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Analysis of splitting mechanisms in the low-lying non-strange negative-parity baryons
Nuclear Physics A393 (1983) 349-371 O North-Holland Publishing Company
ANALYSIS OF SPLITTING MECHANISMS IN THE LOW-LYING NON-STRANGE NEGATIVE-PARITY BARYONSt H.R. FIEBIG* and B. SCHWESINGER** Physics Department, State University of New York, Stony Brook, N Y 11794, USA
Received 1 June 1982 Abstract: We investigate the mass splitting in the negative parity N* and A* spectrum where, experimentally, the spin-orbit force is found to be highly suppressed. A hyperfine interaction between quarks derived from a one-gluon and one-pion exchange model for a spherical chiral bag is used. The interaction is, unlike the usual Fermi-Breit force, fully relativistic and explicitly takes into account confinement effects. We analyze in detail how the force operates in the odd parity N* and A* spectrum and find that the suppression of spin-orbit forces, which is mysterious in constituent quark models, can be explained.
1. Introduction
T h e p r e s e n t c o n c e p t of a b a r y o n b e i n g an i n t e r a c t i n g q u a r k - g l u o n s y s t e m inside a Q C D v a c u u m b u b b l e involves t h e o b l i g a t i o n to u n d e r s t a n d t h e p r o p e r t i e s of t h e o b s e r v e d p a r t i c l e s p e c t r u m on this basis. In o r d e r to test t h e o r e t i c a l n o t i o n s a n d m o d e l s it is r e a s o n a b l e to c o n c e n t r a t e o n a distinct p a r t of t h e h a d r o n m a s s s p e c t r u m . In this p a p e r we w a n t to a d d r e s s the n o n - s t r a n g e s e c t o r of t h e o d d - p a r i t y [70] SU(6) m u l t i p l e t . T h u s w e e n c o u n t e r five e x c i t e d states ( i s o d o u b l e t s ) of t h e n u c l e o n , N*, a n d two e x c i t e d states (isoquartets) of t h e isobar, A*. This s a m p l e of p a r t i c l e s has t h e a d v a n t a g e of an, a p p a r e n t l y , s i m p l e s u b s t r u c t u r e involving o r b i t a l e x c i t a t i o n s of q u a r k s in b a r y o n s i). It serves, t h e r e f o r e , as a s u i t a b l e field to s t u d y t h e forces b e t w e e n q u a r k s as t h e y arise f r o m gluon e x c h a n g e inside a s e c l u d e d r e g i o n of space. In t h e p r e s e n t w o r k we try to a n a l y z e t h e s e forces within t h e b a g - m o d e l f r a m e w o r k a n d s e e k to u n d e r s t a n d c e r t a i n f e a t u r e s in t h e m a s s s p e c t r u m of t h e N* a n d A* n e g a t i v e p a r i t y states. T h e q u a r k c o n t e n t s of t h e s e states, t h e e x p e r i m e n t a l s p e c t r u m of which is given in fig. 1, c o n v e n i e n t l y is d e s c r i b e d b y m e a n s of SU(3) Y o u n g t a b l e a u x . T w o flavors (isospin c o m p o n e n t s ) a n d two spin c o m p o n e n t s yield a t o t a l l y s y m m e t r i c a n d a m i x e d s y m m e t r y p a t t e r n for isospin a n d spin, cf. t a b l e 1. T h e o d d p a r i t y states c o n s i d e r e d a r e i n t e r p r e t e d as h a v i n g o n e q u a r k e x c i t e d to an o r b i t a l p - s t a t e , t h e t w o o t h e r q u a r k s a r e left in t h e l o w e s t s-state. T h e p o s s i b l e o r b i t a l states a r e thus * Supported by Heinrich-Hertz-Stiftung. ** Supported by a fellowship from the Scientific Committee of the NATO via the German Academic Exchange Service (DAAD). t Work supported in part by US DOE Contract DE-AC02-76ER13001. 349
350
H.R. Fiebig, B. Schwesinger / Splitting mechanisms MeV 17001650 (1650)
1675 (1700)
f620 (~650)
1600
1550 (1535) 1500' N i/2-
1710 (1670)
1675 (1670)
1525 (1520)
N 3/2-
N5/2-
A (72-
3/2-
Fig. 1. Experimental spectrum of low-lying negative-parity baryon masses. The positions graphically indicated are compiled from ref. 5). The numbers in brackets are taken from ref. 6).
again restricted to totally symmetric and mixed symmetry patterns. However, the totally symmetric representation belongs to spurious center-of-mass (c.m.) motion and should be excluded. Table 1 shows the isospin (T), spin (S), orbital (L) and total angular momentum of the different patterns. The coupling of the T-, S- and L- representation to a totally symmetric pattern yields seven basis states (cf. sect. 2) which correspond in number and quantum numbers to the states experimentally observed (fig. 1). The quark-quark forces arising from one-gluon exchange are in the non-relativistic constituent quark model given by the Fermi-Breit interaction 2.3). This force has a spin-exchange, a spin-orbit, a tensor term and other spin-independent pieces. As one can see from table 1 all three terms should be operative in the odd-parity N * - d * spectrum. However, there is a clustering into two degenerate groups of particles (fig. 1), which is even more pronounced if one looks at the numbers of the Particle Data Group 6), fig. 1 in brackets. Isgur and Karl have noticed that the mass spectrum of the odd parity N*'s and A*'s can be reproduced in a harmonic oscillator model with Fermi-Breit hyperfine TABLE 1 Characterization of the N* and A* model states by their isospin (T), spin (S) and orbital (L) symmetry
T=" s={ ,=,
_1 s={ ,=, z-~
_3 ,=, t=½ S-~
EP~-~EP Epr~Ep EPF~E£ T={ s:" L=,
_:3 So~ I ,o, Z-~
I L:, Z={ So-~
rmEPEP ~-~IYEP
To" s=" ,=,
PEPEP P F E P NIl2-
*
N312-
N ~
512-
~
A I12-
A .
312
_
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
351
interaction only if one deliberately throws away the L S force terms and keeps the spin-exchange and the tensor pieces 7,8). Indeed, the spin-exchange (Fermi contact) term alone could very well furnish the N*'s with the observed mass split ~ 1 3 0 140 MeV pushing the states with S = 3 up and those with S = ~ down. The two A*'s have the same spin symmetry and thus would remain degenerate. The tensor force then mainly mixes the S = ½ and S = 23-components 7). The effects of L S forces, however, are hardly seen in the N* spectrum, and if the L S force from one gluon exchange were included, it would disrupt the spectrum disastrously. Hence we consider the experimental situation as a clear indication that the spin-orbit pieces of the hyperfine interaction, although present, must be highly suppressed by "some mechanism" for the N*'s. On the other hand, because the tensor force is not operative in S = ½states, any A* split is only to be attributed to L S forces. It seems conceivable that suppression of the spin-orbit force is a genuine confinement effect. Due to confinement the color electromagnetic gluon fields have to vanish outside the hadron, which gives rise to additional terms in the gluon propagator 9,10). Within the framework of Q C D perturbation theory these confinement terms contribute to the quark-quark interaction. We would like to look at these additional terms in the present work. The confinement terms are explicitly calculable in a bag model 11) which we here consider as a suitable starting point for a perturbative approach. It has, furthermore, the advantage that one can keep the theory fully relativistic and time-dependent. However, it is necessary to perform the transformation to a two-component theory for two reasons: only then the full variety of distinct qq forces we wish to analyze, shows up and the elimination of spurious c.m. motion is easy and transparent. The excited baryon spectrum has been calculated in the MIT bag model 12,13). However, the mass splitting pattern found in ref. 13) for the odd parity N* and A* spectrum does not show the experimentally observed gross structure described above. We therefore avoid some of the approximations made there; this is facilitated by transforming to a two-component theory. Our results are not in agreement with the assertion lz-14) that the exchange matrix elements are small. In our calculation these contributions are taken into account (cf. sect. 3). We also restore chiral symmetry, which is broken in the MIT bag, by coupling the pion field to the bag's surface 15,16). The pion field is treated in linear (zeroth order) approximation 17,18) and not only contributes to the nucleon-isobar ground state energy difference but also provides additional splittings in the excited baryon spectrum since the quarks interact by coupling to the axial pion current outside. Finally, in contrast to the work of D e G r a n d and Jaffe, we analyze the qq forces in detail in order to learn about the influence of different force terms on the mass spectrum. In the following section we will first write down the model wave functions for the odd parity N* and A* states and then introduce the quark-quark forces, as derived in ref. 11), and their matrix elements in the model space. In sect. 3 the
352
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
properties of the force are discussed with respect to effects of the boundary, static and non-relativistic approximations and the influence of the pion field. We close with some critical remarks and a concluding section.
2. The basis states and the bag-model qq force Each 3-quark basis state in the N* and A* subspace must be a product of a totally antisymmetric color wave function (singlet) and a totally symmetric wave function in the combined isospin-spin-orbit space
I~, color)® I ~ ,
TJ(SL)) •
(2.1)
The symmetric part of this wave function can be constructed from the isospin, spin and orbital wave functions of definite SU(3) symmetry, as given in table 1. We write symbolically
I~,TJ(SL))= 52 E IT)®IS)®IL)CGsu(3)CGo(3). 0(3) SU(3)
(2.2)
The symbols CG denote appropriate Clebsch-Gordan coefficients by means of which the individual TSL states are coupled to a totally symmetric state with respect to particle exchange (SU(3) sum) and the SL states are coupled to total angular m o m e n t u m J (0(3) sum). The orbital 3-quark wave function ]L> is made up from two ls modes and one l p mode in the bag model 19,20,12). Let us now consider a translation of a threeparticle orbital wave function, say, with all three particles in the same s-state exp [ie. (P1 +/~2 +P2)]s(1)s(2)s(3) = s (1)s (2)s (3) + i e. [PlS (1)s (2)s (3) + s (1)PzS (2)s (3) + s (1)s (2)P3s (3)] + o(e 2), (2.3) where s(i) is a function only depending on ri = Iril and P~ are momentum operators. Since the angular dependence of P~s(i) is given by ri/q, which in spherical components is - Y r , , for l = 1, we observe that the quantity in square brackets in (2.3) belongs to the ~ representation of SU(3) where two particles are in s-states and one is in a p-state. It is, on the other hand, nothing but a translation of the state with three particles in an s-state. This simple observation shows that the totally symmetric configuration of two ls and one l p modes should predominantly be a spurious c.m. state. We therefore only keep the mixed representation for [L). Then, the only way to end up with a ~ symmetry on the left hand side of (2.2) is to couple the isospin and spin pieces to a mixed symmetry representation. Consequently, for the N* states which have ~S] T-symmetry we can have both and ~ in spin space whereas for the A* states which have ~ T-symmetry
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
353
we can have only ~ in spin space. Thus the states of our model space are characterized by their spin configuration and, of course, their total angular momentum. There is only one N*~- state because S = ½ and L = 1 cannot couple to J = 2. Table 1 shows a summary arranged in a way to resemble fig. 1. In a spherical bag model we encounter the cavity modes
~lJ"=Nli[I ,t
• k r
~r.r l+ir
/
r\
/t±l~ktj-~)
~ilm(I2)e-i°~"'/r
(2.4)
for j = l+~, e.g. ref. 13). The bag radius is R and qb~r~(12) denote spin harmonics where 12 is the direction of r, ref. al). For massless quarks we have k~i = tou. The lowest frequencies determined by the linear bag boundary condition 19) are
tos~ = 2.04,
3.81 for/' = ½ top~= 3.20 f o r j =3"
(2.5)
With the spinors tp in (2.4) we are not yet in a position to eliminate the spurious c.m. motion. The reason is that the construction of the model states, as described above, is based on the L S coupling scheme, cf. (2.2), and on the fact that there is only one orbital p-state, cf. (2.3). Both conditions are, of course, not met by the spinors ~. In ref. 11) it was shown how to transform the four-component Dirac spinors by an (approximately) unitary Foldy-Wouthuysen transformation to two-component Pauli spinors keeping only the positive energy solutions. Since the resulting spinors still remain non-degenerate in/" = l + ½ we apply a second (approximately) unitary scaling transformation, which reintroduces the degeneracy in the basis states by means of adding a one-body spin-orbit force to the kinetic energy • tr, N / ~ P~ Hkin=~i (N/~i -}-ITl2"~-I
]~.
(2.6)
Details are again given in ref. 11). The orbital part of the rn = 0 basis states, for s-states denoted by s(r) and for p-states by p , ( r ) , then is N~R -3/2jo(to~r/R ) Yoo(O) = s (t), NoR
-3/2]l(topr/R ) YI~ ([2) = Pc (r) .
