static hyperon properties in a linearizedsu(3)-chiral bag model

Z. Phys. A - Atomic Nuclei 331, 489-498 (1988) Zeitschrift fQr PhysikA

Atomic Nuclei �9 Springer-Verlag 1988

Static Hyperon Properties in a Linearized SU(3)-Chiral Bag Model*

S. Klimt and W. Weise

Institut fiir Theoretische Physik, Universitfit Regensburg, Federal Republic of Germany

Received July 12, 1988

We use a linearized Chiral Bag model to describe the strange octet and decuplet baryons. The approach is canonically extended to spontaneously broken chiral SU(3)L x SU(3)R, and the corresponding Goldstone Bosons are identified with the pseudoscalar meson octet. We include explicit symmetry breaking corrections both for baryons and mesons. The linearized quark-meson intraction is applied in a self-consistent calculation of the masses and, for A, ~* and ~*, of the decay widths. Our special interest is in the influence of the K- and q-cloud (in addition to the r~) on hyperon static properties. We show results for radii, masses, decay widths and renormalization constants as obtained by a fit to the experimental hyperon spectra. The effects of the K- and q-mesons are found to be non-negligible, although supressed by symmetry breaking effects. The effective gluon coupling e is reduced in comparison to the SU(2)L x SU(2)R case. In addition, we discuss the dependence on the bag constant B. It turns out that the lightest hyperon states, A and Z are well described and stable for Bl/4< 130 MeV. The heavier strange baryons have stable solutions also for larger values of B. The bag radii determined at the minimal energies are Ro-~l.15 fm for the octet and Ro~-1.25 fm for the decuplet baryons.

PACS: 12.35; 14.20

I. Introduction

Basic QCD concepts such as quark confinement and asymptotic freedom have been modeled with some success in terms of the MIT bag [1, 23. However, chiral invariance, which is an exact symmetry of QCD with massless quarks, is badly broken by the boundary conditions of the original bag model. The chiral bag with underlying SU(2)L x SU(2)R symmetry E3, 4] overcomes this problem by coupling the confined quarks to the Goldstone pions at the bag boundary. This pion-quark coupling is such that, in the limit of massless u- and d-quarks, the axial current is con- served everywhere in space, including the boundary. A small explicit breaking of chiral symmetry by non- zero quark masses and a non-zero pion mass, m~ ~-140 MeV, are then introduced in accordance with PCAC.

* Work supported in part by BMFT, grant MEP 0234 REA

As one focuses on the physics with strangeness, it is tempting to extend the foregoing formalism to chiral SU(3)L x SU(3)R. However, the s-quark is fairly massive; a strange quark mass m~-250 MeV is typi- cally taken in bag models [5] in order to arrive at the appropriate splitting of baryon ground state masses. Hence the inclusion of strangeness produces substantial deviation from the basic symmetry. This is also evident from the fact that the lightest mesons with strange quark content, the Kaon, K(495) and the ~/(549) have fairly large masses as compared to the pion.

Our intention is to examine the consequences of the inclusion of the K-(and, less importantly the ~/) cloud into a chiral bag model description of the baryons which form the JP=21+-octet and JP =2 +-decuplet representations (L = 0, 56 of SU(6)fl . . . . . . . pin)" Similar approaches have already been undertaken previously (see e.g. [6-8]), but our present treatment goes considerably further in the

490 S. Klimt and W. Weise: Static Hyperon Properties

self-consistent evaluation of quantities such as the decay widths of baryon resonances. Our primary interest is in the hyperons and their resonances. We shall demonstrate that their sizes converge at typical bag radii R ~ 1 fin, unlike the nucleon and the A-isobar which have no strange valence quarks. Such large radii justify a posteriori a linearized coupling of the quarks and pseudoscalar meson fields at the bag boundary. This is not the case for N and A for which only a complete chiral bag description including the full non-linearity of the chiral meson field and its coupling to quarks at the bag surface gives meaningful stable results [9, 10].

This paper is organized as follows: in Sect. 2 we summarize the linearized Chiral Bag model and its extension to SU(3) which is our main concern. In Sect. 3 we present the results Of our self-consistent calculations of hyperon masses, followed by a discus- sion and conclusions in Sect. 4.

