Applying asymptotic expansions at threshold, we compute the two-loop QCD
correction to the short-distance coefficient that governs the leptonic decay
$\psi\to l^+ l^-$ of a S-wave quarkonium state and discuss its impact on the
relation between the quarkonium non-relativistic wave function at the origin
and the quarkonium decay constant in full QCD.Comment: 7 pages, LaTeX, 1 figure included via epsf.st

Beneke, M.

Signer, A.

Smirnov, V. A.

English

arXiv

Applying asymptotic expansions at threshold, we compute the two-loop QCD correction to the short-distance coefficient that governs the leptonic decay relation between the quarkonium non-relativistic wave function at the origin and the quarkonium decay constant in full QCD

Beneke, Martin

Signer, A

Smirnov, V A

CERN Document Server

CERN-TH/97-353hep-ph/9712302December 1997Two-loop Correction to the Leptonic Decayof QuarkoniumM. Beneke and A. SignerTheory Division, CERN, CH-1211 Geneva 23andV.A. SmirnovNuclear Physics Institute, Moscow State University,119889 Moscow, RussiaAbstractApplying asymptotic expansions at threshold, we compute the two-loop QCD cor-rection to the short-distance coecient that governs the leptonic decay  ! l+l of a S-wave quarkonium state and discuss its impact on the relation between thequarkonium non-relativistic wave function at the origin and the quarkonium decayconstant in full QCD.PACS Nos.: 13.20.Gd, 12.38.BxQuarkonium decays played an important role in establishing Quantum Chromodynamics(QCD) as a weakly interacting theory at short distances. Calculations of heavy quarko-nium decays usually proceed under the assumption that the heavy quark-antiquarkbound state is non-relativistic and that the decay amplitude factorizes into the boundstate 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 charmoniumstate J= , because it can decay only through electro-magnetic annihilation or annihila-tion into at least three gluons [1]. Today's understanding of quarkonium bound stateshas rened this picture and allows us to calculate relativistic corrections systematicallyat the expense of introducing further non-perturbative parameters that characterize thebound state. Such calculations can be done most transparently in the framework ofa 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. Besidespotentially large relativistic corrections, the size of radiative corrections to the quark-antiquark annihilation amplitudes has always been a matter of concern. The one-loopradiative corrections to the decays  ! l+l (where l = e; ) [4],  ! light hadrons, ! + light hadrons [5] are large and question the practicability of factorization forcharmonium and, perhaps, even bottomonium. [Here and in the following we use  asa label for any S-wave spin-triplet state, i.e. J= or  0for charmonium and (nS) forbottomonium.]In this Letter we address this question and report on the calculation of two-loopshort-distance corrections to the leptonic decay  ! l+l . This is the rst (and also`easiest') two-loop matching calculation for quarkonium decays and it can also be used toconnect the decay constant of the  meson dened in QCD with the non-relativistic wavefunction at the origin, which appears in NRQCD and potential models. In turn, thiswave function is an important input parameter for the prediction of other quarkoniumdecays and also quarkonium production cross sections.We recall that  decays leptonically through interaction with the electro-magneticcurrent. The partial decay rate, neglecting the tiny lepton masses, is exactly given by ( ! l+l ) =4e2Q2emf2 3M ; (1)where M is the mass of  , emthe ne structure constant and eQthe electric chargeof the heavy quark in units of the electron charge. The  decay constant f is denedthrough the following matrix element of the electro-magnetic current:h (p)jQQj0i = ( i)f M (p): (2)[(p) is the  polarization vector and p the  momentum.] The leptonic decay rates areknown experimentally [6]. For J= we nd fJ= = (405 15)MeV.