(2.7a) (2.7b)
As shown in ref. 11) the wave number of the p-states and the strength K of the spin-orbit force are related to each other, they must be top = 3.40, (2.8) K = --0.06.
354
H.R. Fiebig,B. Schwesinger/ Splitting mechanisms
The wave number of the s-states, of course, remains unchanged, tos = 2.04• The normalized orbital states (2.7a, b),
fr
(2.9)
have to be multiplied with Pauli spinors and with flavor and color functions for a complete description of single quark degrees of freedom. The one-gluon and one-pion exchange force we are considering is derived in ref. 11). For the analysis attempted here we cast the force into a different form, which is also more suitable for later discussions. The entire hamiltonian reads •
1
H = 2 H k i n ( t ) + g Y. [HoGz(i,f)+H,~(i,f)], i
(2.10)
i~i
where//kin is defined in (2.6) and contains the one-body LS force (m = O)
Ills = Kl . 0.p2/.,/-~.
(2.11)
We write the quark-quark interaction as 1 0 HOOE(1, 2) = gas{-- Y~Z (1, 2) + 10.1" 0.2 E X°( 1, 2) +1/0.1X 0.2"~X1(1, 2)+0.1 "Y~ y1(1, 2)+0.2"~ Y1(2, 1)
"+-(0'1°0"2)" ~ X2(1, 2)},
(2.12a)
1
H~(1, 2 ) = - - -16f~ "rl "~'2{½0.1"0.2X°(1, 2) + li(0.a x 0.2)" X~(1, 2) + (0.100"2).X2(1, 2)}.
(2.12b)
The upper indices K = 0, 1, 2 identify Z °, X ° as scalar, X 1 and y1 as vector and X 2 as tensor operators. The rank-2 tensor product of 0.1 and 0.2 is written as 10tI oi]1"2 and each • denotes the usual scalar product of two irreducible rank-K tensor operators 21). In equation (2.12a) the operators Y~Z °, Y, y1 and Y~X K, K = 0, 1, 2, are defined as the sum of all gluon exchange terms appearing in the first column of table 2, respectively. The pion exchange pieces X °, X 1, and X 2 in eq. (2.12b) are listed below the corresponding gluon terms in table 2. The operators in table 2 all involve the velocity operators
=#/4F,
=#/,/y,
(2.13)
where the arrows indicate whether the velocity operator acts on bra or ket. The propagators G are in general two-index objects because they have to rotate the gluon vector potential away from the generating currents in order to satisfy the boundary conditions n . E a = 0 and n x B a = 0 on the surface (a = 1 . . - 8 ) . We encounter the following propagators: (a) free field Go(r1, r2, w), Go =
cos
Irl - r21
Irl - r21
'
for all w,
(2.14)
355
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
(b) transverse electric GTE(ra, r2, to), 1 (GTE)q = -4~'to(ll)i(12)i l=l ~ ~ml(l + 1)
x (I + 1)nl (toR) - toRnt+l (toR) (I + 1)jt(toR) -toRfl+a(toR) ]t(torl)jl(torz) Ylm(g21) Y*m (J'-~2)
for all to,
,
(2.15) (c) transverse magnetic GTM(rb r2, to),
[
-4~rto (Vl ×/1)i(V2 ×/2)j
1
ndtoR)
,
X ~ Y'.--~,,':-~7, .--7--gT~xjt(tor~)jl(torE)Ytm(~l)Yt,,~(l'22), l=l into l ( l + l ) jl(tolX)
forto ¢ 0 (2.16a)
(GTM)~; ='
I1 ~
4Ir
(v o,
k-
+ (V,),
2
+r
The use of two different transverse magnetic propagators, depending on whether the frequency to vanishes or not, originates in a different choice of the gauge in order to perform the limit to -~ 0 as explained in ref. H). Due to the different gauge for to = 0 there arises a longitudinal electric propagator: (d) longitudinal electric GEE(r1, r2, o3 = 0),
1~~
GLE=R -
1=1
4"rr
2l+~
l+l(rl~t(r2~ '
l \ R ] \ - R : Yt~(J2~)Y*m(~2),
forto=Oonly.
(2.17) For any current satisfying the continuity equation the contribution of the transverse magnetic gluon propagator (2.16b) to the color fields vanishes for to = 0 because the divergence of the current vanishes. However, we will later also consider static approximations to the propagators, keeping the time dependence of charge and current distributions. Then, the divergence of the currents does not vanish giving rise to sizeable contributions from GTM(rl, r2, to = 0). For our purposes the propagator of the pion field is needed only on the bag surface. We include this restriction to the surface in the definition of the propagator and have (e) pion surface G~(rl, rE, to), 1
O~ = 6 ( r , - R ) 6 ( r 2 - R ) ~
[
hll)(toR)
,~E\ohll)(toR)/OR
+
,_(2),ttoR ' ) nl
"v ~
:,o ~v*
Ohl2)(toR)/OR] • ,mt,.,,H.,--m, (,..Q2), for all to.
Here h ~1), h Iz) denote spherical Hanckel functions z2).
(2.18)
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
356
TABLE 2 Orbital operators of the one-gluon and one-pion exchange forces HOGE and H= (first column) and reduced matrix elements of these operators for three different l p frequencies (a, b, c), the static (d) and the non-relativistic limit (e)
(ssllX°(1, 2)tlss)
(psi[X°(1, 2)lips)
(psi[X°(1, 2)lisp)
-- 1[(7~ 1 4- 7~1) o G 0 o (7~"2 + 7~2)] 0
19.2
13.1
23.2
~[(Tr 1 4- 71"1)o G T E o ( ~ 2 + "~'2)]
5.8
5.4
X°(1, 2)
3.1
1[(7~ 1 + 7~1) ° G T M o (7~'2 + ~'2)] 0
~[(zT1 x r~l)o Goo(~2 x ~2)] ° 1[(#1 × 7~1)o GEE o(7~"2 X 7~2)] 0 Sum of OGE: ½[(~1+~1)o
s o G,~ o(7r2 + rr2)]
X1(1, 2)
-4.1 -2.0
(0.0)
(0.0)
0.0
0.0
0.0
25.0
18.5
20.2
-12.5
-21.5
-18.7
(ssllXl(1, 2)llss)
(psllX~(1, 2)lips)
(psllX~(1, 2)lisp)
i [(~'1 + #1) o Go o(T~24- 7~'2)] 1
-0.3
-i[(~1 + ~'x) o GTE o (#2 + 7~'2)] 1
-6.7
- i [ ( ~ l + r~l) o G T M o ( ~ 2 + ~ 2 ) ] 1
--8.8
-i[('~1 x ~1)o Goo(#2 x ~2)] 1
-4.3
- i [ ( f f l X ~'1)o GLEO ( ~ 2 X ~ 2 ) ] 1
0.0
Sum of OGE:
-20.0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-i[(~a + r71) o G~. o (r~z + ~ 2 ) ] a
X2(1, 2) --[('~ 1-1- ~'1) o G o o (7~2 -- 7~2)] 2 [(7~1 + '/~1) o G T E o ("/'~24- 7~2)] 2
12.4
(ssllX2(1, 2)ltss)
(psllX2(1, 2)lips) -21.0
-4.2
[('w14- '~1) o G y M o (,t~2 -'l---~2)] 2 --[(~'1 X q~l)O G 0 o (7~2 x qT2)] 2 --[(~1 X 7~1) o GLEO(~2 × 7~2)]2
Sum of OGE: -[(r~l + 61) ° G~ o(~2 + ~'2)] 2
(psllX2(1, 2)lisp) 17.8 - 1.2 --8.0
(0.0)
--3.9
0.0
0.0
-25.2
4.7
16.4
15.5
2a: ~o~,= 3.40
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
357
19.2
20.2
26.9
19.2
9.3
22.8
19.2
13,1
20.0
21.2
25.6
25.6
5.8
7.3
3.6
5.8
4.2
3.0
5.8
5.4
3.2
3.8
6.5
2.0
-1.3
-8.0
-2.2
0.0
(0.0)
(0.0)
-1.7
(0.0)
(0.0)
-2.2
(0.0)
(0.0)
-1.6
(0.0)
(0.0)
-3.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-0.2
0.0
0.0
-2.2
25.0
27.6
27.5
25.0
13.5
15.6
25.0
18.5
19.3
25.0
32.1
22.1
-12.5
-17.8
-15.9
-12.5
-21.1
-18.9
-12.5
-21.5
-19.6
0.0
0.0
0.0
-7.6
4.8
1.1
0.0
-7.6
-6.4
-6.8
-4.3
-2,7
-17.0
-4.8
0.0
-3,5
-4.7
-3.3
-7.0
0.0
0.0
-0.4
-4.7
-21.5
-23.2
14.2
16.0
10.9
12.2
-0.7
0.0
-14.1
16.7
-25.0
20.3
-21.0
16.1
-13.8
19.8
-1.8
-1.4
-5.6
-1.2
-4.2
-1.3
0.0
-0.8
-2.5
-15.5
-4.4
0.0
(0.0)
-3.2
(0.0)
-4.3
(0.0)
-3.0
(0.0)
-6.4
0.0
0.0
0.0
0.0
0.0
-0.4
0.0
-4.3
-15.9
9.6
-30.6
-0.7
-25.2
7.1
-13.8
8.4
20.6
12.8
10.3
15.9
16.4
31.1
0.0
0.0
2b: top = 3.81
2c: OJp= 3.20
2d
2e
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
358
Y~(1, 2)
2)llss>
i[(ffl + 771) ° Go o ( i f 2 - 7?2)] 1 -- ff2)] 1 --i[(~1 + ~ 1 ) o GT M o (~2-- 772)] 1 i[('B', x 771)o Go] 1
-- i[(7'~ 1 "4- 7~1) o GT E o (if2
--i[(ff 1X ffl)' Go(if2" 7?2)] 1 i[(ffi x 771)" GLE] 1 - i [ ( f f l x if1)' GLE(ff2 ' if2)] 1 Sum of O G E :
Y'(1, 2)
i[(21 + 771) ° Go ° ( 2 2 - 7?2)] 1 - i [ ( 2 1 q- ¢~'1)o GTE° (22 -- 7?2) 1
(psi[ Y~+ (1, 2)lips)
(psllY~+ (1, 2)lisp)
56.9 15.2 (48.9) (12.6) 0.0 0.0
-64.5 0.0 77.6 -36.1 -6.0 0.0 0.0
72.2
-29.1
(psll Y ~_(1, 2)lips)
(psll Y 1_(1, 2)l}sp)
-56.9 --15.2
--i[(7~1 + 771) o G T M o ( 2 2 -- 7?2)] 1 i[(7~1 × 7?1)" G o ] 1
(48.9)
-i[(¢71 x 771)" G o ( # 2 ' 7?2)] 1 i[(771 x '~'1) ' GLE] 1
(12.6) 0.0
- - i [ ( ~ I × 771)' G L E ( 2 2 ' 7?2)] 1
0.0
Sum of O G E :
Z°(1, 2)
-72.2
(psll/°(1, 2)liPS)
- [ ( ~ 1 - #~)" Go' (~'2- ~'2)] ° [(7~1 -- ~ 1 ) " G T E ' ( 2 2 -- ~ 2 ) ] 0 [(T~I -- q'~l) ' G T M ' (7"~2-- qT2)] 0
Go(1-2x'
"t7"1-22" ~2)
G 0 ( 2 1 ' ~1)('/~2 ' 7?2)
(169.8) (10.9)
(339.6) (34.7)
GEE(1 - ~'1" 'ff'l - 2 2 ' 22) GLE(21 ' ~1)(~'2 ' ~2) Sum of O G E :
121.7 70.8 2.0 0.0 0.0 134.4
2a (cont'd): wp = 3.40
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
54.5
-43.4
58.7
-84.4
56.9
-75.9
61.4
-60.3
13.6
0.0 27.2
16.2
0.0 135.1
15.2
0.0 48.5
9.2
0.0 17.8
(29.8) -33.6
(61.9) -36.5
(48.9) -27.7
(6.2)
-8.1
(16.9)
-4.5
(12.6)
-4.5
0.0
0.0
0.0
0.0
0.0
-3.3
0.0 - 1 7 . 9
0.0
0.0
0.0
0.0
0.0
-1.1
0.0
0.0
68.1
-57.9
74.9
9.7
72.2
-64.0
70.7
-90.6
(30.7) -30.2 (0.0)
-54.5
-58.7
-56.9
-61.4
-13.6
-16.2
-15.2
-9.2
(29.8)
(61.9)
(48.9)
(30.7)
(6.2)
(16.9)
(12.6)
(0.0)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-68.1
-74.9
-72.2
-70.7
-32.8
-80.8
48.0 (169.8) (362.7) (10.9)
359
(39.9)
83.6 4.2 00 0.0 103.1
2b (cont'd): top = 3.81
-114.0
190.1 (169.8) (334.1) (10.9)
62.7
(33.7)
1.2 0.0 0.0
(10.9)
(34.7)
173.2
2c (cont'd): ~op= 3.20
-122.2
72.1 (169.8) (339.6)
53.9 1.5 7.8 0.5
66.5 (167.5) (383.1) (0.0)
(0.0)
21.9
2d (cont'd)
0.0
55.2 0.0 48.7 0.0 48.2
2e (cont'd)
To reconvert the units, defined in the text, the numbers have to be divided by 77.49 for gluon and 134.3 for pion matrix elements.