2. The Model

We start from the Cloudy Bag version [3] of the linearized Chiral bag model, with underlying chiral SU(3)LX SU(3)R and pseudoscalar meson fields 4~, a = 1 . . . . ,8 (the n, K , / ( and t/fields) coupled to quarks at the bag boundary. The basic assumption is that the meson fields qS, are sufficiently weak so that non- linear effects can be ignored. The corresponding La- grangian is:

= ~MIT 7t- ~q~ "j- ~Pint,

~MIT = [ - ~ ( i ~ - - m) T - B ] . O(R-r) �89 ~T.g)(R-r),

1 s

,W~t=~ (-- Cgivs 2~ ~P dp~/f).b(R--r). (1) a = l

The MIT bag Lagrangian ~'PMIT is characterized by the bag constant B, and 7 j refers to the quark fields inside the bag. The quarks appear in three (u, d, s) flavour components, and their mass matrix is rh = diag (m,, rod, m~) with current quark masses m,, m e and rn~ respectively. The step function 6) ( R - r) defines the region of confinement parametrized by the bag radius R. The b-function term in 2PM~T makes sure that the color flux at the bag boundary r = R vanishes. Furthermore ~L~~ describes the free SU(3) chiral meson fields ~b, (a= 1, . . . , 8) with masses m,. Finally, ~-~i.t introduces the quark-meson couplings at the boundary to first order in ~b~/f, where f is the meson

decay constant to be specified later, and 2, denotes the set of eight standard SU(3)-(Gell-Mann)-matrices.

The quark fields 7 j are expressed in terms of MIT bag eigenstates with wave functions q,(r):

T(r, t ) = ~ [q,,(r) b, e - i " t + ...3. (2a) n

We consider here the octet ground state baryons (N, A, Z, E) and the decuplet (A, Z*, E*, O) with their 0s valence quark content. The corresponding MIT bag wave functions are [53

q ( r ) = ~ ( k - i j ~ (2b)

~ - ~ [a" rJl (kr/R)]

with spherical Bessel functions J0 and Jl. For a quark of mass m, we have ~=mR and k 2 + ~ 2 = O 2 where g2 is determined by the boundary condition

1 - ~ - O = k. cot (k). (3)

The normalization constant N~ is given by

N ~ = ~ 4 ~ JR3 (292((2 - 1)+ 4)]- t/2. (4)

The Hamiltonian of the system as derived from the Lagrangian (1) is [6]"

H = H0 + Hint, H 0 = H u i T -[- H e . (5)

Here Ho incorporates the free MIT bag Hamiltonian in a form which projects into the space {IBo> = B + 10>} of colorless baryons in the _8 and 1_9_0 representations of SU(3):

HMIT = E M~ (6)

where M ~ are the bare baryon masses as given in the MIT model. The free meson Hamiltonian

H e = ~ ~ d 3 q c~q a+q aM, q (7) Me8

includes the pseudoscalar meson octet (n, K, K, ~) with O)q = i/q2 + m~, where m u are the meson masses.

(Here a meson state is ]M(q)>=a~,q]0> with + [aM, q, aM, q,]=b3(q--q') .) The interaction part of H

is given by [6]

Hint = ~ ~d3q(UoJv(q)[n'o,M(q))U+B'aM, q B,B'~56

M e 8

+ herm. conj. (8)

S. Klimt and W. Weise: Static Hyperon Properties 491

where the vertex form factor (Bo] v(q)IB;, M) is evaluated using the meson-quark coupling Lagrangian 5fin t of (1). The result is"

(Bol v(q)IB;, M ( q ) )

: _ i( 4 nR3)(Nr Nr u (q) \ fM ] [/(2r0aac%

3

�9 (Bo[ ~, ai.qAM(i)lB'o} i = l

(9)

where

normalization factor Nr of the s-quark wave function.

