SU(4) flavor symmetry breaking in D-meson couplings to light hadrons

The validity of SU(4)-flavor symmetry relations of couplings of charmed $D$ mesons to light mesons and baryons is examined with the use of $^3{\rm P}_0$ quark-pair creation model and nonrelativistic quark model wave functions. We focus on the three-meson couplings $\pi\pi\rho$, $KK\rho$ and $DD\rho$ and baryon-baryon-meson couplings $NN\pi$, $N\Lambda K$ and $N\Lambda_c D$. It is found that SU(4)-flavor symmetry is broken at the level of 30% in the $DD\rho$ tree-meson couplings and 20% in the baryon-baryon-meson couplings. Consequences of these findings for DN cross sections and existence of bound states D-mesons in nuclei are discussed.


Introduction
Currently there is considerable interest in exploring the interactions of charmed hadrons with light hadrons and atomic nuclei [1]. Particular attention is paid to D-mesons, much discussed over the last few years in connection with D-mesic nuclei [2][3][4] and J/ψ binding to nuclei [5,6]. Presently, there is no experimental information about the DN interaction, a situation that the PANDA@FAIR experiment [7] could remedy in the future. Most of the knowledge on the DN interaction comes from calculations using hadronic Lagrangians motivated by SU (4) extensions of light-flavor chiral Lagrangians [8][9][10][11][12][13][14] and heavy quark symmetry [15,16]. These require as input coupling constants and, in some cases, form factors. For the particular case ofDN reactions (whereD ≡ {D 0 , D − }), ref. [11] found that among all the couplings in the effective Lagrangian, g DDρ and g DDω provide the largest contributions to cross sections and phase shifts for kinetic center of mass (c.m.) energies up to 150 MeV -they also play an important role for the DN interaction [13]. Flavor SU (4) symmetry relates those couplings to couplings in the lightflavor sector, a e-mail: gkrein@ift.unesp.br The studies in refs. [11][12][13] utilized the SU (4) relations above, based on g ππρ = 6.0 and g NNπ = 13.6, which are the values used in a large body of work conducted within the Jülich model [17,18] for light-flavor hadrons. Given the prominent role played by meson-baryon Lagrangians in the study of the DN interaction and associated phenomena, it is of utmost importance to assess the validity of (1) and (2). SU (4) breaking effects on three-hadron couplings were examined recently using a variety of approaches, that include vector meson dominance (VMS) [19,20], Dyson-Schwinger and Bethe-Salpeter equations (DS-BS) of QCD [21], QCD sum rules (QCDSR) [22][23][24], lattice QCD [25], and holographic QCD [26]. In this work we use the quark model with a 3 P 0 quark-pair creation operator [27][28][29]. In this setting, the three-hadron couplings are given by matrix elements of the 3 P 0 operator evaluated with quark-model wave functions. The literature on the 3 P 0 model is too vast to be properly reviewed here, we simply mention that it is being used extensively since the early 1970s to study strong decays and that our calculation of vertices shares similarities with those of nucleon-meson couplings and form factors in [27][28][29][30].

