Chiral effective Lagrangian for heavy-light mesons from QCD

We derive the chiral effective Lagrangian for heavy-light mesons in the heavy quark limit from QCD under proper approximations. The low energy constants in effective Lagrangian is expressed in terms of light quark self-energy. With typical forms of the quark self-energy and running coupling constants of QCD, we estimate the low energy constants in the model and the strong decay widths. A comparison with data and some discussions of the numerical results are presented.


I. INTRODUCTION
It is widely accepted that the dynamics of strong interaction at large distance can be well captured by effective theories which are controlled by certain symmetries and symmetry breaking, for example chiral symmetry breaking, of the fundamental theory of strong interaction -QCD (see, e.g., Ref. [1]). Little progress has been made to establish a direct relation between QCD and effective theory in an analytic way although such a relation is very significant since, to our opinion, almost all the puzzles and problems in hadron dynamics in both matter free space and in-medium can be attributed to our less knowledge about nonperturbative QCD.
In Ref. [2], the authors derived a relation between QCD and chiral perturbation theory (ChPT) including the Nambu-Goldstone bosons, pions, only [3][4][5]. The derivation is based on the standard generating functional of QCD with external bilinear light-quark field sources in the path integral formalism so that, the coefficients in the chiral Lagrangian can be derived from QCD Green functions. Thanks to the developments of lattice QCD and Dyson-Schwinger equation (DSE) methods, although an exact calculation of the nonperturbative QCD Green functions are not feasible so far, this derivation provides a possibility to calculation ChPT from QCD -the fundamental theory of strong interaction [6][7][8][9][10]. The extension of this approach to include other light mesons are also investigated in the literature [11][12][13].
The purpose of this work is to establish a relation between QCD and chiral effective theory for heavy-light mesons [14]. Since in a heavy-light meson, there is a heavy quark and a light quark, its dynamics is controlled by both the heavy quark symmetry [15] and chiral symmetry (see Ref. [16] for a systematic discussion). This system is a good environment for studying the mechanism of chiral symmetry breaking since it contains only one light quark.
In addition to the lowest-lying heavy quark doublet H * huifengfu@jlu.edu.cn † yongliangma@jlu.edu.cn ‡ wangq@mail.tsinghua.edu.cn with J P = (0 − , 1 − ), we introduce the first orbital excitation states, the heavy quark G doublet with J P = (0 + , 1 + ) (both H and G doublets will be specified latter) to our system. Since in these H and G doublets, the light quark clouds have quantum numbers j P = (1/2) + and j P = (1/2) − , respectively, they are regarded as chiral partners to each other and their mass splitting arises from the chiral symmetry breaking due to the quark condensation in the chiral doublet model [17,18]. We explore the quark condensate dependence of the mass splitting between the H and G doublets in the present approach from QCD. We express the low energy constants (LECs) in the effective theory, for example the heavy-light meson mass, heavy-light meson and pion coupling constant, in terms of the light quark self-energy (with the heavy quark mass rotated away) therefore become calculable from fundamental QCD. By used the light-quark self-energy calculated from truncated DSE, we calculate these LECs as a function of quark condensate. We find that, with the decreasing of the quark condensate, the masses of both the H and G doublets decrease. Although this observation agrees with Ref. [19] with a specific choice of the parameters in the model, it disagrees with what was found in Ref. [20] that although the G doublet decreases with the decreasing of quark condensate, the H increases. So far, we cannot give a clarified comment on this discrepancy between effective model calculation and the present QCD approach. What is interesting in the present calculation is that the mass splitting between H and G doublets decreases with the decreasing of quark condensate. This tells us that the mass splitting between chiral partners arises from the quark condensate, at least partially if not all. A more possible interesting point can be drawn from the present work is that, once the quark condensate is influenced by the environment such the medium density and temperature, the LECs in the effective Lagrangian are also affected. In the literature, such modification is referred to as intrinsic density dependence in the dense system as opposite to the density dependence induced by nuclear correlations (see, e.g., Ref. [21] for a recent comprehensive discussion). This paper is organized as follows. In Sec. II, to set up the framework, we write down the chiral effective the-ory of heavy-light mesons that will be derived in the present work. In Sec. III we derive the heavy-light meson Lagrangian from QCD and express the low energy constants in the effective theory in terms of quark selfenergy. The numerical results calculated by using the quark self-energy obtained from a typical DSE are given in Sec. IV. Sec. V is devoted to the our discussion and perspective.

