Analysis of Infrared Divergence in OZI-Forbidden Decays of Orthoquarkonia

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1 Commun. Theor. Phys. (Beijing, China 5 (29 pp. 6 c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 6, June 5, 29 Analysis of Infrared Divergence in OZI-Forbidden Decays of Orthoquarkonia ZHAO Shu-Min,, PANG Xue-Xia, LIU Li-Quan,, and LI Yong-Hui 2 Department of Physics and Technology, Hebei University, Baoding 72, China 2 Basic Department, North China Institute of Aerospace Engineering, Langfang 65, China (Received August 5, 28; Revised December 29, 28 Abstract It is interesting to investigate the OZI-forbidden decays of orthoquarkoknium, J/ψ γπ (γη, γη, J/ψ π + π (π π. At first glance, there is infrared divergence in the above processes. In this work, to clear the mist we use the factorization method with the light cone wavefunction of the final state light meson. We prove straightforwardly that the infrared divergence can be eliminated completely. PACS numbers: 2.38.Bx, 3.25.Gv Key words: OZI-forbidden, infrared divergence, factorization Introduction Generally the OZI [] processes are supposed to dominate the charmonium decays, at least provide a reasonable explanation for the narrow width of J/ψ. Therefore, a more complete calculation of the contribution from the OZI processes based on the general principle of QCD is crucially needed. In the frame work of perturbative QCD, the calculations for OZI-forbidden are carried out, which can deepen our insight into the perturbative QCD. To compute the transition matrix elements, we must deal with the hadronization, which is related to the nonperturbative QCD. In order to get the final results, some model-dependent wavefunctions are employed, which may contaminate the theoretical predictions. More than twenty years ago, Körner et al. [2] did their pioneer work of investigating the OZI-forbidden radiative decays of orthoquarkonia in perturbative QCD. They took the weak-binding approximation for both heavy and light mesons, which is reasonable and much simplifies the calculations. With the technique of scalarizing loop integrals by using covariant helicity projectors, all occurring loop integrals in their work can be integrated analytically. The resulting individual integrals are partly infrared divergent, which is an artifact of their decomposition. They introduced an infrared cutoff ε, and, after combining the total integrals, all terms proportional to ε and lnε were canceled as expected. Because the initial meson is much heavier than the produced light mesons, the 3-momentum of the final mesons are large. Therefore, the factorization method and the light cone wavefunctions are reasonable for the light mesons. With the development of the technique about calculating loop diagrams and the enrichment on the wavefunctions of light mesons, the authors [3] carried out a full one-loop calculation, which involves integrations of 4- and 5-point functions. [4,5] In the numerical calculation, there is infrared divergence, which is not eliminated evidently, though it is not as serious as that in the B-meson decays. The authors [3] only assigned a small mass to the gluon and vary it from 4 to 6 MeV to check if the result is stable. Though the numerical result is reasonable, it is not convincing. In Ref. [6] the authors discussed the infrared divergence of the decays M(Q Q ggg and M(Q Q ggγ. [7] In their work, the infrared (or soft divergence is logarithmic and arises only from the emission of a gluon (photon, whose four-momentum s components are all small. When all components of the four-momentum of the up and down gluons