4.3 Covariant vector meson-vector meson interactions and dynamically generated resonances 83
4.3.2 First iterated solution of the N/D method
In general, the overlap of the artificial unphysical cut and the right-hand cut breaks the unitarity and analyticity, and thus leads to the invalidation of the unitarization formula (3.99). One way to avoid this problem is to employ theN/Dmethod [109, 156, 158, 185] to the coupled-channel systems, see sec. 3.4.3.
For the first iterated solution, see eq. (4.40) and for the matching condition eq (4.41). If we extend this to coupled channels, we have
N(s)i j=V(s)i j, (4.57)
D(s)i j=γ0i j+γ1i j(s−sithr)+ 1
2γ2i j(s−sithr)2+(s−sithr)s2 π
Z ∞ sithr
ds0 ρ(s0)iV(s0)i j (s0−sithr)(s0−s)s02, withγthe subtraction constants and the subscriptiand jthe channel indices whereas matching conditions
10Notice that the pole position has an imaginary part in ref. [116] due to the convoluted propagators in loop functions.
has the form
γ0i j+γ1i j(s−sithr)+ 1
2γ2i j(s−sithr)2=−gi(s)V(s)i j− (s−sithr)s2 π
Z ∞ sithr
ds0 ρ(s0)iV(s0)i j (s0−sithr)(s0−s)s02
≡w(s)i j, (4.58)
where the contributions ofO(s−sthr)3
is neglected. The subtraction constants depend on the cutoff qmaxvia the matching conditions (4.58). Note that for each scattering process, the left-hand branch point associated with thet- andu-channel diagrams are located lower than its own threshold, although they may be above the lowest threshold with the same quantum numbers.
The continuation ofDfunction to the second Riemann sheet has the form11 DII(s)i j=γII0i j+γII1i j(s−sithr)+ 1
2γII2i j(s−sithr)2− (s−sitrh)s2 π
Z ∞ sithr
ds0 ρ(s0)iV(s0)i j
(s0−sithr)(s0−s)s02. (4.59) Likewise, the subtraction constantsγIIs can be determined by matching
γII0i j+γII1i j(s−sithr)+ 1
2γII2i j(s−sithr)2=−gIIi (s)V(s)i j+ (s−sithr)s2 π
Z ∞ sithr
ds0 ρ(s0)iV(s0)i j (s0−sithr)(s0−s)s02
≡wII(s)i j. (4.60)
In this way, we have
γ0i j(II)=w(II)(sthr)i j, γ1i j(II)=w(II)0(sithr)i j,
γ2i j(II)=w(II)00(sithr)i j. (4.61)
The ObtainedDfunctions have the correct analytical properties on both the first and second Riemann sheets, see e.g. in fig. 4.21, where the associated zero points correspond to possible dynamically generated resonances or bound states. For comparison, the real parts of the determinants ofN/Dand of (3.99), denoted by DetN/Dand DetUrespectively, are ploted in fig. 4.21.
Single-channels
In this section, we first consider single-channels. In theN/Dmethod, the determinant of amplitudes is free of any unphysical cut. Thus, it is expected that the poles, due to the presence of left-hand cuts found in the unitarization formula, see e.g. in Figs. 4.16 and 4.17, should be absent in theN/Dmethod.
It is the case for the pole on the first Riemann sheet, see e.g. 4.16, in channel (S,I,J)=(2,0,1), see in fig. 4.19. For (S,I,J)=(0,0,1), the pole on the first Riemann sheet is found much deeper than that in BSE, see in fig. 4.19. It stems from the matching condition (4.58) where the contribution from t-andu-channel diagrams still play a role to let the determinant eventually to vanish at some point. The position of this pole dramatically depends on the coupling and the cutoffwe use and much below the threshold. Therefore, these strongly indicate that it is an artificial pole due to the perturbative treatment of the left-hand cut. However, the pole on the second Riemann sheet for (S,I,J)=(2,0,1) is changed
11The usual continuationTII(s)= 1
(ND)−1−2iρwould introduce unexpected left-hand cuts into unphysical Riemann sheets.
4.3 Covariant vector meson-vector meson interactions and dynamically generated resonances
barely. The fig. 4.20 indicates that the formation of the virtue state stems from thet- andu-channel diagrams via the matching condition (4.58). The pole corresponding to either a bound state or virtual state in channel (S,I,J)=(0,0,1), which is close to the threshold, changes barely. It is not a surprise since the difference between the two methods is ofO(s−sthr)3
.
