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

TL;DR

This paper investigates the extent of SU(4) flavor symmetry breaking in D-meson couplings to light hadrons using quark models, revealing significant symmetry breaking that impacts D-meson interactions and bound state predictions.

Contribution

The study quantifies SU(4) flavor symmetry breaking in D-meson couplings using quark models, providing new insights into the magnitude of symmetry violation.

Findings

SU(4) symmetry broken by about 30% in D-meson couplings

20% symmetry breaking in baryon-baryon-meson couplings

Implications for D-meson interactions and bound states in nuclei

Abstract

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.

Figures

Tables

Table 1. Table 1: Calculated hadron masses ( m calc subscript 𝑚 calc m_{\rm calc} ) and sizes ( α 𝛼 \alpha ). Experimental values for the masses ( m exp subscript 𝑚 exp m_{\rm exp} ) are from the PDG  PDG . All values are in MeV.

	$\pi$	$K$	$D$	$\rho$	$N$	$\Lambda$	${\Lambda }_c$
$m_{\mathrm{calc}}$	138	495	1866	770	958	1115	2195
$m_{\exp }$	138	495	1866	770	940	1115	2286
$\alpha$	359	377	499	275	234	241	253

Table 2. Table 2: Ratios of three-hadron couplings—for exact SU(4) symmetry, all ratios should be equal to 1 (see text).

$P/P^{\prime }$	$\pi /K$	$\pi /D$	$K/D$
${}_{}{}^{3}P_0^{}ℛ\left(0\right)$	1.05	1.26	1.19
${}_{}{}^{3}P_0^{}ℛ\left(-m_{\rho }^2\right)$	0.99	1.07	1.08
Ref. ElBennich:2011py	1.09	0.21	0.19
Ref. Ballon-Bayona:2017bwk	1.11	2.23	2.00
$BP/B^{\prime }{P}^{\prime }$	$N\pi /\Lambda K$	$N\pi /{\Lambda }_{c}D$	$\Lambda K/{\Lambda }_{c}D$
${}_{}{}^{3}P_0^{}ℛ\left(0\right)$	0.89	0.83	0.92
Ref. lc-sr	—	—	0.68

Equations

$$g_{DD\rho }=g_{KK\rho }=\frac{1}{2}{g}_{\pi \pi \rho },$$

$$g_{N{\Lambda }_{c}D}=g_{N\Lambda K}=\frac{3\sqrt{3}}{5}{g}_{NN\pi }.$$

$$⟨h_2{h}_3|{\hat{O}}_{\mathrm{pc}}|h_1⟩\equiv \delta ({\mathbf{P}}_1-{\mathbf{P}}_2-{\mathbf{P}}_3){\mathcal{M}}_{h_1{h}_2{h}_3}(\mathbf{q}),$$

$${\hat{O}}_{\mathrm{pc}}=\gamma \sum _{cfs{s}^{\prime }}\int d^{3}p{\mathbf{\sigma }}_{s^{\prime }s}^c\cdot \mathbf{p}{q}_{s^{\prime }}^{cf†}(\mathbf{p}){\bar{q}}_s^{cf†}(-\mathbf{p}),$$

$$H$$

$$\Phi (\mathbf{r})=\sum _{n=1}^N{c}_n{\phi }_n(\mathbf{r}),\Psi (\mathbf{\rho },\mathbf{\lambda })=\sum _{n=1}^N{c}_n{\phi }_n(\mathbf{\rho }){\phi }_n(\mathbf{\lambda }),$$

$${\phi }_n(\mathbf{x})={\left(\frac{n{\alpha }^2}{\pi }\right)}^{3/4}{e}^{-n{\alpha }^2{\mathbf{x}}^2/2}.$$

$${\mathcal{M}}_{h_1{h}_2{h}_3}(\mathbf{q})={\kappa }_{h_1{h}_2{h}_3}{A}_{h_1{h}_2{h}_3}(\mathbf{q})|\mathbf{q}|Y_{1m}(\hat{\mathbf{q}}),$$

$${\kappa }_{DD\rho }={\kappa }_{KK\rho }=\frac{1}{2}{\kappa }_{\pi \pi \rho }=1,$$

$${\kappa }_{N{\Lambda }_{c}D}={\kappa }_{N\Lambda K}=\frac{3\sqrt{3}}{5}{\kappa }_{NN\pi }=1.$$