360
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
The square brackets in table 2 indicate the decomposition of products of velocity operators and propagators into spherical rank-K tensors according to [ U oG o V] K=°'M=° = eiikenmUiGkm Vt ,
(2.19a)
[Uo G
e3i,,,eiike,tmUiGkm Vt ,
(2.19b)
3 1 x/~ (613 -- g)eijkeilmUiGkm VI,
(2.1 9C)
o V] K=I'M=O =
[UoGoV]
K=z,M=O=
where 8qk is antisymmetric and the notation
8123--'--
[U. G] T
1. For the Y~ and Z ° operators we also use
M
= UiGi3,
(2.19d)
[ U " G V " W ] K=''M=°= UiGi3 V/Wi ,
(2.19e)
[U. G. V] K =0.M=o = UiG~,Vj.
(2.19f)
In the remaining columns of table 2 we have listed the reduced matrix elements of the corresponding operators for the s- and p-states defined in (2.7). We also encounter the symmetric and antisymmetric part of Yt y 1 (1, 2) = Y'(1, 2)+ Y1(2, 1).
(2.20)
A discussion is postponed to the next section. We are now is a position to write down the matrix elements of H in the N* and A* model space. The mass matrix can be written as a sum of a diagonal part which shifts all states by the same amount and a splitting matrix which includes all spin-dependent terms: M[~ Mo
fi°°
0
+ Mo
0 Nj
0
Md~
0
NJ M~':n~
0 M~i~ Ep N,J
(2.21)
I
The elements of the 2 × 2 submatrix in the N* sector are characterized by the spin-symmetry indices in table 1. For the overall J-independent mass shift in the spectrum one obtains Mo
O)p - - COs
OLs
+g-k-[3(ssll E x°llss)
+ 2 +'/~(-2(psll E Z°llps> + )] 1 1 o - 16f~R 3 5.
(2.22)
For convenience we explicitly evaluate the J-dependent recoupling coefficients and
361
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
employ the notation
Ial Ia
(2.23)
b = b ifJ =3. c c ifJ =5
Then we obtain for J - ~, 3 M ~A J
= ~s [(ssll E x ° l l s s ) - 43 (4(psll Y. x°llps>
6R
1 ~ - 5 ( p s l l E x o Ilps))-g~/g
,o17
1 ((psllE g+llps)
+ (psllE Y~ lisp) - 3(psllY. y1 lips) + 3(psllE xaHsp))] + 1 6 f @ . [(ssllX°l[ss)- x/3r(4(psl[X°Hps)
- 5(psllX°llsp)) - q~g
Eil
(psllX~l]sp)
+KOJp -~-
I Eil
.
(2.24)
In the N* sector we have M~
= a~-~{(ssll~ x°llss) +,/3r-(2(psHY~x°lips)
- (ps[lZ x°]]sp)) + 245 [- 2 ](2(psllE x2llps) t_ la6.a
- (psllY x 2 lisp)) - gi ~/g~
(2(psE Y+1 lips)
1 - (psllZ YLlisp))} +~{(ssllX°llss) _ ,/r (4(psllXOllps> _ 5(psllX Ollsp)) 1
- ~45 I- ~](4(ps[[X2llps) - 5 (psl[X 2
~ KOJpV[25], (2.25)
362
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
N,J
Ols {
MUz~z = ~-- -
4~(2(psllYx°llps)
o
(psllYX [[sp))+g ~
-
(3(psllEx'llsp)
2(ps[[• Y~ lips)+ (psllY ya+lisp))}
+~
1
0
{ 5
1 K.pf2 l OR /
-2(psllEX2llsp))
+14~ a g ,4- 0
((psHE gl+llps)
- 2(psllY. y 1 lisp) + 3(psIIE Y! lips))} + ~ { - 21 ~ / }
I i ! l -( -) j (psl~(lllsP)
(5(pslIXZllps) - 4(pslIX211sp)) j
+ 2KtOp -~-
. (2.27)
The parameters in our model are the bag radius R, the color coupling constant as and the pion decay constant f,~ which should, however, come out to be close to ~93 MeV. Although we cannot expect to reproduce the ground-state energy of the nucleon and the isobar, the model should be suitable to obtain the correct energy differences such as the N-A mass split M a - M N = 300 MeV. In terms of reduced matrix elements this reads Ma
C~s - MN = ~--
o 1 6(ssllY~X Ilss) 16f~R 312(sslP(°llss).
This has to be considered as an additional constraint on our parameters.
(2.28)
H.R. Fiebig, B. Schwesinger/ Splitting mechanisms
363
3. Discussion of results The reduced matrix elements of the various force terms are listed in table 2 for three values of top and for the static and non-relativistic limit. In order to make the relative importance of gluon and pion terms more transparent the matrix elements are given in units of a s / 6 R and 1 / ( 1 6 f ~ R 3 ) , respectively, with the parameters being fixed such that gluon and pion exchange interactions build up the N-A mass difference of 2 × 150 MeV to equal amounts, cf. eq. (2.28). Blank positions in table 2 indicate that the corresponding matrix element vanishes a priori (mostly because of some symmetry reason). The sums of all one-gluon exchange terms in table 2 do not include the contributions in brackets which arise from propagation of static electric monopole fields; these violate the boundary conditions and have to be eliminated 12.13.11). Most of them vanish anyhow, except for y 1 and Z ° terms. The latter do not contribute to the splitting in the N*, A* spectrum. The y1 terms however are just part of the spin-orbit force. Thus we encounter a first clue as to why spin-orbit forces are highly suppressed: the omission of static electric monopole terms removes about one third of all spin-orbit interaction energy coming from gluon exchange. In table 2a the matrix elements for the average p-state frequency top = 3.40, introduced to eliminate c.m. motion, constitute our main result. The numbers in the next two columns, tables 2b, c, correspond to the wave numbers for massless quarks in the lpl/2 (top= 3.81) and the lp3/2 (top= 3.20) cavity mode and enable us to compare our results to the work of D e G r a n d and Jaffe 12) and D e G r a n d 13). These numbers also give us a feeling about the frequency dependence of the matrix elements. In table 2d we give the results for a static approximation. Time-dependent charge and current distributions occur only in the exchange matrix elements where to = top-tos appears in the propagators. The approximation is performed by taking the limit to -~ 0 in (2.14)-(2.18) but retaining top = 3.40 and tos = 2.04 everywhere else. In order to avoid the calculation of even more matrix elements, the part of the expression ]o,(1)" GTM(1, 2;to = 0)']~ (2) involving the third term of the transverse magnetic propagator (2.16b), i.e. V 1
t=l
47r
, 2+
trl
2,[r1~ l(r2~l
r 2 ) t - ~ ) \~,. ' Ylm(~"~l) gl*m (~2)
(3.1)
has been integrated by parts. The boundary terms vanish; and the divergences of the currents can be replaced by charge densities using the continuity equation V "]o, = itoo,o.
(3.2)
After these manipulations the force term of (3.1) has the same algebraic structure as the longitudinal electric (LE) terms; and their matrix elements have just been added to them in table 2d. The static approximation hardly effects matrix elements
364
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
involving free propagators. The changes in boundary terms are already bigger but still not very dramatic. We would like to point out that the procedure of performing the static limit has been consistently applied to the propagators only, lim,,_,0 G(1, 2; to), and not to the matrix elements, which would amount to lim,,~0],~(1). G(1, 2; to).],,(2). This would make a difference since, in the latter case, for static currents the contributions from the transverse magnetic gluons vanish exactly, whereas we encounter sizeable T M contributions in table 2. The biggest one by far is the Y~+ T M matrix element which would help to cancel about a third of the spin-orbit energy in static approximations. Table 2e gives the non-relativistic limit of the matrix elements, where the masses of the quarks are taken to be large compared to the frequencies to~ = 7r, top = 4.49. Again there are modest changes when comparing to the extreme relativistic cases (tables 2a-c). Obviously the pion field now decouples from the quarks in the bag because the axial quark current vanishes at r = R. Let us now insert the numbers for the matrix elements from table 2a into the splitting part of the mass matrix (2.23-2.27). The result of this is arranged in table 3. The contributions are grouped into four categories: free: boundary: pion: Thomas:
interaction energy from exchange of free gluons corrections to the gluon exchange energy from the boundary interaction energy from pion exchange contributions from the one-body spin-orbit force (2.11) (Thomas precession).
These four categories are further subdivided into contributions coming from spinexchange, tensor or spin-orbit forces. The parameters are, again, chosen such that gluon and pion fields contribute 150 MeV each to the N-A split and R is taken to be R = 0.8 fm. We do not claim that these parameters should have these values, but we do think they reflect the correct order of magnitude with which different effects participate in the splitting of the odd-parity states. In any case, by keeping the four different categories mentioned before, we are still free to apply any weighting factor on the three participating mechanisms: gluon exchange, pion exchange and Thomas precession. In order to compare with constituent quark models we have included the matrix elements of the Fermi-Breit interaction 3) evaluated for oscillator states. Here, the combination of ets with the oscillator length is fixed to give a 300 MeV N-A split (furnished entirely by gluons). (a) Spin-exchange forces. We first concentrate on the spin-exchange forces (X ° terms). In harmonic-oscillator constituent quark models the split of the (S = 3) and (S =1) nucleons caused by the Fermi-Breit interactions is just a half of their contributions to the nucleon-isobar split; the A* remain degenerate with the N * (S = 3). The same feature also shows up in the bag model. The sum of free and boundary terms from the spin-exchange part of HOGE is 35 MeV for N*- (S = 3),
365
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
TABLE 3 Mass splitting matrix for the N* and A* spectrum in the bag model and the constituent quark model (CQM) Element
Force
Free Boundary
F1--I
11|×
spin-exchange
Pion
Thomas C.Q.M.
[ 22.
+13.
-17.]
[75.]
1/2-] /-2/q×
[-83.
+1.
+18.]
[-150.1
[~]x
[-30.
+6.
L 1A
[M ~N,, E~
tensor
L1/IOJ
spin-orbit
+17.1
[-75.]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [. -. . 3. .J. . . . . . . . . g-~-I
JiJX
spin-exchange
[
--
]
[0.]
[-73.
+21.
-30.]
[-150.]
[ 14.
-25.
-20.
-13.
-113.]
[-75.]
--
]
[0.]
-10.
-17,]
[+75.]
--
]
[0.]
F-1/27 M~
tensor
11/,/40/× L
spin-orbit
s'-exc"ao,e M~SJ~n
0.A
i-O/
x
/;|x
-34.]
[50.]
L1A
tensor
x
[
spin-orbit
+0.
LoJ spin-exchange
MU[5~
i-17 /1/X
[50.
tensor
I~Tx L0A
[
spin-orbit
[-1Ix
[-15.
L0d
-12.
-10.
-17.]
[-25.]