II. CHIRAL EFFECTIVE THEORY OF HEAVY-LIGHT MESONS
To set up our framework, we write down the chiral effective theory of heavy-light mesons that will be studied in this work. We introduce the charmed heavylight meson doublets H and G with quantum numbers J P = (0 − , 1 − ) and J P = (0 + , 1 + ), respectively. In terms of the physical states and the notation of PDG [23], they are expressed as In the following, without specification, we will focus on two light flavors although the extension to three flavors are straight forward. Under chiral transformation, the H doublet transforms under the unbroken SU (2) V subgroup of chiral symmetry as an antidoublet with a being the light flavor indices. The same transformation holds for G doublet. Then the simplest effective Lagrangian describing the interaction between pion and heavy-light meson can be written as [16,22] where where we have decomposed the chiral field U (x) = exp(iπ(x)/f π ) as U = Ω 2 and A µ = i 2 Ω † ∂ µ Ω − Ω∂ µ Ω † and D µ = ∂ µ − iΓ µ with Γ µ = i 2 Ω † ∂ µ Ω + Ω∂ µ Ω † . From PDG [23], one can obtain the spin-averaged masses of H and G doublets as which yields the mass splitting It is believed that the value of the mass splitting between chiral partner arising from the chiral symmetry breaking.
In the chiral doubler structure, the mass splitting ∆m is attributed to the chiral symmetry breaking, i.e., the vacuum expectation value of the sigma field in the linear sigma model [17,18]. Therefore, the magnitude of this mass splitting measures the magnitude of the chiral symmetry breaking, i.e., the larger ∆m, the stronger chiral symmetry breaking.
The Lagrangian (3) is written down with respect to the chiral symmetry and heavy quark symmetry of QCD. Only the terms with the minimal number of derivatives are included. The parameters g H , g G , g HG , m H and m G are free ones at the level of effective theory since they can not be controlled by symmetry argument. We next evaluate these LECs from fundamental QCD.

III. CHIRAL EFFECTIVE LAGRANGIAN FOR HEAVY-LIGHT MESONS FROM QCD
In this section, we devote ourselves to derive the heavylight effective theory (3) from generating functional of QCD.
A. Integrating in the (psuedo-)Nambu-Goldstone boson fields We first focus on the (pseudo-)Nambu-Goldstone boson part in the effective Lagrangian. For this purpose, we introduce an external source J(x) of the composite light quark operators [4,5] into the QCD generating functional where q, Q and G µ are the light-, heavy-quark fields and gluon fields, respectively. In the chiral limit which will be considered through out this work, this generating functional can be rewritten as where M is the mass matrix for the heavy quark fields.
Here and in the following, we use i, j, · · · to represent color indices in the adjoint representation, and I µ i = q λi 2 γ µ q +Q λi 2 γ µ Q to denote the quark composite operator. The external source J(x) can be generally decomposed into scalar, pseudoscalar, vector, and axialvector parts where s(x),p(x),v µ (x), and a µ (x) are Hermitian matrices in the flavor space, and the light quark masses have been absorbed into the definition of s(x). The vector and axial-vector sources v µ (x) and a µ (x) are taken to be traceless. To introduce the pseudoscalar meson field U = Ω 2 = e iπ(x)/fπ into the theory, following Ref. [2], we insert the following constant into the QCD generating functional (8). In Eq. (10), ]. B is a short notation for the bilocal composite light quark fields: where η, ξ, · · · represent spinor indices, a, b, · · · represent flavor indices and α, β, · · · represent color indices in the fundamental representation. N f is the number of light flavors, and tr l denotes tracing over the where The exponential part in Eq. (11) is invariant under the chiral rotation if the external field J(x) undergoes the following transformation In addition, one can find In this work, we concentrate on the normal part of the chiral Lagrangian for heavy-light mesons and ignore the chiral anomaly. In this case, the generating functional can be reexpressed in terms of rotated fields as: where . Now we integrate out the gluon fields, which generates pure Yang-Mills gluon Green's functions µ1···µn (x 1 , · · · , x n ) in the action. By Fierz reordering, we can further diagonalize the color indices of the quark fields, and get (15) where the symbol ψ represents both light and heavy quarks, i.e., ψ = (q, Q).Ḡ σ1···σn ρ1···ρn (x 1 , x 1 , · · · , x n , x n ) is a generalized gluon Green's function. Then, the generating functional becomes

