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

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 $f_0(1500)$.


Introduction
The chiral unitary approach, which has made much progress in the study of pseudoscalar 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 ρρ 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 ρ 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 ρρ 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 rebuked by Ref. [11] in which the non-relativistic assumption was dropped by evaluating exactly the loops with full relativistic ρ propagators in solving the B-S equation for ρρ 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].
Starting with the Lagrangian in Eq.(1) we can immediately obtain the amplitude A t (p 1 , p 2 , p 3 , p 4 ) of ρ + (p 1 )ρ − (p 2 ) → π + (p 3 )π − (p 4 ) corresponding to Fig.1 as In this equation, the i corresponds to the polarization vector of the i-th ρ. Each polarization vector is characterized by its three-momentum p i and third component of the spin σ i . Explicit expressions of these polarization vectors are given by [5] (p, 0) = where β = |p|/E p and γ = 1/ 1 − β 2 . The u-channel π-exchange amplitude A u can be obtained from the expression of A t by exchanging p 3 ↔ p 4 . In this way, And now we write the tree-lavel amplitude for ρρ → ππ with π-exchange In order to obtain the S-wave amplitude in isospin I = 0 channel we need the isospin eigenstates. We have where we have used the convention |ρ + = −|1, 1 and |π + = −|1, 1 of isospin. By taking into account Eq.(7) and the amplitudes in Eq.(6) we can now write the isospin I = 0 amplitude for ρρ → ππ

ρρ → ππ with ω-exchange
One needs the ρωπ coupling which is provided within the framework [14] of the hidden gauge formalism by means of the Lagrangian with where G V = 55M eV and f = 93M eV . At this point we can write down the amplitude of ρ + (p 1 )ρ − (p 2 ) → π + (p 3 )π − (p 4 ) with ω-exchange corresponding to Fig.2 as in the πexchange case < l a t e x i t s h a 1 _ b a s e 6 4 = " K r / 3 P 4 e 0 c p 2 M f e e d f s n k Y f j 1 f g E z g D U 4 C c 7 8 5 v 1 3 U 9 9 5 c 3 8 F 5 4 L 6 + k r t P W P A c 3 w t v / A w Q C C 6 A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K r / 3 P 4 e 0 c p 2 M f e e d f s n k Y f j 1 f g E z g D U 4 C c 7 8 5 v 1 3 U 9 9 5 c 3 8 F 5 4 L 6 + k r t P W P A c 3 w t v / A w Q C C 6 A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 J 9 1 8 I n a G l J s f R e D u 2 E V Z 9 X z 5 j 0 And the u-channel ω-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 π-exchange by exchanging p 3 ↔ p 4 , thus Next we write the tree-level amplitude for ρρ → ππ with ω-exchange Then using Eq. (7) we can get the I = 0 amplitude

ρρ → KK with K-exchange
The ρKK coupling is provided in the same Lagrangian in Eq.(1), so we can immediately write down the amplitude of ρ and the u-channel C u (p 1 , p 2 , p 3 , p 4 ) = C t (p 1 , p 2 , p 4 , p 3 ).
Then we can obtain the tree-level amplitudes for ρρ → KK with K-exchange as the Similar to Eq. (7) we need the isospin I = 0 eigenstate for |KK . We have where we use the convention |K + = −| 1 2 , 1 2 of isospin. By using the isospin wave functions we can obtain for I = 0 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 → 2 √ 6 and m π → m K .
Next we list the tree-level amplitudes for ρρ → KK with K * -exchange as the following: Using Eqs. (7) and (18) we obtain the I = 0 amplitude which can be obtain from Eq.(14) by substituting 1 → √ 6/4 and m ω → m K * .
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, , S = 0, 0, 0. We denote T (00) 00;00 by V for simplicity and we have for ρρ → ππ with π-exchange for ρρ → ππ with ω-exchange for ρρ → KK with K-exchange for ρρ → KK with K * -exchange 3 Results and discussion We label the three channels, ρρ, KK and ππ as 1, 2 and 3, respectively. With the channel transition amplitudes V π , V ω , 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] G is a diagonal matrix made up by the two-point loop function [4,5] with P the total four-momentum of the meson-meson systems and q the four-momentum of one intermediate meson. The channel is labelled by the subindex j. By using dimensional regularization the integration can be recast as or using a momentum cutoff q max as where w = q 2 + m 2 j . The integral can be done algebraically Typical values of the cutoff q 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 where the index (2) indicates the second Riemann sheet of G jj . Then Other potential of coupled-channels like ππ − KK can be found in [1]. Our results are shown in Table 1 for various q max values. For comparison, the results for the ρρ single channel without considering the coupled channel effects as in Ref. [5] are show in the second row. The 3 ∼ 6 rows show the results including one coupled channel with the exchanged meson listed in the first column. For example the π denotes the ρρ − ππ channel with π exchange and so on. The 7-th row gives the results including all three coupled channels of ρρ, ππ andKK.  Table 1: Pole position for coupled-channels The above results show that the influence of vector meson ω and K * exchanges is very small; the largest influence comes from the ρρ−ππ channel coupling by the pion exchange, which shifts up the pole mass and results in a sizable ππ decay width, comparable with relevant PDG values for f 0 (1500) [15]. For the ρρ − KK coupled-channel case we can see that the width is consistent with f 0 (1500) decaying into KK in PDG, which is about 8.9M eV . 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 max = 1.4GeV , the pole mass and partial decay widths to ππ andKK are roughly consistence with PDG values for f 0 (1500). The largest decay channel should be 4π either through ρρ directly or by its cross talk with σσ. Note that due to the binding energy of the molecule as well as the kinetic energy of ρ inside the molecule, the 4π decay width through the decay of ρ inside the ρρ molecule can be smaller than the decay width of a single free ρ meson. Similar effect was pointed out by Refs. [16,17] in their studies of d * (2380) as a ∆∆ molecule which gets a decay width smaller than the decay width of a single free ∆ state. This kind of effect was also observed by the study of other hadronic molecules [18,19].
In summary, the ρρ scattering is revisited by including its coupled-channels of pseudoscalar mesons, i.e., ππ andKK. 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. The ρρ scattering has been extended to the S-wave interactions for the whole vector-meson nonet by two groups [20,21]. We expect similar significant coupled channel effects there.