$$A_{h_1{h}_2{h}_3}(\mathbf{q})=\gamma \sum _{n_1{n}_2{n}_3}{c}_{n_3}^{\ast }{c}_{n_2}^{\ast }{c}_{n_1}(n_1{n}_2{n}_3{)}^{3/4}$$

$$\times f_{h_1{h}_2{h}_3}(n_1,n_2,n_3)e^{-{\mathbf{q}}^2/{\Lambda }_{h_1{h}_2{h}_3}^2(n_1,n_2,n_3)},$$

$$f_{PP\rho }(n_1,n_2,n_3)$$

$$f_{NBP}(n_1,n_2,n_3)$$

$${\Lambda }_{PP\rho }^2(n_1,n_2,n_3)$$

$${\Lambda }_{NBP}^2(n_1,n_2,n_3)$$

$${\overline{m}}_{1,2}=\frac{2m_{1,2}}{m_2+m_1},{\tilde{m}}_{1,2}=\frac{3m_{1,2}}{2m_1+m_2},\Delta \overline{m}=\frac{m_2-m_1}{m_2+m_1},$$

$${\mathcal{R}}_{P/P^{\prime }}({\mathbf{q}}^2)=\frac{A_{PP\rho }({\mathbf{q}}^2)}{A_{P^{\prime }{P}^{\prime }\rho }({\mathbf{q}}^2)},{\mathcal{R}}_{BP/B^{\prime }{P}^{\prime }}({\mathbf{q}}^2)=\frac{A_{NBP}({\mathbf{q}}^2)}{A_{N{B}^{\prime }{P}^{\prime }}({\mathbf{q}}^2)},$$

$$A_{PPV}({\mathbf{q}}^2)={\varphi }_{\mathrm{KF}}{g}_{PPV}{\left(\frac{{\Lambda }_{PPV}^2-m_V^2}{{\Lambda }_{PPV}^2-q^2}\right)}^n|\mathbf{q}|Y_{11}(\hat{\mathbf{q}}).$$

Full text

Introduction. Currently there is considerable interest in exploring the interactions of charmed hadrons with light hadrons and atomic nuclei Briceno:2015rlt . Particular attention is paid to $D$ mesons, much discussed over the last few years in connection with $D$-mesic nuclei Tsushima:1998ru ; GarciaRecio:2010vt ; GarciaRecio:2011xt and $J/\psi$ binding to nuclei Ko:2000jx ; Krein:2010vp . Presently, there is no experimental information about the $DN$ interaction, a situation that the $\overline{\rm P}$ANDA@FAIR experiment Wiedner:2011mf 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 Mizutani:2006vq ; Lin:1999ve ; Hofmann:2005sw ; Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch ; Fontoura:2012mz and heavy quark symmetry Yasui:2009bz ; GarciaRecio:2008dp . These require as input coupling constants and, in some cases, form factors. For the particular case of $\bar{D}N$ reactions (where $\bar{D}\equiv\{\bar{D}^{0},D^{-}\}$), Ref. Haidenbauer:2007jq found that among all the couplings in the effective Lagrangian, $g_{DD\rho}$ and $g_{DD\omega}$ 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 Haidenbauer:2010ch . Flavor SU(4) symmetry relates those couplings to couplings in the light-flavor sector:

[TABLE]

The studies in Refs Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch utilized the SU(4) relations above, based on $g_{\pi\pi\rho}=6.0$ and $g_{NN\pi}=13.6$, which are the values used in a large body of work conducted within the Jülich model Haidenbauer:1991kt ; Hoffmann:1995ie 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) Mat98 ; Lin00a , Dyson-Schwinger and Bethe-Salpeter equations (DS-BS) of QCD ElBennich:2011py , QCD sum rules (QCDSR) Bracco:2011pg ; lc-sr ; qcdsr , lattice QCD Can:2012tx , and holographic QCD Ballon-Bayona:2017bwk . In this work we use the quark model with a ${}^{3}{\rm P}_{0}$ quark-pair creation operator 3P0 . In this setting, the three-hadron couplings are given by matrix elements of the ${}^{3}{\rm P}_{0}$ operator evaluated with quark-model wave functions. The literature on the ${}^{3}{\rm P}_{0}$ 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 3P0 ; Downum:2006re .