K*K*threshold
(S,I,J)=(0,0,1) qmax=0.775 GeV
N/D BSE
1.4 1.5 1.6 1.7 1.8 1.9
-1.0 -0.5 0.0 0.5 1.0 1.5
s (GeV) ReDK*K_*→K*K_*
K*K*threshold
(S,I,J)=(2,0,1) qmax=0.775 GeV
N/D BSE
1.4 1.5 1.6 1.7 1.8 1.9
-1.0 -0.5 0.0 0.5 1.0 1.5
s (GeV) ReDK*K*→K*K*
Figure 4.19: The real part of the determinant of (0,0,1) and (2,0,1) based on theN/Dand algebraic approximation of BSE. The pole due to the left-hand cut is absent for (2,0,1), and is moved to very deep (around 1.49 GeV) below the threshold where the matching (4.58) applicability is in question.
K*K*threshold
(S,I,J)=(2,0,1) g=4.168 qmax=0.775 GeV
N/D BSE
1.4 1.5 1.6 1.7 1.8 1.9
-6 -4 -2 0 2 4
s (GeV)
ReDet
K*K*threshold
(S,I,J)=(2,0,1) g=4.596 qmax=1.0 GeV
N/D BSE
1.4 1.5 1.6 1.7 1.8 1.9
-6 -4 -2 0 2 4
s (GeV)
ReDet
Figure 4.20: The real part of the determinant on the second Riemann sheet of (2,0,1) based on theN/Dand BSE.
Apart from above poles, additional poles are found on the second Riemann sheet for channels (S,I,J)=(0,2,2), (1,3/2,2) and (2,1,2), which are absent in the algebraic approximation of the BSE, see e.g. table 4.8. All of them are located on the unphysical Riemann sheet in the tensor sector. An inspection shows that the discrepancy stems from the badly matching of eq. (4.58) far away from the threshold on the second Riemann sheet in the corresponding channels. The matching condition eq. (4.58) works well sufficiently for theJ =0 andJ=1 channel. However, for the tensor sector, the matching is much worse, especially for the second Riemann sheet, see e.g. in Figs. 4.21 and 4.22, where the DetU represents the determinant of eq. (3.99). The poles in table 4.8 are located around 0.3 GeV away from their threshold, and thus beyond the reliability. As a consequence, we do not relate these poles to physical
resonances.
Table 4.8: The pole positions (in GeV) on the second Riemann sheet evaluated withg=4.596 andqmax=0.875 GeV for the first iterated solution of theN/Dmethod.
Channel Process Threshold [GeV] Pole position [GeV]
(S,I,J)=(0,2,2) ρρ→ρρ 1.55 1.73±0.21i (S,I,J)=(1,3/2,2) ρK∗→ρK∗ 1.67 1.85±0.21i (S,I,J)=(2,1,2) K∗K∗→K∗K∗ 1.78 1.97±0.22i
(S,I,J)=(1,3/2,0)
DetIN/D DetIU
DetIIN/D DetIIU
1.2 1.4 1.6 1.8 2.0
0 1 2 3 4
s (GeV)
ReDet
(S,I,J)=(1,3/2,2)
DetIN/D DetIU DetIIN/D
DetIIU
1.2 1.4 1.6 1.8 2.0
0 5 10 15 20 25 30
s (GeV)
ReDet
Figure 4.21: The real part of the determinant of channel (S,I,J)=(1,3/2,0) and (1,3/2,0) on the first and second Riemann sheet evaluated with the cutoffqmax=0.875 GeV. The superscript I and II indicate on which Riemann sheet are plotted, and the subscript denotes the unitarization formula.
