-
In the framework of FCA, we consider
ˉKK∗(ηK∗) as a cluster, andη(ˉK) interacts with the components of the cluster. The total three-body scattering amplitudeT can be simplified as the sum of two partition functionsT1 andT2 , by summing all diagrams in Fig. 1, starting with the interaction of particle 3 with particle 1(2) of the cluster. The FCA equations can be written in terms ofT1 andT2 , which give the total scattering amplitudeT , and read [25, 26, 51]T1=t1+t1G0T2,
(1) T2=t2+t2G0T1,
(2) T=T1+T2,
(3) where the amplitudes
t1 andt2 represent the unitary scattering amplitudes with coupled channels for the interactions of particle 3 with particle 1 and 2, respectively. The functionG0 in the above equations is the propagator for particle 3 between the particle 1 and 2 components of the cluster, which we discuss below.We calculate the total scattering amplitude
T in the low energy regime, close to the threshold of theηˉKK∗ system or below, where FCA is a good approximation. The on-shell approximation for the three particles is also used.Following the field normalization of Refs. [52, 53], we can write the
S matrix for the single scattering term [Fig. 1(a) and 1(e)] as①S(1)=S(1)1+S(1)2=(2π)4V2δ4(k3+kcls−k′3−k′cls)×1√2w31√2w′3(−it1√2w11√2w′1+−it2√2w21√2w′2),
(4) where
V stands for the volume of a box in which the states are normalized to unity, while the momentumk(k′) and the on-shell energyw(w′) refer to the initial (final) particles.The double scattering contributions are obtained from Fig. 1(b) and 1(f). The expression for the
S matrix for double scattering [S(2)2=S(2)1 ] is given byS(2)=−it1t2(2π)4V2δ4(k3+kcls−k′3−k′cls)×1√2w31√2w′31√2w11√2w′11√2w21√2w′2×∫d3q(2π)3Fcls(q)1q02−|→q|2−m23+iϵ,
(5) where
Fcls(q) is the form factor of the cluster which is a bound state of particles 1 and 2. The information about the bound state is encoded in the form factorFcls(q) in Eq. (5), which is the Fourier transform of the cluster wave function. The variableq0 is the energy carried by particle 3 in the center-of-mass frame of particle 3 and the cluster, and is given byq0(s)=s+m23−m2cls2√s,
(6) where
s is the invariant mass squared of theηˉKK∗ system.For the form factor
Fcls(q) , we take the following expression fors wave bound states only, as discussed in Refs. [52-54]:Fcls(q)=1N∫|→p|<Λ,|→p−→q|<Λd3→p12w1(→p)12w2(→p)×1mcls−w1(→p)−w2(→p)12w1(→p−→q)12w2(→p−→q)×1mcls−w1(→p−→q)−w2(→p−→q),
(7) where the normalization factor
N isN=∫|→p|<Λd3→p(12w1(→p)12w2(→p)1mcls−w1(→p)−w2(→p))2,
with
mcls the mass of the cluster. Note that the width ofK∗ should also be included inFcls(q) [30]. However, as shown below, the masses off1(1285) andK1(1270) are below the threshold ofˉKK∗ andηK∗ , and the effect of the width ofK∗ is small and can be neglected.Similarly, the full
S matrix for the scattering of particle 3 on the cluster is given byS=−iT(2π)4V2δ4(k3+kcls−k′3−k′cls)×1√2w31√2w′31√2wcls1√2w′cls.
(8) By comparing Eqs. (4), (5), and (8), we see that it is necessary to introduce a weight in
t1 andt2 so that Eqs. (4) and (5) include the factors that appear in Eq. (8). This is achieved by,˜t1=t1√2wcls2w1√2w′cls2w′1,˜t2=t2√2wcls2w2√2w′cls2w′2.
Eq. (3) can then be solved to give
T=˜t1+˜t2+2˜t1˜t2G01−˜t1˜t2G20,
(9) where
G0 depends on the invariant mass of theηˉKK∗ system, and is given byG0(s)=12mcls∫d3q(2π)3Fcls(q)q02−|→q|2−m23+iϵ.
(10) -
It is worth noting that the argument of the total scattering amplitude
T can be regarded as a function of the total invariant mass√s of the three-body system, while the arguments of two-body scattering amplitudest1 andt2 depend on the two-body invariant masses√s1 and√s2 .s1 ands2 are the invariant masses squared of the external particle3 with momentumk3 , and particle 1 (2) inside the cluster with momentumk1 (k2 ), which are given bys1=m23+m21+(s−m23−m2cls)(m2cls+m21−m22)2m2cls,s2=m23+m22+(s−m23−m2cls)(m2cls+m22−m21)2m2cls,
where
ml (l=1,2,3) are the masses of the corresponding particles in the three-body system.It is worth mentioning that in order to evaluate the two-body scattering amplitudes
t1 andt2 , the isospin of the cluster should be considered. For the case of theη -(ˉKK∗)f1(1285) system, the clusterˉKK∗ has isospinIˉKK∗=0 . Therefore, we have|ˉKK∗⟩I=0=1√2|(12,−12)⟩−1√2|(−12,12)⟩,
(11) where the kets on the right-hand side indicate the
Iz components of particlesˉK andK∗ ,|(IˉKz,IK∗z)⟩ . For the case of the total isospinIη(ˉKK∗)=0 , the single scattering amplitude is written as [20]⟨η(ˉKK∗)|t|η(ˉKK∗)⟩=(⟨00|⊗1√2(⟨(12,−12)|−⟨(−12,12)|))(t31+t32)(|00⟩⊗1√2(|(12,−12)⟩−|(−12,12)⟩))=(1√2⟨(1212,−12)|−1√2⟨(12−12,12)|)t31(1√2|(1212,−12)⟩−1√2|(12−12,12)⟩)+(1√2⟨(1212,−12)|−1√2⟨(12−12,12)|)t32(1√2|(12−12,12)⟩−1√2|(1212,−12)⟩),
(12) where the notation for the states in the last term is
|(IηˉKIzηˉK,IzK∗)⟩ fort31 and|(IηK∗IzηK∗,IzˉK)⟩ fort32 . This leads to the following amplitudes for single scattering [Fig. 1(a) and 1(e)] in theη -(ˉKK∗)f1(1285) system,t1=tI=1/2ηˉK→ηˉK,t2=tI=1/2ηK∗→ηK∗.
