Pseudoscalar Mesons Scattering off D ∗-mesons with Chiral Perturbation Theory

We have systematically calculated the s-wave pseudoscalar mesons and D*mesons scattering lengths to the third order with chiral perturbation theory, which will be helpful to reveal their strong interaction. We present the scattering lengths (1) in the framework of the heavy meson chiral perturbation theory and (2) in the framework of the infrared regularization.

Many new charmed particles have been discovered over the past few years.Some of them are speculated to be the candidates of the molecular states.The pseudoscalar meson and heavy vector D * -meson might also form the molecular states if there exists strong attractive interaction between them.
Recently there have been some research papers devoted to the scattering of the pseudoscalar meson and heavy pseudoscalar D meson using different methods [9][10][11].Here we will report the pseudoscalar meson and heavy vector D * -meson scattering length with chiral perturbation theory.
Chiral perturbation theory is based on chiral symmetry and spontaneous chiral symmetry breaking.Its expansion parameter is the small momentum of the interacting particles.It is widely applied in many fields such as ππ scattering, γγ → ππ, baryon form factors, πD scattering and so on.In this report we will use heavy meson effective field theory(HMEFT) [12][13][14][15] and infrared regularization method(IR) [16][17][18][19], which are both based on chiral perturbation theory, to study on pseudoscalar meson and D * meson scattering.
We represent that the T -matrices of pseudoscalar meson and D * meson scattering with HMEFT in Sec. 2. The T -matrices with IR are shown in Sec. 3 accompanied by the comparison of the two different methods.Then we will give a short summary in Sec. 4.

The T -matrices with the Heavy Meson Chiral Perturbation Theory
Besides chiral symmetry we will also utilize the heavy quark symmetry which ensures that the system is independent on heavy quark spin and flavor.HMEFT has good power counting, that is the T -matrices at a certain order is only corresponding to definite relevant Feynman diagrams.We will calculate the πD * s-wave scatting length to O( 3 ) at the threshold.For the calculations in detail, see the Ref. [20].
In the calculation we would use the following Lagrangians, where H represents the lightest charmed doublet of D and D * mesons, both u µ and χ − contain the odd numbers of Goldstone bosons, and Γ µ contains even numbers of Goldstone bosons.We would use the above Lagrangians to get the T -matrices.Loop-diagrams in Fig. 1 would generate the ultraviolet divergence in the terms containing L But after wave function renormalization and the renormalization of the following low-energy constants all infinities will be cancelled.Then we will get the terms in the first-, second-and third-bracket are at O( 1 ), O( 2 ) and O( 3 ) respectively.The superscript '3/2' of T (3/2) πD * in the bracket represents the total isospin of the channel.We find that πD in Ref. [9] when the mass difference δ between D and D * approaches 0. There are 11 independent T -matrices in the pseudoscalar meson and D * meson scattering due to the isospin symmetry, and the other 10 T -matrices are listed in Ref. [20].
To get the final numerical results we still need determine the low-energy constants, which can not be determined by chiral theory itself.Instead we can obtain them by fitting the experimental data.However, due to the lack of the relevant data, we resort to the resonance saturation model [21,22].We consider the contribution of the σ(600) singlet and κ(800) octet as shown in Fig. 2(a), then integrate these intermediate particles, and finally get the relevant effective interacting vertices as in Fig. 2(b).Comparing the vertices with those in Lagrangian (3), one gets  For the low-energy constants at the third order, the resonance saturation method would bring large uncertainty.So as in Ref. [9] we take which is obtained by fitting the data of the lattice calculation [23].Other low-energy constants are just assumed to be negligible as done in Refs.[3,24].We list the T -matrices of the πD * scattering length in Table 1.We find that the chiral expansion of the pion-D * channels converges well.The loop contribution in the kaon/eta-D * channels is large due to the large mass of kaon/eta.Luckily it is cancelled mostly by the tree contribution at the same order, which makes the results at O( 3 ) smaller than that at the leading order as expected.The positive real part indicates that the strong interactions are attractive for the following channels: ηD * and a (0) ηD * s .

The T -matrices with the Infrared Regularization Method
IR has both good power counting and correct analyticity since IR formalism includes the higher-order infrared parts of the loop graphs.Relativistic Lagrangians will be used in IR.When calculating loop integral d d k, the integral is replaced by the infrared integral I d d k.The replacement would drop some terms.But the dropped terms could be absorbed by tree diagrams.Thus it is just a redefinition of low-energy constants.
IR was first introduced to study the πN scattering [16].Here we extend it to study the πD * scattering.We obtain the T -matrices with the following relativistic Lagrangians, where (Similar for P * .)If letting the heavy meson mass approach infinity, we verify that the results with IR are the same with those with HMEFT.The difference between IR and HMEFT results from the different loop integrals.So here we only list the numerical comparison of the loop diagrams in Table 2. From Table 2, one can see that the difference is not very large in the numerical results between the two methods.Also we find that the contribution with the intermediate D * meson dominates the loop contribution in both cases of HMEFT and IR.The difference is not very large in the numerical results between the two methods.

Summary
In short summary, we have calculated the pseudoscalar meson and heavy vector meson scattering length to the third order.We found there exists attractive interaction in some channels, which might indicate the existence of molecular states.Hopefully our work will be useful to the future lattice study and the investigation of possible molecular states.

Fig. 1 .Fig. 2 .
Fig. 1.Nonvanishing loop diagrams for the pseudoscalar meson and D * meson scattering lengths to O( 3 ) with HMχPT and IR.The dashed lines, thin solid lines and thick solid lines represent the pseudoscalar Goldstone bosons, D mesons and D * mesons, respectively.

Table 1 .
The threshold T -matrices for the pseudoscalar meson and D * meson scattering order by order in units of fm with HMχPT.

Table 2 .
Comparison of the T -matrices from the loop diagrams for the pseudoscalar meson and D * meson scattering between HMχPT and IR.