Line shape and $D^{(\ast)}\bar D^{(\ast)}$ probabilities of $\psi(3770)$ from the $e^+e^-\to D\bar D$ reaction

We have performed a calculation of the $D\bar D$, $D\bar D^\ast$, $D^\ast\bar D$, $D^\ast\bar D^\ast$ components in the wave function of the $\psi(3770)$. For this we make use of the $^3P_0$ model to find the coupling of $\psi(3770)$ to these components, that with an elaborate angular momentum algebra can be obtained with only one parameter. Then we use data for the $e^+e^-\to D\bar D$ reaction, from where we determine a form factor needed in the theoretical frame work, as well as other parameters needed to evaluate the meson-meson selfenergy of the $\psi(3770)$. Once this is done we determine the $Z$ probability to still have a vector core and the probability to have the different meson components. We find $Z$ about $80\sim85\%$, and the individual meson-meson components are rather small, providing new empirical information to support the largely $q\bar q$ component of vector mesons, and the $\psi(3770)$ in particular.


I. INTRODUCTION
The nature of hadronic resonances is a field of continuous debate [1][2][3][4]. The simple picture of mesons as qq objects and baryons as qqq objects gave an impressive boost to hadron physics and large amount of mesons and baryons were described with this picture [5]. Yet, the advent of a new wave of experiments in the charm and bottom sectors has brought new information that clearly challenges this early picture in many cases [2][3][4]. Even in the light quark sector there are mesonic resonances that clearly cannot be represented as qq states, as the low lying scalar mesons (f 0 (500), f 0 (980), a 0 (980), · · · ) [6][7][8][9]. On the other hand, the elaborate analysis of meson-meson data by means of QCD and large N c argument concluded that low lying vector mesons are largely qq objects [10].
It is unclear whether in the charm or bottom sector one can come to a similar conclusion.
In fact, in Ref. [11] as study was made within the quark model of the meson-meson components of the charmonium vector states, and it was concluded that even the ground state J/ψ had only as survival probability as a vector of about 0.69 when the meson-meson components to which it couples were considered. This makes us think that higher excited vector charmonium states could actually have even smaller qq components.
In the present work we retake this issue for the ψ(3770) vector state using data from the e + e − → DD reactions. We make an elaborate study of the DD, DD * , D * D , D * D * components of this resonance using the 3 P 0 model for hadronization of qq into meson-meson components which requires only one parameter. By means of this and the data of the e + e − → D + D − , D 0D0 reactions we can determine the parameters of the theory that allows us to evaluate the meson-meson selfenergy of the ψ(3770). The data of the e + e − → DD reaction are essential for the reliable calculations of the selfenergy, since the unknown couplings and a form factor entering the calculation are extracted from the data. In fact the form factor is relevant to the evaluation of the meson-meson probabilities and we show that it is tied to the fast fall down of the e + e − → DD cross section above the ψ(3770) peak.
The asymmetry of the ψ(3770) peak observed in the e + e − → DD reactions [12][13][14] has been the subject of the intense discussion (see Ref. [15] for a recent review). In Ref. [15] a work similar to the one we do here, but using only the DD components, which are the most relevant, is done, and the shape of the ψ(3770) peak is tied to a form factor that is introduced in an empirical way. We also implement this form factor in the same form and two different forms to estimate uncertainties. What we find is that the ψ(3770) is largely a qq state and the meson-meson components are small. The Z probability of having a qq vector core for the ψ(3770) is about 80 ∼ 85% and the individual meson-meson components are small. This paper is organized as follows. In Sec. II, we establish the formalism of calculating the cross section for e + e − → DD through the dressed propagator of ψ(3770), and the mesonmeson probabilities in the ψ(3770) wave function. In Sec. III, we present the results on the line shape of ψ(3770) fitting to the experimental data, and then calculate the Z probabilities using the parameters extracted from the fitting. A summary is presented in Sec. IV. The angular momentum algebra employed in the calculations is done explicitly in Appendix A.

