$\rho\rho$ scattering revisited with coupled-channels of pseudoscalar mesons

TL;DR

This paper reexamines $ ho ho$ scattering by including coupled channels of pseudoscalar mesons, revealing that the resulting scalar meson properties align more closely with experimental data, especially for $f_0(1500)$.

Contribution

It introduces the coupled-channel effects of pseudoscalar mesons into $ ho ho$ scattering analysis, refining the understanding of the scalar meson spectrum.

Findings

Pole positions are closer to PDG values for $f_0(1500)$ with coupled channels.

Partial decay widths better match experimental data.

Coupled-channel effects significantly influence scalar meson properties.

Abstract

The $\rho\rho$ scattering has been studied by two groups which both claimed a dynamical generated scalar meson, most likely to be $f_0(1370)$. Here we investigate the influence of coupled-channels of pseudoscalar mesons, i.e., $\pi\pi$ and $\bar KK$, on this dynamical generated scalar state. With the coupled channel effect included, the pole and partial decay widths are found to be more close to PDG values for $f_0(1500)$.

Figures

Tables

Table 1. Table 1: Pole position for coupled-channels

Equations

$${\mathcal{L}}_{VPP}=-ig⟨V^{\mu }[P,{\partial }_{\mu }P]⟩.$$

$$V_{\mu }={\left(\begin{array}{ccc}\frac{{\rho }_0}{\sqrt{2}}+\frac{\omega }{\sqrt{2}}& {\rho }^{+}& K^{\ast +}\\ {\rho }^{-}& -\frac{{\rho }_0}{\sqrt{2}}+\frac{\omega }{\sqrt{2}}& K^{\ast 0}\\ K^{\ast -}& {\bar{K}}^{\ast 0}& \varphi \end{array}\right)}_{\mu },P=\left(\begin{array}{ccc}\frac{{\pi }^0}{\sqrt{2}}+\frac{{\eta }_8}{\sqrt{6}}& {\pi }^{+}& K^{+}\\ {\pi }^{-}& -\frac{{\pi }^0}{\sqrt{2}}+\frac{{\eta }_8}{\sqrt{6}}& K^0\\ K^{-}& {\bar{K}}^0& -\frac{2{\eta }_8}{\sqrt{6}}\end{array}\right).$$

$$A_t(p_1,p_2,p_3,p_4)=\frac{-8g^2}{(p_1-p_3{)}^2-m_{\pi }^2}{ϵ}_1\cdot p_3{ϵ}_2\cdot p_4.$$

$$ϵ(\mathbf{p},0)=\left(\begin{array}{c}\gamma \beta \cos \theta \\ \frac{1}{2}(\gamma -1)\sin 2\theta \cos \varphi \\ \frac{1}{2}(\gamma -1)\sin 2\theta \sin \varphi \\ \frac{1}{2}(1+\gamma +(\gamma -1)\cos 2\theta )\end{array}\right),$$

$$ϵ(\mathbf{p},\pm 1)=\mp \frac{1}{\sqrt{2}}\left(\begin{array}{c}\gamma \beta e^{\pm i\varphi }\sin \theta \\ 1+(\gamma -1)e^{\pm i\varphi }{\sin }^2\theta \cos \varphi \\ \pm i+(\gamma -1)e^{\pm i\varphi }{\sin }^2\theta \sin \varphi \\ \frac{1}{2}(\gamma -1)e^{\pm i\varphi }\sin 2\theta \end{array}\right).$$

$$A_u(p_1,p_2,p_3,p_4)=A_t(p_1,p_2,p_4,p_3).$$

$${\rho }^{+}(p_1){\rho }^{-}(p_2)$$

$${\rho }^0(p_1){\rho }^0(p_2)$$

$$|\rho \rho ,I=0⟩$$

$$|\pi \pi ,I=0⟩$$

$$T_{\pi }^{(0)}=16g^2\left(\frac{{ϵ}_1\cdot p_3{ϵ}_2\cdot p_4}{m_{\pi }^2-t}+\frac{{ϵ}_1\cdot p_4{ϵ}_2\cdot p_3}{m_{\pi }^2-u}\right).$$

