Description of nucléon excitations as decaying states *
Bojan Golli**
Faculty of Education, University of Ljubljana, and J. Stefan Institute, Ljubljana, Slovenia
Abstract. Two methods to describe excited states of baryons as decaying states are presented: the Analytic Continuation in Coupling Constant and the Kohn variational principle for the if-matrix. The methods are applied to a simple model of the zi resonance consisting of the pion coupled to three valence quarks.
The work has been done in collaboration with Vladimir Kukulin and Simon Sirca. 1 Motivation
Baryons are usually computed as bound states neglecting possible decay channels. The inclusion of strongly decaying channels may considerably influence the position of the state as well as some other properties. The aim of the present work is to estimate this effect in a simplified model and to discuss two possible approaches to describe decaying states. The methods determine the position and the width of the resonance, and furthermore, provide a suitable tool to calculate new observables, which cannot be obtained in a bound state calculation, such as non-resonant contributions to production amplitudes. In this work we shall focus on the decay of the A resonance.
2 The model
The decay of the A resonance into the nucléon and the pion is most naturally described in models with chiral symmetry, such as the linear a model (LSM), the chromodielectric model (CDM), the cloudy bag model CBM, etc. Here we use a simplified model which contains the main features of these models. It assumes frozen quark profiles and neglects meson-self interaction. Furthermore, it does not take into account additional scalar fields (sigma mesons in the LSM, chromodielectric field and sigma mesons in the CDM, or the bag potential in the CBM) since their main role is to fix the quark profiles and generate a constant energy shift for all baryons. In the present calculation, the quarks profiles are taken over from the ground state calculation in the LSM[1], We know that the profiles do not change considerably from one model to the other, so this is not a very severe restriction. The inclusion of meson self-interaction may, however, more importantly alter the results.
For the quark-pion interaction we assume the usual pseudoscalar form:
#quark-meson = ¡.'/ j dr3ÇT • TTJaq .	(1)
In models with spontaneous symmetry breaking, such as the LSM, the parameter g is related to the 'constituent' quark mass by Mq = gf%. From 350 MeV< Mq <450 MeV we estimate that physically sensible values for g are 4 < g < 5.
The model is usually solved at the mean field level. We interpret the solution as a coherent state of pions around the three quark core, and generate physical N and A states by the Peierls Yoccoz projection of good spin and isospin. The resulting states are interpreted as a superposition
Talk delivered at the workshop on Few-Quark Problems in Bled, Slovenia, July 8-16, 2000.
E-mail: Bojan.Golli@ijs.si
2	B. Golli
of 3 bare quarks plus 3 quarks with one or more pions coupled, respectively, to nucleón or A quantum numbers:
\$N) = pJ=hT=h |#) = (3 q)N + [{:>»i)xk\J ?:r ? +	+ [(3q)Nirir]?-i +... (2)
\$A) = PJ=l'T=l |#) = (3q)A + [(■■>></) x n\J ú + [(3^]?'? +	+ • • • (3)
In the A channel, the probability of finding one or more pions is higher than in the N channel; as a consequence the A lies higher then the nucleón. In the simplified model we obtain EA — Ejv = 84 MeV and 126 MeV for g = 4.3 and 5 respectively; including meson self interaction and performing self-consistent calculation increases the splitting by some 40 MeV. Hence, the AN splitting due to pions is only roughly one half of the experimental one; an additional hyperfine interaction is needed to bring EA — Ejv to the experimental value (293 MeV). In our simple model we therefore introduce a phenomenological form of the interaction:
H' = eP(3qU	(4)
where P(3q)A is the projector onto components containing 3 quarks coupled to A quantum numbers. Using e = 262 MeV and 235 MeV for g = 4.3 and 5 respectively, increases the splitting to the desired value.
3 The Kohn variational principle for the phase shift
The ansatz for the A resonance is taken in the form
\VA) = c\$A) + J d kV{ko,k) [alt(k)WN)]J=i,T=i
where aj„t(fc) creates a p-wave pion, m,t are the third components of its spin and isospin, ko denotes the pion momentum, while |#jv) and \$a) correspond to the nucleon and the A bound states ((2) and (3)), respectively. Asymptotically, the pion state behaves as
t](ko,r) = k0ji(k0r) - tan S k0 yi(k0r) , r ^ oo .
Here we use standing waves to describe the pion rather than outgoing (and incoming) waves. In fc-space this leads to
V(k0, k) = ,f^8(k - k0) + ^^ , tan5 = ^hx ^ x(h, kQ) V 2	ujk — wo	k0
The variational principle requires that the Kohn functional[2]
'«-^'-mk)^-'**)
remains stationary with respect to variation of c and x{ko-,k), as well as to variation of the intrinsic pion profile in
In the above form only one channel is assumed; if more than one channel is open, tan S is replaced by the K matrix.