II. FORMALISM
Our starting point is the hadronization in the process ψ → D ( * )D( * ) shown in Fig. 1, where we introduce aqq pair with the quantum numbers of the vacuum, and insert it between the quark constituents of ψ(3770), cc. The insertion ofqq is implemented in a 3 P 0 state [16,17], which indicates that the insertedqq has positive parity and zero angular momentum, and sinceq has negative parity we need an orbital angular momentum L = 1 forqq to fix the parity, which makesqq couple to spin S = 1, then S = 1 and L = 1 couple to total angular momentum J = 0. The ψ(3770) according to Ref. [5] corresponds to a D-wave cc state with no radial excitation, a 1 3 D 1 state with J P C = 1 −− . The hadronization in Fig. 1 proceeds as follows: where M corresponds to the following matrix Alternatively, we can write qq in Eq. (3) in terms of their meson components by means of the φ matrix for pseudoscalar mesons with the mixing between η and η taken into account [18], Similarly, the vector matrix corresponding to qq, which is also needed in our calculations, is given by As shown in Eq. (2), where the matrix M could either be the pseudoscalar matrix (which is labeled as P in the following) or the vector matrix (labeled as V ), we can have four different types of hadronization of the ψ(3770) leading to P P , P V , V P and V V . For example, when both M in Eq. (2) are pseudoscalar matrices we have where we have neglected η 2 c which is too heavy to be operative in the meson-meson loop that we shall consider below. It can be noticed that, since the ψ(3770) has isospin zero, the final hadronized combination of D 0D0 + D + D − + D + s D − s has isospin zero. Indeed, recalling the isospin doublets  Eq. (6) can be rewritten in a isospin-zero combination, which is Similarly, we can write the combinations coming from V P , P V and V V Note that the combination (P V ) 4,4 + (V P ) 4,4 that we get has the desired negative C-parity as it corresponds to the ψ(3770) (C D * = −D * in our formalism).
In order to interpret the line shape of the ψ(3770) we follow the steps of Ref. [15]. We consider the propagator of the vector meson R ≡ ψ(3770) with G(p) = The fact that ψ(3770) couples to P P , P V , V P , V V indicates that ψ(3770) will get a selfenergy Π(p) that we depict diagrammatically in Fig. 2. One can keep the covariant form of Π, but as shown in Ref. [15] only the transverse part of the propagator is relevant for the discussion here. We argue in a different way, with the same conclusion. In the loop one has Π ∼ d 4 qG(q)G(p − q) and the relevant part of it that enters the shape is ImΠ, where Indeed, as shown in the Appendix of Ref. [19], the error induced by neglecting the zero component in this case is 0.7%. Hence we need only the spatial component, i , and deal with G ij (p) = δ ij G(p) (i, j = 1, 2, 3). When we dress the propagator with the selfenergy of the diagrams in Fig. 2 we obtain and we must evaluate Π(p). Note that we write M R rather than M ψ because M R is now the bare mass of the resonance. The novelty in the present work with respect to Ref. [15] is that we include the contribution of P V , V P , V V mesons in the selfenergy. They only contribute indirectly to the line shape of the ψ(3770) because ImΠ is zero in all these cases. However, and then ImΠ in the numerator comes only from DD, but ReΠ(p) in the denominator comes from all the channels. Yet, the most novel thing here is that we will evaluate the probability that the ψ(3770) contains P V , V P and V V components in its wave function.
The evaluation of Π requires to relate the strength of the P P , P V , V P and V V couplings to the ψ(3770). This we can do with the help of the 3 P 0 model and the details are given in Appendix A. While the evaluation is involved, requiring elaborate sums of many Clebsch-Gordan (CG) coefficients, the results are very simple and we write the with and F (q) a form factor coming from the integrals of the quark radial wave functions discussed in Appendix A, where A in Eq. (16) is an unknown coefficient to be fitted to the data, and C i are the coefficients listed in Table I.
The former coefficients are for ψ(3770) assumed a 1 3 D 1 state. The terms of the Π(p) selfenergy are evaluated as follows, see Fig. 3. For D + D − , for example, we have The ψ propagator dressed with a D + D − loop as an example.
which gives us The q 0 integration can be done analytically and then we get in the rest frame of the ψ(3770) whereG(p 0 ) has the form with ω 1 (q) = q 2 + m 2 D + , ω 2 (q) = q 2 + m 2 D − . Let us note in passing thatG(p 0 ) has a structure similar to the G(p 0 ) function used in the study of meson-meson interaction [6] except for the extra factor q 2 that makesG(p 0 ) more divergent in the absence of the form factor. However, this form factor makes it convergent and we shall come back to it.
With the former expression forG(p) we can already write the ψ(3770) selfenergy as: Rather than evaluating the form factor F (q) with quark wave function we take an empirical attitude as in Ref. [15], and let the data determine this form factor from the shape of the e + e − → D + D − cross section. Once again we follow Ref. [15] and write where M inv is the e + e − invariant mass, √ s, and g ψe + e − , as in Ref. [15], will also be determined from the strength of the cross section.
