Absorption within the Framework of the Fixed Center Approximation to Faddeev equations
We present a method to evaluate the absorption width in the bound system. Most calculations of this system ignore this channel and only consider the conversion. Other works make a qualitative calculation using perturbative methods. Since the resonance is playing a role in the process, the same resonance is changed by the presence of the absorption channels and we find that a full nonperturbative calculation is called for, which we present here. We employ the Fixed Center Approximation to Faddeev equations to account for rescattering on the cluster and we find that the width of the states found previously for and increases by about 30 MeV due to the absorption, to a total width of about 80 MeV.
pacs:11.10.St; 12.40.Yx; 13.75.Jz; 14.20.Gk; 14.40.Df
The interaction of antikaon () with nucleon () has drawn attention for decades Kaiser:1995eg (); angels (); Oller:2000fj (); Lutz:2001yb (); Oset:2001cn (); Hyodo:2002pk (); Jido:2003cb (); Borasoy:2005ie (); Oller:2006jw (); Borasoy:2006sr (); Ikeda:2012au () since this interaction provides essential elements to understand the strangeness in nuclear hadron physics. The interaction is strongly attractive in the isospin and also the interaction is attractive at low energies. The and are the most important channels for the : these two channels dynamically generate the quasi bound .
To understand the many body kaonic nuclear system, the is the simplest system. The investigation of this system started in the 60’s nogami (). More recently, using the variational approach, a state was found bound by 48 MeV with width of 60 MeV Yamazaki:2002uh (). In Dote:2008in (), the three body system was investigated using the variational approach taking the results of interactions from a chiral model, and they get a much less bound state with, B=19 MeV and =40-70 MeV. In a further paper, the same group quantifies the uncertainties and evaluates the extra width coming from absorption from the pair of nucleons, with the results of B=20-40 MeV and as large as 100 MeV with admitted large uncertainties Dote:2008hw () . Coupled channel three body calculations of the quasi bound system were investigated in Shevchenko:2006xy (); Shevchenko:2007zz () using Faddeev equations, and their results for the binding is 50-70 MeV and the width around 100 MeV. The work of Ikeda:2007nz () studied three body resonances in the system within a framework of the coupled channel Faddeev equation. In this work they found the binding energy 79 MeV with a width of 74 MeV. However, in more recent papers Ikeda:2008ub (); Ikeda:2010tk (), where the energy dependence of the potential is taken from chiral dynamics, the same authors find much smaller binding energies, below 20 MeV. We also calculated the system bayarnpa (); Oset:2012gi () for the and case, using the Fixed Center Approximation (FCA) to the Faddeev equations taking into account channel explicitly and also including the charge exchange diagrams. We get a binding of 26-35 MeV for while around 9 MeV for the case. The widths are in both cases around 50 MeV. A very recent variational calculation Barnea:2012qa () finds that the state is bound by about 16 MeV and the by less than 11 MeV. The widths, without including absorption by a pair of nucleons, are around 40 MeV.
The FCA to the Faddeev equations is an efficient and useful method to investigate many body systems. It relies upon having a pair of particles that interact strongly among themselves and make a cluster that is supposed not to be much disturbed by the interaction of the third particle with it. Then all that is left is the possibility that the third particle interacts with the components of the cluster and this is done non perturbative allowing for multiple scattering of this third particle with any of the components of the cluster. The cluster is then assumed to have the wave function that it would have without the presence of the third particle. Intuitively, one should expect it to work when the third particle is light compared to those of the cluster, or when it is very bound such that it cannot allow excitations of the cluster. There is substantial work to rely on this model. For instance, the three body scattering amplitude Xie:2010ig () was calculated using the FCA to the Faddeev equations and the results of this work are in good agreement with the other theoretical works Jido:2008kp (); MartinezTorres:2008kh () evaluated using variational and Faddeev approaches, respectively. Besides, in Xie:2011uw (), using the same model, the authors give a plausible explanation for the puzzle. As important as to know the success of the FCA, it is to know the limitation of this procedure, and it was found in MartinezTorres:2010ax () that the approximation collapses for the study of resonant states, where there is plenty of phase space for the excitation of intermediate states.
