Direct pion emission in D^*+ → D⁺π decays^*

In the framework of the QPC model [4–21], the decay D^{* +} → D⁺π⁰ occurs via a quark-antiquark pair creation from the vacuum. It is an OZI-allowed process. The decay mechanism is displayed graphically in Fig. 1. Many soft gluons are emitted from the quark and anti-quark legs, which then annihilate into a quark-antiquark pair. Equivalently, the physics scenario can be described as a quark pair being excited out from the vacuum. The ³P₀ model has been widely applied to calculate such hadronic strong decays.

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This process has been calculated by some authors [22] where they only gave the numerical results but not the details of the calculations. We have repeated the calculation and obtained a rate which is consistent with theirs. For readers' convenience, we have collected the relevant formulations about the calculation in terms of the ³P₀ model in Appendix A. The transition operator for the quark pair creation reads

and the hadronic matrix element is determined as

In Eq. 1, i and j are the SU(3)-color indices of the created quark and anti-quark respectively. are for flavor and color singlets, respectively. is the spin wave function. is the ℓth solid harmonic polynomial. γ is a dimensionless constant which denotes the strength of quark pair creation from vacuum and is fixed by fitting data. Following Ref. [23], we take γ = 13.4 in this work. For Eq. 2, the explicit expressions for the wave function of a meson and the hadronic matrix elements are presented in Appendix A.

The helicity amplitude of this process can be extracted from the relation . Then, the decay width corresponding to the process is written in terms of the helicity amplitude as

where we take K_D⁺ = −K_π⁰ = K in the center of mass frame of the D^*+.

Numerically, we take a typical R value for the D meson from Ref. [17] as 2.3 GeV⁻¹ and R = 2.1 GeV⁻¹ for π⁰ from Ref. [23]. With these parameters, the decay width of D^*+ → D⁺π⁰ can easily be obtained and we have got it as 21.9 keV, which coincides with the result given in Ref. [22]. Experimentally, the decay width of this mode is 25.6 ± 1.0 keV [24]. The consistency of the numerical results evaluated in terms of the ³P₀ model with data indicates that the theoretical framework is well established and applicable to describe such processes.

On the other hand, based on the heavy quark effective theory (HQET), one can extract the effective coupling constant of D*Dπ from the previously calculated decay width, which might offer significant information for application of an effective theory. Following Ref. [25], the related effective Lagrangian can be written as

Then we can get the decay width as

with . The transition amplitude of D^*+ → D⁺π⁰ is [25]

where is the polarization vector of D*. From equation (4) we obtain g_D*Dπ = 0.51.

The pion can be directly emitted before D* transits into D, thus the amplitude, in principle, should be added to the process depicted in the above subsection and interferes with it. Just by the qualitative analysis, the one-pion emission is an OZI-suppressed process and moreover, it violates isospin conservation, therefore must be very small compared to the vacuum creation.

It is interesting to investigate such an effect in other processes, i.e. as long as the one-pion emission is not a leading term, how small it would be compared with the leading terms. Below, we will quantitatively investigate direct one-pion emission in D*→ D + π⁰.

The corresponding diagrams for direct pion emission in D^*+ → D⁺π⁰ for which the QCDME is responsible are shown in Fig. 2.

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Fig. 2 is just obtained by distorting Fig. 1, as shown in Fig. 3.

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Here we only draw two gluon field lines, but as well understood, in the QCDME scenario the lines correspond to a field of EN or MN mode, which are by no means free gluons, and the line also does not correspond to a single-gluon propagator. Thus the line denotes a collection of many soft gluons just as shown in Fig. 1.

Now let us calculate the rate contributed by the processes shown in Fig. 2 in the framework of QCDME.

The process of directly emitting a soft π⁰ from D* in the decay D*→ D+π⁰ is dominated by an E1 − M2 coalescence transition. This is an OZI-suppressed process and violates isospin conservation. The transition amplitude is [26]

where the S operator acts on the total spin of the heavy-quark and light-anti-quark system, N and L are the principal quantum number and the orbital angular momentum respectively of the intermediate hybrid state, M_I and E_NL are the mass of the initial meson D* and the energy eigenvalues of the hybrid state respectively, and m is the energy scale of the M2 transition, which we set to be m_c and in our numerical computations. The amplitude reduces to [27, 28]

where is the polarization vector of D*, and R_I, R_F and R_NL are the radial wave functions of the initial, final and intermediate hybrid state, respectively.

The radial wave functions are calculated by solving the relativistic Schrödinger equation [29]. The potentials for the initial and final D^(*) mesons are taken from Ref. [29] and the potential for the intermediate hybrid states is taken from Ref. [30].

The matrix element is of the form [27, 28]

where g_E and g_M are the coupling constants for the color electric field and color magnetic field, and k is the momentum of π⁰.

In order to compare the results with the effective coupling constant g_D*Dπ obtained by using the QPC model, the transition amplitude of D^*+ → D⁺π⁰ can be rewritten as [25]

is the effective coupling constant obtained by means of QCDME, which is then

with

For our numerical analysis, the input parameters are taken from Ref. [24]. Here m_D = 1.869 GeV, m_D* = 2.010 GeV, m_π = 0.135 GeV, m_c = 1.800 GeV, m_u = 0.3 GeV, , f_B* = 0.230GeV, f_K = 0.160 GeV, F_π = 0.093 GeV and . α_s = 0.31 for . Following Ref. [1, 2, 27, 28] we set , with α_E = 0.6 and for a possible error range, according to the literature, we let α_M vary from α_E to 10α_E. The constants and effective coupling constant g obtained in terms of QCDME are listed in Table 1.