It is also useful to separate σ into the contribution of the different channels (D + D − , D 0D0 ). Then we easily write: where where Π i (p) is the contribution to ImΠ(p 2 ) from the D + D − or D 0D0 channel (see Eq. (21)).
Note that in the denominator we have Π(p), meaning that all channels are included here.
A. Meson-meson probabilities in the ψ(3770) wave function Let us write for convenience, as in Ref. [15], which vanishes at √ s = M ψ , and with this choice we can write We can make an expansion around M ψ and have since ReΠ (M ψ ) = 0 and hence This is the typical wave function renormalization [20] and Z is interpreted as the probability to still have the original vector when it is dressed by the meson-meson components. Conversely 1 − Z will be the meson-meson probability of the dressed vector. If ∂ReΠ ∂p 2 is reasonably smaller than 1, one can make an expansion as in Eq. (29), and furthermore we have such that − ∂ReΠ can be interpreted as the meson-meson probability and in particular one can get the contribution of each channel: where Π i is the contribution of i-th channel to Π.

III. RESULTS
In Ref. [15] a form factor was used with ξ = 4(q 2 + m 2 ), that is the equivalent to our F (q) 2 , and Λ was fitted to data. We get similar results using this form factor. In addition, we use two other form factors: and with q on the following form for DD where λ is the usual Källén function, and the parameter R is fitted to the data in both cases. We have thus four parameters, as in Ref. [15], which in our case are M ψ , g ψe + e − , A and R. M ψ is of course very close to the nominal mass of the ψ(3770), g ψe + e − determines the strength of the cross section, A is related to the width of the resonance, and R determines the fall down of the resonance shape above the resonance peak. The parameters are fitted to the data of the cross section for e + e − → DD [12][13][14].
Given the fact that in the Appendix A we found that the form factor comes from an integral of the radial wave function of the quarks, and these are the same, independent of the different spin couplings, we assume this form factor to be the same for the P V , V P and V V cases. In Fig. 4 we show the results for the e + e − → D + D − cross section using the form factor of Eq. (34). The parameters used can be seen in Table II. As we can see, there is a good fit to the data, both above and below the peak, reflecting the asymmetry of the distribution, which does not have a Breit-Wigner form.
We should note that the description of the data is a result of the parametrization, and in particular the fall down of the distribution above the peak is related to the parameter R.
There is nothing fundamental in this interpretation of the asymmetry. However, the data and particularly the fall down above the threshold determine the range of the form factor, and this is important to make the integralG(p) convergent, such that the probabilities that we obtain are a consequence of the peculiar shape of the e + e − → D + D − data. In this sense, the probabilities that we obtain are a prediction based on the e + e − → D + D − data, while those in Ref. [11] were based on a particular quark model.  Table II. It is also interesting to evaluate the e + e − → D 0D0 cross section and compare with the  Table II. data, This is done in Fig. 5. We can see that the agreement with the data is also very good, Note that once the e + e − → D + D − is fitted, we have no freedom for the e + e − → D 0D0 , so the latter one is a prediction of the approach.
In Fig. 6  We observe a good fit in the region above the peak, but not as good as before below it, although still comparable with the bulk of the data. Concerning our main goal, which is the evaluation of the meson-meson probabilities, the fall down of the cross section above the peak is acceptable.

A. Evaluation of the vector and meson-meson probabilities
In Table IV we show the probability of Eqs. (29) and (31) using the form factor of Eq. (34).
What we see is that the probabilities of the D + D * − + c.c or D 0 D * 0 + c.c are practically zero.
However, there is the unpleasant feature that − ∂Π DD  Table III.
, which provides the DD probability as we have seen, is negative, is unexpected and unacceptable. Fortunately, the value is very small, and could be admitted as an uncertainty related to the approximation implicit in Eq. (28). As a consequence of this negative number, the Z probability of having the original vector in the ψ(3770) wave function is bigger than one. Yet, by an amount of 9.3%, which tells us the uncertainties that we have in this approach. It is interesting to note that if we use the form factor of Ref.  Table III.  In view of this, we use a form factor more in agreement with phenomenology, which is the one of Eq. (33). This form factor induces a correction to the width where Γ 0 is the width evaluated at √ s = M ψ . This factor is the Blatt-Weisskopf barrier penetration factor [21], commonly used to write the width in usual Breit-Wigner amplitudes.
In view of this, we can give more credit to the results that come from this factor. The results can be seen in Table V.   Fig. 10.