Three-Hadron Couplings. To evaluate the matrix element of the ${}^{3}{\rm P}_{0}$ quark-pair creation operator, $\hat{O}_{\rm pc}$, it is convenient to employ the “decay frame” of an initial hadron at rest 3P0 ; Downum:2006re , i.e. the transition of a hadron state $|h_{1}\rangle$ into a final two-hadron state $|h_{2}h_{3}\rangle$ is written as

[TABLE]

with ${\bm{q}}={\bm{P}}_{2}=-{\bm{P}}_{3}$, and

[TABLE]

where $\gamma$ gives the strength of the quark-pair creation, $q^{cf{\dagger}}_{s^{\prime}}(\bm{p})$ and $\bar{q}^{cf{\dagger}}_{s}(\bm{p})$ are creation operators with color $c$, flavor $f$, spin projection $s$, and momentum ${{\bm{p}}}$, $\bm{\sigma}^{c}_{s^{\prime}s}=\chi^{\dagger}_{s^{\prime}}\bm{\sigma}\chi^{c}_{s}$, with ${\bm{\sigma}}=(\sigma^{1},\sigma^{2},\sigma^{3})$ being the Pauli matrices, $\chi_{s}$ Pauli spinors, and $\chi^{c}_{s}=-i\,\sigma^{2}\chi^{\ast}_{s}$.

We employ the standard quark-model Hamiltonian Swanson:1992ec :

[TABLE]

where $m_{i}$ are the quark masses and ${\bm{F}}={\bm{\lambda}}/2$, with ${\bm{\lambda}}$ the color SU(3) Gell-Mann matrices and ${\bm{S}}$ 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 $\pi-\rho$ mass splitting Szczepaniak:2000bi . 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 $\Phi$ and baryon $\Psi$ intrinsic wave functions as Swanson:1992ec ; SilvestreBrac:1995gz :

[TABLE]

where the $c_{n}$ are dimensionless expansion parameters and

[TABLE]

Here, $\alpha$ is the variational, ${\bm{r}}={\bm{r}}_{1}-{\bm{r}}_{2}$, ${\bm{\rho}}=({\bm{r}}_{1}-{\bm{r}}_{2})/\sqrt{2}$, and ${\bm{\lambda}}=\sqrt{2/3}[({\bm{r}}_{1}+{\bm{r}}_{2})/2-{\bm{r}}_{3}]$. The matrix element ${\cal M}({\bm{q}})$ can be evaluated analytically; it is given by

[TABLE]

where $Y_{1m}(\hat{\bm{q}})$ are spherical harmonics with $m=1(0)$ for three-meson (nucleon-baryon-meson) couplings, $\kappa$ comes from summing over color, spin, and flavor and is given by

[TABLE]

The amplitude $A_{h_{1}h_{2}h_{3}}({\bm{q}})$ in (8) is given by

[TABLE]

where $f_{h_{1}h_{2}h_{3}}$ are given by ($P$ in $PP\rho$ stands for $\pi,K,D$ and $B$ in NBP for $N,\Lambda,\Lambda_{c}$)

[TABLE]

and the “cut-off” parameters $\Lambda_{h_{1}h_{2}h_{3}}$ are given by

[TABLE]

where

[TABLE]

with $m_{1}=m_{u}=m_{d}$, $m_{2}=m_{s},m_{c}$.

In the limit of SU(4) symmetry, $m_{1}=m_{2}$, $\alpha_{D}=\alpha_{K}=\alpha_{\pi}$ and $\alpha_{\Lambda_{c}}=\alpha_{\Lambda}=\alpha_{N}$, and the ratios

[TABLE]

are all equal to 1, expressing the same symmetry as in (1) and (2). In this limit, $\gamma$ 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 $\Lambda$.