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 andthe long-distance (bound state) scales are 1=(M v) and 1=(M v2), where v is the (small)1characteristic velocity of the  's quark constituents. The short-distance contributionscan be isolated, and calculated in perturbation theory, by matching the vector current inQCD onto a series of operators in NRQCD. Up to corrections of order v4, the matchingrelation is given byh (p)jQQj0i = i(p)"C0 s;mQ!h j yij0i()+C1s;mQ6m2Qh j y~D2ij0i() +O(v4)#; (3)where  (not to be confused with the  meson) and  denote non-relativistic two-spinors,~D the spatial covariant derivative and mQthe heavy quark mass. [More details onnotation and NRQCD can be found in Ref. [3]. Note, however, that we use a relativisticnormalization of states also for the matrix elements in NRQCD.] The matrix elements onthe right-hand side are dened in the  rest frame and (p) is the matrix that performsthe Lorentz boost into this frame. The matching coecients C0and C1are expressedas series in the strong coupling sand account for the short-distance QCD corrections.The matching coecients and matrix elements in NRQCD individually depend on thefactorization scale . C1is dened (as in Ref. [7]) such that C1= 1 +O(s) and [4]C0 s;mQ!= 1 2CFs(mQ)+ c2(mQ=)s2+ : : : ; (4)where CF= (N2c  1)=(2Nc) and Nc= 3 is the number of colours. We now discuss thecalculation of the two-loop matching coecient c2.Since the matching coecient contains only short-distance eects, it can be obtainedby replacing the quarkonium state  on both sides of Eq. (3) by a free quark-antiquarkpair of on-shell quarks at small relative velocity. In terms of this on-shell matrix element,the matching equation can be rewritten asZ2;QCD QCD= C0Z2;NRQCDZ 1J NRQCD+O(v2); (5)where Z2are the on-shell wave function renormalization constants in QCD and NRQCDand   the amputated, bare electro-magnetic annihilation vertices in QCD and NRQCD.The Feynman diagrams for  QCDat two loops are shown in Fig. 1. Since the currentJ =  yi need not be conserved in NRQCD, we allowed for its renormalization, Jbare=ZJJren, on the right-hand side. We then obtain C0by calculating all other quantities inEq. (5) in dimensional regularization and using the modied minimal subtraction scheme(MS scheme) [8].The matching calculation is considerably simplied, if one uses the threshold ex-pansion of Ref. [9] to compute  QCDdirectly as an expansion in v2. The thresholdexpansion is obtained by writing down contributions corresponding to hard (l  mQ),soft (l  mQv), potential (l0 mQv2, li mQv) and ultrasoft (l  mQv2) regions. The2D1D2D3D4D5D6{ D9Figure 1: Diagrams that contribute to  QCD. Symmetric diagrams exist for D2;3;5. The lastdiagram summarizes vacuum polarization contributions from massless fermions (D6), gluons(D7), ghosts (D8) and the massive fermion with mass mQ(D9).contributions from soft, potential and ultrasoft loop momenta can all be identied withdiagrams in NRQCD that appear in the calculation of  NRQCD. Hence, they drop out ofthe matching relation Eq. (5) and it suces to compute the contribution to the thresholdexpansion of the diagrams in Fig. 1, where all loop momenta are hard. [The thresholdexpansion is not only convenient; it also provides an implicit denition of NRQCD in di-mensional regularization. This is necessary, because dimensionally regularized NRQCDis not given by the dimensionally regularized Feynman integrals constructed from thevertices and propagators of NRQCD. In order to avoid that the cut-o for the eectivetheory be treated as larger than mQ, dimensionally regularized NRQCD has to be sup-plemented by a prescription for expanding the Feynman integrands. This prescription isprovided to all orders as part of the diagrammatic threshold expansion method [9].]We briey describe the calculation of the hard contributions to  QCD. [Details of thecalculational method and a solution to the recurrence algorithm for the two-loop integralswill be given in a long write-up of this Letter.] The spinor structure of the on-shell matrixelement in QCD is conventionally parametrized by two form factors, F1and F2, of whichonly the combination F1+ F2is required here. Since terms of order v2are not neededto determine C0[see Eq. (5)], we may set the relative momentum to zero and computethe form factors directly at threshold. The form factors have Coulomb singularities atthreshold and diverge as 1=v2. However, these singularities appear only in the soft,potential and ultrasoft contributions, and the hard contribution is well-dened directly3Colour factor121niteD1C2F932 2764 5224 81128 133296 52ln 212 35(3)8D2C2F 316 4332733192+9712576D3CFCA1532 564 216715384 3192576 2ln 28 21(3)16D4CF(CA  2CF) 0316 216 3932 25121152 32ln 28 31(3)16D5CF(CA  2CF)  932 1964761384+115721152+2ln 26 3(3)4D6CFTFnf 18548 355288 5248D7CFCA19128 5376867874608+952768D8CFCA112817683614608+52768D9CFTF 141348 14596+5272Sum C2F2132 9964 212637384 