$${\mathcal{L}}_{VVP}=\frac{G^{\prime }}{\sqrt{2}}{ϵ}^{\mu \nu \alpha \beta }⟨{\partial }_{\mu }{V}_{\nu }{\partial }_{\alpha }{V}_{\beta }P⟩$$

$$G^{\prime }=\frac{3g^{\prime 2}}{4{\pi }^2{f}_{\pi }}{g}^{\prime }=-\frac{G_V{m}_{\rho }}{\sqrt{2}{f}_{\pi }^2}.$$

$$\begin{array}{rl}{B}_t& =\frac{-G^{\prime 2}}{(p_1-p_3{)}^2-m_{\omega }^2}(p_3\cdot p_4{p}_1\cdot {ϵ}_2{p}_2\cdot {ϵ}_1+p_1\cdot p_2{p}_4\cdot {ϵ}_1{p}_3\cdot {ϵ}_2+p_1\cdot p_4{p}_2\cdot p_3{ϵ}_1\cdot {ϵ}_2\\ & -p_2\cdot p_3{p}_1\cdot {ϵ}_2{p}_4\cdot {ϵ}_1-p_1\cdot p_4{p}_2\cdot {ϵ}_1{p}_3\cdot {ϵ}_2-p_1\cdot p_2{p}_3\cdot p_4{ϵ}_1\cdot {ϵ}_2).\end{array}$$

$$B_u(p_1,p_2,p_3,p_4)=B_t(p_1,p_2,p_4,p_3).$$

$${\rho }^{+}(p_1){\rho }^{-}(p_2)$$

$${\rho }^0(p_1){\rho }^0(p_2)$$

$$\begin{array}{rl}{T}_{\omega }^{(0)}& =\frac{-G^{\prime 2}}{(p_1-p_3{)}^2-m_{\omega }^2}(p_3\cdot p_4{p}_1\cdot {ϵ}_2{p}_2\cdot {ϵ}_1+p_1\cdot p_2{p}_4\cdot {ϵ}_1{p}_3\cdot {ϵ}_2+p_1\cdot p_4{p}_2\cdot p_3{ϵ}_1\cdot {ϵ}_2\\ & -p_2\cdot p_3{p}_1\cdot {ϵ}_2{p}_4\cdot {ϵ}_1-p_1\cdot p_4{p}_2\cdot {ϵ}_1{p}_3\cdot {ϵ}_2-p_1\cdot p_2{p}_3\cdot p_4{ϵ}_1\cdot {ϵ}_2)+(p_3\leftrightarrow p_4).\end{array}$$

$$C_t(p_1,p_2,p_3,p_4)=-4g^2\frac{{ϵ}_1\cdot p_3{ϵ}_2\cdot p_4}{(p_1-p_3{)}^2-m_K^2}.$$

$$C_u(p_1,p_2,p_3,p_4)=C_t(p_1,p_2,p_4,p_3).$$

$${\rho }^{+}(p_1){\rho }^{-}(p_2)$$

$${\rho }^0(p_1){\rho }^0(p_2)$$

$$|K\bar{K}⟩=-\frac{1}{\sqrt{2}}|K^{+}(p_1)K^{-}(p_2)+K^0(p_1){\bar{K}}^0(p_2)⟩.$$

$$T_K^{(0)}=2\sqrt{6}{g}^2\left(\frac{{ϵ}_1\cdot p_3{ϵ}_2\cdot p_4}{m_K^2-t}+\frac{{ϵ}_1\cdot p_4{ϵ}_2\cdot p_3}{m_K^2-u}\right)$$

