QCD prediction for the non- D D ¯ annihilation decay of ψ(3770)
aa r X i v : . [ h e p - ph ] S e p QCD prediction for the non- D ¯ D annihilation decay of ψ (3770) Zhi-Guo He, Ying Fan, and Kuang-Ta Chao Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
To clarify the marked difference between BES and CLEO measurements on the non- D ¯ D decaysof the ψ (3770), a 1 D -dominated charmonium, we calculate the annihilation decay of ψ (3770) inNRQCD. By introducing the color-octet contributions, the results are free from infrared divergences.The color-octet matrix elements are estimated by solving the evolution equations. The S-D mixingeffect is found to be very small. With m c = 1 . ± . ψ (3770) → light hadrons) =467 − KeV. For m c = 1 . D ¯ D decay branching ratio of ψ (3770) could reach about 5%. Our results do not favor eitherof the results of BES and CLEO collaborations, and further experimental tests are urged. PACS numbers: 12.38.Bx, 12.39.Jh, 13.20.Gd,
Heavy quarkonia decays play an important role in un-derstanding quantum chromodynamics (QCD)[1]. Theseinclude not only the determination of the running strongcoupling constant α s from S-wave decays J/ψ → ggg andΥ → ggg , but also the study of factorization from theP-wave annihilation decays, where appear infrared (IR)divergences in P → ggg and P J → gq ¯ q [2, 3]. A tra-ditional way to treat the IR divergences was to use thequark binding energy or the gluon momentum as cut-off to estimate these IR divergences, but this is modeldependent and breaks factorization of short and longdistance processes. In [4], a new factorization schemewas proposed to absorb the IR logarithms by new non-perturbative parameters, the color octet matrix elements.Based on the non-relativistic nature of heavy quarkonia,an effective theory, Non-Relativistic QCD (NRQCD) wasdeveloped [5], in which the inclusive annihilation decayscan be calculated in a systematic way by double expan-sions in terms of α s and v , the relative velocity of quarksin heavy quarkonium. In [6, 7, 8], the authors calculatedQCD radiative corrections to the light hadron (LH) de-cays of P-wave charmonium in NRQCD, and showed ex-plicitly the cancelation of infrared divergences at the nextto leading order (NLO). In [9] a more complete and pre-cise NLO calculation for the P-wave decay perturbativecoefficients in NRQCD is given (see also [10]). At NLO in α s , the NRQCD predictions for the relative decay ratesof χ cJ → LH are consistent with more updated data (seeChapter 4 of [1]). Moreover, the relativistic corrections ofS and P-wave electromagnetic quarkonium decays havebeen given at order v [11]. As for the D-wave, in [12, 13]calculations of D J → ggg decays were given but suf-fered from IR divergences; while in [8] only the leadingorder (LO) color-octet contribution to D J → LH wasgiven. So for the D-wave a complete calculation for theIR cancelation and radiative correction in NRQCD is ap-parently needed.Phenomenologically, for the D ( J P C = 1 −− ) char-monium state ψ (3770), there is a long-standing puz-zle in its non- D ¯ D decays that the ψ (3770) might havesubstantial decays not into D ¯ D and D + D − . BES earlier reported two results based on different analy-sis methods: Br( ψ (3770) → non- D ¯ D ) = (14 . ± . ± . ψ (3770) → non- D ¯ D ) = (16 . ± . ± . σ ( e + e − → ψ (3770) → non- D ¯ D ) = − . ± . +0 . − . nb. Very recently, with the first direct mea-surement on the non- D ¯ D decay, BES gives σ ( e + e − → ψ (3770) → non- D ¯ D ) = (0 . ± . ± .