attaching to the on-shell heavy quark become small, the contributions represented by Figs. (a and (b are both infrared divergent. However, the infrared divergences emerging from the two diagrams can be canceled each other. [6] In Fig. (c, if the middle gluon is soft, the other two final-state gluons must both carry large momenta, and some of their components are at the order of M. That large momentum must flow through the heavy-quark propagator to which the soft gluon attaches, and consequently, it cuts off the potential infrared divergence. The controlling momentum tells us that an infrared divergence can never arise from a soft gluon attaching to a propagator that is off shell by order M. In this work, we study the infrared problem of the decays J/ψ γπ and J/ψ ππ, which is not the same with that of the decays J/ψ ggg and J/ψ ggγ. In the OZI-forbidden radiative decays J/ψ γπ, the photon in the final state is hard, but one of the off-shell Supported by National Natural Science Foundation of China under Grant No. 8477, the Nature Science Fund of Hebei Province under Grant No. A259, and Fund of Education Department of Hebei Province under Grant No zhaosm@hbu.edu.cn lzht44@63.com

2 2 ZHAO Shu-Min, PANG Xue-Xia, LIU Li-Quan, and LI Yong-Hui Vol. 5 gluons may be soft, which causes the infrared divergence. Because the analytic calculation of the general four- and five-point loop functions is almost impossible, the numerical calculation is necessary. However, the single D(E loop functions treated by LoopTools [8] is infrared divergent. In order to eliminate the infrared divergence visibly, we use the factorization method to the hadronization, [9] which is related to the non-perburbative QCD. Fig. The Feynman diagram describing the OZI-forbidden decay V ggγ. In the decays J/ψ ππ, one or two of the three gluons that are off-shell can be soft, which produces the infrared divergence. The calculation of this decay is more difficult than that of the OZI-forbidden radiative decays. Then the numerical calculation is necessary, which makes the elimination of infrared divergence vague. To clear the mist, we carry out our study in a form that is easy to see the elimination of the infrared divergence. The paper is organized as follows. After the introduction, we present our formulation to eliminate the infrared divergence in Sec. 2. Section 3 is devoted to a simple discussion and our conclusion. 2 Fomulation In Ref. [3], we derive an effective Lagrangian [] at the quark level. In order to get the decay rates, one has to evaluate the hadronic matrix elements. It is well known that the hadronization happens at the energy scale of Λ QCD, which is related to the non-perturbative QCD. Because the produced meson is light and its threemomentum is larger than Λ QCD, the light cone wavefunction isreasonable for description of the light meson. To obtain the decay amplitude, one needs to evaluate the hadronic matrix elements [,2] P V eff J/ψ, V eff = g eff ɛ αβµν P α V εβ J/ψ pµ ε ν γ, ( where V eff is the effective vertex for J/ψ γp and P stands for a pseudoscalar. P V = PV = M J/ψ represents the four-momentum of J/ψ in its center-of-mass frame. ε β J/ψ and ε ν γ are the polarizations of J/ψ and the emitted photon respectively, and p is the relative momentum between the photon and pseudoscalar. The g eff is an effective coupling and should be derived by evaluating the corresponding Feynman diagrams and loop integrations. This expression is at hadron level, while we treat the decay at quark level to see the elmination of the infrared divergence obviously. 