$$\begin{array}{rl}{D}_t& =\frac{-G^{\prime 2}/2}{(p_1-p_3{)}^2-m_{K^{\ast }}^2}(p_3\cdot p_4{p}_1\cdot {ϵ}_2{p}_2\cdot {ϵ}_1+p_1\cdot p_2{p}_4\cdot {ϵ}_1{p}_3\cdot {ϵ}_2+p_1\cdot p_4{p}_2\cdot p_3{ϵ}_1\cdot {ϵ}_2\\ & -p_2\cdot p_3{p}_1\cdot {ϵ}_2{p}_4\cdot {ϵ}_1-p_1\cdot p_4{p}_2\cdot {ϵ}_1{p}_3\cdot {ϵ}_2-p_1\cdot p_2{p}_3\cdot p_4{ϵ}_1\cdot {ϵ}_2).\end{array}$$

$$D_u(p_1,p_2,p_3,p_4)=D_t(p_1,p_2,p_4,p_3).$$

$${\rho }^{+}(p_1){\rho }^{-}(p_2)$$

$${\rho }^0(p_1){\rho }^0(p_2)$$

$$\begin{array}{rl}{T}_{K^{\ast }}^{(0)}& =\frac{-\sqrt{6}{G}^{\prime 2}/4}{(p_1-p_3{)}^2-m_{K^{\ast }}^2}(p_3\cdot p_4{p}_1\cdot {ϵ}_2{p}_2\cdot {ϵ}_1+p_1\cdot p_2{p}_4\cdot {ϵ}_1{p}_3\cdot {ϵ}_2+p_1\cdot p_4{p}_2\cdot p_3{ϵ}_1\cdot {ϵ}_2\\ & -p_2\cdot p_3{p}_1\cdot {ϵ}_2{p}_4\cdot {ϵ}_1-p_1\cdot p_4{p}_2\cdot {ϵ}_1{p}_3\cdot {ϵ}_2-p_1\cdot p_2{p}_3\cdot p_4{ϵ}_1\cdot {ϵ}_2)+(p_3\leftrightarrow p_4).\end{array}$$

Full text

1 Introduction

The chiral unitary approach, which has made much progress in the study of pseudo-scalar meson-meson [1] and meson-baryon [2, 3] interactions, has been used to study the interaction of vector mesons among themselves. The first such study of the $S$-wave $\rho\rho$ interactions found that the $f_{0}(1370)$ and the $f_{2}(1270)$ could be dynamically generated [4]. The work found that the strength of the attractive interaction in the tensor channel is much stronger than that in the scalar channel, hence leads to a tighter bound tensor state than the corresponding scalar one.

The work [4] based on the assumption that the three momenta of the $\rho$ is negligibly small compared to its large mass. This assumption was questioned by a recent work [5] which pointed out the importance of relativistic effect for energies around $f_{2}(1270)$ well below $\rho\rho$ threshold. The $N/D$ method [6, 7, 8, 9, 10] was used to get the partial wave amplitudes which result a pole for the scalar state similar to Ref. [4] but no pole for any tensor state in contradiction with Ref. [4]. However, this conclusion was not agreed upon by Ref. [11] in which the non-relativistic assumption was dropped by evaluating exactly the loops with full relativistic $\rho$ propagators in solving the B-S equation for $\rho\rho$ scattering. Both scalar state and tensor state associated with $f_{0}(1370)$ and $f_{2}(1270)$, respectively, were found in consistence with the conclusion of Ref. [4].

From the studies of above two groups, obviously, for the energies around $f_{2}(1270)$ well below $\rho\rho$ threshold, there is strong model dependence for the interaction of two far off-mass-shell $\rho$ mesons. For the scalar state closer to the $\rho\rho$ threshold, the two groups got similar result rather model independently. In this paper we shall study the influence of coupled-channels of pseudoscalar mesons, i.e., $\pi\pi$ and $\bar{K}K$, on this dynamical generated scalar state. In the $\rho\rho-K\bar{K}$ coupling we consider the case of $K$ and $K^{*}$ exchange, while in the $\rho\rho-\pi\pi$ coupling we consider the case of $\pi$ and $\omega$ exchange.