(13) In the
ˉK -(ηK∗)K1(1270) system, the clusterηK∗ can only have isospinIηK∗=1/2 . Therefore, for the total isospinIˉK(ηK∗)=0 , the scattering amplitude is written as [20]⟨ˉK(ηK∗)|t|ˉK(ηK∗)⟩=1√2(⟨1212|⊗⟨(12,−12)|−⟨12−12|⊗⟨(12,12)|)(t31+t32)1√2(|1212⟩⊗|(12,−12)⟩−|12−12⟩⊗|(12,12)⟩)=(1√2⟨(1212,−12)|−1√2⟨(12−12,12)|)t31(1√2|(1212,−12)⟩−1√2|(12−12,12)⟩)+1√2(1√2⟨(00,0)+1√2⟨(00,0)|)t321√2(1√2|(00,0)⟩+1√2|(00,0)⟩).
(14) This leads to the following amplitudes for single scattering in the
ˉK -(ηK∗)K1(1270) system,t1=tI=1/2ˉKη→ˉKη,t2=tI=0ˉKK∗→ˉKK∗.
(15) We see that only the transition
ˉKK∗→ˉKK∗ withI=0 gives a contribution, since the total isospinIˉK(ηK∗)=0 and theη meson has isospin zero. -
An important ingredient in the calculations of the total scattering amplitude for the
ηˉKK∗ system using FCA are the two-bodyηK ,ηK∗ , andˉKK∗ unitarizeds wave interactions from the chiral unitary approach. These two-body scattering amplitudes are studied with the dimensional regularization procedure, and they depend on the subtraction constantsaηK ,aηK∗ andaˉKK∗ , and also on the regularization scaleμ . Note that there is only one parameter for the dimensional regularization procedure, since any change inμ is reabsorbed by the change ina(μ) througha(μ′)−a(μ)=ln(μ′2μ2) , so that the scattering amplitude is scale independent. In this work, we use the parameters from Refs. [21, 49, 50]:aηK=−1.38 andμ=mK forIηK=1/2 ;aηK∗=−1.85 andμ=1000 MeV forIηK∗=1/2 ;aˉKK∗=−1.85 andμ=1000 MeV forIˉKK∗=0 . With these parameters, we get the masses off1(1285) andK1(1270) at their estimated values.In Figs. 2(a) and (b), we show the numerical results for
|tI=0ˉKK∗→ˉKK∗|2 and|tI=1/2ηK∗→ηK∗|2 , respectively, where we see clear peaks for thef1(1285) andK1(1270) states. -
To connect with the dimensional regularization procedure, we choose the cutoff
Λ such that the two-body loop function at the threshold coincides in both methods. Thus, we takeΛ=990 MeV such thatf1(1285) is as obtained in Refs. [55, 56], while forK1(1270) , we takeΛ=1000 MeV. The cutoff is tuned to get a pole at1288−i74 for theK1(1270) state.In Figs. 3 and 4, we show the respective form-factors for
f1(1285) andK1(1270) , where we takemcls=1281.3 MeV forf1(1285) and 1284 MeV forK1(1270) , as obtained in Ref. [49]. In FCA, we keep the wave function of the cluster unchanged in the presence of the third particle. In order to estimate the uncertainties of FCA due to this "frozen" condition, we admit that the wave function of the cluster could be modified by the presence of the third particle. To do so, we perform calculations with different cutoffs. The results, shown in Figs. 3 and 4, are obtained withΛ=890 ,990 and1090 MeV forf1(1285) , while forK1(1270) we takeΛ=900 , 1000 and 1100 MeV.Figure 3. Forms-factor Eq. (7) as a function of
q=|→q| for the cutoffΛ=890 (dashed),990 (solid), and1090 MeV (dotted) forf1(1285) as theˉKK∗ bound state.Figure 4. As in Fig. 3, but for
K1(1270) as theηK∗ bound state. The dashed, solid and dotted curves are forΛ=900 , 1000, and 1100 MeV, respectively.In Fig. 5, we show the real (solid line) and imaginary (dashed line) parts of
G0 as a function of the invariant mass of theη -(ˉKK∗)f1(1285) system forΛ=890 ,990 and1090 MeV.Figure 5. (color online) Real (solid line) and imaginary (dashed line) parts of
G0 for theη -(ˉKK∗)f1(1285) system andΛ=890 (blue),990 (red) and1090 MeV (green).The results for
G0 of theˉK -(ηK∗)K1(1270) system are shown in Fig. 6, where the real (solid line) and imaginary (dashed line) parts are forΛ=900 ,1000 and1100 MeV.From Figs. 5 and 6, it can be seen that the imaginary part of
G0(s) is not sensitive to the value of the cutoff, while the real part slightly changes with the cutoff.
Prediction of possible exotic states in the ηˉKK∗ system
- Received Date: 2019-10-28
- Accepted Date: 2019-12-20
- Available Online: 2020-05-01
Abstract: We investigate the