B. Integrating in the heavy-light meson fields
We next introduce the heavy-light meson fields into the system by considering the heavy quark symmetry. Following the standard heavy quark effective theory (HQET) ( see,.e.g., Ref. [16] for a pedagogical discussion), we introduce the velocity-dependent heavy quark field by using the following substitution where v µ is related to the heavy quark momentum p µ by p µ = M v µ + k µ with k µ the residue momentum. Introducing the projection operator (1 ± / v)/2, one can decompose the heavy quark field as The N v field is the large component of the quark field which survives in the heavy quark limit whereas the N v field is the small component of the quark field which disappears in the heavy quark limit. Applying Eq. (17) to To proceed, we introduce a bilocal auxiliary field Φ ab ηξ by inserting the following constant into Eq. (16): where ψ = (q, N v ) in the sense that only the contribution from N v -the large component of the heavy qaurk field -is considered. 1 The generating functional then becomes where we have replaced the bilocal quark fields with Φ in the spirit of the heavy quark limit, i.e., the replacement is correct in the heavy quark limit. The δ function can be further expressed in the Fourier representation 1 The small component of the heavy quark field can be easily included in our approach along Ref. [24].
Then the generating functional becomes wherē , only the "positive projected" part (the part projected by where Π qQ Since we only interest in the leading contribution from heavy quark expansion, Π qQ − has no contributions here. So eventually, we can keep only the positive projected parts of the Φ and Π fields for their heavy flavor components in the generating functional. Now, we can integrate out quark fields q and N v and obtain where we have defined the functional trace Tr taking over the flavor space, spinor space and coordinate space and I 1 and I 4 are the following matrices in the flavor space where A similar decomposition holds for Φ. Tracing back to Eq. (22) and from the termψ (x)Π(x, x )ψ (x ), one can find that Π directly couples to quark-anti-quark pairs and has the same transformation properties as the composite quark fields, so that it is reasonable to identify Π 2 and Π 3 as the bosonic interpolating fields for heavy-light mesons. However, since Π is a bilocal field, to get a local effective lagrangian, we need a suitable localization on Π 2 and Π 3 fields. Here we take the following localization conditions which are essentially the point coupling between quarks [25] It is easy to check that Φ and Π have the following properties:

So thatΠ
. Then the generating functional can be written as where the δ(O † − O) term has been reexpressed as with Ξ ab (x) being new auxiliary fields Ξ ab (x) and Now, by integrating out the fields Φ, Ξ, Π 1 and Π 4 , we can obtain the action, denoted as S[U, Π 2 ,Π 2 ], for the chiral effective theory with heavy-light mesons The heavy-light meson effective theory can be derived by expanding the action S with respect to the Goldstone Boson fields U and the heavy-light meson fields Π 2 ,Π 2 .

C. The action in the large Nc Limit
The action defined in the previous subsection is not practically useful. In order to calculate the coefficients in the chiral Lagrangian, we make a further approximation, namely, keeping the leading order in the large N c expansion. Under large N c limit, the effective action is simply the corresponding classical action, so we have where Φ c , Ξ c , Π 1c , Π 4c are classical fields satisfying the saddle point equations and Γ I term has been ignored because it is of O(1/N c ) [2].
These saddle point equations provide important information. For instance, equations δS δΠ1c = 0 and δS δΦ1c = 0 generate the coupled equations where the term involving Ξ has been omitted because it vanishes once the external sources J are turned off [2]. When J Ω is turned off, the coupled equations (35) and (36) are nothing but the DSE for the quark propagator with Π 1c being the self-energy for light quarks. So that we rewrite Π 1c (x, y) asΣ(x−y)I 1 . Along the same procedure, the DSE for the heavy quark propagator which depends on the self-energy of heavy quark, Π 4c , can be obtained. However, since the contribution from the heavy quark self-energy is less significant than the light ones, we will simply ignore it. The direct calculation of the low energy constant in the effective Lagrangian from action (33) is not so easy, if not impossible, because the fields Φ c and Ξ c are functionals of U , Π 2 andΠ 2 through the saddle point equations (34). To proceed, we follow Ref. [6] and keep only the "Tr ln" term in the action in the spirit of the dynamical perturbation. Then the action becomes It should be noted that although the contribution from gluon field is not explicit appeared in the action, it does contribute through the quark self-energyΣ, which requires the application of the DES.