2. Elimination of Infrared Divergence in Decay J/ψ γ + π For example, we study in Fig. 2(a particularly to confirm that it is infrared safe. The amplitude of Fig. 2(a can be written as M a = ε µ (γeq c g3t 4 a T b T b T a v c (p 2 γ α (/p + /k /q + m c (/p /q + m c γ µ u c (p ū q (p 3 γ β (/k + /p 3 + m q γ α v q (p 4 (2π 4 k 2 (p + p 2 + k q 2 [(p + k q 2 m 2 c][(p q 2 m 2 c][(k + p 3 2 m 2 q], (2 where p, p 2, p 3, and p 4 stand for the four-momenta of Q, Q, q, and q respectively. k is the four-momentum of the photon. The initial Dirac state v c, u c constitute the initial vector meson J/ψ. Correspondingly, the final Dirac state ū q, v q make up of the product π, which is a neutral pseudoscalar meson. With the transformation k k, p + p 2 + k q k 2, the denominators in Eq. (2 can be written as k 2 k2 2 [(k 2 p 2 2 m 2 c][(k + p 3 2 m 2 q][(p q 2 m 2 c]. (3 In order to make the infrared divergence apparent, we use the weak-binding approximation to simplify the propagators. By the approximation, which is reasonable, the heavy quarks Q, Q (c and c for J/ψ and light quarks q, q in the decay

3 No. 6 Analysis of Infrared Divergence in OZI-Forbidden Decays of Orthoquarkonia 3 product-light orthoquarkonium are set to possess equal momenta and be on their mass shells, i.e p Q = p Q, p q = p q and p 2 Q = m2 Q, p2 q = m 2 q. [3] Using the relation (k 2 p 2 2 m 2 c = k p 2 2 m 2 c 2k 2 p 2 = k 2 2 2k 2 p 2, (4 (p 3 k 2 m 2 q = p k 2 2k p 3 m 2 q = k 2 (k + k 2 k = k k 2, (5 the denominators (3 of the integral become k 2k2 2 (k 2p 2 (k k 2 [(p q 2 m 2 c]. (6 With the relation p +p 2 = k +k 2 +q, it is easy to see that there is surficial infrared divergence in the amplitude M a, when the internal four-momentum k or k 2 becomes soft. Fig. 2 The Feynman diagram describing the OZI-forbidden decay V γ + P. The pseudoscalar meson π is described by the light cone distribution amplitudes, and we can perform the replacement [3] ū ρ Γ ρθ v θ if P 4 Γ θρ = (γ β (/k + /p 3 + m q γ α θρ, { ( due iup y+iūp x Γ θρ /p γ 5 Φ(u µ P γ 5 Φ p (u σ µν p µ z ν Φ } σ(u 6 θρ, (7 where µ p is defined as m b r p x/2 [3] with r p x(µ = 2m 2 π m b (µ(m u + m d (µ Λ QCD m b. (8 Φ(u is the leading twist light cone wavefunction, while Φ p (u and Φ σ (u are higher twist light cone wavefunctions. The normalization of the light cone wavefunctions are defined as Φ(udu =, In the end, we get a trace [ Tr γ β (/k + /p 3 + m q γ α Φ p (udu =, Φ σ (udu =. (9 ( (/p γ 5 Φ(u µ P γ 5 Φ p (u σ µν p µ z ν Φ ] σ(u. ( 6 It is easy to see that the terms of Φ(u and Φ p (u have no IR divergence with k and p 3 = xp, where p is the momentum of the final state meson π. In the decay J/ψ γπ, m q is the mass of the light quark u or d, which is very small and can be neglected safely. Therefore, the contribution of the light cone wavefunction Φ σ (u is zero. In general, the leading twist wavefunction is enough, then from the above study, we have confirmed the elimination of the infrared divergence completely in the OZI-forbidden radiative decays V γp. 2.2 Elimination of Infrared Divergence in Decay J/ψ ππ The situation in the decay of J/ψ ππ is not the same with that of J/ψ γπ. To prove the elimination of the infrared divergence of the decay J/ψ ππ is more difficult, which must take into account of the initial states wavefunction. The infrared divergences in two diagrams with the same topology eliminate, and the two diagrams