In this case we find Z ∼ 0.854. This is a reasonable number, but in view of the results in Table V with the former fit, we can settle the value of Z within 0.80 ∼ 0.85, which is a reasonable range of uncertainty.  Table VI.  Table VI.
This result is very valuable and we consider it the most important output of the work.
There is a continuous debate about the nature of the hadron resonances and it is long since the ideal picture of mesons as pure qq and baryons as qqq has been abandoned. With the advent of hadrons in the charm and bottom sectors, the evidence for more complex structures is appalling [2,3]. Yet, in spite of this, an elaborate study combining elements of  the σ, f 0 (980), · · · are completely off the qq picture, the vector mesons are largely qq states [10]. Our result comes handy when some calculations could make us lose confidence in this picture. Indeed, in Ref. [11], where a calculation within a quark model was done to assess the relevance of the meson-meson components in the vector mesons, even the J/ψ was found to have a Z probability of only 65%, implying that more massive ψ vectors could have an even smaller Z probability. The result of the present paper incorporating the features of the ψ(3770) shape in the e + e − → DD reactions, demanded the presence of a form factor that has a consequence the small meson-meson probabilities and the large Z value.

IV. SUMMARY AND DISCUSSION
We have performed an evaluation of the meson-meson components in the ψ(3770) wave function, considering P P , P V , V P and V V components. We found that the determination of such probabilities was much tied to the shape of the e + e − → DD reaction, which we described in terms of the ψ(3770) selfenergy due to the meson-meson components. Indeed, the shape of the cross section for this reaction determined the range of a form factor that was determined in the evaluation of the meson-meson probabilities of the ψ(3770) wave function.
Within uncertainties we found that the Z probability of a vector component in the ψ (3770) is of the order of 80 ∼ 85% and the individual meson-meson components are small. This finding is very important, extracting from this phenomenological study the same conclusion obtained from QCD and large N c behavior, plus meson-meson scattering data, that vector mesons are largely qq objects [10]. This is also in line with Z evaluation for the ρ with a different method which gives Z ∼ 0.75, even with such a large width for the decay to two pions [22].
We do the same to couple the spin and orbital angular momentum of the qq vacuum state 3 P 0 in Fig. 1, as done in Refs. [23,24], and we combine this state, |1S 3 , with the L = 1 state Y 1,M 3 (r) to give J = 0, implying M 3 + S 3 = 0, i.e., M 3 = −S 3 , which allows us to rewrite Eq. (A4) as follows In addition we have the spatial matrix element, where the c,c quark states are in their ground state. Then we have where q is the exchanged momentum between the two mesons produced after the hadronization, and e iq·r can be expanded as The coupling rule for spherical harmonics permits an easy way of combining three spherical harmonic functions as we show in the following equation, where two of them come from Eq. (A6) and the other one, Y * l,µ (r), from Eq. (A7). After integrating over the full solid angle, we arrive at [25] dΩ Y * lµ (r)Y 1,−S 3 (r)Y 2,M 3 (r) = 15 4π(2l + 1) where for parity reasons 2 + 1 + l must be even, hence, l = 1, 3, but l = 1 is required to have a P -wave coupling of J/ψ to DD at the end, such that we obtain (where we use C(2, 1, 1; 0, 0, 0) = − 2 5 ) Since j 1 (qr) goes as qr for small values of qr, M E(q) grows linearly q for small q, and for that reason we rewrite M E(q) as where the factor 3j 1 (qr) qr goes to 1 as qr approaches 0 and is a smooth function, such that the integral in Eq. (A10) is a smooth function of q for small q, the typical form of the form factors and the form that we will take for our empirical form factors. We can write Now we use S 3 = M 1 + M 2 −M and the above equation can be rewritten as, In Eq. (A14) there are four CG coefficients that depend on s. In order to get an expression with three CG coefficients to be written in terms of Racah coefficients we proceed as follows.
Firstly, we need to permute some indices in the fourth CG coefficient in Eq. (A14) as Ref. [25], where W is a Racah coefficient [25]. Similarly, we need to permute indices of the third CG coefficient in Eq. (A14) and the first CG coefficient in Eq. (A16) before we move on to the next combination, We combine now the three CG coefficients from Eqs. (A17), (A18) and the fifth CG coefficient in Eq. (A14) [25], and since the phase does not depend on s, we can write such that Eq. (A14) can be rewritten as W j can be expressed explicitly as follows, with the condition thatM = M 1 . Furthermore, the other two CG coefficients in Eq. (A20) can be rewritten as