Finally, the t/meson with its SU(3) flavour content tl = (u ~ + d d - 2 s g)/]/ 6 (ignoring t / - i7' mixing) couples diagonally to both non-strange and strange quarks, as shown in Fig. lc. Hence for the BB't 1 vertex functions the normalization factors Nr o for non-strange quarks and N~= r~,R for strange quarks appear in com- binations with the appropriate statistical weight de- pending on the total strangeness of the baryon in question.

u(q) = 3j~ (qR)/(qR) (lO)

(with u(0) = 1) is the form factor of the bag. The normalization constants N~ and Nr for the valence quark wave functions of the bag model baryons B e and B~) are given in (4), with ~ = 0 for (massless) u- and d- quarks and ~ = m~ R for the strange quark. The meson decay constants f~ are given by their empirical values:

f~ = 93.3 MeV

for the pion and

fK = 119 MeV

for the kaon. For the t/meson we use f , =f~ in absence of any more detailed information. The remaining bar- yonic matrix element in (9) involves the spin-flavour operator a. q2M in which 2M refers to the appropriate combination of flavour SU(3)-matrices for a given meson M. Using standard reduction techniques one finds:

3

(BoJ ~ a,. q2M(i)IB;) i = 1

= f~ m'r tu pu I TroT) 2 q,(S' m's 1 r I Sms), Y

(11)

where t M is the meson isospin and #M its 3-compo- nent. The spins and isospins of the baryons B o and B; are denoted by (S, ms), (T, roT) and (S', re's), (T', re'r), respectively. The reduced matrix elements f~ M have the values listed in Table 1.

The vertex function (9-11) summarizes all the rele- vant quark-meson couplings illustrated in Fig. l a c. The pion couples to u- and d-quarks only, in which case tM = 1 and the 2M reduce to SU(2)-isospin matrices. The kaon couplings (with tM=l /2 ) connect strange and non-strange quarks as shown in Fig. lb. Note that in this case the explicit chiral symmetry breaking by the non-zero strange quark mass m e appears in the kaon-baryon vertex function through the

3. Baryon Self Energies and Decay Widths

The surface coupling of the bare (MIT bag) baryon states ]Bo} to the pseudoscalar meson clouds gener- ates dressed baryons ]B} which have the following expansion in terms of the bare valence quark state IBo) and mixed states with three valence quarks plus one or more mesons:

B~):I: Bo

I B } = ~ [ B o } + ~ [a(1)JB'o,M} Bbe~ .

M, M ' , . . . e8

+ a (2) IB'o M, M'} +.. .] . (12)

Here the 'spectroscopic factor'

Z~-=](BIBo}[ 2 (13)

measures the valence quark content of the physical dressed baryon IB}.

We are interested in the meson cloud contribu- tion to the baryon mass, i.e. in the baryon self energy defined by ZB= (BI H IB}- (Bo[ Ho IBo} =(Bo]gin t lB) / ] /~2 . Using (13) together with the Dyson equation for the dressed state IB) the resulting self-energy is

1 2;B=(Bo[ Hi,,t MB__Ho__AHintA HintlBo}, (14)

where MB denotes the physical (dressed) baryon mass and M ~ is the bare mass. The operator A projects out all but the pure IBo} states and guarantees that only irreducible self-energy graphs will arise from (14). We adopt the treatment given in [6] and approximate the eigenstates of Ho+AHin tA by one-meson- one-baryon states IBm, M}, assuming (H o + AHI,,t A) IB'o, M}~- MB, [B'o, M } so that one finally obtains:

I<Bol v(q)IBm), M}I 2

B ' e ~ M e 8

(15)


The diagram corresponding to this expression is shown in Fig. 2. Self consistency is imposed in (15) by taking the full masses MB and Mw instead of the unperturbed M ~ and M~ in the denominator of (15). Finally one obtains, using (8), (11) and (15):

t,', m \ JM /

oo q4 122 (qR) ( 1 6 )

" o ~ dq c%(M _Mw_coq+ie) ,

where the sum extends over B'=(N, A, Z, E, A, Z*, ~*, f2) and M =(re, K, K, ~/). (The sum over spin-3- as well as isospin-z- components has already been performed.)

The renormalized coupling constants fSB'M are obtained by evaluating the vertex corrections due to one-meson-loops (according to Fig. 3); again, self- consistency of the baryon masses and decay widths is imposed. One uses

f , WM/fOwM = ZfWV (17)

and finds for Z t the expression

(zBIB'M)- 1

(fOcM,)(fOo~)(fOoM,) (N~ N~,~i(.~xR3)2 = 1 + ~ fOB'M \ fM" ] [c, D]