4 4 ZHAO Shu-Min, PANG Xue-Xia, LIU Li-Quan, and LI Yong-Hui Vol. 5 have such a relation u d, ū d. The initial state J/ψ is a heavy meson, so the leading twist light cone distribution wavefunction is enough. We can perform the replacement C α(yc i j 2β (x J/ψ = δ ij due iup y+iūp x{ f ψ m ψ /ε ψ φ (u + } ν 4N c 2 σµ if ψ (ε ψµ p ψν ε ψν p ψµ φ (u. ( βα The contribution in Fig. 3(a can be written in the following form: A B = d 4 y d 4 y 4eik y i 2 2 y 2 i i 2 (2π k 2 (2π 4e ik k2 2 (p 3 + p 5 C i 2 β( { ( ig s Tisγ a ν i ( ig s T /k + /p /p 3 /p 5 m srγ b µ i } ( ig s T d c /p /p 3 /p 5 m rjγ λ c βα { Cα( j n d q 2η (y 2 ( ig s Tnwγ a 4 k ν (y y 2 i } (2π 4eik ( ig s T b /k m wlγ µ q ητ ql k 2τ(y q ρ (( ig s Tkmγ d λ ρθ qθ( m ( gstr 6 (T a T b T d Tr(T a T b T d 4N c 2 π (p 3 + p 5 2 [(p p 3 p 5 2 m 2 c] (2π 4Tr [γν (/k + /p /p 3 /p 5 + m c γ µ (/p /p 3 /p 5 + m c γ λ (f ψ m ψ /ε ψ f ψ /ε ψ /p ψ ] Tr [/p D γ 5 γ ν ( /k /p 4 γ µ /p U γ 5 γ λ φ(xφ(y + /p D γ 5 γ ν γ µ γ 5 γ λ ( µ π m q φ(xφ p (y + γ 5 γ ν γ µ /p U γ 5 γ λ ( µ π m q φ p (xφ(y] k 2 [(k + p 4 2 m 2 q](k + p 4 + p 6 2 [(k + p p 3 p 5 2 m 2 c], (2 where p D, p U stand for the four-momenta of the final light mesons π, π respectively. The contribution of the leading twist wavefunction is dominant, so considering twist-three light cone wavefunction of π is enough. It is easy to see that the terms with twist-three light cone wavefunctions are suppressed by a factor 2m q /M J/ψ (p = p = M D U J/ψ/2. Because the mass of the current quark (u,d is almost zero, it is reasonable to neglect the terms with m q. When k, Eq. (2 can be simplified as ( A B gstr 6 (T a T b T c Tr(T a T b T c 2 4N π c (p 3 + p 5 2 [(p p 3 p 5 2 m 2 c] (2π 4Tr [γν (/p /p 3 /p 5 + m c γ µ (/p /p 3 /p 5 + m c γ λ (f ψ m ψ /ε ψ f ψ /ε ψ /p ψ ] Tr [/p D γ 5 γ ν ( /p 4 γ µ /p U γ 5 γ λ φ(xφ(y] k 2 [(k + p 4 2 m 2 q](k + p 4 + p 6 2 [(k + p p 3 p 5 2 m 2 c]. (3 Let we see the denominators (p 3 + p 5 2 [(p p 3 p 5 2 m 2 c]k 2 [(k + p 4 2 m 2 q](k + p 4 + p 6 2 [(k + p p 3 p 5 2 m 2 c] with the relation p 3 = xp U, p 4 = ( xp U, p 5 = yp D, and p 6 = ( yp D. We can find that the denominators have the symmetry of p U p D, x y. Contract the two traces in Eq. (3. Tr [γ ν (/p /p 3 /p 5 + m c γ µ (/p /p 3 /p 5 + m c γ λ (f ψ m ψ /ε ψ f ψ /ε ψ /p ψ ]Tr [/p D γ 5 γ ν ( /p 4 γ µ /p U γ 5 γ λ φ(xφ(y] = 32φ(xφ(yf ψ m ψ (x (2p D p U p U ɛ ψ m 2 c 3p D ɛ ψ p 2 U m2 c + 4(A p U 2 p D ɛ ψ 2A p D A ɛ ψ p 2 U + A2 p D ɛ ψ p 2 U + 4A p D A p U p U ɛ ψ 2A 2 p D p U p U ɛ ψ, (5 where A = p p 3 p 5. The corresponding diagram can be obtained by the relation p U p D, x y. The amplitude of the corresponding diagram reads as ( A BC gstr 6 (T a T b T d Tr(T a T b T d 2 4N π c (p 3 + p 5 2 [(p p 3 p 5 2 m 2 c ] (2π 4Tr [γν (/p /p 3 /p 5 + m c γ µ (/p /p 3 /p 5 + m c γ λ (f ψ m ψ /ε ψ f ψ /ε ψ /p ψ ]Tr[/p D γ 5 γ ν ( /p 4 γ µ /p U γ 5 γ λ φ(xφ(y] k 2 [(k + p 4 2 m 2 q](k + p 4 + p 6 2 [(k + p p 3 p 5 2 m 2 c] (4

5 No. 6 Analysis of Infrared Divergence in OZI-Forbidden Decays of Orthoquarkonia 5 ( = gstr 6 (T a T b T d Tr(T a T b T d 4N c 2 π (p 3 + p 5 2 [(p p 3 p 5 2 m 2 Q ] (2π 4 k 2 [(k + p 4 2 m 2 q](k + p 4 + p 6 2 [(k + p p 3 p 5 2 m 2 c] 32f ψ m ψ (y (2p U p D p D ɛ ψ m 2 c 3p U ɛ ψp 2 D m2 c + 4(A p D 2 p U ɛ ψ 2A p U A ɛ ψ p 2 D + A2 p U ɛ ψ p 2 D + 4A p U A p D p D ɛ ψ 2A 2 p U p D p D ɛ ψ. (6 The leading twist light cone distribution wavefunction of π are symmetry in the regain (,, such as Φ(x = 6x( x. From the above formulas, we can find that the sum of the infrared divergent terms of the two amplitudes A B and A BC is zero with the conditions (p U + p D ɛ ψ =, p 2 U = p2 D, and A p U = A p D. Then we can apparently see that the sum of infrared divergences in the two diagrams disappears. The other Feynman diagrams are the same too. In the end, the sum of all Feynman diagrams in the decay J/ψ ππ is infrared safe. Fig. 3 The Feynman diagram describing the OZI-forbidden decay V PP.