7332576+ (3)CFCA1132 49192 2848111152+2092576 2ln 23  4(3)CFTFnf 18548 355288 5248CFTF 141348 14596+572Table 1: Coecient of (s=)2(eEm2Q=(42)) 2for the hard contribution to the diagramsof Fig. 1 evaluated at threshold q2= 4m2. D2;3;5include a factor of 2 to account for thecorresponding symmetric diagrams. For SU(3) the colour factors are CF= 4=3, CA= 3,TF= 1=2. The number of light (massless) quark avours is denoted by nf.at threshold in dimensional regularization. The loop integrals simplify considerably, oncethe relative momentum is set to zero, since they then depend only on a single scale. Wethen project on the form factor F1+ F2and reduce all integrals to integrals withoutnumerators. These integrals can be further reduced to `simple' integrals and two non-trivial two-loop integrals by means of recurrence relations derived from integration byparts in the loop momenta [10]. The solution to the recurrence relations is the dicultpart of the calculation. The remaining non-trivial two-loop integrals can be calculatedexplicitly using standard Feynman parameters. The results obtained for the diagramsof Fig. 1 are listed in Table 1.After summing all the diagrams, multiplying by the two-loop QCD on-shell wavefunction renormalization constant [11], and performing (one-loop) coupling and massrenormalization the result still contains poles in  = (4   d)=2 (where d is the space-time dimension). Since the wave function renormalization constant Z2;NRQCDin NRQCDequals 1 up to higher-order terms in v2, not needed here, these poles are attributed to4an anomalous dimension of J , which rst arises at the two-loop order. As a consequencethe matrix element h j yij0i() is factorization scale-dependent. We dene it in theMS scheme and obtain the anomalous dimension for the NRQCD vector current J :J=d lnZJd ln=  CF(2CF+ 3CA)26s2+O(3s): (6)The scale dependence is compensated by the scale dependence of the two-loop matchingcoecient c2(mQ=) in Eq. (4). Separating the dierent colour group factors, the nalresult for c2(mQ=) in the MS scheme is:c2(mQ=) = C2Fc2;A+ CFCAc2;NA+ CFTFnfc2;L+ CFTFc2;H; (7)c2;A= 2"16ln m2Q2! 7936+ ln 2#+238 (3)2; (8)c2;NA= 2"14ln m2Q2!+89144 56ln 2# 15172 13(3)4; (9)c2;L=1118; (10)c2;H=  229+229: (11)Here (3) = 1:202 : : : and we have taken all fermions with masses less than mQasmassless, which is a good approximation even for mQ= mb, the bottom quark mass, inwhich case we neglect mc, the charm quark mass. The coecient c2;Lproportional to nf,the number of light fermions, has been recently obtained by Braaten and Chen [7]. Notealso that the C2F-term of the form factors F1;2close to threshold has been calculated byHoang [12], using the absorptive parts of the form factors obtained in Ref. [13]. Becausethe result in Ref. [12] contains hard and soft (potential, ultrasoft) contributions, it is notpossible to extract the matching coecient c2;Afrom Ref. [12]. [The structures in c2;Athat cannot arise from the small loop momentum regions agree with Hoang's result.]The size of the two-loop correction to C0[Eq. (4)] given by Eqs. (7){(11) is enormous.We dene the (scale-dependent) non-relativistic decay constant ash j yij0i() = ( i)fNR ()M i; (12)in analogy with Eq. (2). The non-relativistic decay constant is related to the  wavefunction at the origin by M (fNR )2= 12 j	(0)j2. Using Eq. (3), we obtainf = 1 8s(mQ)3  (44:55  0:41nf)s2+ : : :!fNR (mQ): (13)With s(mc)  0:35 and s(mb)  0:21 the second-order correction exceeds the rst-order correction even for the bottomonium states. For charmonium, the second-order5term is almost twice as large as the already sizeable rst-order correction. [Note thatthe BLM estimate of the two-loop correction [7] is far o the exact two-loop result.]Perturbative matching at the scale  = mQdoes not seem to work. Can the factorizationof short- and long-distance eects still be useful?A novel, and perhaps unexpected, aspect at the two-loop level is the factorizationscale dependence of the non-relativistic decay constant and, hence, the quarkonium wavefunction at the origin. The scale dependence is large, especially due to the non-abelianterm in Eq. (6). This scale dependence of the wave function indicates the limitation ofthe non-relativistic potential model approach already at leading order in v2, since thewave functions obtained from solving the Schrodinger equation are scale-independent.Despite this shortcoming it may be argued that the wave function/non-relativistic