Meson nucleus bound systems are good laboratories to investigate the finite density QCD. The existence of the bound states in nuclear systems, which are called kaonic nuclei, is theoretically expected because of the strong attractive interactions between the and the nucleon. Models range from empirical ones with a depth of around 200 MeV at normal nuclear matter density Friedman:1999rh (), to those imposing chiral symmetry constraints that lead to an attraction of around 40-50 MeV Koch:1994mj (); Lutz:1997wt (); Ramos:1999ku (); SchaffnerBielich:1999cp (); Cieply:2001yg (). However there is no experimental evidence on this system. In spite of experimental claims, which have been proved to be unfounded (see Ref. Ramos:2008zza () for a review on the issue). The large width of the predicted states compared to the binding could justify why such states are not being found Hirenzaki:2000da (). The is the prototype of the nuclei and to observe the kaonic nuclei in experiments, the precise knowledge of the width of this state is important. In this sense, it is important to recall that in all previous works of the system the source of the width is only the conversion channel. The absorption channels are not considered explicitly, although in some case it is estimated perturbatively Dote:2008hw (). Early experiments on -nucleus absorption at rest were done in the 1970s. A precise measurement of the total two nucleon absorption of stopped mesons in deuterium was reported in Veirs:1970fs () using deuterium bubble chambers. This is the first detailed measurement of two nucleon absorption branching ratio and these results were compared with the predictions of isospin invariance for strong interaction. The ratio of the rates and was in good agreement with the results of isospin invariance for strong interaction. A Similar experiment was done using a helium bubble chamber Katz:1970ng ().
absorption by two nucleons in nuclei, , has been considered before in the selfconsistent evaluation of the nucleon optical potential of Ramos:1999ku (); Oset:2000eg (). More recently it has also been evaluated in a different way in Sekihara:2012wj (). The nucleus optical potential evaluated in Ramos:1999ku () has been checked versus kaonic atoms in Hirenzaki:2000da () with good agreement with experiment. The discrepancies seen for the atom have been revolved recently with very precise measurement in Okada:2007ky (); Bazzi:2009zz () which agree with the early predictions of Hirenzaki:2000da (). Also the study done in Baca:2000ic () showed that a best fit to the data could be obtained with a potential that differed from the predicted one of Ramos:1999ku () at the level of 20 . This agreement with data gives us confidence that the input used in Ramos:1999ku () for the absorption is realistic and we use this input here to evaluate the absorption width of the state.
The article is organized as follows. In Section II, the calculation of the three body amplitude including the charge exchange mechanisms is summarized using the Faddeev equations under the FCA. In Section III, The explicit derivation of the absorption by two nucleons is given. The results of the absorption both for spin-0 and spin-1 are shown in Section IV.
Ii Calculation of the amplitude
Here we will summarize the derivation of the three body amplitude within the framework of the FCA to the Faddeev equations taking into account the charge exchange contributions. In order to investigate the three-body amplitude of the system, we have two possible spin states, and . Let us first concentrate on the calculation of the case which is done in detail in bayarnpa (). At the end we want total isospin- for three-body amplitude. The wave function for this state is
with the basis of . For the total amplitude with total isospin- we have
First, we start from the amplitude where a interacts with either of the two protons and the diagrammatic representation of this amplitude is shown in Fig. 1, where the shaded ellipse includes all multiple scattering of the where the interacts first with the first nucleon. We call the result of this diagram and hence the total amplitude including the diagrams where the interacts first with the second nucleon is . Looking in detail at the ellipse in Fig. 1 we have three partition functions (see Fig. 2) that fulfill the following coupled channel equations
with the factor standing for the form factor of the bound cluster. The diagrammatic representation of the partition functions is shown in Fig. 2. Calculating the three equations in Eq. (3) we get as below
Now we need to evaluate the second term of Eq. (2). This term does not have charge exchange and its diagrammatic representation is shown in Fig. 3. Taking into account the equivalent diagram where interacts first with the second nucleon we obtain
Using Eq. (2), the final result of the amplitude for case is
We are going to formulate the absorption by two nucleons. Take for instance absorption by a pair of nucleons. The corresponding diagrams for this process are given by Fig 4
The S-matrix elements for these diagrams are given as follows:
where is the meson baryon Yukawa vertex, stands for the spin- octet baryons, represents the spin- decuplet baryons, is equal to for the vertex and is equal to for the vertex, where is the spin transition operator from spin- to spin-. Here is the wave function in momentum space of the pair of initials nucleons and and are the initial and final momenta, respectively. The momentum stands for that of the produced hyperon.