2 Formalism

2.1 $\rho\rho\to\pi\pi$ with $\pi$-exchange

We investigate the coupled channel effect based on a chiral covariant framework [5]. We follow the formalism of the hidden gauge interaction which provides the $\rho\pi\pi$ coupling by means of the Lagrangian [12, 13]

[TABLE]

where the symbol $\langle\ldots\rangle$ stands for the trace in the $SU(3)$ space with the coupling constant $g=M_{V}/2f_{\pi}$ with $M_{V}$ the mass of vector meson and $f_{\pi}=93MeV$ the pion decay constant. The matrices $V_{\mu}$ and $P$ are given by

[TABLE]

To get the three different isospin amplitudes for $\rho\rho\to\pi\pi$ we need the knowledge of the transitions $\rho^{+}(p_{1})\rho^{-}(p_{2})\to\pi^{+}(p_{3})\pi^{-}(p_{4})$, $\rho^{+}(p_{1})\rho^{-}(p_{2})\to\pi^{0}(p_{3})\pi^{0}(p_{4})$, etc.

Starting with the Lagrangian in Eq.(1) we can immediately obtain the amplitude $A_{t}(p_{1},p_{2},p_{3},p_{4})$ of $\rho^{+}(p_{1})\rho^{-}(p_{2})\to\pi^{+}(p_{3})\pi^{-}(p_{4})$ corresponding to Fig.1 as

[TABLE]

In this equation, the $\epsilon_{i}$ corresponds to the polarization vector of the $i$-th $\rho$. Each polarization vector is characterized by its three-momentum $\mathbf{p}_{i}$ and third component of the spin $\sigma_{i}$. Explicit expressions of these polarization vectors are given by [5]

[TABLE]

where $\beta=|\mathbf{p}|/E_{p}$, $\gamma=1/\sqrt{1-\beta^{2}}$, $\theta$ and $\phi$ are polar angle and azimuthal angle of $\mathbf{p}$, respectively. The $u$-channel $\pi$-exchange amplitude $A_{u}$ can be obtained from the expression of $A_{t}$ by exchanging $p_{3}\leftrightarrow p_{4}$. In this way,

[TABLE]

And now we write the tree-lavel amplitude for $\rho\rho\to\pi\pi$ with $\pi$-exchange

[TABLE]

In order to obtain the $S$-wave amplitude in isospin $I=0$ channel we need the isospin eigenstates. We have

[TABLE]

where we have used the convention $\lvert\rho^{+}\rangle=-\lvert 1,1\rangle$ and $\lvert\pi^{+}\rangle=-\lvert 1,1\rangle$ of isospin. By taking into account Eq.(2.1) and the amplitudes in Eq.(2.1) we can now write the isospin $I=0$ amplitude for $\rho\rho\to\pi\pi$

[TABLE]

where $t=(p_{1}-p_{3})^{2}$ and $u=(p_{1}-p_{4})^{2}$.

2.2 $\rho\rho\to\pi\pi$ with $\omega$-exchange

One needs the $\rho\omega\pi$ coupling which is provided within the framework [14] of the hidden gauge formalism by means of the Lagrangian

[TABLE]

with

[TABLE]

where $G_{V}=55MeV$ and $f_{\pi}=93MeV$. At this point we can write down the amplitude of $\rho^{+}(p_{1})\rho^{-}(p_{2})\to\pi^{+}(p_{3})\pi^{-}(p_{4})$ with $\omega$-exchange corresponding to Fig.2 as in the $\pi$-exchange case

[TABLE]

And the $u$-channel $\omega$-exchange amplitude $B_{u}(p_{1},p_{2},p_{3},p_{4})$ can be obtained from the expression of $B_{t}$ as the case in $\pi$-exchange by exchanging $p_{3}\leftrightarrow p_{4}$, thus

[TABLE]

Next we write the tree-level amplitude for $\rho\rho\to\pi\pi$ with $\omega$-exchange

[TABLE]

Then using Eq.(2.1) we can get the $I=0$ amplitude

[TABLE]

2.3 $\rho\rho\to K\bar{K}$ with $K$-exchange

The $\rho KK$ coupling is provided in the same Lagrangian in Eq.(1), so we can immediately write down the amplitude of $\rho^{+}(p_{1})\rho^{-}(p_{2})\to K^{+}(p_{3})K^{-}(p_{4})$ with $K$-exchange corresponding to Fig.3

