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Masayasu Harada, Yong-Liang Ma, Daiki Suenaga, Yusuke Takeda, Relation between the mass modification of heavy–light mesons and the chiral symmetry structure in dense matter, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 11, November 2017, 113D01, https://doi.org/10.1093/ptep/ptx140
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Abstract
We point out that the study of the density dependences of the masses of heavy–light mesons gives some clues to the chiral symmetry structure in nuclear matter. We include the omega meson effect as well as the sigma meson effect at mean-field level on the density dependence of the masses of heavy–light mesons with chiral partner structure. It is found that the omega meson affects the masses of the heavy–light mesons and their antiparticles in opposite ways, while it affects the masses of chiral partners in the same way. This is because the |$\omega$| meson is sensitive to the baryon number of the light degrees included in the heavy–light mesons. We also show that the mass difference between chiral partners is proportional to the mean field of sigma, reflecting the partial restoration of chiral symmetry in the nuclear matter. In addition to the general illustration of the density dependence of the heavy–light meson masses, we consider two concrete models for nuclear matter, the parity doublet model and the skyrmion crystal model in the sense of the mean-field approximation.
1. Introduction
In the nucleon sector, by using an effective model with parity doublet structure of baryons, it has been found that |$\sim 70\%$| of nucleon mass comes from chiral symmetry breaking [1,2]. However, when the baryon as a topological soliton in the hidden local symmetry Lagrangian is immersed in the dense matter, which is treated as skyrmion matter, it was found that the chiral invariant mass comprises |$\sim 60\%$| of the nucleon mass [4,5], which is roughly the same as that obtained based on the renormalization group analysis of the hidden local symmetry Lagrangian with baryons [3]. In this paper, we study the medium-modified mass splitting of heavy–light mesons with chiral partner structure, in which it is widely accepted that the mass splitting arises from the spontaneous breaking of chiral symmetry [8,9]. This type of study can be tested in the planned experiments at J-PARC, FAIR, and so on.
Studying the properties of heavy–light mesons in medium is also expected to give clues for understanding the chiral symmetry structure (see, e.g., Ref. [10] for a review). The medium-modified heavy–light meson spectrum has been studied by several groups in the literature [11–17]. In Ref. [15], it was shown that the |$D$| meson (|$J^P=0^-$|) is mixed with a |$D^\ast$| meson (|$J^P=1^-$|) in the spin–isospin-correlated matter, in which the mixing strength reflects the strength of the correlation. In Refs. [11,16,17], by regarding the |$D_0^\ast$| (|$J^P=0^+$|) and |$D_1$| (|$J^P=1^+$|) mesons as chiral partners to the |$D$| and |$D^\ast$| mesons, it was shown that the mass splitting of the chiral partner is reduced at high density and temperature. In particular, in Ref. [17], by replacing the chiral field for pions interacting with the heavy mesons with its mean-field value obtained in the nuclear matter created by the skyrmion crystal approach [4], it was shown that the masses of |$D$| and |$D^\ast$| increase with density while the masses of |$D^\ast_0$| and |$D_1$| decrease, and that their masses approach the average value. In other words, the degenerated mass (actually, the difference between the degenerated mass and the heavy quark mass) agrees with the chiral invariant mass, which is given by the average at vacuum. However, in the analyses of Refs. [15,17], only the pion is included in the light hadron sector, and effects of other mesons are not included. In particular, the analysis in Ref. [12] shows that the |$\omega$| meson increases the mass of the |$D$| meson, while it decreases that of the |$\bar{D}$| meson.