6 6 ZHAO Shu-Min, PANG Xue-Xia, LIU Li-Quan, and LI Yong-Hui Vol. 5 3 Discussion and Conclusion In this work, we study the elimination of the infrared divergence in the OZI-forbidden decays of orthoquarkonia in perturbative QCD. In the process J/ψ γπ, we use the weak-binding approximation to simplify the propagators, which makes the infrared divergence apparent. The weak-binding approximation is reasonable as discussed in Refs. [2,3]. The final-state meson is a pseudoscalar, which can be used to simplify the discussion. Because the final state meson is light and relativistic, the light cone wavefunction is reasonable. With the factorization method, we study the part of two virtual gluons forming the pseudoscalar. After careful study, we find that all terms of infrared divergences disappear at last. The situation of J/ψ ππ is more difficult than that of J/ψ π γ, and their ways of eliminating the infrared divergences are different. Each diagram of the process J/ψ γπ is infrared safe, but almost every diagram in the decay J/ψ ππ is infrared divergent. It is interesting to find that the infrared divergences in the two diagrams of the same topology annihilate each other. The concomitant diagram can be obtained from Fig. 3 by using the following symmetry u d, ū d. In this decay, we discuss the light cone wavefunction of the leading twist without considering the higher twist wavefunctions, which is suppressed by the factor m q /M J/ψ. The fcctor is very small, so the term of the higher twist light cone wavefunctions can be neglected safely. It is well known that the calculation of the integrals of general 4- and 5-point loop functions is very difficult. One can not evaluate them in analytic form, which makes numerical computation necessary. There are three reasons, which cause the infrared divergences emerge in numerical evaluation. First, the internal four-momentum k is very small; Second, the masses of gluons on internal line is zero; Third, the light cone distribution variable x and y are in the range near or. The light cone wavefunctions can reduce the degree of infrared divergence, but the infrared divergence can not be eliminated completely. One uses LoopTools to compute loop functions. If there is gluon (photon propagator in D(E-function, one should assign the small mass to the gluon (photon. Though the final numerical result is reasonable, the method of eliminating infrared divergence is not compellent. Here, with the factorization method we study the infrared problem in analytic form. It is easy to find that the infrared divergence can be eliminated safely, which guarantee that the numerical computation is reasonable. It opens a window to study the infrared divergence in the OZI-forbidden decays, which involve multi-point functions. Acknowledgments We benefit greatly from the discussions with Professor X.Q. Li and Dr. T. Li. References [] S. Okubo, Phys. Lett. 5 (963 63; G. Zweig, CERN Report NO. 882/TH (964 4; NO. 849/TH (964 ( [2] J.G. Körner, J.H. Kühn, M. Krammer, and H. Schneider, Nucl. Phys. B 229 ( [3] G. Li, T. Li, X.Q. Li, et al., Nucl. Phys. B 727 (25 3; T. Li, S.M. Zhao, and X.Q. Li, arxiv: [4] A. Denner and S. Dittmaier, Nucl. Phys. B 658 (23 75; A. Denner, Nucl. Phys. B 59 ( [5] G. t Hooft and M. Veltman, Nucl. Phys. B 53 ( ; A. Denner, U. Nierste, and R. Scharf, Nucl. Phys. B 367 ( [6] G.T. Bodwin, B. Braaten, and G.P. Lepage, Phys. Rev. D 5 ( [7] M. Krammer, Phys. Lett. B 74 ( [8] T. Hahn, hep-ph/ [9] M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Nucl. Phys. B 59 (2 33; N.H. Fuchs and M.D. Scadron, Phys. Rev. D 2 ( ; V.L. Chernyak and A.R. Zhitnitsky, Nucl. Phys. B 2 ( [] B. Guberina and J.H. Kühn, Nuovo Cimento Lett. 32 (98 295; J.H. Kühn, in Proceedings of the Workshop on Doris Experiments, DESY, - February 98; J.H. Kühn, J. Kaplan, and E.G.O. Safiani, Nucl. Phys. B 57 ( [] S. Chung, Phys. Rev. D 48 ( [2] A. Deandrea, G. Nardulli, and A.D. Polosa, hepph/243; A. Deandrea, G. Nardulli, and A.D. Polosa, Phys. Rev. D 68 (23 342, hep-ph/ [3] M. Beneke and M. Neubert, Nucl. Phys. B 675 (23 333; M. Beneke, G. Buchalla, M. Neubert, and C.T. Sachrajda, Nucl. Phys. B 59 (2 33; M. Beneke and Th. Feldmann, Nucl. Phys. B 592 (2 3.