[TABLE]

and the $u$-channel

[TABLE]

Then we can obtain the tree-level amplitudes for $\rho\rho\to K\bar{K}$ with $K$-exchange as the following

[TABLE]

Similar to Eq.(2.1) we need the isospin $I=0$ eigenstate for $\lvert K\bar{K}\rangle$. We have

[TABLE]

where we use the convention $\lvert K^{+}\rangle=-\lvert\frac{1}{2},\frac{1}{2}\rangle$ of isospin. By using the isospin wave functions we can obtain for $I=0$

[TABLE]

with $t$ and $u$ the usual Mandelstam variable. We can see that the Eq.(19) is similar to the Eq.(8). The former can be obtained from the latter just by substituting $16\to 2\sqrt{6}$ and $m_{\pi}\to m_{K}$.

2.4 $\rho\rho\to K\bar{K}$ with $K^{*}$-exchange

As for the $\rho KK^{*}$ coupling, we use the Lagrangian in Eq.(9). Then we get the amplitude for $\rho^{+}(p_{1})\rho^{-}(p_{2})\to K^{+}(p_{3})K^{-}(p_{4})$ with $K^{*}$-exchange corresponding to the Fig.4 as

[TABLE]

and the $u$-channel

[TABLE]

Next we list the tree-level amplitudes for $\rho\rho\to K\bar{K}$ with $K^{*}$-exchange as the following:

[TABLE]

Using Eqs.(2.1) and (18) we obtain the $I=0$ amplitude

[TABLE]

which can be obtain from Eq.(14) by substituting $1\to\sqrt{6}/4$ and $m_{\omega}\to m_{K^{*}}$.

2.5 Partial-wave decomposition

In term of these amplitudes with isospin $I=0$, we can calculating the partial-wave amplitudes in the $\ell SJI$ basis  [5] , denoted as $T^{(JI)}_{\ell S;\bar{\ell}\bar{S}}(s)$ for the transition $(\bar{\ell}\bar{S}JI)\to(\ell SJI)$

[TABLE]

with $M=\sigma_{1}+\sigma_{2}$ and $\bar{M}=\bar{\sigma}_{1}+\bar{\sigma}_{2}$. And $N$ accounts for identical particles, for example

[TABLE]

By using Eq.(24) we can calculate the partial-wave projected tree-level amplitudes of Eqs.(8), (14), (19) and (23) with quantum number $I,\ell,S=0,0,0$. We denote $T^{(00)}_{00;00}$ by $V$ for simplicity and we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

3 Results and discussion

We label the three channels, $\rho\rho$, $K\bar{K}$ and $\pi\pi$ as 1, 2 and 3, respectively. With the channel transition amplitudes $V_{\pi}$, $V_{\omega}$, $V_{K}$ and $V_{K^{*}}$ given in last section, we calculate the full amplitude and its pole positions by using the Bethe Salpeter equation in its on-shell factorized form [4, 5]

[TABLE]

$G$ is a diagonal matrix made up by the two-point loop function [4, 5]

[TABLE]

with $P$ the total four-momentum of the meson-meson systems and $q$ the loop momentum. The channel is labelled by the subindex $j$. By using dimensional regularization the integration can be recast as

[TABLE]

with

[TABLE]

or using a momentum cutoff $q_{\text{max}}$ as

[TABLE]

where $w=\sqrt{q^{2}+m^{2}_{j}}$. The integral can be done algebraically

[TABLE]

Typical values of the cutoff $q_{\text{max}}$ are around 1 GeV. $G_{jj}(s)$ has a right-hand cut above the threshold $2m_{j}$. In order to make an analytical extrapolation to second Riemann sheet we make use of the continuity property

[TABLE]

where the index $(2)$ indicates the second Riemann sheet of $G_{jj}$. Then

[TABLE]

Other potential of coupled-channels like $\pi\pi-K\bar{K}$ can be found in [1]. Our results are shown in Table 1 for various $q_{\text{max}}$ values. For comparison, the results for the $\rho\rho$ single channel without considering the coupled channel effects as in Ref. [5] are show in the second row. The $3\sim 6$ rows show the results including one coupled channel with the exchanged meson listed in the first column. For example the $\pi$ denotes the $\rho\rho-\pi\pi$ channel with $\pi$ exchange and so on. The 7-th row gives the results including all three coupled channels of $\rho\rho$, $\pi\pi$ and $\bar{K}K$.