In this paper, we study the effects of the |$\omega$| meson as well as the |$\sigma$| meson on the density dependence of the effective masses of heavy–light mesons. We show that the effect of the |$\sigma$| meson increases the masses of the (|$D$|, |$D^\ast$|) heavy quark doublet, while it decreases the masses of the chiral partners, i.e., the (|$D^\ast_0$|, |$D_1$|) doublet; this is similar to the analysis in Refs. [16,17]. On the other hand, the effect of the |$\omega$| meson increases the masses of both doublets. Nevertheless, the difference between the masses of chiral partners decreases proportionally to the mean-field value of the |$\sigma$| meson, which reflects the partial chiral symmetry restoration. As a result, the masses of the (|$D$|, |$D^\ast$|) and (|$D^\ast_0$|, |$D_1$|) doublets approach a certain degenerate value. Unlike the previous analysis, the degenerate value does not agree with the average value at vacuum, which is the chiral invariant mass of these doublets. In the following analysis, after general consideration, we consider two concrete models, the parity doublet model [18] and skyrmion crystal model based on hidden local symmetry [4], to give quantitative results.
2. Framework
To make a numerical estimation, we use |$m_G=2.40\,$|GeV, |$m_H = 1.97$| GeV, and |$\Delta_M= m_G - m_H = 430 \mbox{MeV}$| for the masses, in addition to |$m_\omega = 783$| MeV, |$m_\pi = 137$| MeV, and |$f_\pi =92.1\,$|MeV. For the other parameters, we use |$\sigma_{\pi N} = 45\,$|MeV, |$\vert g_{\omega DD} \vert = 3.7$|, estimated in Appendix A, and |$\vert g_{\omega NN} \vert = 6.23$|,2 which lead to |$\vert g_{\omega NN} g_{\omega DD} \vert = 23 $|, as reference values. We note that the |$D$| meson includes the anti-light quark and the |$\bar{D}$| meson the light quark. Therefore, it is natural to consider that the |$\bar{D}$| meson is affected by Pauli blocking in a dense medium, which is represented by the effect of the mean field of the |$\omega$| meson. Then, for concreteness, we take |$g_{\omega NN} g_{\omega DD} < 0 $| below. When we study the case with |$g_{\omega NN} g_{\omega DD} > 0 $|, we just exchange |$H$| with |$\bar{H}$| and |$G$| with |$\bar{G}$| in the following discussion.
We plot the density dependence of the masses in Fig. 1. This shows that the masses of the |$H$| and |$\bar{H}$| doublets as well as those of the |$G$| and |$\bar{G}$| doublets are split by the |$\omega$| contribution.
From the |$\omega$| contribution combined with the |$\sigma$| contribution, the mass of the |$G$| doublet (indicated by the dashed red curve) decreases with increasing density, and the |$\bar{H}$| mass (the dotted blue curve) increases. On the other hand, the |$H$| mass (the solid black curve) and the |$\bar{G}$| mass (the dot-dashed green curve) are rather stable. As a result, the |$G$| mass tends to degenerate with the mass of the |$H$| doublet at certain high densities. If one measures the mass of |$H$| only, one might think that the chiral invariant mass would be almost the same as the mass of the |$H$| doublet. However, the actual chiral invariant mass is larger than the |$H$| mass at vacuum, which can be obtained by averaging the masses of the particles (|$G$| and |$H$|) and antiparticles (|$\bar{H}$| and |$\bar{G}$|), as shown in Fig. 2. We should note that the sums of the particle and antiparticle masses are actually independent of the sign of |$g_{\omega DD} g_{\omega NN}$|.
So the mass difference is expected to give a clue to the chiral condensate. In the mean-field approximation, it is actually proportional to the mean field |$\langle \sigma \rangle$|, as shown in Fig. 3. This figure clearly shows that, with increasing nuclear matter density, the chiral symmetry is (partially) restored.
To check the |$\pi$|–|$N$| sigma term dependence of the effective masses, we vary the value of |$\sigma_{\pi N}$| as |$30$| and |$60$| MeV, which are plotted in Fig. 4. This shows that the difference between the masses of |$H$| and |$G$| as well as that between |$\bar{H}$| and |$\bar{G}$| decreases more rapidly for larger values of |$\sigma_{\pi N}$|. As a result, the chiral symmetry is restored more rapidly for larger |$\sigma_{\pi N}$|.