Coupled-channels
The bound state in channel (S,I,J)=(0,0,0), which is absent in the BSE, is obtained in the coupled-channel version, see in fig. 4.23. It is consistent with the extreme non-relativistic limit in ref. [116] and with results of the previous section. One may wonder the reliability of this pole since it is deeply below theφφthreshold where theD(s)55is matched via eq. (4.58). To show that this pole is reasonable, we calculate its residues which stand for its coupling to various channels. They are defined as
Ti j = gigj
s−spole, (4.62)
withithe channel index and gi is the effective coupling to channeli. A calculation shows that the coupling of the bound state to theρρchannel dominates and the channels with higher thresholds are negligible. In addition, a pole located at 1.68±0.02iGeV is found on an unphysical Riemann sheet which is associated to a possible resonances. It corresponds to the pole 1.726±0.028iGeV which is assigned to f0(1710). It is not surprising since the channelK∗K¯∗dominates whose threshold is close to the pole position and thus the non-relativistic limit makes sense. It is worth stressing that its position largely
4.3 Covariant vector meson-vector meson interactions and dynamically generated resonances
(S,I,J)=(2,1,0)
DetIN/D DetIU DetIIN/D
DetIIU
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
-2 -1 0 1 2 3 4
s (GeV)
ReDet
(S,I,J)=(2,1,2) DetIN/D DetIU
DetIIN/D
DetIIU
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
-5 0 5 10 15 20 25
s (GeV)
ReDet
Figure 4.22: The real part of the determinant of channel (S,I,J)=(2,1,0) and (2,1,0) on the first and second Riemann sheet evaluated with the cutoffqmax=0.875 GeV. The superscript I and II indicate on which Riemann sheet are plotted, and the subscript denotes the unitarization formula.
ρρthreshold (S,I,J)=(0,0,0)
1.2 1.3 1.4 1.5 1.6 1.7
-0.2 0.0 0.2 0.4 0.6
s (GeV)
Det
ρρthreshold
(S,I,J)=(0,0,2)
1.3 1.4 1.5 1.6 1.7 1.8
-20 0 20 40 60 80 100 120
s (GeV)
Det
Figure 4.23: The determinant of channel (S,I,J)=(0,0,0) and (0,0,2) on the first Riemann sheet evaluated with the cutoffqmax=0.875 GeV.
depends on the coupling and cutoff. So far, the scalar and vector sectors of (S,I)=(0,0) are consistent with the non-relativistic result in ref. [116]. In ref. [116], two bound states, assigned to f2(1270) and f20(1525), are reported in the tensor sector. However, no bound state is found in this channel fig.4.23, which agrees with the SU(2) relativistic result in the previous section. Instead, a resonance located at 1.93±0.15iGeV is found. As discussed in section 4.3.2, the matching condition (4.58) does not work very well on the unphysical Riemann sheet for the tensor sectors. From table 4.9, it is easy to see that all poles in the tensor sector are found on the unphysical Riemann sheet, in particular for the dominating channel. The relative distant positions of the poles from thresholds (of the dominating channels) indicate that these poles are “artificial". Hence, we do not relate them to physical resonances. For the channel (S,I)=(1,1/2), a pole around 1.64 GeV is found, see e.g. table 4.9, which corresponds to the bound state at 1.643±0.047iGeV in ref. [116]12. However, the bound state in the tensor sector reported in ref. [116] is not found in this work, see e.g. fig. 4.24.
ρK*threshold (S,I,J)=(1,1/2,0)
N/D BSE
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
s (GeV)
ReDet
ρK*threshold (S,I,J)=(1,1/2,2)
N/D BSE
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
-20 -15 -10 -5 0 5 10
s (GeV)
ReDet
Figure 4.24: The real part of the determinant of channels (S,I,J) = (1,1/2,0) and (1,1/2,2) on the physical Riemann sheet evaluated with the cutoffqmax=0.875 GeV. A bound state is found in the scalar sector for the first iterated solution ofN/D, while no bound state is there in the tensor sector.
To investigate the dependence of the pole position on the coupling and cutoff, we search the poles with different choices ofgandqmax. A list of the poles in the region we are interested in is given in table 4.9.
Resonances in the scalar and vector sectors having small width is due to that we neglect the width of ρandK∗. A convolution ofGfunctions over the mass distribution of vector mesons will give a width to bound states and resonances related to theρ→ππandK∗→πKdecays. In addition, box diagrams with four intermediate pseudoscalars give a width to resonances/bound states due to decaying to two pseudoscalars [115, 116]. However, to that end, we have to introduce model-dependent form factors and the width strongly depends on the cutoffwe use in form factors. However, the real part of pole positions is merely affected. In this section, only the possible dynamically generated resonances are considered.
Thus, we will not introduce the convolution of loop functions and the box diagrams to broaden the width of resonances, which is sensitive to form factors.