Study of the anomalous cross-section lineshape of e^+e^-\to D\bar D at \psi(3770) with an effective field theory

We study the anomalous cross-section lineshape of $e^+e^-\rightarrow D\bar D$ with an effective field theory. Near the threshold, most of the $D\bar D$ pairs are from the decay of $\psi(3770)$. Taking into account the fact that the nonresonance background is dominated by the $\psi(2S)$ transition, the produced $D\bar D$ pair can undergo final-state interactions before the pair is detected. We propose an effective field theory for the low-energy $D\bar D$ interactions to describe these final-state interactions and find that the anomalous lineshape of the $D\bar D$ cross section observed by the BESII collaboration can be well described.

from all of the R i in reality. As a reasonable approximation, one can include the contributions from the vector mesons in the vicinity of the considered energy region but neglect those far off-shell vector mesons. In the energy region of the BES data from 3.74 GeV to 3.8 GeV, one can expect that ψ(3770) plays the most important role among all of the R i , whereas the contributions from all the other R i can be treated as background. As shown in Ref. [3], the contribution from ψ(2S) dominates the background, whereas the contributions from other states are negligible. Therefore, we only include the contributions from the resonances ψ(3770) and ψ(2S) and neglect those from the other resonances. Namely, ψ(2S) would be the main background near the DD threshold.
In VMD [13,14], the coupling between the vector meson and a virtual photon can be described as where V µ is the vector meson field, A µ is the photon field, and M V is the mass of the vector meson. Setting the electron mass to m e ≈ 0, the coupling can be obtained as where Γ ee is the electron-position decay width of V µ and α = 1/137 is the fine-structure constant.
Once the DD pair is produced from the decay of the vector meson ψ(3770) or ψ(2S), the pair can undergo final-state interactions through the rescattering processes DD → DD → · · · → DD, which can be described by the effective field theory. In the energy region of interest, the three-vector momentum p of the D(D) meson is small. Thus, it is possible to construct an effective field theory for the low-energy DD interactions by making use of the expansion of the small momentum p. Because the mass of ψ(3770) is just above the threshold of DD, we need to include ψ(3770) explicitly in the formulation. Near the threshold, the D(D) meson can be treated as nonrelativistic. Thus, the interaction Lagrangian for the DD system with the quantum number J P C = 1 −− can be constructed as where ψ is the field operator of ψ(3770); D (D † ) annihilates a D(D) meson; D † (D) creates a D(D) meson; τ i is the Pauli matrix, and the ellipsis denotes other contact terms with more derivatives that are higher order terms. The first term in L (DD) 2 accounts for the interaction in the isospin singlet channel, and the second term accounts for the isospin triplet channel. The contributions from other resonances, which are not included in the Lagrangian, can be saturated into the contact terms L (DD) 2 . Therefore, we take the coefficients such as f 1 , f 3 to be complex, where the imaginary parts of these terms come from the width of the saturated resonances and the DD annihilation effect. With isospin symmetry, we only have to consider the terms for the isospin singlet channel in L (DD) 2 to study the DD final-state interactions because the DD pair comes from the decay of ψ(3770) and ψ(2S) in our approach. Now we come to the discussion of the power counting of this effective field theory. The tree-level diagrams for the DD elastic scattering are shown in Fig. 1. Near the DD threshold, the denominator of ψ(3770) the ψ(3770) propagator can be expressed as where p is the magnitude of the three-vector momentum of the D(D) meson in the overall center-ofmass frame, Γ non-DD ψ denotes the non-DD decay width of ψ(3770), and M ψ is the mass of ψ(3770). The DD decay width of ψ(3770) will be included through the summation of the D meson loops in the following. Because ψ(3770) is close to the threshold of DD, we expect that P (ψ) is at O(p −2 ). Taking the momentum power of the ψDD vertex into account, we find that Fig. 1(a) is at O(p 0 ). From the naive power counting, the leading contact terms have two derivatives; hence, these terms are at O(p 2 ). However, in this naive power counting, we have assumed that the coefficients of the contact terms, i.e., f 1 , f 3 , · · · , are at order of O(p 0 ). In some cases, especially when there are bound states or resonances near the threshold, the coefficients of the contact terms can be enhanced. For example, in a N N interaction, the S-wave contact terms C S scale as O(p −1 ) [15]. Another example is the N N interaction in 3 P 0 , where the leading contact term C3 P0 can be promoted to O(p −2 ) [16]. It is interesting to study whether the same enhancement mechanism takes place in the DD interactions because the resonance ψ(3770) is located near the DD threshold. If f 1 is promoted to O(p −2 ) as C3 P0 in a N N interaction, then the corresponding tree diagram shown in Fig. 1 , which is the same as Fig. 1(a). However, because we do not know the power of f 1 at the beginning, we then assume that the leading contributions to DD elastic scattering come from both Fig. 1(a) and (b). We will use the experiment data to determine f 1 and see whether this contact term is enhanced. Accordingly, the DD scattering amplitude in the specific channel (J P C = 1 −− , I = 0) can be obtained by summing the bubble diagrams as shown in Fig. 2, which is equivalent to solving the Lippmann-Schwinger equation T = V + V GT with the DD potential truncated at the leading order. Figure 3 illustrates the final-state interactions between the produced DD. Because we first assume f 1 is enhanced, which indicates the interaction between DD is strong or the DD scattering length is large, we will use the power divergent substraction (PDS) scheme proposed by Ref. [15] to describe the large-scattering-length system in our calculations. The loop integrals that we will encounter in Fig. 3 can The bubble diagrams for the DD interactions where the potential is truncated at the leading order.