29) nb andBr( ψ (3770) → non- D ¯ D ) = (13 . ± . ± . D ¯ D decay of ψ (3770). Meanwhile, a numberof experiments to search for the exclusive hadronic non- D ¯ D decays of ψ (3770) have been done by BES[18] andCLEO[19], but no significant signals are found.At least two kinds of non- D ¯ D decays of ψ (3770) havebeen observed. The hadronic transitions ψ (3770) → π + π − J/ψ was first observed by BES with a branchingratio of (0 . ± . ± . ψ (3770) → π + π − J/ψ ) = (0 . ± . ± . π π J/ψ and ηJ/ψ modes were also seen witheach having a branching ratio of about one half of thatof π + π − J/ψ [22]. These results are within the rangeof theoretical predictions based on the QCD multipoleexpansion for hadronic transitions [20]. With the totalwidth of 23 . ± . ψ (3770) [23], the width ofall hadronic transitions is about 100 −
150 KeV. Anotherkind of non- D ¯ D decays of ψ (3770) are the E1 transi-tions ψ (3770) → γ + χ cJ (J=0,1,2), and their widths aremeasured by CLEO to be 172 ± , ± , <
21 KeVfor J=0,1,2 respectively [24], which are in good agree-ment with predicted values 199, 72, 3.0 KeV in a QCD-inspired potential model calculation with relativistic cor-rections [25] (see also [26, 27]). The width of all E1 transi-tions ψ (3770) → γ + χ cJ (J=0,1,2) is about 250 ± D ¯ D de-cay width and branching ratio of ψ (3770).To clarify the puzzle of ψ (3770) non- D ¯ D decay, in thisletter we will give a complete infrared safe NLO QCDcorrections to the annihilation decay rate of the ψ (3770)in the framework of NRQCD. Since v ∼ α s ( m c ) ≈ . ψ (3770) can be viewed as a 1 D dominated statewith a small admixture of 2 S , and expressed as (seee.g. [25, 26]) | ψ (3770) i = cos θ | D i + sin θ | S i , | ψ (3686) i = − sin θ | D i + cos θ | S i , (1)where θ is the S-D mixing angle and it is about (12 ± ◦ by fitting the leptonic decay widths of ψ (3770) and ψ (3686). Then the LH decay width of ψ (3770) isΓ( ψ (3770) → LH ) = cos θ Γ(1 D → LH ) +sin θ Γ(2 S → LH ) + IF, (2)where IF stands for the S-D interference term. The cal-culation of S-wave decay at order α s and leading order in v is trivial, and it givesΓ(2 S → LH ) = | R S (0) | π α s ( π − m c , (3)where R S (0) is the 2 S wave function at the origin.The S-D interference term IF in Eq.(2) is infrared finiteat leading order in v and α s , and can be obtained bycombining the 1 D → g with 2 S → g amplitudes IF = 2 sin θ cos θ −
240 + 71 π ) α s m c R S (0) √ π r π R ′′ D (0) , (4)where R ′′ D (0) is the second derivative of the 1 D wavefunction at the origin.We now proceed with the calculation of the main part,the D-wave quarkonium LH decay. In NRQCD, the in-clusive annihilation decay of D at leading order in v is factorized asΓ( D J → LH ) = 2Im f ( D [1] J ) H D + X J =0 f ( P [8] J ) H P J + 2Im f ( S [8]1 ) H S + 2Im f ( S [1]1 ) H S , (5)where Im f ( n ) is the imaginary part of the Q ¯ Q → Q ¯ Q scattering amplitude, and can be calculated perturba-tively . And the corresponding non-perturbative matrixelements are H D = h H |O ( D ) | H i m c , H P J = h H |O ( P J ) | H i m c ,H S = h H |O ( S ) | H i m c , H S = h H |O ( S ) | H i m c , (6)where H is ψ (1 D ). Those four-fermion operators of S-wave and P-wave are defined in [5], and here we only givethe definition of the D-wave four-fermion operator (thenormalization of the color singlet four-fermion operatorsagree with those in [9]): O ( D ) = 310 N c ψ † T i χχ † T i ψ, (7)where T i = σ j S ij and S kl = ( − i ) ( ←→ D i ←→ D j − ←→ D δ ij ).We calculate the short distance coefficients at order α s ,and details of our calculation will be given elsewhere. TheS-wave and P-wave short-distance coefficients have beencalculated in [9], and our calculated results agree withtheirs. The D-wave short distance coefficients presentedhere are new, and they are2Im f ( S [1]1 ) = 40 α s ( π − , (8a) 2Im f ( S [8]1 ) = α s
