ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 4, с. 538-546
ЯДРА
CAN THE Л(1405) RESONANCE BE SEEN IN NEUTRON SPECTRA
FROM THE K- + d REACTION?
©2014 J. Revai*
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia; Wigner Research Center for Physics, Budapest, Hungary Received January 29, 2013
Neutron spectra from the low-energy reaction K- + d ^ n + T + n were calculated using coupledchannel Faddeev equations. The possibility of tracing the signature of the Л(1405) resonance in the spectra was investigated for four different phenomenological models describing KN—nT interactions. We found, that while in the direct spectra kinematic effects mask completely the peak corresponding to the resonance, the deviation spectrum method [J. Esmaili et al., Phys. Rev. C 83, 055207 (2011)] is able to eliminate kinematics and differentiate between different models of Л(1405).
DOI: 10.7868/80044002714040126
Работа посвящается моему старому другу и верному соратнику на поприще физики малочастичных систем Володе Беляеву по случаю его славного 80-летнего юбилея
1. INTRODUCTION
The A(1405) resonance plays a central role in low-energy antikaon—nuclear physics. Since it is assumed to be a manifestation of attraction between negative antikaons and nucleons, its observability and properties are crucial for the possible existence of antikaonic nuclear clusters. The origin and structure of the resonance, its one- or two-pole nature together with positions and widths of the poles are subject to vivid discussions among the representatives of different opinions (mainly chiral perturbation theory versus phenomenology). For a review of the "state of art" see, e.g., [1,2] and references therein. From experimental point of view, the clarification of this problem is hindered by the fact, that this resonant state cannot be reached in two-body reactions involving stable particles, but in processes having three or more particles in final states only. The simplest of these is the reaction
y K- + d or K- + p + n, K- + d ^ (n + X)/=o,i + n, (1)
X (n + E)=i + p,
E-mail: revai.janos@wigner.mta.hu
which has the advantage that its dynamics can be treated exactly in the framework of the coupled particle-channels Faddeev approach. The formalism was applied for KNN system for calculation of three-body quasi-bound states [3, 4] or K-d scattering length [5—7]. The break-up reaction (1) was studied in the early papers [8, 9] (together with the K- d scattering length), but the main emphasis was put on the possible signal of a resonance in the AN—EN subsystem close to the EN threshold. Probably, at that time the A(1405) topic was not as hot as nowadays.
The only available experimental data on neutron spectra from reaction (1) are those of Tan [10] from a bubble chamber experiment. What he gets is a large peak near the origin with no distinguishable structure in it. It will be seen later that the peak corresponds to our "direct" spectrum P(En).
Since Faddeev equations are, strictly speaking, valid in the framework of non-relativistic quantum mechanics, our calculation is done for the center-of-mass energy range E^- = 0—50 MeV, having in mind stopped or slowed down antikaons. The recent papers [11, 12] are devoted to the possible observation of the A(1405) in reaction (1) for much higher energies. The dynamics is treated there in single-plus-double-scattering approximation, which might be justified in the case of high-energy an-tikaons. However, as it was shown in [2] and [13], the authors of [11, 12] make unjustified approximations and assumptions, which lead them to incorrect, too optimistic conclusions about the observability of the A(1405) in the considered reaction. As for [14],
there the same method was applied for much lower energies, similar to ours, whithout any justification.
The paper is organized as follows: in Section 2 we give a brief description of the applied formalism, Section 3 contains the details and description of the input of the calculation. Our results and their discussion are presented in Section 4, while our conclusions are done in Section 5.
2. FORMULATION OF THE PROBLEM
The calculation is based on the coupled-channels Faddeev-type AGS treatment of the KNN-nZN system. The approach has been already described in detail in several papers [3—5], here we briefly recall it in order to introduce the notations. The operator AG S equations for the transition operators Uij read
Uij = (1 - öij
-»-I r0
+ £ TsGoU
s = i
sj
(2)
where i,j,s = 1,2,3 are the usual pair—spectator indices,
Tj = Vj + Vj GoTj,
are the two-particle T operators and G0(z) = (z — — H0)-1 is the free Green operator.