Let us now connect to meson-exchange models. A typical three-meson vertex function, as it appears in that approach in the $PN$ potentials (with $P=K,\,\bar{K},\,\bar{D},\,D$) Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch , is given by (in the decay frame)

[TABLE]

Here $\phi_{\rm KF}$ is a kinematical factor involving the energies of the hadrons, $g_{PPV}$ is the Lagrangian coupling constant, and there is also form factor with a cutoff mass $\Lambda_{PPV}$, where $n=1$ or $n=2$ Haidenbauer:1991kt ; Hoffmann:1995ie . Here, the value of $g_{PPV}$ refers to the case when the vector meson $V$ is on its mass shell. Then $q^{2}=(q^{0})^{2}-{\bm{q}}^{2}=m^{2}_{V}$ and the form factor is $1$. For low-energy elastic $PN$ scattering, the exchanged $\rho$ (and $\omega$) meson is far from its mass shell; the momentum transfer $q^{2}$ is small and negative, i.e. $q^{2}=(q^{0})^{2}-{\bm{q}}^{2}\equiv-{\bm{q}}^{2}$ with ${\bm{q}}^{2}\gtrsim 0$. Therefore, it is common practice to use the static approximation $q^{2}=-{\bm{q}}^{2}$ in the form factors. We note that for the $DN$ ($\bar{D}N$) processes studied in Refs. Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch ; Fontoura:2012mz up to kinetic c.m. energy of $150$ MeV, the highest c.m. momentum is $400~{}{\rm MeV}/c$. The cutoff mass in the form factors is another source of symmetry breaking in the meson-exchange potentials. However, in the $DN$ ($\bar{D}N$) interactions in Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch those masses were simply taken over from the corresponding $\bar{K}N$ ($KN$) interactions, for $\rho$- as well as for $\omega$ exchange. Thus, they drop out in the ratio (17).

The situation with baryon exchange is much more complicated, as different baryons are exchanged in the $\bar{D}N$ and $DN$ reactions. The separation of kinematical effects and the coupling strength, as in (18), cannot be easily done. Indeed in $\bar{D}N$ ($KN$) elastic scattering only $B=\Lambda_{c}$ ($\Lambda$) exchange contributes while for $DN$ ($\bar{K}N$) there is only $N$ exchange, and only in the transitions $DN\leftrightarrow\pi\Lambda_{c}$ ($\bar{K}N\leftrightarrow\pi\Lambda$). Furthermore, for heavy baryons like $\Lambda_{c}$ 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 Swanson:1992ec : $m_{l}=375~{}{\rm MeV}$, $m_{s}=650~{}{\rm MeV}$, $\alpha_{c}=0.857$, $\alpha_{s}=0.84$, $b=0.154~{}\text{GeV}^{2}$, $\sigma=70~{}\text{MeV}$. We take $m_{c}=1657~{}{\rm MeV}$ to fit the $D$ meson mass. Tab. 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 $\Lambda_{c}$. In particular, the $\rho-\pi$ and $N-\Lambda$ mass splittings are well described. In addition, $m_{\Sigma}-m_{\Lambda}=82$ MeV and $m_{\Sigma_{c}}-m_{\Lambda_{c}}=135$ MeV, also in fair agreement with data PDG . Since the corresponding effects on the $\Sigma$ and $\Sigma_{c}$ wave functions have a very small effect on the coupling constants, we consider only those couplings involving $\Lambda$ and $\Lambda_{c}$. We take $m_{u}=m_{d}$ and so $m_{\rho}=m_{\omega}$.

The ratios ${\cal R}({\bm{q}}^{2})$ are shown in Fig. 1; we recall, $PP\rho$ couplings enter graphs with $\rho$ exchange and $NBP$ couplings in graphs with baryon $B$ exchanges. Fig. 1 reveals that SU(4) breaking, at ${\bm{q}}^{2}=0$ and ${\bm{q}}^{2}=-m^{2}_{\rho}$, is relatively modest. At ${\bm{q}}^{2}=0$, the largest SU(4) breaking, not unexpectedly, is in $DD\rho$, of the order of $30$% compared to $\pi\pi\rho$ coupling, and $20$% compared to $KK\rho$. Moreover, in agreement with phenomenology, there is almost no SU(3) breaking in $KK\rho$. At the $\rho$ pole (${\bm{q}}^{2}=-m^{2}_{\rho}$) the breaking is also small, at most $10$% in $DD\rho$ coupling. The ratios of $NBP$ couplings are presented in the bottom panel of the figure. As can be seen, the SU(4) breaking at ${\bm{q}}^{2}~{}=~{}0$ is at most $20$% in the $N\Lambda_{c}D$ vertex compared to the $NN\pi$ coupling and $10$% compared to the $N\Lambda K$. The SU(3) symmetry breaking, i.e. in the $N\Lambda K$ coupling, is of the order of $10$%, also compatible with phenomenology. Interestingly, for ${\bm{q}}^{2}\approx-0.9$ GeV/c, i.e close to the nucleon pole (for orientation, shown by the vertical line in the bottom panel of Fig. 1), the $N\Lambda_{c}D$ coupling is 3 times smaller than the $NN\pi$ coupling, while the ratio of the $N\Lambda K$ to $N\Lambda_{c}D$ couplings is around 1.8. This is to be compared with the value 0.68 in lc-sr . However, such possible SU(4) breaking far into the time-like region might not be relevant for low-energy $\bar{D}N$ scattering because, according to Haidenbauer:2007jq , the contribution of $\Lambda_{c}$ exchange to the $\bar{D}N$ cross section is very small anyway.