We next check the dependence on the value of |$\vert g_{\omega DD} g_{\omega NN}\vert$| in Fig. 5, by taking 30% deviation from the estimated value. This shows that the masses change more rapidly for larger values of |$\vert g_{\omega DD} g_{\omega NN}\vert$|.
3. Model analysis
After the above general discussion, let us study the density dependences of the effective masses based on some specific models. Here we use the nuclear matter described by the parity doublet model [18] and by the skyrmion crystal model based on hidden local symmetry [4].
3.1 Parity doublet model
In Ref. [18], the parity doublet model based on the linear |$\sigma$| model [1,2], in which an excited nucleon with negative parity, |$N^\ast(1535)$|, is regarded as the chiral partner to the ordinary nucleon, was extended by including a six-point interaction for the |$\sigma$| field and interactions to the |$\omega$| and |$\rho$| mesons based on hidden local symmetry, to study the nuclear matter. It was shown that, for a wide range of chiral invariant masses for the nucleon, the model reproduces the saturation density, binding energy, incompressibility, and symmetry energy. In Refs. [19] and [20], it is shown that the ratio of the mean field |$\langle \sigma \rangle$| at normal nuclear matter density to that at vacuum obtained for the chiral invariant mass of a nucleon |$m_{0}=500$| MeV is consistent with the experimental value of that for the pion decay constant [21,22]. Here we use the density dependences of |$\langle \sigma \rangle$| and |$\langle \omega \rangle$| obtained from the model with |$m_{0} = 500$| MeV.
We show the resultant density dependence of the masses in Fig. 6. Here we use |$\vert g_{\omega DD} \vert = 3.7$|, estimated in Appendix A, as a typical value. This shows that the density dependence of all masses in the very-low-density region |$\rho_B/\rho_0 \lesssim 0.3$| is similar to that in Fig. 1, reflecting that both |$\langle \sigma \rangle$| and |$\langle \omega \rangle$| in the parity doublet model are consistent with those obtained in the linear density approximation, as can be seen in Ref. [19]. However, around the density region |$\rho_B /\rho_0 \sim 0.3$|, the mass of the |$H$| doublet (black curve) starts to decrease and that of the |$G$| doublet (green curve) to increase, unlike the linear density approximation. In this model, the mean field |$\langle \omega \rangle$| is proportional to the density, like the linear density approximation in Eq. (9). Then, the different density dependence of the masses originates from |$\langle \sigma \rangle$|. In Fig. 7, we plot the difference in the two masses of the |$H$| and |$G$| doublets. This shows that the difference decreases more slowly than that in the linear approximation shown in Fig. 3.
3.2 Skyrme model
In Refs. [4,5], the skyrmion crystal model is used to study the qualitative structure of nuclear matter by regarding the skyrmion matter as nuclear matter in the sense of the large-|$N_c$| limit of QCD. A robust conclusion drawn in the skyrmion crystal approach is that, when the density of the nuclear matter is increased, the skyrmion matter undergoes a topological phase transition to matter made of half-skyrmions, in which the space average of the chiral condensate vanishes, although it is locally non-zero and the chiral symmetry is still broken [23]. Since the half-skyrmion phase is not observed in nature, the critical density should be higher than the normal nuclear density. Recently, the description of nuclear matter from the skyrmion crystal and the implication of the topological phase transition in the equation of state of neutron stars have seen great progress (see, e.g., Ref. [7] for a recent review).