Three-hadron couplings
To evaluate the matrix element of the 3 P 0 quark-pair creation operator,Ô pc , it is convenient to employ the "decay frame" of an initial hadron at rest [27][28][29][30], i.e. the transition of a hadron state |h 1 into a final two-hadron state |h 2 h 3 is written as with q = P 2 = −P 3 , and where γ gives the strength of the quark-pair creation, q cf † s (p) andq cf † s (p) are creation operators with color c, flavor f , spin projection s, and momentum p, σ c s s = χ † s σχ c s , with σ = (σ 1 , σ 2 , σ 3 ) being the Pauli matrices, χ s Pauli spinors, and χ c s = −i σ 2 χ * s . We employ the standard quark-model Hamiltonian [31], where m i are the quark masses and F = λ/2, with λ the color SU (3) Gell-Mann matrices and S the spin 1/2 vector. Notwithstanding the inability of the model to describe all features associated with the Goldstone-boson nature of the pion, nonetheless it mimics some of the effects of dynamical chiral symmetry breaking, notably the π-ρ mass splitting [32]. As in QCD itself, the only source of SU (4) breaking in (5) is the quark-mass matrix and hence the breaking in the couplings comes solely from the hadron wave functions. The Schrödinger equation is solved as a generalized matrix problem using a finite basis of Gaussian functions with the eigenvalues determined by the Rayleigh-Ritz variational principle. Reasonable values for the masses of the ground states of the hadrons of interest can be obtained by expanding the meson Φ and baryon Ψ intrinsic wave functions as [31,33] where the c n are dimensionless expansion parameters and Here, α is the variational, r = r 1 − r 2 , ρ = (r 1 − r 2 )/ √ 2, and λ = 2/3[(r 1 +r 2 )/2−r 3 ]. The matrix element M(q) can be evaluated analytically; it is given by where Y 1m (q) are spherical harmonics with m = 1(0) for three-meson (nucleon-baryon-meson) couplings, κ comes from summing over color, spin, and flavor and is given by The amplitude A h1h2h3 (q) in (8) is given by where f h1h2h3 are given by (P in P P ρ stands for π, K, D and B in NBP for N , Λ, Λ c ) f P P ρ (n 1 , n 2 , n 3 ) = 64 9π and the "cut-off" parameters Λ h1h2h3 are given by where In the limit of SU (4) symmetry, m 1 = m 2 , α D = α K = α π and α Λc = α Λ = α N , and the ratios are all equal to 1, expressing the same symmetry as in (1) and (2). In this limit, γ must be the same for all couplings, which seems a reasonable assumption, as they involve the same light-quark pair creation. Symmetry-breaking effects are contained in the factors f , c n and Λ.
Let us now connect to meson-exchange models. A typical three-meson vertex function, as it appears in that approach in the P N potentials (with P = K,K,D, D) [11][12][13], is given by (in the decay frame) Here φ KF is a kinematical factor involving the energies of the hadrons, g P P V is the coupling constant in the Lagrangian, and there is also a form factor with a cutoff mass Λ P P V , where n = 1 or n = 2 [17,18]. Here, the value of g P P V refers to the case when the vector meson V is on its mass shell. Then q 2 = (q 0 ) 2 − q 2 = m 2 V and the form factor is 1. For low-energy elastic P N scattering, the exchanged ρ (and ω) meson is far from its mass shell; the momentum transfer q 2 is small and negative, i.e. q 2 = (q 0 ) 2 − q 2 ≡ −q 2 with q 2 0. Therefore, it is common practice to use the static approximation q 2 = −q 2 in the form factors. We note that for the DN (DN ) processes studied in refs. [11][12][13][14] up to kinetic c.m. energy of 150 MeV, the highest c.m. momentum is 400 MeV/c. The cutoff mass in the form factors is another source of symmetry breaking in the meson-exchange potentials. However, in the DN (DN ) interactions in [11][12][13] those masses were simply taken over from the correspondingKN (KN ) interactions, for ρ as well as for ω exchange. Thus, they drop out in the ratio (17).
The situation with baryon exchange is much more complicated, as different baryons are exchanged in theDN and DN reactions. The separation of kinematical effects and the coupling strength, as in (18) (KN ↔ πΛ). Furthermore, for heavy baryons like Λ c an extrapolation to the pole is rather questionable as the quark-model is not expected to work at such high momenta. Despite these drawbacks, we include here our baryon results for illustration purposes.