D. Chiral Effective Lagrangian
Now, we are ready to derive the chiral effective Lagrangian (3) for the heavy-light mesons. We only con-sider the (0 − , 1 − ) and (0 + , 1 + ) states, so the two nonzero elements in Π 2 take the form where q = 1, 2 denotes light flavor indices and H and G fields are given by Eq. (1). Expanding the action S (37) with respect to U , Π 2 andΠ 2 generates the chiral effective Lagrangian. Since U field is attached to the rotated external source, we actually take derivatives on the action S with respect to J Ω . So where the subscript 0, 1, 2, 3, · · · denotes the order in the expansion in terms of J Ω , Π 2 andΠ 2 . In the expansion (39), the S 0 term, a constant of the action S, has no physical significant and, the first order term S 1 vanishes due to the saddle point equation. Therefore, the leading order contribution comes from S 2 . Due to symmetry arguments, only the following terms survive: The first term generates the nonlinear sigma model of the Nambu-Goldstone bosons which has been extensively discussed in Ref. [2]. We will not go to the details here. The second term of S 2 generates the model of heavy-light mesons which is interested in the present work. We denote the second term in S 2 as S 22 , then for the fields H andH, we have where we have used the equationHi / ∂H =Hiv · ∂H and / vH = H. As indicated in Ref. [6] to keep the invariance of the vector part of the chiral transformation, the self-energy should take the formΣ(x − y) = Σ(∇ 2 )δ(x − y) with Σ being the self-energy of the light quark field and ∇ ≡ ∂ − iV Ω as the covariant derivative. This form retains the correct chiral transformation properties in the theory. Through a derivative expansion, we obtain (up to the first order) where we have used the relation H/ v = −H. From Eq. (42) one can easily see that S 22 includes the mass term, the kinetic term and a part of the interaction term of the H fields.
We next consider the part of the S 3 which generates the interaction between the heavy-light meson field H and the light Goldstone boson field upto the leading order of the chiral counting. This can be obtained as Recalling that S Ω and P Ω are of O(p 2 ) while A µ Ω and V µ Ω are of O(p) in the chiral counting, we neglect S Ω and P Ω term. Through a derivative expansion we obtain Summing up Eqs. (42) and (44), one can obtain the expression of the constant m H and g H as with Z H being the wave-function renormalization factor It is interesting to note that the summing of the S 22 and S 3 terms yield the same coefficients for the Tr H(x)iv · ∂H(x) term and Tr H(x)v ·V ΩH (x) term. This means that the vector part of the chiral symmetry is reserved in our approach, agrees with the pattern of the chiral symmetry breaking in QCD. By using the same argument, we obtain the parameters of the heavy-light hadrons in positive parity sector. The expressions are with Z G being the renormalization factor of G field And, we obtain the coupling between the parity partner fields H and G as In our the above formula, Σ(−p 2 ) stands for self-energy of light quarks and needs to be calculated from QCD. This is an improvement compared to the work in Ref. [17]. In addition, our procedure of deriving the chiral Lagrangian allows a systematic improvements in the calculation.