40 M e V for A*- and - 3 5 M e V for N~- (S = 1), which is a b o u t half of the gluon contributions to the N-A split fixed here at 150 M e V . Surprisingly the s p i n - e x c h a n g e forces f r o m pion e x c h a n g e also show this feature: - 1 7 M e V for N~*- (S =3) and for za~' and - 1 1 3 M e V for N j* (S =½). T h e r e f o r e any ratio of gluon e x c h a n g e to
366
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
pion exchange contributions will give roughly a 150 MeV split between N*- (S = 3), Aj-N~ (S = 1), provided they contribute 300 MeV to the N-A split. (b) The tensor forces. The spin-exchange forces alone give a remarkably good fit to the experimental splittings shown in fig. 1. The only quantities which don't fit are the wave functions of the N1/2-, * which are measured to mix by an angle of approximately 30 ° [for further details see ref. 7)]. For this reason Isgur and Karl 7,8) also included the tensor force of the Fermi-Breit interaction excluding deliberately spin-orbit terms. This procedure then resulted in a surprisingly good fit to large parts of the baryon spectrum. Tensor forces can only be operative if at least one S = 3 state is involved in the matrix element. Therefore the tensor forces only show up in the diagonal element M N~, J and in the off-diagonal elements M ~ . In the latter gluons and pions contribute roughly equal amounts but they only add up to about half of the value given by the constituent quark model. Even if the entire N-A split were attributed to gluon exchange the bag model is different from constituent quark models in that respect. As can be seen in table 3 the reason for the difference are the boundary terms. Table 2 then shows that the transverse magnetic contributions carry the largest share. In the diagonal element the pion tensor forces are negligible, whereas the gluon contributions of the bag model and constituent quark model are practically equal (Remember that the gluons in the bag are fixed to contribute 150 MeV to the N-za split in contrast to the 300 MeV for the constituent quark model.) We may therefore state that the mixing in the N*- from tensor forces alone is quite independent of the ratio between gluon and pion contributions and smaller than the corresponding constituent quark model value. (c) Spin-orbit forces. It is quite easy to see why the spin-orbit terms of the Fermi-Breit interactions in constituent quark models are incompatible with a good • fit to the experimental spectrum: m M N~J they contribute as much as 8 x 75 MeV to a split in the upper N* (S = 3), whereas experimentally these states are nearly degenerate! The numbers in the other matrix elements are less spectacular though still large. The bag model on the other hand does quite well here, since the pion field does not contribute anything and Thomas precession terms together with the boundary terms cancel all contributions from the free gluons. One should recall that we have already eliminated a larger portion of spin-orbit interactions because static color electric gluon monopole fields violate the boundary conditions. Thus, color confinement effects, the spin-orbit forces from Thomas precession, and certainly also the pions, work together to suppress spin-orbit forces. Of course, the extent to which the spin-orbit forces are suppressed depends on the choice of the parameters as, f~ and R. Cancellations are also present in the other N* matrix elements. In the off-diagonal element they actually even overshoot. As to the A* we see that these cancellations do not occur: the free and boundary terms again are as big as pionic contributions and the Thomas precession effects add up splitting the A* by 162 MeV. This split shows the correct ordering, A*/2- is heavier than
H.R. Fiebig,B. Schwesinger/ Splitting mechanisms
367
A1/2-, although it is too large. As pointed out by Close and Dalitz 23) this split is one of the major problems for constituent quark models: the N* spectrum can only be fitted by eliminating spin-orbit forces in which case A*/2 and A*/2 remain degenerate. We have shown here, that there are no such difficulties in a bag-model description. (d) Comparison to other bag-model calculations. To our knowledge the only attempt to calculate the spectra of odd-parity states in a bag model was in the work of DeGrand and Jaffe 12.13). The results in ref. 12) did not agree with experimental findings because c.m. excitations of quarks in the bag were not eliminated. This was improved on in a second work 13) by identifying the c.m. motion with a certain linear combination of basis states. The procedure there, however, can work at best approximately because the lpl/2 and lp3/2 p-states were left non-degenerate. Further in ref. 13) all processes involving both (non-degenerate) p-states simultaneously were ignored and finally all exchange processes were neglected because they were found to be small in the previous work 12) [see also ref. 14)]. If we insert only the exchange terms of tables 2a, b, c into the mass matrix we find that they contribute a fourth of the spin exchange forces and half of the tensor forces in the off-diagonal element. Roughly half of the spin-orbit interaction comes from exchange terms. In order to clarify the discrepancy and to check our numbers, we have included the spurious state in our basis and calculated the gluon exchange energy for the only J = 23-state that can be formed from two quarks in lsl/2 states and one in a lpl/2 state. This is given by 3R <] =
2c~
IHoo
lJ = b =
+
-
'
"
"
= l- ~{,/r(psllE x°llps> + ~4g-,/](psllE Y+Hps>} - l~{5~- 2v~(psllY xZllsp) - ~(psllE Y+llsp>
+ 2,/~ - ~ (psllY g °lisp>}.
(3.3)
Comparing this to the corresponding expression of ref. 12), we can identify 0A s - - 1
2
~ r .... = (---~_21) M,,(0, 0),
(3.4a)
~rp, ps = (0As-- 1)(0Ap+ 1) Ms,(0, 0) '
(0As-
)(0A.
"
A~rp,sp = (0A'- 1)(0Ap+ ~ 8 (0, 0) ,
(0A
(3.4b) '
(3.4c)
'
where M~(0, 0), M~p(0, 0) and 6 (0, 0) are defined in ref. 12). In deriving (3.4a)-(3.4c) we have corrected for the errors of the Foldy-Wouthuysen transformation [which
368
H.R. Fiebig, B. Schwesinger / Splitting mechanisms MeV
1700-
1600
1500
1400' NI/2-
Ns/2-
Ns/2-
L~l/2-
Z~3/2-
Fig.
2. Calculated N* and A* mass spectrum normalized to the nucleon g r o u n d state MN = 940 M e V (. . . . ), R = 0.8 fm. Calculated s p e c t r u m without anti-symmetric s p i n - o r b i t terms ( . . . . ), R = 0.7 fm. E x p e r i m e n t a l spectrum f r o m ref. 5) ( ).
gives the frequency-dependent factors 11)] and for a missing factor of 2 in 8(0, 0) [erratum in ref. 13)]. Insertion of the corresponding numbers of table 2b yields perfect agreement, and indeed the exchange matrix elements in 8(0, 0) are small. This, then, means that the exchange matrix elements in the calculation of ref. 12) were small because the c.m. excitation was mixed into their states causing an accidental cancellation. (e) Mass spectrum. The diagonalization of the splitting matrix in table 3 doesn't yield a good fit to the experimental spectrum, fig. 2. This is entirely due to the fact that the cancellations from boundary, pion and Thomas precession terms overshoot the free gluon exchange contributions in the off-diagonal element. This results in a strongly suppressed mixing of the N1"/2 and a too-strong mixing and splitting of the N*/2 since the tensor force terms cancel in the first case and add up in the second. With the parameters c~, f= fixed to give 150 MeV for the N-A split each at R = 0.8 we have O~s= 1.9, f,~ = 83.6 MeV. Both numbers include factors originating from corrections to the Foldy-Wouthuysen transformation; the unrenormalized numbers for o~s and 1/f~ would be smaller by a factor (oJ~-1)2/(~o~ - 1)2= 0.46, cf. ref. 11). This choice of parameters gives M0 = 675 MeV for the shift of the states (2.21) relative to the nucleon at 940 MeV; this is too small by about 50 MeV. A smaller radius of the bag would give a bigger shift but in order to keep the pion contributions small we would have to choose very small values of f=. Too-large pion contributions on the other hand are ruled out, because then gluon contributions to the N-A split would be small, which in turn would result in too-small X-X* and -~--=* splittings. Pions are not contributing there.
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
369
The major shortcomings in our calculation originate from approximations in order to eliminate the c.m. motion. Boundary terms were neglected twice in the Foldy-Wouthuysen and the scaling transformation 11). The former gives rise to additional factors to the matrix elements, an example of which is given in equations (3.4a, b, c). Since in general these factors depend on the states involved in the matrix elements, they cannot simply be absorbed in the parameter combinations as~R, 1 / f E R 3. In the space of states considered here, these effects act like an additional spin-orbit force. Further spin-orbit contributions will be encountered if the scaling transformation is also applied to the interaction terms 11). The size of these effects can be estimated by inserting the wp dependent matrix elements of tables 2a, b, c in the mass matrix (2.22-2.27). The variations are considerably smaller than the spin-orbit terms from direct electric contributions and also smaller than corresponding boundary terms from the transverse magnetic gluons. Therefore we do not think that these approximations endanger our explanation of why spin-orbit forces are suppressed in odd-parity nucleons, although they certainly will modify the calculated spectrum. Further errors more difficult to estimate arise from the fact that the pion field couples only to the surface, because just there major approximations were involved. Certainly, the pion exchange energy will also be sensitive to different bag radii for different states and to a deformation of the bag in order to balance the bag pressure for the lp3/2 states. These corrections might improve the quality of the spectrum, as well as a fit of ~s, f~ and R would do. However, we believe that the parameters as and f~ have a more fundamental meaning. Another source of errors might be hidden in spurious contributions from c.m. motion. Although our basis states were constructed by exciting purely intrinsic degrees of freedom this only means that c.m. motion in excited states vanishes to the same extent as it vanishes in the ground state. Since HOGE (as well as the Fermi-Breit interaction) contains terms which depend on the c.m. momentum (e.g. P 1 +P2 = 31-(p1 q - P 2 - 2 P 3 ) + 2 p . . . . ) we will clearly have to expect contributions from them. These terms are all contained in (a non-relativistic reduction of) the operator ~(tra -or2)" E yl_ (1, 2),
(3.5)
which is part of HomE. In a similar way it follows also that ½i(crl × 1111'2)'XI(1, 2),
(3.6)
which is part of H= (2.12b) depends on c.m. operators. N,J Matrix elements of these antisymmetric spin-orbit operators occur in M ~ A,J and M~E ? , which are the numbers causing the major discrepancies to the experimental spectrum. (There is also a term of the structure of (3.6) in Home (2.12a) giving minor contributions to M ~ . ) Although these forces also contain purely intrinsic pieces it is legitimate to estimate the effects of spurious parts by neglecting all contributions to the mass matrix from X 1 and Y~_ terms (the oscillator matrix
370
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
elements of these terms are less than half the bag-model value). The resulting spectrum is shown in fig. 2, in order to show how well the experiment can be fitted; the radius is R = 0.7 fm and gluon and pion exchange contribute equal amounts to the N-A split. This is only a rough estimate, however, it shows that the reason for missing the experimental numbers might well be buried in the treatment of c.m. motion, which apparently is still insufficient. 4. Conclusions The odd-parity states of non-strange baryons have been investigated in the chiral bag model. The main motivation was to understand why spin-orbit forces are highly suppressed in negative-parity excited nucleons and show up in the excited isobars. Considerable effort was necessary in order to simultaneously maintain the relativistic features of the bag model and to eliminate c.m. excitations of the quarks. The present approach has led to the construction of one-gluon and one-pion exchange forces which incorporate correct bag boundary conditions on gluon and pion fields. Calculation with these forces is very similar to working with constituent quark models. We have shown that the bag model contains all features necessary to explain experimental findings on spin-orbit forces. The main contributions to the suppression of spin-orbit forces are found to originate from confinement of the gluon fields. Deviations from the 1/r behaviour of the gluon propagator at larger distances, which here is simulated by the bag boundary conditions on color electric and magnetic fields, affect mainly the spin-orbit terms, at least in the space of N* and A* states considered here. The spin-orbit forces coming from Thomas precession of the relativistically moving quarks then cancel to a large extent the spin-orbit forces from one-gluon exchange in the odd-parity nucleons and add to them in the isobar states. Forces from the exchange of pions outside the bag show the same features. In the odd-parity states the spin-exchange and tensor forces pieces operate in even quantitatively the same manner as their gluon counterparts. As to the spin-orbit forces, they again are quite negligible in the nucleon states and contribute like the gluon exchange to the odd-parity J = ½ and J = 3 isobar splitting. These features explain the "mystery of the missing spin-orbit force" without commitment to special values for parameters, which are the color coupling constant, the pion decay constant and the bag radius. The similarity to constituent-quark-model calculations allows a direct comparison. Constituent quark models using a Fermi-Breit interaction without spinorbit terms give roughly the same results apart from the A* split. This is quite surprising since the bag-model approach is extremely relativistic and incorporates important effects from confinement. Therefore one is lead to believe that the agreement is just accidental. On the other hand, the remarkable success of the
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
371
c o n s t i t u e n t q u a r k m o d e l in fitting a vast a m o u n t of b a r y o n p r o p e r t i e s s e e m s to c o n t r a d i c t an a c c i d e n t a l a g r e e m e n t . A s t h e t h e o r i e s st an d now, t h e b a g - m o d e l a p p r o a c h n o t only has t h e i n d i s p u t a b l e asset of b e i n g m o r e f u n d a m e n t a l , it also is c a p a b l e of e x p l a i n i n g t h e e x p e r i m e n t a l findings c o n c e r n i n g s p i n - o r b i t forces, w h i ch in c o n s t i t u e n t q u a r k m o d e l s a p p e a r to b e a mystery. W e are g r a t e f u l to G . E . B r o w n for i n t e r e s t i n g us in t h e p r o b l e m . T h e a u t h o r s also a c k n o w l e d g e h e l p f u l discussions with J. D u r s o an d t h a n k M. H a r v e y f o r c h e c k i n g phases.