Table 1. Coupling constant and (in units of GeV⁻³), with α_M set to be α_E, 3α_E, 10α_E, and 30α_E separately, and the value of m set to .

	αM = αE	αM = 3αE	αM = 10αE	αM = 30αE
	5.677	9.833	17.952	31.094
	0.00145	0.00251	0.00459	0.00794

In order to explore possible validity ranges of QCDME, we increase the maximum value of α_M to 30α_E, whereupon reaches 0.00794. It is thought that the size of heavy quarkonia might be much smaller than 1/Λ_QCD, whereas, for D(D*), because of the existence of a light quark those sizes may be larger. But some model dependent calculations show the mean square root radius R to be R = 1.94 GeV⁻¹ for J/ψ in terms of the relativistic-like Schrödinger equation [29], whereas Kuang et al. obtained R = 2.13 GeV⁻¹[28]. For the case of the D meson, R = 2.3 GeV⁻¹, which is slightly larger than the expected value for J/ψ, but not by much, and it is obviously smaller than 1/Λ_QCD ∼ 5 GeV⁻¹. Therefore, we believe that the errors brought up in this scheme are tolerable. With possible errors, this result is 60 times smaller than that obtained by the QPC model [24, 31].

In Table 2 we list the decay widths Γ(D^*+ → D⁺π⁰) from the QCDME contribution, and see that when α_M takes the maximum value 30α_E, the decay width of Γ(D^*+ → D⁺π⁰) from the QCDME contribution is 5.31 × 10⁻³ keV. This result is more than three orders of magnitude smaller than the result obtained by using the QPC model(21.9 keV).

Table 2. Decay width of Γ(D^*+ → D⁺π⁰) (in unit of keV) from QCDME contribution, for α_M we take α_E, 3α_E, 10α_E, and 30α_E respectively.

	αM = αE	αM = 3αE	αM = 10αE	αM = 30αE
Γ(D*+ → D+π0) with	4.43 × 10−5	5.31 × 10−4	4.44 × 10−4	5.31 × 10−3
Γ(D*+ → D+π0) with m = mc	1.77 × 10−4	1.33 × 10−4	1.78 × 10−3	1.33 × 10−3

Our numerical results for the decay mode D*→ Dπ⁰ confirm that direct pion emission is not the leading term and the contributions determined by the QCDME must be much smaller than those induced by other mechanisms.

Let us try to understand the smallness of the contribution from direct pion emission, using data to show that our estimate is reasonable. There are two suppression factors: one-pion emission violates isospin conservation, and QCDME is an OZI-suppressed mechanism. Compared with vacuum quark pair creation, it must be small and we see that its contribution to the decay width is in the range of about 10⁻⁴ − 10⁻⁵ keV. The decay ψ(2S)→ J/ψ + π⁰ is also an isospin violating and OZI-suppressed process, and its branching ratio is 1.27 × 10⁻³ [24], slightly larger than our estimate for D*→ Dπ⁰. That further suppression factor comes from the fact that the transition ψ(2S)→ J/ψ + π⁰ is an S-wave process, whereas D^*+ → D⁺π⁰ is a P-wave process whose decay width is proportional to the three-momentum k, which is small, so the decay width Γ(D^*+ → D⁺π⁰) suffers from a P-wave suppression.

We may also look at ψ(2S) as an example for direct π emission. ψ(2S)→ π⁰ + h_c(1P) can only occur via a direct pion emission, so is completely determined by the QCDME mechanism, and its partial width is about 0.26 keV. Since it is an S-wave process, it does not suffer from P-wave suppression. In ψ(2S) decays, the mode ψ(2S) → η_c + π⁰ is not seen, but ψ(2S) → η_c + π⁺π⁻π⁰ has been measured, and its branching ratio is not too small (< 1.0 × 10⁻³), because the direct emission of three pions does not violate isospin conservation.

Equivalently, the QCDME can be replaced by chiral perturbation theory, for example the transition of ϒ(nS)→ ϒ(mS) + π⁺π⁻ has been studied in terms of the chiral theory [32].

By the discussion above, we conclude that for the decays of mesons containing a heavy quark and a light anti-quark, the contribution from QCDME is much smaller than that caused by the QPC mechanism, thus when evaluating the decay rates, one need only take into account the contribution from the QPC mechanism and the direct pion emission can be safely ignored.

In conclusion, we confirm the validity of the QCDME and determine its application region. Since the framework applies only to direct pion emission, if there are other mechanisms to contribute, such as quark-pair creation from vacuum to D^*→ D+π⁰, or ϒ(5S)→ BB̅^*→ ϒ(1S)+π⁺π⁻ where an intermediate state of the BB̅^* portal is open, i.e the available energy is above the production threshold of BB̅^*, then the direct emission induced by the QCDME is no longer the leading term and can only contribute a tiny fraction.

In the ³P₀ model, for a two-body decay process A → BC, the total wave function of a meson can be written as

which satisfies the normalization condition: . The subscripts 1 and 2 refer to the quark and anti-quark within meson A. k_A is the momentum of meson A. , , are the spin, flavor and color parts respectively. ψ_{nA LA MLA}(k₁,k₂) is the spatial part of a meson wavefunction in the momentum representation. When the mesons concerned are in the ground states, the simple harmonic oscillator (HO) wavefunction is employed, which reads as

where k = (m₂k₁ − m₁k₂)/(m₁ + m₂) is the relative momentum between the quark 1 (with mass m₁) and the anti-quark 2 (with mass m₂) within a meson. The R value is fixed by fitting experimental data.

The explicit matrix element is depicted as

where is a spatial integral, reading as