Taking the initial system at rest, we write the propagator as below
where with , with the mass of the system and , where we have subtracted 1/3 of binding energy, (), to this single nucleon. Since the values of are around 550 MeV/c, neglecting the Fermi motion of the initial nucleons induces errors of the order of in Eq. (10), which is about 1 .
After redefinition of , the square of the total matrix element summed over the spins of the , or , and averaged over the spin of the nucleons is obtained as
where multirho (); Bayar:2011qj (), with and . The formula for is obtained assuming that is the same for the two nucleons, such that . This neglects recoil effects on the nucleon or the kaons. Effects from this corrections were estimated in bayarnpa () and affected mildly the binding energy but not the width, and we assume it to be the case for the absorption width too.
Using the coefficients in Table I and Eq. (14) in Ref. Oset:2000eg (), we have for the coefficients of the Yukawa vertex
with and . Similarly, for the vertex we have
The coefficient is given in Table 2 in Ref. Oset:2000eg () and the coupling is given by,
Thus, the coefficients for the Yukawa vertices are
It is useful to relate the matrix with the cross section. Using as the matrix for the transition from and the cluster to nucleon hyperon , we calculate the cross section as
It is interesting to relate this cross section with the imaginary part of the amplitude for the diagrams of Fig. 5. Borrowing the optical theorem we rewrite the cross section in terms of the imaginary part of the amplitude, , as follow:
In analogy to bayardnn (), we can convert the absorption diagram of Fig. 5 into the ”many-body” diagram of Fig. 6, where the nucleon on which the virtual is absorbed is converted into a ”hole” line. The purpose of following this path is that we can now include the absorption process by making a modification of the meson baryon loop function, , to include the ”ph” excitation in the propagator. Then, we can reevaluate the amplitude in the coupled channels nonperturbative Bethe Salpeter equations and use the new amplitude in the amplitudes of Eqs. (7) and (8) which sum the multiple scattering series of the system. In this way we perform fully nonperturbatively the implementation of the absorption in the system.
Since we are only concerned about the absorption process it suffices to evaluate , but later on we shall also look at the real part. This is easy since is the same as removing the modulus squared of the amplitude in Eq. (11). This amplitude, needed for the physical process, is regained when we evaluate the amplitudes in the chiral unitary approach substituting by (See Ref. bayardnn () for details). Thus we can immediately write
where is the matrix defined before, specifying the final hyperon, removing the modulus squared of . The sum for 1-3 of runs over for type and for type , and the sum 1-2 of runs over of type and of type (see Fig. 6).
where , , are given in Eqs. (12) and (15), respectively. Note that when removing the amplitude in the integral of Eq. (11), the remaining integral simply gives the wave function in coordinate space in the origin, .
For we take the same form of wave function as in bayardnn () for the cluster
but with parameters suited to the wave function in the cluster. For this, the parameter is the chosen such as to get the cluster with a relative distance of around 2 fm as found in Dote:2008in (); Dote:2008hw () for the present problem. We accomplish this with fm. The momentum of the baryons are given by
and in order to account for relativistic corrections, accompanied by is given by
Since we get different contributions for in the or intermediate states, we are now forced to recalculate the amplitudes using the charge basis of the coupled channels, rather than the isospin base normally used.
Furthermore, as seen in Eqs. (3), we also need to calculate the amplitude. This is a channel that has only one Feynman diagram as shown in Fig. 7. In order to calculate the amplitude we use the same potential with Oset:1997it (). The channel is renormalized in the same way as the , as one can see, comparing Fig. 7 (a) with Fig. 5 (b). Hence, one can use for the channel .
We have been concerned about the consideration of the absorption channel in the width of the system. With a bit more of effort we can equally calculate the effects of the real part associated to the absorption (dispersive part). For this it is sufficient to substitute of Eqs. (17), (18) by given by
where is the energy of the initial and the first nucleon of the system and is the energy of the second nucleon. We calculate them splitting equally the binding energy B between the three hadrons then
The integration in Eq.(23) can be performed analytically and we find
We are also interested in learning the effects of making the nonperturbative approach that we have done compared to one typical perturbative approach. We can make use of the same formalism that we have followed using an easy algorithm: The perturbative calculation contains only one bubble of nucleon hole-. We can regain the one bubble results by substituting with a parameter running from zero to 1. We then take the tangent of the curve in at the origin of . The result of this linear extrapolation at gives us the perturbative results for the amplitude.