Typical results for the phase shift are displayed in Fig. 1 and compared to the experimental values. By varying e it is possible to reproduce the experimental position of the resonance; using g = 4.3 (e = 273 MeV) the width (i.e. the slope of the curve) is well reproduced while for g = 5 (e = 252 MeV) the width is too large. These results are obtain by optimizing	if we do
not vary the intrinsic pion profile but take it over from the bound state calculation the results change only very slightly provided the value of e is changed by a few MeV. Hence, the properties of the A do not change significantly when the decay channel is open; the main effect is that the energy drops by some 10 MeV (10 MeV for g = 4.3 and 13 MeV for <7 = 5).
Description of nucleón excitations as decaying states
3.0
$33
2.5 2.0 1.5 1.0 0.5 0.0
1 1 1 1 1 1 1	1 1 | 1 1 1 1 1	1 1 1	1 1 1 1 1 1 1 1 1 1 • -
-			0 • -
			*
_		0°	* .
_	_ 0	*	-
	•		
	0		
—	*		—
-	0		-
_	0		_
_	0		_
	0		
	0		
	i		
	0		
-	0		-
-	0		-
- t>			■
- •			—
. 0			-
" 0			-
1 .00* rtm .......	III.....	1 1 1	1.........
	2		3 .
			oj0/mw
Fig. 1. The phase shift in the P33 channel: o are the experimental values, • values from the variational calculation using y = 4.3 and e = 273 MeV, and * those for y = 5 and e = 253 MeV.
4 The Analytic Continuation in Coupling Constant
Consider the scattering of a non-relativistic particle on an attractive potential V(r) which possesses a quasi bound state in the continuum. Introduce a parameter (coupling constant) A:
H = Hkin + XV(r) .
For sufficiently large A, A > 1 the state becomes bound. Let's denote the threshold value as Ath-The method [3] is based on the fact that it much easier to solve the bound state problem than the continuum case. It consists of the following steps:
•	Determine Ath and calculate E as a function of A for A > Ath •
•	Introduce a variable x = — Ath! calculate k(x) = \y/—2mE in the bound state region.
•	Fit k(x) by a polynomial:
k(x) = i(co + C\X + C2X2 Construct a Padé approximant:
ciq + a\x +
C2MX
2 M\
k(x) = i ■
• a,mx
M
1 + b\x
bMXM
(5)
Analytically continue k(x) to the region A < Ath (i.e. to imaginary x) where k(x) becomes complex.
Determine the position and the width of the resonance as analytic continuation in A:
1 , Er = -— Re contai« 2m
1 9 r = ^2 -— Im contA^ifc . 2m
(6)
This method does not provide only the position and the width of the resonance; the matrix element of an operator O between the resonant state \\Pr) and a bound state |#) can be calculated as
<<Fr|0|#) = contA^i<<F6(A)|C>|#> .
4 B. Golli
In our implementation of the method, we relate the coupling constant A to the parameter of the phenomenological hyperfine interaction:
XV(r) eP(3qU , x = y/eth - e	(7)
where eth is the value of e at the threshold: -Ev^eyJ — En = m,. For sufficiently high e, the real part of the energy eventually reaches the experimental position of the resonance; this value of e then corresponds to A = 1 of the original formulation of the method.
In our very preliminary calculation we treat the pion non-relativistically. For e < eth we calculate
k{x) = iy/2m«{Eth-E), E = EA(x)-EN,
fit k(x) using a Pade approximant (5) and continue k(x) to the resonance region. The energy difference, E& — En, and the width of the resonance are then obtained by (6). The 'physical value' of x (and e from (7)) is determined as ReE(= Ea> — En) reaches the experimental value 293 MeV. The corresponding value of lmE(= F) then predicts the width of A and is to be compared with the experimental value ~ 120 MeV.
Fig. 2 shows the behaviour of Ea> — En and F as functions of x for two vales of g. For higher order of the Pade approximant, M > 3, the method becomes numerically instable and the determination of E and F is no more reliable. For g = 4.3 and M = 1 and 2, the experimental splitting is reached for x2 « 230 MeV (and corresponding e = 300 MeV). This yields F « 60 MeV which is only half of the experimental value, most probably due to the non-relativistic treatment. For g = 5 the value of F is larger (in accordance with Fig. 1) but its determination is less reliable.
In order to be able predict reliable results it is necessary formulate the approach relativisti-cally and to understand the origin of numerical instabilities for higher M.
(a)	(b)
Fig. 2. AN splitting and A width (in MeV) as functions of x (in units -\/MeV) for y = 4.3 (a), and y = 5 (b).
References
1.	B. Golli and M. Rosina, Phys. Lett. B 165 (1985) 347; M. C. Birse, Phys. Rev. D 33 (1986) 1934.
2.	B. Golli, M. Rosina, J. da Providencia, Nucl. Phys. A436 (1985) 733
3.	V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 69A (1978) 251, V.M. Krasnopolsky and V.I.Kukulin, Phys. Lett, 96B (1980) 4, N. Tanaka et al. Phys. Rev. C59 (1999) 1391