decayconstant obtained from potential models corresponds { if to anything { to the wavefunction evaluated at a scale typical for the bound state and not mQ. This point ofview is also evident if the wave function is computed non-perturbatively using latticeNRQCD, in which case the ultraviolet cut-o/factorization scale is also much smallerthan mQ. We consider  = 1GeV as an adequate bound state scale for bottomoniumand charmonium, as the applicability of perturbation theory prevents us from taking yetsmaller scales. We then nd, for bottomonium (mb= 5GeV, nf= 4):f(nS)=0@1 8s(mb)3  1:74 s(mb)!2+ : : :1AfNR(nS)(1GeV)  0:81 fNR(nS)(1GeV):(14)The numerical factor 0.81 is stable against variations of the scale of the coupling atxed factorization scale 1GeV. At this factorization scale the second-order correction isnumerically insignicant. Although we do not know whether the three-loop correctionwould also be small at the low factorization scale, we tend to consider Eq. (14) as anaccurate prediction. This prediction could be tested, if fNR(nS)(1GeV) were accuratelyknown, for example from NRQCD lattice simulations. The large scale dependence raisesthe theoretical question (the answer to which we postpone to the long write-up) as towhether it is necessary to resum the logarithms ln(m2Q=2) to all orders.The situation is less favourable for charmonium states. Since mc 1:5GeV, thesize of the second-order correction is altered little for scales  > 1GeV. We have notsucceeded to nd a trustworthy interpretation of Eq. (13) and conclude that the fac-torization of short-distance and long-distance eects may not be useful in practice forcharmonium. Since the leptonic decay is the simplest conceivable decay, this puts intoquestion the possibility to obtain universal relations between various charmonium decaysand production processes through the use of NRQCD [3,14]. This pessimistic conclu-sion may be biased by our use of the MS factorization scheme. It is conceivable thatother factorization schemes or relations between physical observables, from which thewave function at the origin is eliminated, exhibit better convergence properties of theirperturbative series. A denitive conclusion on this issue can be obtained only once asecond quarkonium decay is computed to two-loop order.6While this paper was being written, Czarnecki and Melnikov [15] also considered thetwo-loop matching of the electro-magnetic current, also using the asymptotic expansionmethod of [9]. After correction of a trivial normalization error (A. Czarnecki and K.Melnikov, private communication), their result agrees with ours.Acknowledgements. We thank G. Buchalla for useful discussions and A. Hoang for cor-respondence. The work of V.S. has been supported by the Russian Foundation for BasicResearch, project 96{01{00726, and by INTAS, grant 93{0744.References[1] T. Appelquist and H.D. Politzer, Phys. Rev. Lett. 34, 43 (1975); A. De Rujula andS.L. Glashow, Phys. Rev. Lett. 34, 46 (1975).[2] W.E. Caswell and G.P. Lepage, Phys. Lett. B167, 437 (1986).[3] G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51, 1125 (1995) [Erratum:ibid. D55, 5853 (1997)].[4] R. Barbieri, R. Gatto, R. Korgerler and Z. Kunszt, Phys. Lett. B57, 455 (1975).The one-loop result can also be inferred from earlier QED calculations: R. Karplusand A. Klein, Phys. Rev. 87, 848 (1952).[5] P.B. Mackenzie and G.P. Lepage, Phys. Rev. Lett. 47, 1244 (1981).[6] Particle Data Group, R.M. Barnett et al., Phys. Rev. D54, 1 (1996).[7] E. Braaten and Y.-Q. Chen, OHSTPY-HEP-T-18 [hep-ph/9710357].[8] W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D18, 3998 (1978).[9] M. Beneke and V.A. Smirnov, CERN-97-315 [hep-ph/9711391].[10] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192, 159 (1981).[11] D.J. Broadhurst, N. Gray and K. Schilcher, Z. Phys. C52, 111 (1991).[12] A.H. Hoang, Phys. Rev D56, 7276 (1997).[13] R. Barbieri, J.A. Mignaco and E. Remiddi, Nuovo Cim. 11A, 824 (1972).[14] For reviews, see: E. Braaten, S. Fleming and T.C. Yuan, Annu. Rev. Nucl. Part.Sci. 46, 197 (1996); M. Beneke, in: The strong interaction, from hadrons to partons,p. 549, Stanford 1996 [hep-ph/9703429].[15] A. Czarnecki and K. Melnikov, TTP97-54 [hep-ph/9712222].7

Two-loop Correction to the Leptonic Decay of Quarkonium

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Two-loop Correction to the Leptonic Decay of Quarkonium

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