As we stated in the Introduction, the plays a key role in the absorption. As mentioned before, we recalculate the amplitudes in the charge base. In the case of the and , there are ten coupled channels which are Oset:1997it (). As discussed above, we also need the amplitude in pure . For this we take the component and use the six coupled channels, Oset:1997it (). In the coupled channels reevaluation of the amplitudes for we add , to , , respectively. For we add to , but for isospin symmetry reasons which we have discussed above. In all the channels we use the same numeric value for , (=93 MeV) as in Oset:1997it (). For the form factor of Eq. (4) we use the deuteron type form factor with the reduced size of the two N system found in Dote:2008hw ().
As discussed in the former section we also perform the calculations using the full , that contains the real part associated to the absorption mechanism. In addition we also evaluate the results in the perturbative case for the absorption mechanism.
In Figs. 8 and 9 we show the absolute squared of the matrix for scattering on the cluster , including absorption, for the cases of and . As we can see from Fig. 8, the most striking thing is a substantial increase in the width, that goes from about 50 MeV to about 80 MeV, and which is due to absorption. The centroid of the distribution is also a bit displaced to lower energies, but what concerns us now is that the absorption on two nucleons has increased the width by about 30 MeV. The order of magnitude is similar to what was estimated in Dote:2008in (); Dote:2008hw (), but here we have done a full nonperturbative evaluation of this magnitude. In the case of in Fig. 9 we observe that the shape of the distribution has been distorted considerably due to the consideration of the absorption and the centroid has not changed appreciably. From this distorted shape one can also estimate that the width has increased in about 30 MeV, like in the previous case. The width of the state is estimated from the width of the curve at half the value of its maximum, even if the shape is not that of a Breit Wigner distribution.
We should mention that we have also included the effect of the intermediate states in the calculation. Here also the amplitudes have been modified due to absorption, since they come from the coupled channels calculation. The contribution of this channel has been evaluated as in Bayar:2011qj () and its effect is small, like also found there.
In the figures we can see the effect of considering the real part of . We observe that the inclusion of the real part does not practically modify the results. One appreciates a small increase in the width by about 5 MeV for the case of . We also show in the figures the result obtained in the nonperturbative calculation. This has been done as discussed in the former section but just in the case of taking only , which we found to be the relevant part. For , we can see that, with respect to the nonperturbative calculation the strength at the peak in is reduced by about 20 and the width is increased by about 5 MeV.
The results for are similar, the only difference being that when the full is considered the width decreases by about 5 MeV in this case and a small shift on the peak to bigger energies by the same amount is produced.
We have made a detailed calculation of the contribution to the width of the bound states from absorption on two nucleons. The evaluation is done nonperturbatively in two aspects: First, the amplitudes are reevaluated in the unitary coupled channels approach taking into account the absorption of the . Second the resulting amplitudes are used in the nonperturbative formula of the Fixed Center Approximation that takes into account the rescattering of the kaons on the nucleons of the cluster.
The result of these calculations is that the width of the states with , is increased by about 30 MeV to values of the total width of 75-80 MeV. The relative simplicity of the formalism has allowed us to address some issues for the first time, as the nonperturbative evaluation of the absorption width and the effects of the real (dispersive) part associated to absorption.
With the large width obtained and the small values of the binding, 15-30 MeV, to which the different groups are converging Dote:2008hw (); Ikeda:2010tk (); Barnea:2012qa (), we are facing a situation of states with much larger width than binding, which makes the experimental observation problematic. Further calculations taking advantage of the steps given in the present paper, but using different formalisms, would be most welcome.
We thank A. Gal and T. Hyodo for useful comments. This work is partly supported by DGICYT contract number FIS2011-28853-C02-01, the Generalitat Valenciana in the program Prometeo, 2009/090. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity Study of Strongly Interacting Matter (acronym HadronPhysics3, Grant Agreement n. 283286) under the Seventh Framework Programme of EU.
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