simultaneously within 10% of their central states, see Fig. 3 as an example of the radial wave func- values, and take the largest variation as the uncertainty. tions of ψ(2S ), the non-relativistic wave functions still With the errors, most predictions are much larger than the dominate in the small q region, but there is a node struc- experimental data. The only exception is the channel ture in each curve where the wave function changes sign. ψ(2S ) → τ+τ−, which has large uncertainties1). We note The contributions in the low q region may cancel each that a calculation of the J/ψ leptonic decay in lattice other, and the wave functions for large q (vQ) may give QCD with fully relativistic charm quarks was reported in sizable contributions, resulting in large relativistic correc- Ref. [52], and gave Γ(J/ψ → e+e−) = 5.48(16) keV , con- tions. sistent with the experimental data. This indicates that the There are other methods for considering the relativist- disagreement of our results with the experimental data ic effects in heavy quarkonium decays. For example, may be due to the lack of QCD corrections. Bodwin et al. [39] and Brambilla et al. [51] computed the We note that in a recent paper, Soni et al. [53] calcu-

1) The reason is that the ψ(2S ) mass is only a slightly heavier than that of two τ, so the phase space of this channel is very sensitive to the variation of parameters.

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vary from 20% to 28%. These predictions agree with those in literature. For example, Bodwin et al. [39] pre- 2 dicted the relativistic effect of 13.2% for vb ∼ 0.10 using 4 NRQCD up to the vb accuracy, and a lattice QCD prediction indicated that the relativistic effects are about 15%- 2 25% [22] for Υ(1S ) and Υ(2S ) up to the vb accuracy. We should point out that the above large relativistic effects are specific for bottomonium leptonic decays Υ(nS ) → ℓ+ℓ−, and are not universal for processes in- volving a bottomonium. In the di-lepton decays, the amp- ∫litude[ is proportional] to the wave function as 3⃗ ψ ⃗ − ⃗q2 ψ ⃗ d q 5(q) 3M2 3(q) , 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 the overlapping integral of the wave functions Fig. 3. (color online) Radial wave functions of ψ(2S ). ∫ 3 for the initial and final states d ⃗q ψA · ψB. Because the lated the quarkonium leptonic decay using the Cornell wave functions are large in the small q region, the potential in a non-relativistic version and with the pQCD product of two wave functions is suppressed in the large correction up to NLO. Their results for charmonium are q region compared to the case with one wave function, neither consistent with the experimental data nor with our and the contributions from the relativistic terms are results, while for bottomonium their results are compar- greatly suppressed. able with ours (see below). Also, Badalian et al. [54] cal- In Table 3, we also give the theoretical uncertainties, culated the decay rates of the ψ(1S − 4S ) leptinic decays which are obtained by varying all parameters simultaneously within 10% of the central values, and the largest with the QCD correction at NLO using the Cornell poten- variations are taken as the errors. Our relativistic results tial and the semi-Salpeter equation, and obtained 5.47 agree well with the experimental data. We also note that keV, 2.68 keV, 1.97 keV, and 1.58 keV , respectively, + − for Υ(1S ) → e e , PDG gives two different results: dir- which are smaller than our charmonium results. These ectly listed is Γ = 1.34 keV, but a branching ratio studies indicate that the relativistic corrections and QCD ee Br = 2.38% is also given, leading to ΓΥ(1S )→e+e− = 1.29 corrections are large for the charmonium system. keV using the full width ΓΥ(1S ) = 54.02 keV [50]. The 6.3 Bottomonium leptonic decay widths second value is the same as our relativistic result. Simil- arly, in the case of Υ(4S ) → e+e−, PDG directly lists We present the non-relativistic and relativistic results Γee = 0.272 keV [50], but from the branching ratio also of the bottomonium leptonic decay widths in Table 3. given in PDG, we get Γee = 0.322 keV. We hope PDG Similarly to charmonium, the relativistic corrections are will update the data in the near future. sizable. For the ground state Υ(1S ), the relativistic effect Table 3 shows that all relativistic results for is about 14%, and for the excited states Υ(2S − 5S ), they Γℓℓ(1S − 5S ) are consistent with the experimental data.

Table 3. 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].