The configuration space jx^^), in which the operators act, contains apart from the usual Jacobi momentum variables xj, yi a discrete index vi = = (a, ai). It is a combination of the particle composition index a
a = {1, 2, 3} = {KNN2, nXiN2,nNi£2}
and an isospin label ai = (IiI) corresponding to the isospin Ii of the particle pair i and the total isospin I :
Oi
[tj tk ] ti
This choice of the isospin labels corresponds to the "isospin representation", which is useful when isospin-conserving pair interactions are used. Another possibility is the equivalent "charge state" or "particle" representation, characterized by the third component of the particle isospins:
Ö0 ~ {tlz ,t2z ,t3z }■
Here we use i = 0 for the "particle" representation instead of a Faddeev index. For the channel a = 1, for example, the possible values of a0 are:
ao ~ {K°UiU2, K-PlU2, K-ni,p2}■
"Isospin" and "particle" representations can be transformed into each other with the help of orthogonal matrices
Щ = (aoi\aoj >
composed from 6j symbols for i,j = 1,2,3 and from Clebsch—Gordan coefficients for i = 0, j = 1,2,3 indices. Using isospin conserving separable interactions of the form
^ = £ ^ 5II' Krt (ti I (4)
' 6 4 z z
Viv[
the Ti(z) operators can be written as
Ti(z) = £ \gVi)rVi<(z)(gViI (5)
Vivi
with rv.v>, (z) being the usual (c-number) matrix, defined as:
(t(z))-V = (^Ii -(g&IGoWgj). (6)
i i i i i
Due to isospin conservation of our interactions, the coupling constant matrix A is diagonal in (IiI), but it has non-diagonal elements in the particle labels (aa'), responsible for the change of identity of the particles (KN ^ n£). As for the matrix elements of Go, they do not change particle identities, thus Go is diagonal in a. If we take averaged masses for particles within an isospin multiplet, it is also diagonal in pair-and total isospin indices (IiI). However, if physical (unequal) masses are used, Go will be diagonal only in the "particle" representation, while in the "isospin" one it will acquire non-diagonal elements both in Ii and I, proportional to the mass differences.
Equations (2) for the transition operators take the form with j = 1
Un = (1 - 6i1)G- + £ £ \gVs )TVsK (Xsv, \ (7)
s = i v's
with (Xt, \ = (gV |GoUsl.
Introducing the functions Xj (yj) = (Xj\$0), where yj is the momentum of the spectator particle corresponding to the pair j and \$0) = PK) is the initial state with the deuteron wave function ) and the momentum of the incident kaon PK, we get the set of integral equations:
Кг(yi) = (l - öii)(gii\Фо> +
(8)
+ £ £ J ZviVj (yi, yj)TVjv'. (Z - yj/2ßj,ki) j = ivj vj
X Xj (yj)dyj
with
ZviVj (yi, yj ) = (дЬг \ Go(z) \ gj >.
(9)
(3)
The size of the system (9) can be reduced by introducing symmetric or antisymmetric combinations of
i
x
X functions with respect to interchange of baryon numbering. The baryon spins do not enter the formalism explicitly, therefore the total baryon spin S remains unchanged in the process (it is a conserved quantum number in the case, when the third particle is a pseudoscalar meson). For a given S value the total antisymmetry, required by the Pauli principle, has to be ensured by the space—isospin part. Thus for spin S = 0 (K-pp system) we have to work with the symmetric combinations of X's, while for S = 1 (our K-d system) the antisymmetric combinations are needed. As a result, the labelling vi = (aai) is changed to ¡ia = (a,aa), where a denotes a pair of interacting particles irrespectively to the original particle channel and aa denotes the corresponding isospin value. Thus, we are left with Xaaa (ya) with a taking the values KN, NN, EN, and nE (nN is missing since we neglected the nN interaction, see next section).
The break-up transition operator U01 can be expressed in terms of the Ui1s as:
Uoi = ^ (C/n + U2l + Usi)
and the break-up amplitude reads
^bu = <$/|Uoi |$o>- (10)
For the reaction under consideration the properly antisymmetrized final state is
|$f > = Ixnx, yN; =
= -yldxTrS^yAia^TrSi) - |x7rS2)yiVi;0'7rS2))-
The break-up amplitude can be expressed in terms of the X functions as
, YN ; = -9nS (XnS ) X
(ii:
x [TnS,KN (z - V2N/2^N,ns)XKN (yN) + + TnS,ns(z - y"N)Xns(yN)] -
- #31 g^N (uyN + VXns) X x TZN,SN (z - |xn£ - WyN\2/2^,SN) X
X X^N (XnS - WyN ),
where BN1 is an isospin recoupling matrix (Eq. (3)), u, v, and w are mass coefficients of the transformation between the Jacobi momentum sets. In Eq. (11) we omitted the isospin labels, thus the quantities are vectors or matrices in isospin space. The on-shell amplitude for a given neutron energy En depends on t and the isospin labels
A(En ,t,anz ) = Aeu(xns, Yn ; ),
IxttsI = \/2(EnxN - En)/x7rS; t = cos(x7rS,yN).
The physically observable final state corresponds to a certain particle composition, not to a definite isospin state. Therefore the amplitude has to be transformed into the a0 representation, using the suitable B matrix of Eq. (3):
A(En, t, aa) = J2 (Bo23)CT0;^s A(En, t, a^),
where a0 can be equal to {n+X-n, n°TPu, n}. The neutron spectrum is proportional to the differential cross section
P(E t )
d^xnE düyN dEn
= (27T)4/X7rs/XAf,7rS№,AfAf 7r^N \A(En, t, <70)|2,
Pk
while the inclusive neutron spectrum (no other particles are detected) is given by
i
P(En) = j dtP(En,t,ao)
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