IV. NUMERICAL RESULTS
We have obtained the chiral effective Lagrangian for heavy-light mesons and the integral forms of low energy constants in terms of the dynamical quark mass Σ(−p 2 ). To have a quantitative idea, we make some numerical calculation in this section.
It is clear that, the integral forms of the low energy constants have an ultraviolet cutoff Λ c which should be of the order of the scale of the chiral symmetry breaking. To obtain a reasonable prediction, we fix it to be ≈ 0.9 GeV. In order to calculate the LECs, we need to know the light quark self-energy Σ(−p 2 ). Since the light-quark mass functions calculated by the DSEs and by the Lattice QCD are in agreement to each other [26], we adopt the DSE method (which is much more easier for us to handle) to calculate the light quark self-energy. Here, as a typical form, we take the following differential form [6] with boundary conditions where α s is the running coupling constant of QCD, Λ is an ultraviolet cut-off. In the following calculation, we take Λ = (10 3 Λ QCD ) 2 .
Since the low energy behavior of the QCD running coupling constant is yet unclear to us, as a comparison, we take two models for α s . One is a segmented form of the coupling constant [6], which has a finite infrared limit (Model A): where N f = 2 and Λ QCD = 0.25 GeV; a, b and d are the parameters of the model, and c is an independent parameter keeping the strong coupling continuous at the boundary via the relation c = (a−1/d)/(2+d) 2 . We have found that the values of d = 4 and b = 3 give reasonable sets of results, so we fix them and focus on the variation of the results on the parameter a alone. The other form of the strong coupling constant is taken from the refined Gribov-Zwanziger formulism [27], which has a vanishing infrared limit (Model B):  To show the parameter dependence of the self-energy Σ(−p 2 ), we illustrate in Fig. 2 the Σ(−p 2 ) obtained by solving the DSE with the two models with parameter choice relevant to the present work. Remember that we have taken the chiral limit. So the results exhibit desired behavior of dynamical chiral symmetry breaking.
Given the self-energy, according to Eqs. (45) and (46), we obtained the masses of the heavy-light mesons and the coupling constants of the heavy-mesons-pion interactions. These results as well as the quark condensate qq calculated with different parameters are shown in Tables I-II. As expected, we find that the mass splitting between the chiral partners increases as the quark condensate increases. Our results directly shows how the dynamics of the fundamental theory affects the LECs of the effective theory. For the mass splitting, we find that both Model A (with parameter a = 0.79) and Model B (with parameter a 0 = 0.58) can give the results which agree with experimental data (see Eq. (6)). However, for the coupling constants, we find Model B with parameter a 0 = 0.58 is more favorable. In this case, the coupling g H = 0.408, which directly determines the width of the D * → Dπ decay. By using the expression with P π being the three momentum of the decay products, we obtain the decay width Γ(D * + → D 0 π + ) = 0.032 MeV which is roughly comparable to the experimental result 0.056 MeV [23]. In addition to the above intramultiplet transition, one can also calculate the intermultiplet transitions. The coupling constant responsible to this transition is calculated as g HG = 0.432. From this value, the intermultiplet onepion transition D * 0+ 0 → D + π − is obtained as which is at the same order as the experimental value Γ D * 0 = 267 ± 40 MeV.  In this paper, we derive the chiral effective Lagrangian for heavy-light mesons from QCD. The relationship between the LECs and the quark self-energy is obtained. By using a typical DES of the light quark self-energy and the running coupling constant of QCD, we calculated the values of the LECs in the Lagrangian. With properly chosen parameters, the numerical results are roughly consistent with the experimental data. In this sense, we say that our results are obtained from QCD.
As this moment, we attribute the deviation between our numerical results and the data as follows: First, as the leading order calculation in heavy quark expansion and large N c expansion, the results suffer from high order corrections from both 1/N c terms and 1/m Q terms. Second, the non-'Tr ln' term in the action has been omitted in our calculation for simplicity, however these terms generate contributions with complicated forms which in general cannot be described by the self-energy alone.
The scheme developed here for deriving the chiral effective Lagrangian for heavy-light mesons and calculating the LECs from QCD can be straightforwardly extended to include the 1/m Q corrections, higher order derivatives as well as excited heavy-light mesons. This is beyond the scope of the present work and will be reported elsewhere. In addition, it might be possible to extend the present work the exotic heavy hadrons such as the tetraquark state. Such kind of study is in progress.
The last but not the least point we want to mention is that, the results shown here illustrate the quark condensate dependence of the LECs in the effective theory. When the change of the quark condensate due to the environment is know, one can easily translate our results to the change of the LECs therefore obtain the intrinsic environment, such as the density, dependence of the effective theory.