[Mq

�9 ~ 1 ~/2Sc+ 11~-T '+ 11/2 r~ + 1

TD tv T'J (So 1

. (_l)W+s~+r'+r~+,M,+*,~

�9 P . ~dq q4 u2(qR)

o (Oq (Ms -- Mc -- COq)(Mw - Mo - coo)' (18)

which involves a principal value integral over intermediate baryon and meson energy denominators together with the spin and isospin couplings of the various baryons and mesons. N~, N~, are the normalization factors which refer to the active quarks in the intermediate baryon states which couple to the meson M'. Since the model space is truncated to the 10-

Table 1. SU(6)-coupling constants f~ , fn~ and f~ ~ normalized to f~ = 5

B ~ B ' ~ N A Z ,E A X* E* 0

fBOB, rc

N A Z

A

Z*

Z*

5

r

]/~ 5

-2 1/g

fBOB,K N --3 --1 - -[ /8

A - 4 -7 73

E 1 5 - ] / 8

-%5 E* 2 - 2 l / ~ f2 --4 ]/@6

--4

fBOB,~t N

A

Z

A

Z*

f2

- 2 2 /8

- 2

-2 -1/g

S. Klimt and W. Weise: Static Hyperon Properties

o,d \~.:~ .........

u,dA:O

s\~=msR (B=O) {u.d)

\'N" . . . . . . . . K(K) /

Is) ~=0 (~=m~R)

u.d,~O

. . . . . . . . . q

u,d/~--O 7 s ~ . . . . . . . . q

Fig. 1. Quark-meson vertices, illustrating different mass parameters ~=mR of in/out-going quarks for different SU(3)-mesons: a) qqrc- vertex; b) qqK- and qq/<-vertex; c) qqq-vertex

B,M B B',M B, B,M B

Fig. 2. Second order diagram determining the baryon self-energy ZB. The baryon masses are denoted by MB, M~,

M M M /s / / /

/ / , / _ " / ."" ">(M' i / / - / ~,,

B o B ' : g o" B ' + B "' "< ~' B'+... f f~ f~ f~ f~

Fig. 3. Renormalization of the baryon-meson vertex

and _8-representations and since chiral as well as SU(6)spinxn . . . . . -symmetry is explicitly broken, the Ward-2dentity Z~ = Ze does not hold and one expects deviations offBB,M/fffB,~t from unity.

In general the self-energy 2; n of (16) is a complex quantity. This is true in particular for the decuplet states A (1236), S* (1385) and 3 " (1533). Their empirical decay widths are

F(A ~ rcN)=(125 +_5) MeV,

F(Z* ~ rcA, rcS) = (37 +- 4) MeV,

F(S* --. zcS) =(10+_2) MeV.

These widths can be calculated in the present model according to

MB = MB ~ + 9te Z R + i .~m Z R

i = M ~ " - 5 r~ (19)

493

where

F B = -- 2.qm Zn(MB). (20)

Note that the width is defined at the resonance posi- tion, i.e. with .~rrt)2 B evaluated self-consistently at the full mass MB.

We proceed now as follows. First, the bare mass MB ~ is obtained from the familiar MIT bag model [5] (compare (2)):

i = l R

+ ~ 2 I(~i, ~i)R <a~. ~rj>,. (22) i< j

Here the sum extends over the three valence quarks (labelled i) with mass parameters ~i = miR. The bare mass includes the quark kinetic energy f2/R and the approximate zero point energy - Z o / R together with the volume energy of the bag. The hyperfine splitting is parametrized by the coupling strength e and breaks the SU(2)spin degeneracy among octet and decuplet states. At the 'bare mass' level, it stems from effective one-gluon exchange [5]. The corresponding integral I is a smooth function of the quark mass parameters ~, ~j. One finds with high accuracy the approximate form

I(~ i, ~j)-~ (0.58-0.07 (~i+ ~s)) (~Rm) GeV. (22)

The full baryon mass MB with inclusion of the meson cloud is then determined by adding the self energy $8 as in (19). Finally, a center-of-mass correction 8 B is included, so that one has

9te MB = M ~ + 9le ZB-- 8B (MB). (23)

In determining e B we are guided by previous work on bag models [11] and follow the simple approach proposed by Jaffe [22] which we generalize to baryons other than the nucleon according to

0.75 Mnuoleon en(MB)-- R 9te MB (24)

Typical values are eB-~ 100-200 MeV for the various baryons considered here.