References 1) F.E. Close, An introduction to quarks and partons (Academic Press, New York, 1979) 2) H.A. Bethe and E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer, Berlin, 1957) 3) A. De Rfijula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147 4) Proc. IV. Int. Conf. on baryon resonances, Toronto, 14-16 July 1980; D. Gromes, Hyperfine interactions in baryons, p. 195 L.J. Reinders, Spin-orbit forces in the baryon spectrum, p. 203 5) R.E. Cutkosky, C.P. Forsyth, J.B. Babcock, R.L. Kelly and R.E. Hendrick, Pion-nucleon partial wave analysis, in ref. 4), P. 19 6) Particle Data Group, Rev. Mod. Phys. 52, No. 2, Part II, Apr. 1980 7) N. Isgur and G. Karl, Phys. Lett 72B (1977) 109; Phys. Rev. D19 (1979) 2653 8) N. Isgur and G. Karl, Phys. Rev. D18 (1978) 4187; L.J. Reinders, J. Phys. G4 (1978) 1241 9) T.D. Lee, Phys. Rev. D19 (1979) 1802 10) F.E. Close and R.R. Horgan, Nucl. Phys. B164 (1980) 413 11) B. Schwesinger, Nucl. Phys. A393 (1983) 335 12) T.A. DeGrand and R.L. Jaffe, Ann. of Phys. 100 (1976) 425 13) T.A. DeGrand, Ann. of Phys. 101 (1976) 496 14) T.A. DeGrand, Thinking about baryon excited states with bags, in ref. 4), P. 209 15) C.G. Callan, R.F. Dashen and D.J. Gross, Phys. Rev. D19 (1979) 1826 16) G.E. Brown and M. Rho, Phys. Lett. 82B (1979) 177 17) R.L. Jaffe, Lectures atthe 1979 Erice Summer School, Ettore Majorana, MIT preprint CTP 814 (1979) 18) V. Vento, M. Rho, E.M. Nyman, J.H. Jun and G.E. Brown, Nucl. Phys. A345 (1980) 413 19) A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. D10 (1974) 2599 20) T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 21) A.R. Edmonds, Angular momentum in quantum mechanics (Princeton University, 1974) 22) M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover Publ. Inc., New York, 1970) 23) F.E. Close and R.H. Dalitz, in Low and intermediate energy kaon-nucleon physics, ed. E. Ferrari and G. Violini (D. Reidel, Dordrecht, Holland) pp. 411~,18
ANALYSIS OF SPLITTING MECHANISMS IN THE LOW-LYING NON-STRANGE NEGATIVE-PARITY BARYONSt H.R. FIEBIG* and B. SCHWESINGER** Physics Department, State University of New York, Stony Brook, N Y 11794, USA
Received 1 June 1982 Abstract: We investigate the mass splitting in the negative parity N* and A* spectrum where, experimentally, the spin-orbit force is found to be highly suppressed. A hyperfine interaction between quarks derived from a one-gluon and one-pion exchange model for a spherical chiral bag is used. The interaction is, unlike the usual Fermi-Breit force, fully relativistic and explicitly takes into account confinement effects. We analyze in detail how the force operates in the odd parity N* and A* spectrum and find that the suppression of spin-orbit forces, which is mysterious in constituent quark models, can be explained.
1. Introduction
T h e p r e s e n t c o n c e p t of a b a r y o n b e i n g an i n t e r a c t i n g q u a r k - g l u o n s y s t e m inside a Q C D v a c u u m b u b b l e involves t h e o b l i g a t i o n to u n d e r s t a n d t h e p r o p e r t i e s of t h e o b s e r v e d p a r t i c l e s p e c t r u m on this basis. In o r d e r to test t h e o r e t i c a l n o t i o n s a n d m o d e l s it is r e a s o n a b l e to c o n c e n t r a t e o n a distinct p a r t of t h e h a d r o n m a s s s p e c t r u m . In this p a p e r we w a n t to a d d r e s s the n o n - s t r a n g e s e c t o r of t h e o d d - p a r i t y [70] SU(6) m u l t i p l e t . T h u s w e e n c o u n t e r five e x c i t e d states ( i s o d o u b l e t s ) of t h e n u c l e o n , N*, a n d two e x c i t e d states (isoquartets) of t h e isobar, A*. This s a m p l e of p a r t i c l e s has t h e a d v a n t a g e of an, a p p a r e n t l y , s i m p l e s u b s t r u c t u r e involving o r b i t a l e x c i t a t i o n s of q u a r k s in b a r y o n s i). It serves, t h e r e f o r e , as a s u i t a b l e field to s t u d y t h e forces b e t w e e n q u a r k s as t h e y arise f r o m gluon e x c h a n g e inside a s e c l u d e d r e g i o n of space. In t h e p r e s e n t w o r k we try to a n a l y z e t h e s e forces within t h e b a g - m o d e l f r a m e w o r k a n d s e e k to u n d e r s t a n d c e r t a i n f e a t u r e s in t h e m a s s s p e c t r u m of t h e N* a n d A* n e g a t i v e p a r i t y states. T h e q u a r k c o n t e n t s of t h e s e states, t h e e x p e r i m e n t a l s p e c t r u m of which is given in fig. 1, c o n v e n i e n t l y is d e s c r i b e d b y m e a n s of SU(3) Y o u n g t a b l e a u x . T w o flavors (isospin c o m p o n e n t s ) a n d two spin c o m p o n e n t s yield a t o t a l l y s y m m e t r i c a n d a m i x e d s y m m e t r y p a t t e r n for isospin a n d spin, cf. t a b l e 1. T h e o d d p a r i t y states c o n s i d e r e d a r e i n t e r p r e t e d as h a v i n g o n e q u a r k e x c i t e d to an o r b i t a l p - s t a t e , t h e t w o o t h e r q u a r k s a r e left in t h e l o w e s t s-state. T h e p o s s i b l e o r b i t a l states a r e thus * Supported by Heinrich-Hertz-Stiftung. ** Supported by a fellowship from the Scientific Committee of the NATO via the German Academic Exchange Service (DAAD). t Work supported in part by US DOE Contract DE-AC02-76ER13001. 349
350
H.R. Fiebig, B. Schwesinger / Splitting mechanisms MeV 17001650 (1650)
1675 (1700)
f620 (~650)
1600
1550 (1535) 1500' N i/2-
1710 (1670)
1675 (1670)
1525 (1520)
N 3/2-
N5/2-
A (72-
3/2-
Fig. 1. Experimental spectrum of low-lying negative-parity baryon masses. The positions graphically indicated are compiled from ref. 5). The numbers in brackets are taken from ref. 6).
again restricted to totally symmetric and mixed symmetry patterns. However, the totally symmetric representation belongs to spurious center-of-mass (c.m.) motion and should be excluded. Table 1 shows the isospin (T), spin (S), orbital (L) and total angular momentum of the different patterns. The coupling of the T-, S- and L- representation to a totally symmetric pattern yields seven basis states (cf. sect. 2) which correspond in number and quantum numbers to the states experimentally observed (fig. 1). The quark-quark forces arising from one-gluon exchange are in the non-relativistic constituent quark model given by the Fermi-Breit interaction 2.3). This force has a spin-exchange, a spin-orbit, a tensor term and other spin-independent pieces. As one can see from table 1 all three terms should be operative in the odd-parity N * - d * spectrum. However, there is a clustering into two degenerate groups of particles (fig. 1), which is even more pronounced if one looks at the numbers of the Particle Data Group 6), fig. 1 in brackets. Isgur and Karl have noticed that the mass spectrum of the odd parity N*'s and A*'s can be reproduced in a harmonic oscillator model with Fermi-Breit hyperfine TABLE 1 Characterization of the N* and A* model states by their isospin (T), spin (S) and orbital (L) symmetry
T=" s={ ,=,
_1 s={ ,=, z-~
_3 ,=, t=½ S-~
EP~-~EP Epr~Ep EPF~E£ T={ s:" L=,
_:3 So~ I ,o, Z-~
I L:, Z={ So-~
rmEPEP ~-~IYEP
To" s=" ,=,
PEPEP P F E P NIl2-
*
N312-
N ~
512-
~
A I12-
A .
312
_
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
351
interaction only if one deliberately throws away the L S force terms and keeps the spin-exchange and the tensor pieces 7,8). Indeed, the spin-exchange (Fermi contact) term alone could very well furnish the N*'s with the observed mass split ~ 1 3 0 140 MeV pushing the states with S = 3 up and those with S = ~ down. The two A*'s have the same spin symmetry and thus would remain degenerate. The tensor force then mainly mixes the S = ½ and S = 23-components 7). The effects of L S forces, however, are hardly seen in the N* spectrum, and if the L S force from one gluon exchange were included, it would disrupt the spectrum disastrously. Hence we consider the experimental situation as a clear indication that the spin-orbit pieces of the hyperfine interaction, although present, must be highly suppressed by "some mechanism" for the N*'s. On the other hand, because the tensor force is not operative in S = ½states, any A* split is only to be attributed to L S forces. It seems conceivable that suppression of the spin-orbit force is a genuine confinement effect. Due to confinement the color electromagnetic gluon fields have to vanish outside the hadron, which gives rise to additional terms in the gluon propagator 9,10). Within the framework of Q C D perturbation theory these confinement terms contribute to the quark-quark interaction. We would like to look at these additional terms in the present work. The confinement terms are explicitly calculable in a bag model 11) which we here consider as a suitable starting point for a perturbative approach. It has, furthermore, the advantage that one can keep the theory fully relativistic and time-dependent. However, it is necessary to perform the transformation to a two-component theory for two reasons: only then the full variety of distinct qq forces we wish to analyze, shows up and the elimination of spurious c.m. motion is easy and transparent. The excited baryon spectrum has been calculated in the MIT bag model 12,13). However, the mass splitting pattern found in ref. 13) for the odd parity N* and A* spectrum does not show the experimentally observed gross structure described above. We therefore avoid some of the approximations made there; this is facilitated by transforming to a two-component theory. Our results are not in agreement with the assertion lz-14) that the exchange matrix elements are small. In our calculation these contributions are taken into account (cf. sect. 3). We also restore chiral symmetry, which is broken in the MIT bag, by coupling the pion field to the bag's surface 15,16). The pion field is treated in linear (zeroth order) approximation 17,18) and not only contributes to the nucleon-isobar ground state energy difference but also provides additional splittings in the excited baryon spectrum since the quarks interact by coupling to the axial pion current outside. Finally, in contrast to the work of D e G r a n d and Jaffe, we analyze the qq forces in detail in order to learn about the influence of different force terms on the mass spectrum. In the following section we will first write down the model wave functions for the odd parity N* and A* states and then introduce the quark-quark forces, as derived in ref. 11), and their matrix elements in the model space. In sect. 3 the
352
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
properties of the force are discussed with respect to effects of the boundary, static and non-relativistic approximations and the influence of the pion field. We close with some critical remarks and a concluding section.