NR−Re modes NR Re Re Exp Υ → + − + . + . + . (1S ) e e 1.47 0 23 0 19 0 9 1.3400.018 (1.290.09) −0.20 1.29−0.16 14.0−1.6% + − Υ(1S ) → τ τ +0.22 +0.18 +1.1 1.400.09 1.46−0.20 1.28−0.16 14.0−1.5% + − Υ(2S ) → e e +0.123 +0.104 +0.0 0.6120.011 0.771−0.125 0.629−0.088 22.6−3.2% + − Υ(2S ) → τ τ +0.120 +0.101 +0.0 0.640.12 0.766−0.123 0.625−0.086 22.6−3.3% + − Υ(3S ) → e e +0.088 +0.070 +7.8 0.4430.008 0.541−0.088 0.450−0.065 20.2−2.5% + − Υ(3S ) → τ τ +0.086 +0.068 +7.7 0.470.10 0.538−0.087 0.448−0.064 20.2−2.8% Υ → + − + . + . + . (4S ) e e 0.429 0 083 0 058 20.8 6 6% 0.2720.029 (0.3220.056) −0.059 0.355−0.050 −7.2 Υ → τ+τ− + . + . + . (4S ) 0.427 0 081 0 056 6 5 − −0.057 0.353−0.050 20.8−7.1% + − Υ(5S ) → e e +0.048 +0.048 +1.1 0.310.07 0.380−0.069 0.296−0.038 28.4−7.9% + − Υ(5S ) → τ τ +0.047 +0.047 +1.0 − 0.378−0.068 0.295−0.038 28.4−7.8%

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Our predictions also agree with the lattice QCD predic- cancellation between the numerator and denominator. + − Υ tion [22], Γ(Υ(1S ) → e e ) = 1.19(11) keV and Even though all central values of the ratios RττnS are smal- Γ(Υ(2S ) → e+e−) = 0.69(9) keV, with the NRQCD predic- ler than 1, they are consistent with the existing experi- tion [26], Γ(Υ(1S ) → e+e−) = 1.25 keV, and with the NR- mental data within errors. QCD prediction with NNNLO pQCD corrections [30], To cancel the model dependence of the theoretical Γ Υ → + − = . . α +0.01 µ predictions, we give in Table 5 and Table 6 the ratios ( (1S ) e e ) 1 08 0 05( s)−0.20( ) keV. Γ(ψ(nS ) → e+e−)/Γ(J/ψ → e+e−) and Γ(Υ(nS ) → ℓ+ℓ−)/ 6.4 Lepton flavor university Γ(Υ(1S ) → ℓ+ℓ−). For the Υ(nS ) decay, we obtain the To test the lepton flavor university, we give the ratios same central values for the e and τ final states, so we only ψ Υ + − + − nS nS present the ratio Γ(Υ(nS ) → ℓ ℓ )/Γ(Υ(1S ) → ℓ ℓ ) in Ta- Rττ and Rττ in Table 4. Their definitions are similar, for + − example, ble 5 and Table 6, which are calculated using the e e fi- + − nal states listed in Table 3. ψ Γ ψ → τ τ nS ( nS ) Γ(ψ(2S )) Rττ = . (29) Table 5 shows that the ratio is larger but close Γ(ψ → µ+µ−) Γ(J/ψ) nS to the experimental data, while the ratios for highly ex- ψnS The deviation of the ratio Rττ from the lepton flavor cited states are much larger than the experimental data. universality indicates the presence of new physics bey- Table 6 shows the bottomonium leptonic decay ratios. In ond the Standard Model. the row ‘Exp1’, the value of Γee(1S ) = 1.340 0.018 keV Table 4 shows the ratios calculated with the ‘Re’ val- is used, which is directly listed in PDG. In the row ues. The uncertainties of the ratios are from the variation ‘Exp2’, Γee(1S ) = 1.29 0.09 keV is used, which is calcu- of the input parameters. In the case of charmonium, the Υ → + − ψ lated using the branching ratio of (1S ) e e given in ratios RττnS are quite different from each other, since the PDG. For Υ(4S ), the results outside the brackets were ob- charmonium mass is a bit higher than of the two τ . For tained using Γee(4S ) = 0.272 0.029 keV from PDG, while the same reason, we get a large uncertainty. For bot- the results inside the brackets used Γee(4S ) = 0.322 0.056 tomonium, since the τ mass is much smaller than the bot- keV obtained from the PDG branching ratio. It can be tomonium mass, we get almost the same values for all ra- seen that all our theoretical predictions are consistent Υ tios RττnS Their uncertainty is also very small due to the with the experimental data.

+ − ψnS Γ(ψnS →τ τ ) ΥnS Table 4. Ratios Rττ = + − and Rττ . The experimental data are from PDG [50], with the statistical and systematic uncertainties added Γ(ψnS →µ µ ) together.