generally be reduced to where E = p 2 /M D is the total kinematic energy of the DD system. It is clear that this result is convergent in D = 4 but divergent in D = 3. With the PDS scheme, we have to remove the D = 3 pole in the above result by adding the counterterm Hence the subtracted integral in D = 4 reads Notice that I P DS = I at µ = 0, which is simply the result in the minimal subtraction (MS) scheme. We can choose µ to be the typical momentum scale of the D(D) meson, which is p ≤ 300 MeV in our calculations.
We can then write down the amplitude for e + e − → DD as To be more specific, the amplitude for process Fig. 3(a) reads with where γ is the Dirac gamma matrix; k 1 and k 2 are the incoming momenta of the electron and positron, respectively, and p 1 and p 2 are the outgoing momenta of D andD, respectively. G ψ cannot be simply interpreted as the width of ψ(3770) because this term is a complex number. If we set f 1 = 0 and µ = 0, The amplitude for Fig. 3(b) can be written as with where the PDG [17] value for the ψ(2S) mass M ψ(2S) can be adopted, and f ψ(2S) can be extracted by Eq.
To proceed, we denote the cross section for e + e − → DD as σ B (s), which does not include the initial state radiation (ISR) effect. In reality, for a given energy √ s, the actual c.m. energy for the e + e − annihilation is √ s ′ = s(1 − x) due to the ISR effect, where xE beam is the total energy of the emitted photons. To order α 2 radiative correction in the e + e − annihilation, the observed cross section σ obs at BESII can be related to our result σ B through [18] σ obs (s) = (1 + δ V P ) where (1 + δ V P ) = 1.047, and the function f (x, s) is given by , Before fitting the BESII data with Eq. (13), we first discuss our treatment of Γ non-DD ψ . It seems impossible to determine Γ non-DD ψ definitely in our fitting because Γ non-DD ψ is always accompanied by f 1 in our formula, and any change of Γ non-DD ψ can be compensated by tuning f 1 . The experimental results on the non-DD branching ratio of ψ(3770) decay are still controversial [19][20][21]. In contrast, the next-to-leading-order (NLO) pQCD calculation expects the non-DD decay branching ratio to be at most approximately 5% [22]. Meanwhile, an effective Lagrangian approach estimates that the D meson loop rescatterings into non-DD light vector and pseudoscalar mesons leads to approximately 1% non-DD branching ratios [23]. A similar calculation by Ref. [24] also confirms such a nonperturbative phenomenon. One also notices that so far, most of the well-measured non-DD decay modes of ψ(3770) are found to be rather small. Namely, their branching ratios are either at the order of 10 −3 − 10 −4 , or only an upper limit is set [17]. Taking all these facts into account and for the purpose of studying the dominant DD channel, we set Γ non-DD ψ to be zero in our fitting as a leading approximation. We have checked that the fitting results are approximately unchanged even though we set the non-DD branching ratio of the ψ(3770) decay to be at the order of several percent.  The fitted parameters and fitting qualities with µ = δ, m π , 300 MeV are shown in Table I. For comparison, we also show the result with µ = 0, which corresponds to the value in the MS scheme. The result shows that the fitted parameters are insensitive to the choice of µ. Moreover, the real part of f 1 is large, at the order of (M D /δ) 2 , which is consistent with our previous assumption. In contrast, the imaginary part of f 1 is not well determined. Note that the NLO term f 1 has a comparable magnitude to that of the leading order term. This result suggests that the effective field theory expansion may not be convergent. Thus, the fitting results may not be quantitatively reliable. To have a better understanding of our results, we investigate the dependence of f 1 on the scattering length a as that was done in Ref. [15]. For the P -wave DD elastic scattering, we denote the Feynman amplitude as iA cos θ, where θ is the angle between the incoming and outgoing momenta in the c.m. frame. Then, the correlation between A and the P -wave phase shift δ is With the effective range expansion and taking the case of P -wave scattering (ℓ = 1), we then obtain .