Results
We use the quark-model parameters of [31]: m l = 375 MeV, m s = 650 MeV, α c = 0.857, α s = 0.84, b = 0.154 GeV 2 , σ = 70 MeV. We take m c = 1657 MeV to fit the D-meson mass. Table 1 shows the results; convergence is achieved with N = 11 Gaussian functions. Clearly, the model fits well the experimental values of the masses, the largest discrepancy is 4% in the mass of Λ c . In particular, the ρ-π and N -Λ mass splittings are well described. In addition, m Σ − m Λ = 82 MeV and m Σc − m Λc = 135 MeV, also in fair agreement with data [34]. Since the corresponding effects on the Σ and Σ c wave functions have a very small effect on the coupling constants, we consider only those couplings involving Λ and Λ c . We take m u = m d so that m ρ = m ω .
The ratios R(q 2 ) are shown in fig. 1; we recall, P P ρ couplings enter graphs with ρ exchange and NBP couplings in graphs with baryon B exchanges. Figure 1 reveals that SU (4) breaking, at q 2 = 0 and q 2 = −m 2 ρ , is relatively modest. At q 2 = 0, the largest SU (4) breaking, not unexpectedly, is in DDρ, of the order of 30% compared to ππρ coupling, and 20% compared to KKρ. Moreover, in agreement with phenomenology, there is almost no SU (3) breaking in KKρ. At the ρ pole (q 2 = −m 2 ρ ) the breaking is also small, at most 10% in DDρ coupling. The ratios of NBP couplings are presented in the bottom panel of the figure. As can be seen, the SU (4) breaking at q 2 = 0 is at most 20% in the NΛ c D vertex compared to the NNπ coupling and 10% compared to the NΛK. The SU (3) symmetry breaking in the NΛK coupling is of the order of 10%, i.e. also compatible with phenomenology. Interestingly, for q 2 ≈ −0.9 GeV/c, i.e. close to the nucleon pole (for orientation, shown by the vertical line in the bottom panel of fig. 1), the NΛ c D coupling is 3 times smaller than the NNπ coupling, while the ratio of the NΛK to NΛ c D couplings is around 1.8. This is to be compared with the value 0.68 in [23]. However, such possible SU (4) breaking far into the time-like region might not be relevant for low-energyDN scattering because, according to [11], the contribution of Λ c exchange to theDN cross section is very small anyway.
Physically, the SU (4) breaking originates from the different extensions of the hadron wave functions. In fig. 2,  Fig. 2. Normalized light-quark radial distributions in mesons and baryons. we plotted the normalized light-quark radial distribution functions in the hadrons of interest -the Fourier transform of h|q † (q)q(q)|h . The distributions get more compact (shorter-ranged) for heavier hadrons as the binding increases due to smaller kinetic energies of the heavy quarks. This implies a smaller P -ρ overlap and thereby a smaller coupling. For NBP fig. 2 shows that the B-P overlap increases because the large-r part of the light quark distribution in B is cut off by the one from P , which explains the increased values of the couplings for heavier baryons. Figure 2 makes the physics transparent and explains the modest effects on the couplings. We have also computed the coupling constants g P P ρ and g NBP of the Lagrangians in [11] by matching the 3 P 0 transition amplitude M h1h2h3 in (3) to the one calculated with those Lagrangians. The matching is done at tree level at q 2 = 0 [27][28][29][30]. Taking the typical values for γ of the literature, γ = 0.4-0.5 [30], the matching leads to g ππρ = 5.85-7.32 and g NNπ = 10.83-13.54, that are in very good agreement with phenomenology, and g KKρ = 2.79-3.49, g DDiρ = 2.34-2.90, g NΛK = 12.65-15.81, g NΛcD = 13.56-16.95. In table 2 we collected the ratios of these couplings and quoted results from the literature. The ratios include isospin factors, as in (1) and (2) -for exact SU (4) symmetry, the ratios are 1. The value for g DDiρ agrees well with VMD [19,20], QCDSR [22], and lattice QCD [25], and agrees within a factor of 2 with DS-BS [21] and holographic QCD [26].

Summary
We used a 3 P 0 quark-pair creation model with nonrelativistic quark-model wave functions to investigate the effects of SU (4) symmetry breaking in the DDρ and NΛ c D couplings, the most relevant ones for theDN and DN interactions [11,13]. The quark masses in the Hamiltonian (5) are the only source of SU (4) breaking. The predictions of the model are reliable for lowmomentum transfers in the vertices. The pattern found for SU (4) breaking for momenta q 2 ≈ 0 in the P P ρ amplitudes is A DDρ < A KKρ < A ππρ , while for NBP it is A NΛcD > A NΛK > A NNπ . Since the DDρ (and DDω) coupling is more important for theDN cross section than the NΛ c D (and NΣ c D) coupling, at least in the calculations in [11][12][13][14], our results indicate that the use of SU (4) symmetry for the coupling constants could be a reasonable first approximation, in line with other studies in the literature [19,20,23,25,26]. Clearly, for estimating the impact of our findings for the SU (4) breaking on DN cross sections, and also binding energies of D-mesic nuclei, further detailed studies are required. Finally, we note that the symmetry breaking pattern we found for P P ρ couplings is opposite to that in ref. [21], but it agrees with the one in the holographic QCD calculation in [26]. We found also an opposite ratio for NΛK/NΛ c D to the one in [23]. Further studies are needed for full clarification. Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.