The set of bag parameter (~, B, Zo) together with the strange quark mass m~ is then fixed by imposing the pressure equilibrium condition

Qg~e MB(R) 0 ~-R --IR=Ro (25)


which determines the proper bag radius Ro. Such an equilibrium radius always exists for the

'bare ' bag with a reasonable choice of B and Z o. The situation changes drastically, however, when the self-energy generated by the meson cloud is added. In particular, it is well known that, due to the strong attraction generated by the pion cloud, nucleon and A do not reach an equilibrium radius R 0 in the linearized chiral bag model: non-strange baryons require a full treatment in a non-linear chiral bag [9], with inclusion of Dirac sea polarization effects [10]. In such elaborate models there is evidence that the resulting mass MB(R) is a smooth function of the bag radius R. The proposed interpretation is that for baryons with massless (chiral) valence quark content, the bag radius is not a physical observable ("Cheshire cat" picture, see [13, 14]). In the extreme limit R ~ 0, the chiral bag turns into a Skyrme type soliton [-15- 17]. What all such models of the nucleon (hybrid chiral bag or chiral Soliton) have nevertheless in common is the fact that the radius of the baryon number distribution inside the nucleon turns out to be <r~) t/2 ---(0.5-0.6) fm [17].

Our aim here is to give a description of the hyperons and their resonances, rather than the nucleon and the A. The latter nevertheless enter indirectly, as intermediate states in the coupled equations which deter- mine the hyperon self-energies. In this case we fix the bag radius for nucleon and A at R = 0.8 fm. Then their baryon number distributions, represented by the valence quarks in this linearized model, have r.m.s. radii between 0.5 and 0.6 fm, in accordance with non- linear chiral models.

We shall then demonstrate that for certain ranges of bag parameters (c~, B, Zo), the hyperons do in fact reach stable bag radii Ro in a linear model, unlike the N and A. This is partly a consequence of the explicit chiral symmetry breaking by the strange quark mass ms, and partly due ot the reduced pressure as generated by the meson cloud in the presence of strangeness, as we shall now see.

3. Results

In this section we present results for the radii R o of the octet and decuplet states as they are determined by the condition OM(R)/OR = 0 at R = Ro. The masses M(R) and, for the decuplet baryons, the decay widths are determined self-consistently by solving the non- linear system of equations (19) and (23) which involves the (generally complex) self-energies of N, A, Z, Z, A, Z*, E* and f2. The following four parameters are then adjusted in order to produce an optimal fit to the mass spectrum: the bag parameter B, the scale

O.7

0.6

0.5

0./+

0.3

0.2

0.1

0.8

0.4.

0

-0./+

-0.8

300 lc

~200

100

I

0.11

i i i i i i I i I

0.12 0.13 / 0.1/+ 0,15.1 ,,,,. . f

/ . J / . f

/ / . J

b I I I I I i I I I i I

0.11 0.12 0.13 0.1/+ 0.15

C

I I I I I I I I I I I

0,11 0.12 0.t3 0.1/+ 0.15 B v~' [GeV]

Fig. 4a-c. Optimized parameters ~, Z0 and m, vs. bag constant B in the MIT model (no meson cloud, - . . . . ), the SU(2) (only pion cloud, - . - . -) and SU(3) chiral bag (solid line). The parameters are obtained by a fit to the hyperon masses

Zo of the zero-point energy, the color fine structure constant c~ and the strange quark mass m~.

We note that the overall results do not depend significantly on whether or not we include the N and A in the fit. This indicates that B (and Zo as well) cannot be fixed just by fitting the ground state spectra. This result is valid for the Chiral Bag as well as for the MIT model. Consequently, we see no reason to favor a special choice of B and/or Zo. We show results for B 1/4 chosen between 115 MeV and 155 MeV. We find good fits to the spectrum for this wide range of bag constants B (with a mean deviation A M ~- 10 MeV, i.e. less than 1% between calculated and experimental baryon masses).

3.1. Parameter Choices

Figure 4a -c present the optimal set of parameters c~, ms, B and Z o resulting from fits to baryon masses. They are plotted here as functions of the bag pressure

s. Klimt and W. Weise: Static Hyperon Properties 495

B. These results are given for the full SU(3)-case with K- and ~/-mesons included, as compared to the reduced ' S U ( 2 ) ' c a s e in which K- and q-contributions are omitted. For comparison, the corresponding parameter sets for the MIT bag (without any meson clouds) is also shown.

