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Two-Loop Correction to the Leptonic Decay of Quarkonium

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Two-Loop Correction to the Leptonic Decay of Quarkonium CERN-TH/97-353 hep-ph/9712302 December 1997 Two-lo op Correction to the Leptonic Decay of Quarkonium M. Beneke and A. Signer Theory Division, CERN, CH-1211 Geneva 23 and V.A. Smirnov Nuclear Physics Institute, Moscow State University, 119889 Moscow, Russia Abstract Applying asymptotic expansions at threshold, we compute the two-lo op QCD cor- + rection to the short-distance co ecient that governs the leptonic decay ! l l of a S -wave quarkonium state and discuss its impact on the relation b etween the quarkonium non-relativistic wave function at the origin and the quarkonium decay constant in full QCD. PACS Nos.: 13.20.Gd, 12.38.Bx Quarkonium decays played an imp ortant role in establishing Quantum Chromo dynamics (QCD) as a weakly interacting theory at short distances. Calculations of heavy quarko- nium decays usually pro ceed under the assumption that the heavy quark-antiquark b ound state is non-relativistic and that the decay amplitude factorizes into the b ound state wave function at the origin and a short-distance quark-antiquark annihilation am- plitude. One can then explain the small width of the S -wave spin-triplet charmonium state J= , b ecause it can decay only through electro-magnetic annihilation or annihila- tion into at least three gluons [1]. Today's understanding of quarkonium b ound states has re ned this picture and allows us to calculate relativistic corrections systematically at the exp ense of intro ducing further non-p erturbative parameters that characterize the b ound state. Such calculations can be done most transparently in the framework of a non-relativistic e ective eld theory (NRQCD) [2,3] that implements the factoriza- tion of contributions to the (partial) decay widths from di erent length scales. Besides p otentially large relativistic corrections, the size of radiative corrections to the quark- antiquark annihilation amplitudes has always b een a matter of concern. The one-lo op + radiative corrections to the decays ! l l (where l = e; ) [4], ! light hadrons, ! + light hadrons [5] are large and question the practicability of factorization for charmonium and, p erhaps, even b ottomonium. [Here and in the following we use as 0 a lab el for any S -wave spin-triplet state, i.e. J= or for charmonium and (nS ) for b ottomonium.] In this Letter we address this question and rep ort on the calculation of two-lo op + short-distance corrections to the leptonic decay ! l l . This is the rst (and also `easiest') two-lo op matching calculation for quarkonium decays and it can also b e used to connect the decay constant of the meson de ned in QCD with the non-relativistic wave function at the origin, which app ears in NRQCD and p otential mo dels. In turn, this wave function is an imp ortant input parameter for the prediction of other quarkonium decays and also quarkonium pro duction cross sections. We recall that decays leptonically through interaction with the electro-magnetic current. The partial decay rate, neglecting the tiny lepton masses, is exactly given by 2 2 2 4e f Q em + ( ! l l )= ; (1) 3M where M is the mass of , the ne structure constant and e the electric charge em Q of the heavy quark in units of the electron charge. The decay constant f is de ned through the following matrix element of the electro-magnetic current: h (p)jQ Qj0i =(i)f M (p): (2) [ (p)isthe p olarization vector and p the momentum.] The leptonic decay rates are known exp erimentally [6]. For J= we nd f = (405 15) MeV. J= The decay constant parametrizes the strong interaction e ects and contains long- and short-distance contributions. For quarkonium the short-distance scale is 1=M and 2 the long-distance (b ound state) scales are 1=(M v ) and 1=(M v ), where v is the (small) 1 characteristic velo city of the 's quark constituents. The short-distance contributions can b e isolated, and calculated in p erturbation theory,by matching the vector currentin 4 QCD onto a series of op erators in NRQCD. Up to corrections of order v , the matching relation is given by " ! m Q i y ; h (p)jQ Qj0i = (p) C h j j0i() s 0 i # m Q ; C s 1 y 2 4 ~ + h j D j0i()+ O(v ) ; (3) i 2 6m Q where (not to b e confused with the meson) and denote non-relativistic two-spinors, ~ D the spatial covariant derivative and m the heavy quark mass. [More details on Q notation and NRQCD can b e found in Ref. [3]. Note, however, that we use a relativistic normalization of states also for the matrix elements in NRQCD.] The matrix elements on the right-hand side are de ned in the rest frame and (p) is the matrix that p erforms the Lorentz b o ost into this frame. The matching co ecients C and C are expressed 0 1 as series in the strong coupling and account for the short-distance QCD corrections. s The matching co ecients and matrix elements in NRQCD individually dep end on the factorization scale . C is de ned (as in Ref. [7]) such that C =1+O( ) and [4] 1 1 s ! 