108 (36 N f π + α s (5( −
657 + 67 π )+ N f (642 − N f − π + 72 ln
2) + 144 β N f ln µ m c )) , (8b)2Im f ( P [8]0 ) = 5 α s π + α s (9096 − N f + 63 π + 2520 ln β ln µ m c + 96 N f ln m c µ Λ )) , (8c)2Im f ( P [8]1 ) = 5 α s (4107 − N f − π + 48 N f ln m c µ Λ )648 , (8d)2Im f ( P [8]2 ) = α s
648 (432 π + α s (12561 − N f − π + 1008 ln β ln µ m c + 240 N f ln m c µ Λ )) , (8e)2 Imf ( D [1]1 ) = (321 π − − ln µ Λ m c ) α s , (8f)where β = N c − N f , N c = 3, N f is the number offlavors of light quarks. µ and µ Λ are renormalizationand factorization scales respectively. We consider tenprocesses to get the short distance coefficients in Eq.[8],including gg, ggg, q ¯ q , and q ¯ qg final states. The contri-butions of q ¯ q and q ¯ qg processes are labeled by the powersof N f .After calculating the short distance coefficients, wecome to determine the long-distance matrix elements.In the P-wave charmonium decay, at leading orderin v there are two four-fermion operators H H H D , H P , H S , H S , where H P = h H |O ( P ) | H i m c = h H |O ( P ) | H i m c = h H |O ( P ) | H i m c ,and these relations can be derived by considering the E1transition from D to P J . In NRQCD, H D is relatedto the wave function’s second derivative at the origin,while for the other three, in the absence of lattice sim-ulations and phenomenological inputs, we will resort tothe operator evolution equation method suggested in [5],where the authors give the result of the matrix elementsin the P-wave decay. Here we derive the following matrixelements in the D-wave case H P = 59 8 C F β ln( α s ( µ Λ ) α s ( µ Λ ) ) H D , (9a) H S = C F B F β ) ln ( α s ( µ Λ ) α s ( µ Λ ) ) H D , (9b) H S = C F N c ( 83 β ) ln ( α s ( µ Λ ) α s ( µ Λ ) ) H D , (9c)where C F = , B F = . We choose the region of validityof the evolution equation: the lower limit µ Λ = m c v andthe upper limit µ Λ of order m c .With both the obtained short distance coefficients andlong distance matrix elements, we predict the LH decaywidth of D . The renormalization proceeds by usingthe M S scheme for the coupling constant α s and theon shell scheme for the charm quark mass. For conve-nience, we take the factorization scale µ Λ to be the sameas the renormalization scale µ of order m c . We choosethe pole mass m c = 1 . , v = 0 . , µ Λ = m c v, µ Λ =2 m c , α s (2 m c ) = 0 . , N f = 3 , Λ QCD = 390MeV , H D = | R ′′ D (0) | πm c = 0 . × − GeV[28]. At O ( α s ), the LH de-cay involves three subprocesses ( P ) → gg, ( P ) → gg, ( S ) → q ¯ q , and the decay width is estimated to beΓ( D → LH ) = 0 . . (10)At O ( α s ), there will be seven more subprocesses( S ) , → ggg, ( P ) → ggg, ( P J ) → q ¯ qg, ( D ) → ggg involved, and the result turns to beΓ( D → LH ) = 0 . . (11) TABLE I: Subprocess decay rates of D charmonium, where v = 0 . , µ Λ = 2 m c , α s (2 m c ) = 0 . S ) → LH S ) → LH
18 33( P ) → LH