ψ S ψ S ψ S ψ S Rττ2 Rττ3 Rττ4 Rττ5

Ours +0.210 +0.073 +0.026 +0.039 0.391−0.391 0.668−0.237 0.767−0.112 0.819−0.091 Υ S Υ S Υ S Υ S Υ S Rττ1 Rττ2 Rττ3 Rττ4 Rττ5 Ours +0.001 +0.003 +0.002 +0.001 +0.000 0.992−0.006 0.994−0.003 0.996−0.002 0.995−0.002 0.997−0.002 CLEO [12] 1.02 0.07 1.04 0.09 1.05 0.13 − − BABAR[14] 1.005 0.035 − − − − PDG [50] 1.05 0.06 1.04 0.20 1.05 0.24 − −

Table 5. Ratio Γ(ψ(nS ) → e+e−)/Γ(J/ψ → e+e−).

Γ(ψ(2S )) Γ(ψ(3S )) Γ(ψ(4S )) Γ(ψ(5S )) Γ(J/ψ) Γ(J/ψ) Γ(J/ψ) Γ(J/ψ) Ours +0.019 +0.016 +0.013 +0.013 0.495−0.017 0.340−0.017 0.259−0.016 0.210−0.016 Exp [50] 0.42±0.02 0.15±0.02 0.086±0.042 0.10±0.02

+ − + − Table 6. Ratio Γ(Υ(nS ) → ℓ ℓ )/Γ(Υ(1S ) → ℓ ℓ ). ‘Exp1’ are the experimental data with Γee(1S ) = 1.340 0.018 keV, ‘Exp2’ are the experimental

data with Γee(1S ) = 1.29 0.09 keV. For Υ(4S ), Γee(4S ) = 0.272 0.029 (0.322 0.056) keV for the result inside (outside) the brackets. Γ(Υ(2S )) Γ(Υ(3S )) Γ(Υ(4S )) Γ(Υ(5S )) Γ(Υ(1S )) Γ(Υ(1S )) Γ(Υ(1S )) Γ(Υ(1S )) Ours +0.008 +0.003 +0.004 +0.003 0.488−0.009 0.349−0.008 0.275−0.000 0.229−0.001 Exp1 [50] 0.457±0.014 0.33±0.01 0.203±0.024 (0.240±0.045) 0.23±0.06 Exp2 [50] 0.47±0.04 0.34±0.03 0.21±0.04(0.25±0.06) 0.24±0.07

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7 Summary enough for charmonium. An improvement of the Cornell potential may be needed, and more importantly, the per- In this paper, we studied the leptonic decays of heavy turbative QCD corrections may have larger effect in char- vector quarkonia. For the charmonium decays, not all monium decays than in bottomonium decays. Since the states are consistent with the experimental data, while for BS equation is an integral equation, the QCD corrections the bottomonium decays, almost all S wave states are in from the gluon ladder diagrams are already included, but good agreement with the data. other QCD corrections in the kernel or in the quark Theoretical results of the ratios Γ(ψ(nS ) → e+e−)/ propagators may need to be improved in future calcula- Γ(J/ψ → e+e−) and Γ(Υ(nS ) → ℓ+ℓ−)/Γ(Υ(1S ) → ℓ+ℓ−) tions. were given in Ref. [55], where the potential model was The Bethe-Salpeter method provides a strict way to used including the v2 relativistic corrections and pQCD deal with the relativistic effects. In this framework, we Q found that the relativistic corrections are large and im- corrections at NLO. These results are comparable with ψ → ℓ+ℓ− ours, i.e. the charmonium leptonic decay widths are not portant for the leptonic decays (nS ) and Υ → ℓ+ℓ− ψ − consistent with the experimental data and the bottomoni- (nS ) . For the (1S 5S ) leptonic decays, the re- +3 +5 +6 +11 um leptonic widths are in good agreement with the data. lativistic effects are 22−2%, 34−5%, 41−6%, 52−13% and +14 This situation was also observed in Ref. [56]. It seems 62−12%, respectively. Therefore, for the highly excited that the same theoretical tool cannot provide satisfactory states ψ(nS ), the relativistic corrections give dominant results for both the charmonium and bottomonium sys- contributions. For the Υ(1S − 5S ) decays, the relativistic +1 +0 +8 +6 +2 tems [57]. There are several possible reasons for this dif- effects are 14−2%, 23−3%, 20−2%, 21−7% and 28−7%, re- ference in our study. It may be that the instantaneous ap- spectively. Thus, relativistic effects should be considered proximation works well for bottomonium, but is not good for a sound prediction of the heavy quarkonium decays.

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