As expected, inclusion of meson cloud effects re- duces the required values of the color fine structure constant e considerably: hyperfine splittings are now shared by gluonic effects and the spin-dependence induced by the meson fields. For B~/4= 130 MeV, we obtain c~=0.52 (with ~, K and t/clouds) and e=0 .56 (with pion cloud only). The required values of e decrease with increasing bag pressure. However, the overall variations are smooth: changes c~ + 0.i induce variations of the baryon masses A M < 2 0 MeV for any value of B.

The value of the input strange quark mass m s as required in the fit to the hyperon mass spectrum is of some interest. In the pure MIT bag one finds a large value around m ~ 2 5 0 MeV, almost constant as a function of the bag parameter B. In the SU(3) chiral bag model the corresponding input strange mass rn~ tends to be smaller and decreases with increasing B, so that the resulting values m~ ~ 200 MeV (for B ~/4~ - 140 MeV) closely resemble the typical mass m~-~(170_+25) McV deduced from current algebra considerations.

3.2. Masses and Radii

Next, let us examine the baryon masses MB as a function of the bag radius R. Figure 5 shows MB(R) for the octet and decuplet hyperons and resonances with a bag constant B 1/4 = 130 MeV. The heavier hyperons systematically develop stable minima for radii Ro between 1.1 fm and 1.25 fm which shift to still slightly larger values when only the pion cloud is incorporated (see Fig. 5b). For the A and 2; hyperons the curves MB(R) are relatively flat, but minima still occur around R 0 -~ 1.1 fm. However, increasing the bag parameter B much beyond B1/4=130 MeV will lead to instability of the A and Z (whereas the heavier baryons remain stable).

It is instructive to decompose the resulting strange baryon masses into contributions from their valence quark energies, from the various meson clouds and from the effective gluon-exchange hyperfine interaction. This is shown in Table 2 for a bag parameter B 1/4 = 130 MeV. For the A and Z hyperons, the general result is that gluon exchange effects are somewhat larger than meson cloud effects but roughly of the same order of magnitude. Taken together the pion cloud and gluon exchange lead to downward shifts

1.4

1.6 'L3

STRANGE BARYON MASSES M (R)

FULL SU (3) [HIRAL MODEL

1.5,

[

i I i i 018 110 1.'2 I&-

R [frn]

1.9

1.8

1.7

1.5

1.4 I i 1.10 i r i I I

0.8 1.2 1.4 R[fm]

STRANGE BARYON MASSES M(R)

(PION CLOUD ONLY)

1.5

IE 1A

1.3

!

1"2I I I i ~ i i A I

b 1.1 0.8 1.0 1.2 1.4 R [fm]

2.1

,2.0

~c 19

1.8

I 1.7

0.8 110 112 1./~ 116 R [fm]

Fig. 5a and b. Strange baryon masses MB(R) as a function of bag radius R for a bag constant B " = 130 MeV. a Full broken SU(3)L x SU(3)R scheme with the complete octet meson cloud (~, K, /<, t/) included; b for comparison: reduced scheme in which only the pion cloud is incorporated

of the A and Z masses by - 1 9 0 MeV and - 1 4 5 MeV, respectively. The additional effect of the kaon cloud is to shift these masses further down by about - 2 5 MeV.

For smaller values of the bag radius R the mesonic self-energy part of the masses increases. For R=0.8 fm the pion cloud alone produces a shift of - 193 MeV and - 161 MeV for the A and Z, respectively.

The t /meson cloud has a comparatively small influence not only for the A and Z, but for all octet and decuplet states. It should be noted that the center- of-mass correction is substantial (more than 10% of


Table 2. Various contributions to the masses of strange baryons at the optimal bag radii Ro as indicated: Valence quark energy Mn ~ (see (21) with e=0) ; gluon exchange hyperfine energy (last term on r.h.s, of (22)); self-energies Z induced by pion, kaon and t/ cloud, respectively, according to (16); center-of-mass correction e~ of (24). All energies are given in MeV. The input parameters are: B~/4= 130 MeV, c~ = 0.52, Z0 = 0, m~ = 220 MeV, m, = md = 0. Resulting total masses Ms are compared with experimental values (in brackets)