184 410( P ) → LH P ) → LH D ) → LH ( D --- > L H )( M e V ) mcNLOLO FIG. 1: Renormalization scale µ -dependence of the decaywidth of charmonium D to light hadrons. Here NLO meansLO contribution+NLO correction Our result shows that in NRQCD factorization the NLOQCD correction is even larger than the LO result. Thenumerical values for all subprocesses are listed in Table1. If we choose µ Λ = m c , α s ( m c ) = 0 . µ dependence of the decay rate isshown in Fig.(1). We see that the µ -dependence at O ( α s )is rather mild when µ > . m c . For simplicity we take µ = 2 m c , where the logarithm term ln µ m c = 0 . With the pole mass m c = 1 . , α s (2 m c ) = 0 . | R ′′ D (0) | = 0 . , and | R S (0) | = 0 . , θ = 12 ◦ , we find that the three terms on the right handside of Eq.(2) contribute 417, 5.3, 44 KeV respectively tothe LH decay of ψ (3770), and result inΓ( ψ (3770) → LH ) = 467KeV . (12)Our result shows that the D-wave LH decay is dominant,and the S-D mixing only has a very small effect on the ψ (3770) LH decay. One important uncertainty of ourprediction is associated with the long-distance matrix el-ements, especially the color-octet matrix elements. Usingthe same evolution equation method in χ cJ decays, wefind the ratio of color octet to color singlet P wave de-cay matrix elements agree with the lattice calculation[29]to within about 20% and with the phenomenologicalvalues[7, 10] to within about 30%. This might indicate,though not compellingly, that the uncertainty related tothe matrix elements calculated using the evolution equa-tion in the D-wave decays are also about (20-30)% or(with more confidence) less than 50%. Other uncertain-ties such as the relativistic corrections and higher orderQCD radiative corrections are beyond the scope of thepresent study. On the other hand, however, we find thedecay rate to be sensitive to the value of the charm quarkmass. If we choose the pole mass m c = 1 . ± . α s ( µ ) = α s (2 m c ), and fix other parameters as before,then our prediction becomesΓ( ψ (3770) → LH ) = 467 − KeV( ± , (13)Br( ψ (3770) → LH ) = (2 . − . . )%( ± . (14)For a small mass m c = 1 . ψ (3770) can reach 805 KeV( ± ± ψ (3770) inour estimation based on the calculation at leading orderin v and next-to leading order in α s in NRQCD.Together with the partial decay width of 350-400 KeVobserved for hadronic transitions and E1 transitions ofthe ψ (3770), the predicted annihilation (LH) decay widthin Eq.(13) will make the total non- D ¯ D decay width of ψ (3770) to be about 820-870 KeV for m c = 1 . m c = 1 . D ¯ D decay width, corresponding to abranching ratio of about 5% of the ψ (3770) decay.In summary, we have given a rigorous theoretical pre-diction for the LH decay of ψ (3770), based on NRQCDfactorization at NLO in α s and LO in v . By intro-ducing the color-octet contributions, the results are freefrom infrared divergences. We find that for the ψ (3770)the D-wave contribution is dominant, and the effect of S-D mixing is very small. Numerically, our results do notfavor either of the two experimental results measured byBES and CLEO collaborations. We hope our theoreti-cal result can serve as a clue to clarify the long-standingpuzzle of the ψ (3770) non- D ¯ D decay. We urge doingmore precise measurements on both inclusive and exclu-sive non- D ¯ D decays of ψ (3770) in the future. If theirtotal branching ratio can be as large as 10%, it will be areal challenge to our current understanding of QCD, andnew decay mechanisms have to be considered.This work was supported in part by the National Nat-ural Science Foundation of China (No 10675003, No 10721063). [1] For a review, see N. Brambilla et al., hep-ph-0412158.[2] R. Barbieri, R. Gatto and E. Remiddi, Phys. Lett. B61 ,465 (1976), R. Barbieri, M. Caffo and E. Remiddi, Nucl.Phys.
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