The above results show that the influence of vector meson $\omega$ and $K^{*}$ exchanges is very small; the largest influence comes from the $\rho\rho-\pi\pi$ channel coupling by the pion exchange, which shifts up the pole mass and results in a sizable $\pi\pi$ decay width, comparable with relevant PDG values for $f_{0}(1500)$ [15]. For the $\rho\rho-K\bar{K}$ coupled-channel case we can see that the width is consistent with $f_{0}(1500)$ decaying into $K\bar{K}$ in PDG, which is about $8.9MeV$. When taking into account all the three channels, the pole position is close to the results by considering only the pion exchange contribution. With $q_{\text{max}}=1.4GeV$, the pole mass and partial decay widths to $\pi\pi$ and $\bar{K}K$ are roughly consistence with PDG values for $f_{0}(1500)$. The largest decay channel should be $4\pi$ either through $\rho\rho$ directly or by its cross talk with $\sigma\sigma$. Note that due to the binding energy of the molecule as well as the kinetic energy of $\rho$ inside the molecule, the $4\pi$ decay width through the decay of $\rho$ inside the $\rho\rho$ molecule can be smaller than the decay width of a single free $\rho$ meson. Similar effect was pointed out by Refs. [16, 17] in their studies of $d^{*}(2380)$ as a $\Delta\Delta$ molecule which gets a decay width smaller than the decay width of a single free $\Delta$ state. This kind of effect was also observed by the study of other hadronic molecules [18, 19].

In summary, the $\rho\rho$ scattering is revisited by including its coupled-channels of pseudoscalar mesons, i.e., $\pi\pi$ and $\bar{K}K$. It is found that the coupled-channel effect is important and shifts up the pole mass of the dynamically generated scalar state significantly. It makes the state to be more consistent with $f_{0}(1500)$ rather than $f_{0}(1370)$ as favored by the previous studies [4, 5] without including these coupled channels. This leads to a nicely consistent picture with a recent dispersive study [20] where a new parametrization for the scalar pion form factors is derived by fitting it to LHCb data on $\bar{B}^{0}_{s}\to J/\psi\pi\pi$, and find an $f_{0}(1500)$ at mass $1465\pm 18MeV$ coupling strongly to $\rho\rho$ (or $\sigma\sigma$). The $\rho\rho$ scattering has been extended to the S-wave interactions for the whole vector-meson nonet by two groups [21, 22]. Both propose $f_{0}(1710)$ to be $K^{*}\bar{K}^{*}$ dynamically generated state. We expect similar significant coupled channel effects there. By including its coupled-channels of pseudoscalar mesons, the $K^{*}\bar{K}^{*}$ dynamically generated state could be $f_{0}(1790)$ suggested by the BES data [26, 27, 28] instead of $f_{0}(1710)$ . The $f_{0}(1370)$, $f_{0}(1500)$ and $f_{0}(1710)$ have been studied before in quarkonia-glueball mixing picture in Refs. [23, 24, 25], trying to pin down partial contributions of glueball, nonstrange and strange quakonia in these scalar mesons. With the new configuration of meson-meson dynamically generated states, the structure of these scalars should be richer than previous assumptions and deserve further exploration by expanding the configuration space.

Acknowledgments

We thank Li-Sheng Geng, Feng-Kun Guo, Ulf-G. Meissner and Eulogio Oset for helpful discussions. This project is supported by NSFC under Grant No. 11621131001 (CRC110 cofunded by DFG and NSFC) and Grant No. 11835015.