B R0 B [fm] M g , c~ = 0 Eoo E ,S ~ Z "r Z" -- e B MB(Mexp)

A 1.09 1458 - 137 --61 --26 - 5 -- 106 1123(1116) Z 1.11 1453 -- 103 --42 --24 - 8 -- 99 1177(1193)

1.13 1570 -- 120 - - 12 --23 - 6 - 88 1321 (1321) N* 1.23 1417 110 - 4 0 --13 --3 - 78 1393(1385) ~* 1.24 1545 95 - 16 - 15 --4 -- 71 1534(1533) I2 1.25 1673 81 0 - 14 - 5 - 65 1670(1672)

the total A and Z masses). It combines with the phe- nomenological zero point energy - Z o / R and should be considered as a rough measure of the overall uncer- tainties in the model.

For the lowest strangeness S = - 2 baryon, the E(1321), the relative amount of pion and kaon cloud contributions to the mass is reversed as compared to the A and N: the kaon cloud is twice as strong as the pion cloud, but both are dominated by the gluon exchange hyperfine shift. As one moves to the decuplet states (the N*, ~* and s the relative importance of the pion and kaon cloud effects generally decreases, and the (positive) hyperfine shift is now governed by gluon exchange. For the ~* and O- gluon exchange and meson cloud corrections together with center-of-mass effects almost cancel, so that their masses are determined already by the valence quark energies.

3.3. Decay Widths

The decay widths for Z * ~ rcA, zcS and ~ * ~ 7zZ are determined by the imaginary parts of the corresponding self-energies (16) according to (20). It is then instructive to present these widths as a function of the bag radius R and compare them with empirical values. The results are shown in Fig. 6 together with the decay width obtained for the A in our model. We note the characteristic decrease of the decay widths with increasing bag radius. For the hyperon resonances N*, ~* the empirical widths are reproduced for radii which are slightly smaller than the values Ro given in Table 2 but there is a remarkable degree of overall consistency for both real and imaginary parts of the N* and ~* energies.

We have also examined the A ~ rcN decay width in a similar way and find that the empirical width F(A ~ z N ) ~ - 1 1 0 MeV is reproduced for bag radii R ~0.9 fm; it is interesting to see that the corresponding r.m.s, radius of the baryon number (valence quark) distribution happens to be close to those obtained

200

150

100

50

>~ :E

'-- 9o

60

I I I I I t I

.8 ~ 1 3 ~ 1 . 3 R[frnl

30 ~ / / / / / / / / / / / / / / / / / / / / / ' / ~ / "

[ I [ I I I I

50 .8 .9 1. 1.1 1.2 1.3 R [ f ro ]

\

[ I I I I I I

.8 .9 1. 1.1 1.2 1.3 R[frn]

Fig. 6. Decay widths F = - - 2 . 3 m Z for A ~ N , S * - ~ A , rcZ and ~* ~ ~Z as a function of bag radius R, calculated self-consistently according to (16, 19, 20). The empirical values are indicated for comparison

in the more sophisticated non-linear chiral models of the A-isobar in which the bag radius is not of direct observable relevance.

3.4. Vertex Renormalization Z 1 and Spectroscopic Factors Z 2

The present approach describes the baryons with strangeness as a core with three valence quarks (including one or more strange quarks) surrounded by a cloud of pseudoscalar mesons. In order to see whether a linearized model and perturbation theory

S. Klimt and W. Weise: Static Hyperon Properties 497

Table 3. Values of the 'spectroscopic factors' Z z = I ( B I B o ) ] 2 (see (12)) taken at R = R o for various baryons. The input is the same as the one which leads to the result of Table 2

B A Z' N X* ~* s

Z2(R0) 0.88 0.88 0.95 0.79 0.87 0.97

Table 4. Values of the vertex renormalization factors fBB,u/fOwM B B" BB'M = ~ 2 Z 2 / Z 1 as obtained from (18) taken at R = R0. The input

is the same as the one which leads to the results of Table 2

B S B' --* A X 3 X* 3* (2

fBB, ff fOB, ~

A 1.05 1.07 X 1.05 1.03 1.05

1.02 0.99 X* 1.16 1.15 1.06 3* 1.00 0.99

fBB,r.jfO~,r

A 1.02 1.08 X 1.03 1.08 3 1.02 1.08 1.02 X* 1.00 1.04 3* 1.14 1.16 1.06 f2 1.02 1.00