2. The basis states and the bag-model qq force Each 3-quark basis state in the N* and A* subspace must be a product of a totally antisymmetric color wave function (singlet) and a totally symmetric wave function in the combined isospin-spin-orbit space
I~, color)® I ~ ,
TJ(SL)) •
(2.1)
The symmetric part of this wave function can be constructed from the isospin, spin and orbital wave functions of definite SU(3) symmetry, as given in table 1. We write symbolically
I~,TJ(SL))= 52 E IT)®IS)®IL)CGsu(3)CGo(3). 0(3) SU(3)
(2.2)
The symbols CG denote appropriate Clebsch-Gordan coefficients by means of which the individual TSL states are coupled to a totally symmetric state with respect to particle exchange (SU(3) sum) and the SL states are coupled to total angular m o m e n t u m J (0(3) sum). The orbital 3-quark wave function ]L> is made up from two ls modes and one l p mode in the bag model 19,20,12). Let us now consider a translation of a threeparticle orbital wave function, say, with all three particles in the same s-state exp [ie. (P1 +/~2 +P2)]s(1)s(2)s(3) = s (1)s (2)s (3) + i e. [PlS (1)s (2)s (3) + s (1)PzS (2)s (3) + s (1)s (2)P3s (3)] + o(e 2), (2.3) where s(i) is a function only depending on ri = Iril and P~ are momentum operators. Since the angular dependence of P~s(i) is given by ri/q, which in spherical components is - Y r , , for l = 1, we observe that the quantity in square brackets in (2.3) belongs to the ~ representation of SU(3) where two particles are in s-states and one is in a p-state. It is, on the other hand, nothing but a translation of the state with three particles in an s-state. This simple observation shows that the totally symmetric configuration of two ls and one l p modes should predominantly be a spurious c.m. state. We therefore only keep the mixed representation for [L). Then, the only way to end up with a ~ symmetry on the left hand side of (2.2) is to couple the isospin and spin pieces to a mixed symmetry representation. Consequently, for the N* states which have ~S] T-symmetry we can have both and ~ in spin space whereas for the A* states which have ~ T-symmetry
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
353
we can have only ~ in spin space. Thus the states of our model space are characterized by their spin configuration and, of course, their total angular momentum. There is only one N*~- state because S = ½ and L = 1 cannot couple to J = 2. Table 1 shows a summary arranged in a way to resemble fig. 1. In a spherical bag model we encounter the cavity modes
~lJ"=Nli[I ,t
• k r
~r.r l+ir
/
r\
/t±l~ktj-~)
~ilm(I2)e-i°~"'/r
(2.4)
for j = l+~, e.g. ref. 13). The bag radius is R and qb~r~(12) denote spin harmonics where 12 is the direction of r, ref. al). For massless quarks we have k~i = tou. The lowest frequencies determined by the linear bag boundary condition 19) are
tos~ = 2.04,
3.81 for/' = ½ top~= 3.20 f o r j =3"
(2.5)
With the spinors tp in (2.4) we are not yet in a position to eliminate the spurious c.m. motion. The reason is that the construction of the model states, as described above, is based on the L S coupling scheme, cf. (2.2), and on the fact that there is only one orbital p-state, cf. (2.3). Both conditions are, of course, not met by the spinors ~. In ref. 11) it was shown how to transform the four-component Dirac spinors by an (approximately) unitary Foldy-Wouthuysen transformation to two-component Pauli spinors keeping only the positive energy solutions. Since the resulting spinors still remain non-degenerate in/" = l + ½ we apply a second (approximately) unitary scaling transformation, which reintroduces the degeneracy in the basis states by means of adding a one-body spin-orbit force to the kinetic energy • tr, N / ~ P~ Hkin=~i (N/~i -}-ITl2"~-I
]~.
(2.6)
Details are again given in ref. 11). The orbital part of the rn = 0 basis states, for s-states denoted by s(r) and for p-states by p , ( r ) , then is N~R -3/2jo(to~r/R ) Yoo(O) = s (t), NoR
-3/2]l(topr/R ) YI~ ([2) = Pc (r) .
(2.7a) (2.7b)
As shown in ref. 11) the wave number of the p-states and the strength K of the spin-orbit force are related to each other, they must be top = 3.40, (2.8) K = --0.06.
354
H.R. Fiebig,B. Schwesinger/ Splitting mechanisms
The wave number of the s-states, of course, remains unchanged, tos = 2.04• The normalized orbital states (2.7a, b),
fr
(2.9)
have to be multiplied with Pauli spinors and with flavor and color functions for a complete description of single quark degrees of freedom. The one-gluon and one-pion exchange force we are considering is derived in ref. 11). For the analysis attempted here we cast the force into a different form, which is also more suitable for later discussions. The entire hamiltonian reads •
1
H = 2 H k i n ( t ) + g Y. [HoGz(i,f)+H,~(i,f)], i
(2.10)
i~i
where//kin is defined in (2.6) and contains the one-body LS force (m = O)
Ills = Kl . 0.p2/.,/-~.
(2.11)
We write the quark-quark interaction as 1 0 HOOE(1, 2) = gas{-- Y~Z (1, 2) + 10.1" 0.2 E X°( 1, 2) +1/0.1X 0.2"~X1(1, 2)+0.1 "Y~ y1(1, 2)+0.2"~ Y1(2, 1)
"+-(0'1°0"2)" ~ X2(1, 2)},
(2.12a)
1
H~(1, 2 ) = - - -16f~ "rl "~'2{½0.1"0.2X°(1, 2) + li(0.a x 0.2)" X~(1, 2) + (0.100"2).X2(1, 2)}.
(2.12b)
The upper indices K = 0, 1, 2 identify Z °, X ° as scalar, X 1 and y1 as vector and X 2 as tensor operators. The rank-2 tensor product of 0.1 and 0.2 is written as 10tI oi]1"2 and each • denotes the usual scalar product of two irreducible rank-K tensor operators 21). In equation (2.12a) the operators Y~Z °, Y, y1 and Y~X K, K = 0, 1, 2, are defined as the sum of all gluon exchange terms appearing in the first column of table 2, respectively. The pion exchange pieces X °, X 1, and X 2 in eq. (2.12b) are listed below the corresponding gluon terms in table 2. The operators in table 2 all involve the velocity operators
=#/4F,
=#/,/y,
(2.13)
where the arrows indicate whether the velocity operator acts on bra or ket. The propagators G are in general two-index objects because they have to rotate the gluon vector potential away from the generating currents in order to satisfy the boundary conditions n . E a = 0 and n x B a = 0 on the surface (a = 1 . . - 8 ) . We encounter the following propagators: (a) free field Go(r1, r2, w), Go =
cos
Irl - r21
Irl - r21
'
for all w,
(2.14)
355
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
(b) transverse electric GTE(ra, r2, to), 1 (GTE)q = -4~'to(ll)i(12)i l=l ~ ~ml(l + 1)
x (I + 1)nl (toR) - toRnt+l (toR) (I + 1)jt(toR) -toRfl+a(toR) ]t(torl)jl(torz) Ylm(g21) Y*m (J'-~2)
for all to,
,
(2.15) (c) transverse magnetic GTM(rb r2, to),
[
-4~rto (Vl ×/1)i(V2 ×/2)j
1
ndtoR)
,
X ~ Y'.--~,,':-~7, .--7--gT~xjt(tor~)jl(torE)Ytm(~l)Yt,,~(l'22), l=l into l ( l + l ) jl(tolX)
forto ¢ 0 (2.16a)
(GTM)~; ='
I1 ~
4Ir
(v o,
k-
+ (V,),
2
+r
The use of two different transverse magnetic propagators, depending on whether the frequency to vanishes or not, originates in a different choice of the gauge in order to perform the limit to -~ 0 as explained in ref. H). Due to the different gauge for to = 0 there arises a longitudinal electric propagator: (d) longitudinal electric GEE(r1, r2, o3 = 0),
1~~
GLE=R -
1=1
4"rr
2l+~
l+l(rl~t(r2~ '
l \ R ] \ - R : Yt~(J2~)Y*m(~2),
forto=Oonly.
(2.17) For any current satisfying the continuity equation the contribution of the transverse magnetic gluon propagator (2.16b) to the color fields vanishes for to = 0 because the divergence of the current vanishes. However, we will later also consider static approximations to the propagators, keeping the time dependence of charge and current distributions. Then, the divergence of the currents does not vanish giving rise to sizeable contributions from GTM(rl, r2, to = 0). For our purposes the propagator of the pion field is needed only on the bag surface. We include this restriction to the surface in the definition of the propagator and have (e) pion surface G~(rl, rE, to), 1
O~ = 6 ( r , - R ) 6 ( r 2 - R ) ~
[
hll)(toR)
,~E\ohll)(toR)/OR
+
,_(2),ttoR ' ) nl
"v ~
:,o ~v*
Ohl2)(toR)/OR] • ,mt,.,,H.,--m, (,..Q2), for all to.
Here h ~1), h Iz) denote spherical Hanckel functions z2).
(2.18)
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
356
TABLE 2 Orbital operators of the one-gluon and one-pion exchange forces HOGE and H= (first column) and reduced matrix elements of these operators for three different l p frequencies (a, b, c), the static (d) and the non-relativistic limit (e)
(ssllX°(1, 2)tlss)
(psi[X°(1, 2)lips)
(psi[X°(1, 2)lisp)
-- 1[(7~ 1 4- 7~1) o G 0 o (7~"2 + 7~2)] 0
19.2
13.1
23.2
~[(Tr 1 4- 71"1)o G T E o ( ~ 2 + "~'2)]
5.8
5.4
X°(1, 2)
3.1
1[(7~ 1 + 7~1) ° G T M o (7~'2 + ~'2)] 0
~[(zT1 x r~l)o Goo(~2 x ~2)] ° 1[(#1 × 7~1)o GEE o(7~"2 X 7~2)] 0 Sum of OGE: ½[(~1+~1)o
s o G,~ o(7r2 + rr2)]
X1(1, 2)
-4.1 -2.0
(0.0)
(0.0)
0.0
0.0
0.0
25.0
18.5
20.2
-12.5
-21.5
-18.7
(ssllXl(1, 2)llss)
(psllX~(1, 2)lips)
(psllX~(1, 2)lisp)
i [(~'1 + #1) o Go o(T~24- 7~'2)] 1
-0.3
-i[(~1 + ~'x) o GTE o (#2 + 7~'2)] 1
-6.7
- i [ ( ~ l + r~l) o G T M o ( ~ 2 + ~ 2 ) ] 1
--8.8
-i[('~1 x ~1)o Goo(#2 x ~2)] 1
-4.3
- i [ ( f f l X ~'1)o GLEO ( ~ 2 X ~ 2 ) ] 1
0.0
Sum of OGE:
-20.0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-i[(~a + r71) o G~. o (r~z + ~ 2 ) ] a
X2(1, 2) --[('~ 1-1- ~'1) o G o o (7~2 -- 7~2)] 2 [(7~1 + '/~1) o G T E o ("/'~24- 7~2)] 2
12.4
(ssllX2(1, 2)ltss)
(psllX2(1, 2)lips) -21.0
-4.2
[('w14- '~1) o G y M o (,t~2 -'l---~2)] 2 --[(~'1 X q~l)O G 0 o (7~2 x qT2)] 2 --[(~1 X 7~1) o GLEO(~2 × 7~2)]2
Sum of OGE: -[(r~l + 61) ° G~ o(~2 + ~'2)] 2
(psllX2(1, 2)lisp) 17.8 - 1.2 --8.0
(0.0)
--3.9
0.0
0.0
-25.2
4.7
16.4
15.5
2a: ~o~,= 3.40
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
357
19.2
20.2
26.9
19.2
9.3
22.8
19.2
13,1
20.0
21.2
25.6
25.6
5.8
7.3
3.6
5.8
4.2
3.0
5.8
5.4
3.2
3.8
6.5
2.0
-1.3
-8.0
-2.2
0.0
(0.0)
(0.0)
-1.7
(0.0)
(0.0)
-2.2
(0.0)
(0.0)
-1.6
(0.0)
(0.0)
-3.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-0.2
0.0
0.0
-2.2
25.0
27.6
27.5
25.0
13.5
15.6
25.0
18.5
19.3
25.0
32.1
22.1
-12.5
-17.8
-15.9
-12.5
-21.1
-18.9
-12.5
-21.5
-19.6
0.0
0.0
0.0
-7.6
4.8
1.1
0.0
-7.6
-6.4
-6.8
-4.3
-2,7
-17.0
-4.8
0.0
-3,5
-4.7
-3.3
-7.0
0.0
0.0
-0.4
-4.7
-21.5
-23.2
14.2
16.0
10.9
12.2
-0.7
0.0
-14.1
16.7
-25.0
20.3
-21.0
16.1
-13.8
19.8
-1.8
-1.4
-5.6
-1.2
-4.2
-1.3
0.0
-0.8
-2.5
-15.5
-4.4
0.0
(0.0)
-3.2
(0.0)
-4.3
(0.0)
-3.0
(0.0)
-6.4
0.0
0.0
0.0
0.0
0.0
-0.4
0.0
-4.3
-15.9
9.6
-30.6
-0.7
-25.2
7.1
-13.8
8.4
20.6
12.8
10.3
15.9
16.4
31.1
0.0
0.0
2b: top = 3.81
2c: OJp= 3.20
2d
2e
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
358
Y~(1, 2)
2)llss>
i[(ffl + 771) ° Go o ( i f 2 - 7?2)] 1 -- ff2)] 1 --i[(~1 + ~ 1 ) o GT M o (~2-- 772)] 1 i[('B', x 771)o Go] 1
-- i[(7'~ 1 "4- 7~1) o GT E o (if2
--i[(ff 1X ffl)' Go(if2" 7?2)] 1 i[(ffi x 771)" GLE] 1 - i [ ( f f l x if1)' GLE(ff2 ' if2)] 1 Sum of O G E :
Y'(1, 2)
i[(21 + 771) ° Go ° ( 2 2 - 7?2)] 1 - i [ ( 2 1 q- ¢~'1)o GTE° (22 -- 7?2) 1
(psi[ Y~+ (1, 2)lips)
(psllY~+ (1, 2)lisp)
56.9 15.2 (48.9) (12.6) 0.0 0.0
-64.5 0.0 77.6 -36.1 -6.0 0.0 0.0
72.2
-29.1
(psll Y ~_(1, 2)lips)
(psll Y 1_(1, 2)l}sp)
-56.9 --15.2
--i[(7~1 + 771) o G T M o ( 2 2 -- 7?2)] 1 i[(7~1 × 7?1)" G o ] 1
(48.9)
-i[(¢71 x 771)" G o ( # 2 ' 7?2)] 1 i[(771 x '~'1) ' GLE] 1
(12.6) 0.0
- - i [ ( ~ I × 771)' G L E ( 2 2 ' 7?2)] 1
0.0
Sum of O G E :
Z°(1, 2)
-72.2
(psll/°(1, 2)liPS)
- [ ( ~ 1 - #~)" Go' (~'2- ~'2)] ° [(7~1 -- ~ 1 ) " G T E ' ( 2 2 -- ~ 2 ) ] 0 [(T~I -- q'~l) ' G T M ' (7"~2-- qT2)] 0
Go(1-2x'
"t7"1-22" ~2)
G 0 ( 2 1 ' ~1)('/~2 ' 7?2)
(169.8) (10.9)
(339.6) (34.7)
GEE(1 - ~'1" 'ff'l - 2 2 ' 22) GLE(21 ' ~1)(~'2 ' ~2) Sum of O G E :
121.7 70.8 2.0 0.0 0.0 134.4
2a (cont'd): wp = 3.40
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
54.5
-43.4
58.7
-84.4
56.9
-75.9
61.4
-60.3
13.6
0.0 27.2
16.2
0.0 135.1
15.2
0.0 48.5
9.2
0.0 17.8
(29.8) -33.6
(61.9) -36.5
(48.9) -27.7
(6.2)
-8.1
(16.9)
-4.5
(12.6)
-4.5
0.0
0.0
0.0
0.0
0.0
-3.3
0.0 - 1 7 . 9
0.0
0.0
0.0
0.0
0.0
-1.1
0.0
0.0
68.1
-57.9
74.9
9.7
72.2
-64.0
70.7
-90.6
(30.7) -30.2 (0.0)
-54.5
-58.7
-56.9
-61.4
-13.6
-16.2
-15.2
-9.2
(29.8)
(61.9)
(48.9)
(30.7)
(6.2)
(16.9)
(12.6)
(0.0)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-68.1
-74.9
-72.2
-70.7
-32.8
-80.8
48.0 (169.8) (362.7) (10.9)
359
(39.9)
83.6 4.2 00 0.0 103.1
2b (cont'd): top = 3.81
-114.0
190.1 (169.8) (334.1) (10.9)
62.7
(33.7)
1.2 0.0 0.0
(10.9)
(34.7)
173.2
2c (cont'd): ~op= 3.20
-122.2
72.1 (169.8) (339.6)
53.9 1.5 7.8 0.5
66.5 (167.5) (383.1) (0.0)
(0.0)
21.9
2d (cont'd)
0.0
55.2 0.0 48.7 0.0 48.2
2e (cont'd)
To reconvert the units, defined in the text, the numbers have to be divided by 77.49 for gluon and 134.3 for pion matrix elements.