2 m 2C (m ) Q F s Q s C ; =1 +c (m =) + :::; (4) 0 s 2 Q 2 where C =(N 1)=(2N ) and N =3 is the number of colours. We now discuss the F c c c calculation of the two-lo op matching co ecient c . 2 Since the matching co ecient contains only short-distance e ects, it can b e obtained by replacing the quarkonium state on b oth sides of Eq. (3) by a free quark-antiquark pair of on-shell quarks at small relativevelo city. In terms of this on-shell matrix element, the matching equation can b e rewritten as 1 2 Z = C Z Z + O (v ); (5) 2;QCD QCD 0 2;NRQCD NRQCD J where Z are the on-shell wave function renormalization constants in QCD and NRQCD 2 and the amputated, bare electro-magnetic annihilation vertices in QCD and NRQCD. The Feynman diagrams for at two lo ops are shown in Fig. 1. Since the current QCD y J = need not b e conserved in NRQCD, we allowed for its renormalization, J = i bare Z J ,on the right-hand side. We then obtain C by calculating all other quantities in J ren 0 Eq. (5) in dimensional regularization and using the mo di ed minimal subtraction scheme ( MS scheme) [8]. The matching calculation is considerably simpli ed, if one uses the threshold ex- 2 pansion of Ref. [9] to compute directly as an expansion in v . The threshold QCD expansion is obtained by writing down contributions corresp onding to hard (l m ), Q 2 2 soft (l m v ), p otential (l m v , l m v ) and ultrasoft (l m v ) regions. The Q 0 Q i Q Q 2 D D 1 2 D D 3 4 D D { D 5 6 9 Figure 1: Diagrams that contribute to . Symmetric diagrams exist for D . The last QCD 2;3;5 diagram summarizes vacuum p olarization contributions from massless fermions (D ), gluons 6 (D ), ghosts (D ) and the massive fermion with mass m (D ). 7 8 Q 9 contributions from soft, p otential and ultrasoft lo op momenta can all b e identi ed with diagrams in NRQCD that app ear in the calculation of . Hence, they drop out of NRQCD the matching relation Eq. (5) and it suces to compute the contribution to the threshold expansion of the diagrams in Fig. 1, where all lo op momenta are hard. [The threshold expansion is not only convenient; it also provides an implicit de nition of NRQCD in di- mensional regularization. This is necessary, b ecause dimensionally regularized NRQCD is not given by the dimensionally regularized Feynman integrals constructed from the vertices and propagators of NRQCD. In order to avoid that the cut-o for the e ective theory be treated as larger than m , dimensionally regularized NRQCD has to be sup- Q plemented by a prescription for expanding the Feynman integrands. This prescription is provided to all orders as part of the diagrammatic threshold expansion metho d [9].] We brie y describ e the calculation of the hard contributions to . [Details of the QCD calculational metho d and a solution to the recurrence algorithm for the two-lo op integrals will b e given in a long write-up of this Letter.] The spinor structure of the on-shell matrix element in QCD is conventionally parametrized bytwo form factors, F and F , of which 1 2 2 only the combination F + F is required here. Since terms of order v are not needed 1 2 to determine C [see Eq. (5)], we may set the relative momentum to zero and compute 0 the form factors directly at threshold. The form factors have Coulomb singularities at 2 threshold and diverge as 1=v . However, these singularities app ear only in the soft, p otential and ultrasoft contributions, and the hard contribution is well-de ned directly 3 1 1 Colour factor nite 2 2 2 2 35 (3) 9 5 133 5 ln 2 27 81 2 D C 1 F 32 64 24 128 96 12 8 2 733 3 43 971 2 D C + 2 F 16 32 192 576 2 2 2 21 (3) 15 5 715 319 ln 2 D C C 3 F A 32 64 16 384 576 8 16 2 2 2 31 (3) 3 39 251 3 ln 2 D C (C 2C ) 0 4 F A F 16 16 32 1152 8 16 2 2 3 (3) 761 1157 ln 2 9 19 + + D C (C 2C ) 5 F A F 32 64 384 1152 6 4 2 1 5 355 5 D C T n 6 F F f 8 48 288 48 2 19 53 6787 95 D C C + 7 F A 128 768 4608 768 2 1 1 361 5 + D C C 8 F A 128 768 4608 768 2 1 13 145 5 D C T + 9 F F 4 48 96 72 2 2 21 99 637 733 2 Sum C + (3) F 32 64 12 384 576 2 2 2 11 49 4811 209 ln 2 C C + 4 (3) F A 32 192 8 1152 576 3 2 1 5 355 5 C T n F F f 8 48 288 48 1 13 145 5 C T + F F 4 48 96 72 2 2 2 2 E Table 1: Co ecientof( = ) (e m =(4 )) for the hard contribution to the diagrams s Q 2 2 of Fig.
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