1.01

0.98

f BB' q / f BOB" tl

A Z 3

X* 3 "

Q

1.09 1.02

1.13

1.05 1.02 1.01

1.01 1.10 1.01

are at all applicable it is important to investigate the normalization Z z of the valence core part [Bo) in the full baryon wave function, (12). We present the calculated values of Z2 for a bag parameter B t/4= 130 MeV, the one wMch turned out to give an overall optimal fit, in Table 3. It can be seen clearly that almost all of the 'spectroscopic factors' are close to or above 0.9. The only exception is the X* for which Z 2 ~0.8. On the other hand, for the nucleon and the A one finds Z2 < 0.7, indicating the inadequacy of perturbation theory and of the linearized chiral bag for systems with u- and d-quarks only. The explicit breaking of chiral symmetry by the strange quark mass ms appears to induce a significant stabilizing effect. The relatively small deviations of Z 2 from unity for the strange baryons justify a posteriori the linearized chiral approach together with perturbation theory; in this approach, baryons with strangeness are still dominantly composed of their three valence quarks, unlike the nucleon and A.

The meson-baryon vertex renormalization constants Z1 of (18) reflect, by their deviation from the corresponding propagator renormalization constants Z2, the degree of explicit chiral SU(3)L X SU(3)R and flavour symmetry breaking. We show in Table 4 typical values for the ratios of renormalized to bare cou-

0 B B' BB'M plings, fBB,M/f~B,M=Z~2Z~/Z1 as obtained with the same optimal parameter sets as the ones which leads to the self-energies in Table 2. As one can see, the renormalized coupling is increased over the bare coupling, but generally by less than 10%.

4. Conclusions

We have performed calculations of masses and widths for the octet and decuplet baryons with strangeness using a SU(3)L X SU(3)R chiral bag model with linearized coupling of the valence quarks to the pseudoscalar meson octet at the bag boundary.

Our results are summarized as follows:

i) The A and X hyperons are stable against the meson cloud pressure for bag constants B1/*---130 MeV or smaller; for the heavier strange baryons (S, X*, 3" and f2) the range of stability extends further to B~140 MeV and above. An optimal fit to the baryon mass spectrum is obtained for the choice of B = 130 MeV and the following additional set of input parameters:

color fine structure constant: e ~ 0.52, zero point energy parameter: Zo ~0, strange quark mass: ms ~ 220 MeV.

We emphasize that the nucleon and the A-isobar are generally unstable in such a linearized chiral model.

ii) The equilibrium bag radii R o determined by pressure balance and 8M(R)/OR=O at R=Ro turn out to be Ro l l . 15 fm for the octet baryons (A, X, 3) and R 0 ~ 1.25 fin for the decuplet states (X*, 3*, f2).

iii) The decay widths of the S* and 3" calculated self-consistently at these radii are not far from their empirical values.

iv) Strangeness is special: the explicit breaking of chiral SU(3)L x SU(3)R by the strange quark mass ms is a substantial stabilizing factor and of crucial importance to the results just mentioned.

v) In this approach, baryons with strangeness emerge with a wave function which has a dominant (more than 80%) valence quark content with small admixtures of [3 quarks + meson) components.

We conclude that the notion of a stable (and relatively large) confinement (or bag) radius apparently makes sense for systems with one or more strange valence quarks, whereas this is known not to be the


case for baryons with pure u - and d-valence quark content, such as the nucleon and the A (1232).

The model presented here can be considered as a limiting case, for sufficiently large bag radii, of the more sophisticated non-perturbative approaches such as the 'chiral hyperbag' [18] which maintain the full non-linear structure of the SU(3) chiral Lagrangian in the quark-bag and the meson (soliton) sector. In particular, it is shown in [18] that, when strange quarks are present, bag radii tend to be bigger (R > 1 fro) than for non-strange baryons, as suggested previously in [19].

We would like to thank G.E. Brown and M. Rho for helpful discus- sions.

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S. Klimt, W. Weise Institut fiir Theoretische Physik Universit/it Regensburg Universit~itsstrasse 31 D-8400 Regensburg Federal Republic of Germany

static hyperon properties in a linearizedsu(3)-chiral bag model

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