360
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
The square brackets in table 2 indicate the decomposition of products of velocity operators and propagators into spherical rank-K tensors according to [ U oG o V] K=°'M=° = eiikenmUiGkm Vt ,
(2.19a)
[Uo G
e3i,,,eiike,tmUiGkm Vt ,
(2.19b)
3 1 x/~ (613 -- g)eijkeilmUiGkm VI,
(2.1 9C)
o V] K=I'M=O =
[UoGoV]
K=z,M=O=
where 8qk is antisymmetric and the notation
8123--'--
[U. G] T
1. For the Y~ and Z ° operators we also use
M
= UiGi3,
(2.19d)
[ U " G V " W ] K=''M=°= UiGi3 V/Wi ,
(2.19e)
[U. G. V] K =0.M=o = UiG~,Vj.
(2.19f)
In the remaining columns of table 2 we have listed the reduced matrix elements of the corresponding operators for the s- and p-states defined in (2.7). We also encounter the symmetric and antisymmetric part of Yt y 1 (1, 2) = Y'(1, 2)+ Y1(2, 1).
(2.20)
A discussion is postponed to the next section. We are now is a position to write down the matrix elements of H in the N* and A* model space. The mass matrix can be written as a sum of a diagonal part which shifts all states by the same amount and a splitting matrix which includes all spin-dependent terms: M[~ Mo
fi°°
0
+ Mo
0 Nj
0
Md~
0
NJ M~':n~
0 M~i~ Ep N,J
(2.21)
I
The elements of the 2 × 2 submatrix in the N* sector are characterized by the spin-symmetry indices in table 1. For the overall J-independent mass shift in the spectrum one obtains Mo
O)p - - COs
OLs
+g-k-[3(ssll E x°llss)
+ 2
(2.22)
For convenience we explicitly evaluate the J-dependent recoupling coefficients and
361
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
employ the notation
Ial Ia
(2.23)
b = b ifJ =3. c c ifJ =5
Then we obtain for J - ~, 3 M ~A J
= ~s [(ssll E x ° l l s s ) - 43 (4(psll Y. x°llps>
6R
1 ~ - 5 ( p s l l E x o Ilps))-g~/g
,o17
1 ((psllE g+llps)
+ (psllE Y~ lisp) - 3(psllY. y1 lips) + 3(psllE xaHsp))] + 1 6 f @ . [(ssllX°l[ss)- x/3r(4(psl[X°Hps)
- 5(psllX°llsp)) - q~g
Eil
(psllX~l]sp)
+KOJp -~-
I Eil
.
(2.24)
In the N* sector we have M~
= a~-~{(ssll~ x°llss) +,/3r-(2(psHY~x°lips)
- (ps[lZ x°]]sp)) + 245 [- 2 ](2(psllE x2llps) t_ la6.a
- (psllY x 2 lisp)) - gi ~/g~
(2(psE Y+1 lips)
1 - (psllZ YLlisp))} +~{(ssllX°llss) _ ,/r (4(psllXOllps> _ 5(psllX Ollsp)) 1
- ~45 I- ~](4(ps[[X2llps) - 5 (psl[X 2
~ KOJpV[25], (2.25)
362
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
N,J
Ols {
MUz~z = ~-- -
4~(2(psllYx°llps)
o
(psllYX [[sp))+g ~
-
(3(psllEx'llsp)
2(ps[[• Y~ lips)+ (psllY ya+lisp))}
+~
1
0
{ 5
1 K.pf2 l OR /
-2(psllEX2llsp))
+14~ a g ,4- 0
((psHE gl+llps)
- 2(psllY. y 1 lisp) + 3(psIIE Y! lips))} + ~ { - 21 ~ / }
I i ! l -( -) j (psl~(lllsP)
(5(pslIXZllps) - 4(pslIX211sp)) j
+ 2KtOp -~-
. (2.27)
The parameters in our model are the bag radius R, the color coupling constant as and the pion decay constant f,~ which should, however, come out to be close to ~93 MeV. Although we cannot expect to reproduce the ground-state energy of the nucleon and the isobar, the model should be suitable to obtain the correct energy differences such as the N-A mass split M a - M N = 300 MeV. In terms of reduced matrix elements this reads Ma
C~s - MN = ~--
o 1 6(ssllY~X Ilss) 16f~R 312(sslP(°llss).
This has to be considered as an additional constraint on our parameters.
(2.28)
H.R. Fiebig, B. Schwesinger/ Splitting mechanisms
363
3. Discussion of results The reduced matrix elements of the various force terms are listed in table 2 for three values of top and for the static and non-relativistic limit. In order to make the relative importance of gluon and pion terms more transparent the matrix elements are given in units of a s / 6 R and 1 / ( 1 6 f ~ R 3 ) , respectively, with the parameters being fixed such that gluon and pion exchange interactions build up the N-A mass difference of 2 × 150 MeV to equal amounts, cf. eq. (2.28). Blank positions in table 2 indicate that the corresponding matrix element vanishes a priori (mostly because of some symmetry reason). The sums of all one-gluon exchange terms in table 2 do not include the contributions in brackets which arise from propagation of static electric monopole fields; these violate the boundary conditions and have to be eliminated 12.13.11). Most of them vanish anyhow, except for y 1 and Z ° terms. The latter do not contribute to the splitting in the N*, A* spectrum. The y1 terms however are just part of the spin-orbit force. Thus we encounter a first clue as to why spin-orbit forces are highly suppressed: the omission of static electric monopole terms removes about one third of all spin-orbit interaction energy coming from gluon exchange. In table 2a the matrix elements for the average p-state frequency top = 3.40, introduced to eliminate c.m. motion, constitute our main result. The numbers in the next two columns, tables 2b, c, correspond to the wave numbers for massless quarks in the lpl/2 (top= 3.81) and the lp3/2 (top= 3.20) cavity mode and enable us to compare our results to the work of D e G r a n d and Jaffe 12) and D e G r a n d 13). These numbers also give us a feeling about the frequency dependence of the matrix elements. In table 2d we give the results for a static approximation. Time-dependent charge and current distributions occur only in the exchange matrix elements where to = top-tos appears in the propagators. The approximation is performed by taking the limit to -~ 0 in (2.14)-(2.18) but retaining top = 3.40 and tos = 2.04 everywhere else. In order to avoid the calculation of even more matrix elements, the part of the expression ]o,(1)" GTM(1, 2;to = 0)']~ (2) involving the third term of the transverse magnetic propagator (2.16b), i.e. V 1
t=l
47r
, 2+
trl
2,[r1~ l(r2~l
r 2 ) t - ~ ) \~,. ' Ylm(~"~l) gl*m (~2)
(3.1)
has been integrated by parts. The boundary terms vanish; and the divergences of the currents can be replaced by charge densities using the continuity equation V "]o, = itoo,o.
(3.2)
After these manipulations the force term of (3.1) has the same algebraic structure as the longitudinal electric (LE) terms; and their matrix elements have just been added to them in table 2d. The static approximation hardly effects matrix elements
364
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
involving free propagators. The changes in boundary terms are already bigger but still not very dramatic. We would like to point out that the procedure of performing the static limit has been consistently applied to the propagators only, lim,,_,0 G(1, 2; to), and not to the matrix elements, which would amount to lim,,~0],~(1). G(1, 2; to).],,(2). This would make a difference since, in the latter case, for static currents the contributions from the transverse magnetic gluons vanish exactly, whereas we encounter sizeable T M contributions in table 2. The biggest one by far is the Y~+ T M matrix element which would help to cancel about a third of the spin-orbit energy in static approximations. Table 2e gives the non-relativistic limit of the matrix elements, where the masses of the quarks are taken to be large compared to the frequencies to~ = 7r, top = 4.49. Again there are modest changes when comparing to the extreme relativistic cases (tables 2a-c). Obviously the pion field now decouples from the quarks in the bag because the axial quark current vanishes at r = R. Let us now insert the numbers for the matrix elements from table 2a into the splitting part of the mass matrix (2.23-2.27). The result of this is arranged in table 3. The contributions are grouped into four categories: free: boundary: pion: Thomas:
interaction energy from exchange of free gluons corrections to the gluon exchange energy from the boundary interaction energy from pion exchange contributions from the one-body spin-orbit force (2.11) (Thomas precession).
These four categories are further subdivided into contributions coming from spinexchange, tensor or spin-orbit forces. The parameters are, again, chosen such that gluon and pion fields contribute 150 MeV each to the N-A split and R is taken to be R = 0.8 fm. We do not claim that these parameters should have these values, but we do think they reflect the correct order of magnitude with which different effects participate in the splitting of the odd-parity states. In any case, by keeping the four different categories mentioned before, we are still free to apply any weighting factor on the three participating mechanisms: gluon exchange, pion exchange and Thomas precession. In order to compare with constituent quark models we have included the matrix elements of the Fermi-Breit interaction 3) evaluated for oscillator states. Here, the combination of ets with the oscillator length is fixed to give a 300 MeV N-A split (furnished entirely by gluons). (a) Spin-exchange forces. We first concentrate on the spin-exchange forces (X ° terms). In harmonic-oscillator constituent quark models the split of the (S = 3) and (S =1) nucleons caused by the Fermi-Breit interactions is just a half of their contributions to the nucleon-isobar split; the A* remain degenerate with the N * (S = 3). The same feature also shows up in the bag model. The sum of free and boundary terms from the spin-exchange part of HOGE is 35 MeV for N*- (S = 3),
365
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
TABLE 3 Mass splitting matrix for the N* and A* spectrum in the bag model and the constituent quark model (CQM) Element
Force
Free Boundary
F1--I
11|×
spin-exchange
Pion
Thomas C.Q.M.