Physically, the SU(4) breaking originates from the different extensions of the hadron wave functions. In Fig. 2, we plotted the normalized light-quark radial distribution functions in the hadrons of interest—the Fourier transform of $\langle h|q^{\dagger}({\bm{q}})q({\bm{q}})|h\rangle$. 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 into smaller $P-\rho$ overlap and thereby a smaller coupling. For the $NBP$, 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. Fig. 2 makes the physics transparent and explains the modest effects on the couplings.

We have also computed the coupling constants $g_{PP\rho}$ and $g_{NBP}$ of the Lagrangians in Haidenbauer:2007jq by matching the ${}^{3}{\rm P}_{0}$ transition amplitude ${\cal M}_{h_{1}h_{2}h_{3}}$ in (3) to the one calculated with those Lagrangians. The matching is done at tree level at ${\bm{q}}^{2}=0$ 3P0 ; Downum:2006re . Taking the typical values for $\gamma$ of the literature, $\gamma=0.4-0.5$ Downum:2006re , the matching leads to $g_{\pi\pi\rho}=5.85-7.32$ and $g_{NN\pi}=10.83-13.54$, that are in very good agreement with phenomenology, and $g_{KK\rho}=2.79-3.49$, $g_{DD\rho}=2.34-2.90$, $g_{N\Lambda K}=12.65-15.81$, $g_{N\Lambda_{c}D}=13.56-16.95$. In Tab. 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 $g_{DD\rho}$ agrees well with VMD Mat98 ; Lin00a , QCDSR Bracco:2011pg , and lattice QCD Can:2012tx , and agrees within a factor of 2 with DS-BS ElBennich:2011py and holographic QCD Ballon-Bayona:2017bwk .

Summary. We used a ${}^{3}{\rm P}_{0}$ quark-pair creation model with nonrelativistic quark-model wave functions to investigate the effects of SU(4) symmetry breaking in the $DD\rho$ and $N\Lambda_{c}D$ couplings, the most relevant for the $\bar{D}N$ and $DN$ interactions Haidenbauer:2007jq ; Haidenbauer:2010ch . The quark masses in the Hamiltonian (5) are the only source of SU(4) breaking. The predictions of the model are reliable for low-momentum transfers in the vertices. The pattern found for SU(4) breaking for momenta ${\bm{q}}^{2}\approx 0$ in the $PP\rho$ amplitudes is $A_{DD\rho}<A_{KK\rho}<A_{\pi\pi\rho}$, while in the $NBP$ it is $A_{N\Lambda_{c}D}>A_{N\Lambda K}>A_{NN\pi}$. Since the $DD\rho$ (and $DD\omega$) coupling is more important for the $\bar{D}N$ cross section than the $N\Lambda_{c}D$ (and $N\Sigma_{c}D$) coupling, at least in the calculations in Haidenbauer:2007jq ; Haidenbauer:2008ff ; Haidenbauer:2010ch ; Fontoura:2012mz , 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 Mat98 ; Lin00a ; Can:2012tx ; lc-sr ; Ballon-Bayona:2017bwk . 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 $PP\rho$ couplings is opposite to that in Ref. ElBennich:2011py , but it agrees with the one in the holographic QCD calculation in Ballon-Bayona:2017bwk . We found also an opposite ratio for ${N\Lambda K}/{N\Lambda_{c}D}$ to the one in lc-sr . Further studies are needed for full clarification.

Work partially supported by the Brazilian agencies CNPq, Grants No. 305894/2009-9 (GK) and 150659/2015-6 (CEF) and FAPESP Grant No. 2013/01907-0 (G.K.).