In the present analysis, we calculate the mean fields |$\langle \sigma \rangle$| and |$\langle \omega \rangle$| in the skyrmion crystal model, and substitute the values into Eqs. (6) and (8) to obtain the density dependence of the charmed meson masses. We plot in Fig. 8 the density dependence of the effective masses of charmed mesons by using |$\langle \sigma\rangle$| and |$\langle \omega\rangle$| with |$g_{\omega DD} ={} - 3.7$| calculated by the skyrmion crystal model based on hidden local symmetry [4]; other parameters are the same as those used in the plot of Fig. 1. In this model, we find that both |$G$| and |$\bar{G}$| masses decrease with density while both |$H$| and |$\bar{H}$| increase with density. Because the density dependence of the |$\bar{H}$| mass and the |$G$| mass is stronger than that of the |$\bar{G}$| mass and the |$H$| mass, |$H$| and |$G$| as well as |$\bar{H}$| and |$\bar{G}$| become degenerate at density |$\sim 1.2 \rho_0$|, at which the skyrmion phase transits to the half-skyrmion phase. This is because the mass difference between |$H$| and |$G$| as well as |$\bar{H}$| and |$\bar{G}$| is proportional to |$\langle \sigma \rangle$|; this vanishes in the half-skyrmion phase. Moreover, we find that the degenerated masses of |$H$| and |$G$| and those of |$\bar{H}$| and |$\bar{G}$| linearly depend on density in the half-skyrmion phase. The reason for this is that |$\langle \omega \rangle$| is a linear function of density and this linear dependence agrees with Eq. (9). We plot in Fig. 9 the density dependence of |$\langle \sigma \rangle$| and |$\langle \omega \rangle$|. Note that, as stressed above, the analysis shows just a qualitative structure, and the degeneracy of chiral partners does not imply chiral restoration but is due to the vanishing of the space average of the chiral condensate in half-skyrmion matter.
4. Summary and discussion
In this work, by regarding the (|$D^\ast_0$|, |$D_1$|) heavy quark doublet as the chiral partner of the (|$D$|, |$D^\ast$|) doublet, we have explicitly shown that the effect of the |$\omega$| meson decreases the masses of both doublets, while the (|$\bar{D}^\ast_0$|, |$\bar{D}_1$|) and (|$\bar{D}$|, |$\bar{D}^\ast$|) meson masses are increased. We explicitly point out that the |$\omega$| meson effect is significant for understanding the density dependence of effective hadron masses in medium. Even though the qualitative dependence is model dependent, the tendency that the masses of the heavy–light mesons and their antiparticles are split due to the |$\omega$| meson effect is robust. We hope that these medium-modified masses of the heavy–light mesons can be detected in future experiments at J-PARC and FAIR through the strong and weak channels, such as |$\psi(3770) \to D\bar{D}$|, |$J/\psi \to \bar{D} e^+ \nu_e$|, and so on (see, e.g., Ref. [25]).
We would like to note that the result of the omega meson effect on the |$D$| and |$\bar{D}$| mesons is consistent with the result obtained in Ref. [12]. In our analysis, we further introduce |$p$|-wave excited |$D$| and |$\bar{D}$| mesons by using the chiral doubling model, and we have found that the difference between the masses of chiral partners decreases in proportion to the mean-field value of the |$\sigma$| meson, which reflects the partial chiral symmetry restoration even if the |$\omega$| meson contribution is present.
In our calculation, we simply take the mean-field approach. An extension of the present work to include some loop contributions is reported in Ref. [26]. In the present work, we have only discussed medium-modified charmed mesons. The results presented here are intact for their bottom cousins, except that the average mass |$m$| should be taken as the value of bottom mesons.
Appendix. Estimation of |$g_{\omega DD}$|
Acknowledgements
The work of M.H. was supported in part by a JSPS Grant-in-Aid for Scientific Research (C) No. 16K05345. Y.-L.M. was supported in part by the National Science Foundation of China (NSFC) under Grant Nos. 11475071, 11547308 and the Seeds Funding of Jilin University. D.S. was supported in part by a Grant-in-Aid for JSPS Research Fellow, No. 17J05638.
Funding
Open Access funding: SCOAP|$^3$|.
Footnotes
1In Ref. [17], another term for the pionic interaction is included. In the present analysis we do not explicitly include the term, since one-pion interaction terms do not contribute to the following analysis.
2There are several values listed in the literature. Here, we use a value obtained in an analysis of nuclear matter based on the parity doublet model in Ref. [18], in which the saturation density, the binding energy, and the incompressibility are reproduced. Table I in that paper includes some errors, and the value |$\vert g_{\omega NN} \vert = 6.23$| is the corrected value obtained for the chiral invariant mass |$m_0 = 700$| MeV.