For simplicity and only illustrating some aspects of the effective field theory, we ignore the ψ(3770) and consider a DD effective theory with only the contact terms. Accordingly, the tree-level amplitude for the P -wave scattering can be written as For the isospin I = 0 channel, we have the coefficient of the leading contact term C 2 = 8f 1 . The full amplitude can then be obtained by summing over all the bubble diagrams as shown in Fig. 2. The amplitude becomes Using the fact that the amplitude A should be independent of the arbitrary subtraction scale µ, we can determine the µ dependence of the coupling constants C 2n (µ) Note that dC2 dµ = 0. One can see that, different from the S-wave scattering that was considered in Ref. [15], the coefficient of the leading contact term C 2 is independent of µ for the P -wave scattering. This fact makes the PDS approach fail to improve the convergence of the effective field expansion for the P -wave scattering. By comparing Eqs. (17) and (19) with each other, we obtain For the I = 0 channel, we have f 1 = C 2 /8 = −6πM D a, which suggests that f 1 can be large if the P -wave scattering length a is sizeable. It is also interesting to notice that, if a ∼ 1 Λp 2 , by choosing p = m π and the cutoff scale Λ = 1 GeV, we will have f 1 ∼ 1900 GeV −2 , which is close to our fitted value.
Our fitting results for the DD cross-section lineshape are presented in Fig. 4, where we only show the result with µ = δ because the other choice of µ gives similar lineshapes.
With the fitted parameters, we can obtain the width and electron-positron decay width of ψ(3770) as the following: The corresponding values with different choices of µ are listed in Table II. In Note that g ψ(2S)DD is more than two times larger than our previous result. The fitted lineshape and exclusive contributions from ψ(2S) and ψ(3770) are presented in Fig. 6. Unsurprisingly, this figure shows that the contribution from ψ(2S) is larger than that in the previous fitting in Fig. 4 because of the larger coupling constant g ψ(2S)DD . The distorted lineshape can be explained by the interference between ψ(3770) and ψ(2S), which is constructive at E cm < M ψ but destructive at E cm > M ψ . This observation can help us conclude that a large g ψ(2S)DD will favor a larger value for M ψ , i.e., a larger mass for ψ(3770) than the present PDG average. We also find that, in this fitting, the fitted χ 2 is sensitive to M ψ . For example, the best fit gives χ 2 ≈ 41 when we fix M ψ = 3.78. By adopting the PDG values [1,17] for M ψ , the yield of χ 2 can be even larger. Furthermore, such a large g ψ(2S)DD suggests that we need to include the contact term f 1 in the DD interaction to saturate the contribution from ψ(2S). With this aspect taken into account, we can affirm that the fitting result with f 1 = 0 is not self-consistent. In general, the inclusion of the f 1 term seems to be necessary to yield a reasonable value for g ψ(2S)DD and, at the same time, determine M ψ in a range closer to the PDG average [1,17].
In summary, we have proposed an effective field theory for low-energy DD interactions in which we have included the resonance ψ(3770) and an additional small scale δ. It is found that the coefficient of the contact term f 1 will be enhanced to be O(p −2 ). Therefore, the leading DD interaction potential in this specific channel would come from the S-channel ψ(3770) exchange and the contact term f 1 . With the leading DD potential, we then sum the bubble diagrams to describe the DD final-state interaction as shown in Fig. 3. We find that we can describe the anomalous cross-section lineshape of e + e − → DD observed by the BESII Collaboration [2] using the effective field theory. This approach should be useful for our further understanding of the ψ(3770) non-DD decays, which could share the same dynamic origin as the DD cross-section lineshape anomaly as emphasized in Refs. [23,25].
We also test the effects of the contact term f 1 and find that, without this term, the extracted value of g ψ(2S)DD is too large to make the fitting self-consistent. Nevertheless, the fitted ψ(3770) mass is significantly larger than that in PDG [1,17]. Our study also suggests that the subthreshold ψ(2S) plays an important role in our understanding of the DD interactions. A better determination of g ψ(2S)DD should be strongly encouraged.