[ 22.
+13.
-17.]
[75.]
1/2-] /-2/q×
[-83.
+1.
+18.]
[-150.1
[~]x
[-30.
+6.
L 1A
[M ~N,, E~
tensor
L1/IOJ
spin-orbit
+17.1
[-75.]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [. -. . 3. .J. . . . . . . . . g-~-I
JiJX
spin-exchange
[
--
]
[0.]
[-73.
+21.
-30.]
[-150.]
[ 14.
-25.
-20.
-13.
-113.]
[-75.]
--
]
[0.]
-10.
-17,]
[+75.]
--
]
[0.]
F-1/27 M~
tensor
11/,/40/× L
spin-orbit
s'-exc"ao,e M~SJ~n
0.A
i-O/
x
/;|x
-34.]
[50.]
L1A
tensor
x
[
spin-orbit
+0.
LoJ spin-exchange
MU[5~
i-17 /1/X
[50.
tensor
I~Tx L0A
[
spin-orbit
[-1Ix
[-15.
L0d
-12.
-10.
-17.]
[-25.]
40 M e V for A*- and - 3 5 M e V for N~- (S = 1), which is a b o u t half of the gluon contributions to the N-A split fixed here at 150 M e V . Surprisingly the s p i n - e x c h a n g e forces f r o m pion e x c h a n g e also show this feature: - 1 7 M e V for N~*- (S =3) and for za~' and - 1 1 3 M e V for N j* (S =½). T h e r e f o r e any ratio of gluon e x c h a n g e to
366
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
pion exchange contributions will give roughly a 150 MeV split between N*- (S = 3), Aj-N~ (S = 1), provided they contribute 300 MeV to the N-A split. (b) The tensor forces. The spin-exchange forces alone give a remarkably good fit to the experimental splittings shown in fig. 1. The only quantities which don't fit are the wave functions of the N1/2-, * which are measured to mix by an angle of approximately 30 ° [for further details see ref. 7)]. For this reason Isgur and Karl 7,8) also included the tensor force of the Fermi-Breit interaction excluding deliberately spin-orbit terms. This procedure then resulted in a surprisingly good fit to large parts of the baryon spectrum. Tensor forces can only be operative if at least one S = 3 state is involved in the matrix element. Therefore the tensor forces only show up in the diagonal element M N~, J and in the off-diagonal elements M ~ . In the latter gluons and pions contribute roughly equal amounts but they only add up to about half of the value given by the constituent quark model. Even if the entire N-A split were attributed to gluon exchange the bag model is different from constituent quark models in that respect. As can be seen in table 3 the reason for the difference are the boundary terms. Table 2 then shows that the transverse magnetic contributions carry the largest share. In the diagonal element the pion tensor forces are negligible, whereas the gluon contributions of the bag model and constituent quark model are practically equal (Remember that the gluons in the bag are fixed to contribute 150 MeV to the N-za split in contrast to the 300 MeV for the constituent quark model.) We may therefore state that the mixing in the N*- from tensor forces alone is quite independent of the ratio between gluon and pion contributions and smaller than the corresponding constituent quark model value. (c) Spin-orbit forces. It is quite easy to see why the spin-orbit terms of the Fermi-Breit interactions in constituent quark models are incompatible with a good • fit to the experimental spectrum: m M N~J they contribute as much as 8 x 75 MeV to a split in the upper N* (S = 3), whereas experimentally these states are nearly degenerate! The numbers in the other matrix elements are less spectacular though still large. The bag model on the other hand does quite well here, since the pion field does not contribute anything and Thomas precession terms together with the boundary terms cancel all contributions from the free gluons. One should recall that we have already eliminated a larger portion of spin-orbit interactions because static color electric gluon monopole fields violate the boundary conditions. Thus, color confinement effects, the spin-orbit forces from Thomas precession, and certainly also the pions, work together to suppress spin-orbit forces. Of course, the extent to which the spin-orbit forces are suppressed depends on the choice of the parameters as, f~ and R. Cancellations are also present in the other N* matrix elements. In the off-diagonal element they actually even overshoot. As to the A* we see that these cancellations do not occur: the free and boundary terms again are as big as pionic contributions and the Thomas precession effects add up splitting the A* by 162 MeV. This split shows the correct ordering, A*/2- is heavier than
H.R. Fiebig,B. Schwesinger/ Splitting mechanisms
367
A1/2-, although it is too large. As pointed out by Close and Dalitz 23) this split is one of the major problems for constituent quark models: the N* spectrum can only be fitted by eliminating spin-orbit forces in which case A*/2 and A*/2 remain degenerate. We have shown here, that there are no such difficulties in a bag-model description. (d) Comparison to other bag-model calculations. To our knowledge the only attempt to calculate the spectra of odd-parity states in a bag model was in the work of DeGrand and Jaffe 12.13). The results in ref. 12) did not agree with experimental findings because c.m. excitations of quarks in the bag were not eliminated. This was improved on in a second work 13) by identifying the c.m. motion with a certain linear combination of basis states. The procedure there, however, can work at best approximately because the lpl/2 and lp3/2 p-states were left non-degenerate. Further in ref. 13) all processes involving both (non-degenerate) p-states simultaneously were ignored and finally all exchange processes were neglected because they were found to be small in the previous work 12) [see also ref. 14)]. If we insert only the exchange terms of tables 2a, b, c into the mass matrix we find that they contribute a fourth of the spin exchange forces and half of the tensor forces in the off-diagonal element. Roughly half of the spin-orbit interaction comes from exchange terms. In order to clarify the discrepancy and to check our numbers, we have included the spurious state in our basis and calculated the gluon exchange energy for the only J = 23-state that can be formed from two quarks in lsl/2 states and one in a lpl/2 state. This is given by 3R <] =
2c~
IHoo
lJ = b =
+
-
'
"
"
= l
+ 2,/~
(3.3)
Comparing this to the corresponding expression of ref. 12), we can identify 0A s - - 1
2
~ r .... = (---~_21) M,,(0, 0),
(3.4a)
~rp, ps = (0As-- 1)(0Ap+ 1) Ms,(0, 0) '
(0As-
)(0A.
"
A~rp,sp = (0A'- 1)(0Ap+ ~ 8 (0, 0) ,
(0A
(3.4b) '
(3.4c)
'
where M~(0, 0), M~p(0, 0) and 6 (0, 0) are defined in ref. 12). In deriving (3.4a)-(3.4c) we have corrected for the errors of the Foldy-Wouthuysen transformation [which
368
H.R. Fiebig, B. Schwesinger / Splitting mechanisms MeV
1700-
1600
1500
1400' NI/2-
Ns/2-
Ns/2-
L~l/2-
Z~3/2-
Fig.
2. Calculated N* and A* mass spectrum normalized to the nucleon g r o u n d state MN = 940 M e V (. . . . ), R = 0.8 fm. Calculated s p e c t r u m without anti-symmetric s p i n - o r b i t terms ( . . . . ), R = 0.7 fm. E x p e r i m e n t a l spectrum f r o m ref. 5) ( ).
gives the frequency-dependent factors 11)] and for a missing factor of 2 in 8(0, 0) [erratum in ref. 13)]. Insertion of the corresponding numbers of table 2b yields perfect agreement, and indeed the exchange matrix elements in 8(0, 0) are small. This, then, means that the exchange matrix elements in the calculation of ref. 12) were small because the c.m. excitation was mixed into their states causing an accidental cancellation. (e) Mass spectrum. The diagonalization of the splitting matrix in table 3 doesn't yield a good fit to the experimental spectrum, fig. 2. This is entirely due to the fact that the cancellations from boundary, pion and Thomas precession terms overshoot the free gluon exchange contributions in the off-diagonal element. This results in a strongly suppressed mixing of the N1"/2 and a too-strong mixing and splitting of the N*/2 since the tensor force terms cancel in the first case and add up in the second. With the parameters c~, f= fixed to give 150 MeV for the N-A split each at R = 0.8 we have O~s= 1.9, f,~ = 83.6 MeV. Both numbers include factors originating from corrections to the Foldy-Wouthuysen transformation; the unrenormalized numbers for o~s and 1/f~ would be smaller by a factor (oJ~-1)2/(~o~ - 1)2= 0.46, cf. ref. 11). This choice of parameters gives M0 = 675 MeV for the shift of the states (2.21) relative to the nucleon at 940 MeV; this is too small by about 50 MeV. A smaller radius of the bag would give a bigger shift but in order to keep the pion contributions small we would have to choose very small values of f=. Too-large pion contributions on the other hand are ruled out, because then gluon contributions to the N-A split would be small, which in turn would result in too-small X-X* and -~--=* splittings. Pions are not contributing there.
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
369
The major shortcomings in our calculation originate from approximations in order to eliminate the c.m. motion. Boundary terms were neglected twice in the Foldy-Wouthuysen and the scaling transformation 11). The former gives rise to additional factors to the matrix elements, an example of which is given in equations (3.4a, b, c). Since in general these factors depend on the states involved in the matrix elements, they cannot simply be absorbed in the parameter combinations as~R, 1 / f E R 3. In the space of states considered here, these effects act like an additional spin-orbit force. Further spin-orbit contributions will be encountered if the scaling transformation is also applied to the interaction terms 11). The size of these effects can be estimated by inserting the wp dependent matrix elements of tables 2a, b, c in the mass matrix (2.22-2.27). The variations are considerably smaller than the spin-orbit terms from direct electric contributions and also smaller than corresponding boundary terms from the transverse magnetic gluons. Therefore we do not think that these approximations endanger our explanation of why spin-orbit forces are suppressed in odd-parity nucleons, although they certainly will modify the calculated spectrum. Further errors more difficult to estimate arise from the fact that the pion field couples only to the surface, because just there major approximations were involved. Certainly, the pion exchange energy will also be sensitive to different bag radii for different states and to a deformation of the bag in order to balance the bag pressure for the lp3/2 states. These corrections might improve the quality of the spectrum, as well as a fit of ~s, f~ and R would do. However, we believe that the parameters as and f~ have a more fundamental meaning. Another source of errors might be hidden in spurious contributions from c.m. motion. Although our basis states were constructed by exciting purely intrinsic degrees of freedom this only means that c.m. motion in excited states vanishes to the same extent as it vanishes in the ground state. Since HOGE (as well as the Fermi-Breit interaction) contains terms which depend on the c.m. momentum (e.g. P 1 +P2 = 31-(p1 q - P 2 - 2 P 3 ) + 2 p . . . . ) we will clearly have to expect contributions from them. These terms are all contained in (a non-relativistic reduction of) the operator ~(tra -or2)" E yl_ (1, 2),
(3.5)
which is part of HomE. In a similar way it follows also that ½i(crl × 1111'2)'XI(1, 2),
(3.6)
which is part of H= (2.12b) depends on c.m. operators. N,J Matrix elements of these antisymmetric spin-orbit operators occur in M ~ A,J and M~E ? , which are the numbers causing the major discrepancies to the experimental spectrum. (There is also a term of the structure of (3.6) in Home (2.12a) giving minor contributions to M ~ . ) Although these forces also contain purely intrinsic pieces it is legitimate to estimate the effects of spurious parts by neglecting all contributions to the mass matrix from X 1 and Y~_ terms (the oscillator matrix
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elements of these terms are less than half the bag-model value). The resulting spectrum is shown in fig. 2, in order to show how well the experiment can be fitted; the radius is R = 0.7 fm and gluon and pion exchange contribute equal amounts to the N-A split. This is only a rough estimate, however, it shows that the reason for missing the experimental numbers might well be buried in the treatment of c.m. motion, which apparently is still insufficient. 4. Conclusions The odd-parity states of non-strange baryons have been investigated in the chiral bag model. The main motivation was to understand why spin-orbit forces are highly suppressed in negative-parity excited nucleons and show up in the excited isobars. Considerable effort was necessary in order to simultaneously maintain the relativistic features of the bag model and to eliminate c.m. excitations of the quarks. The present approach has led to the construction of one-gluon and one-pion exchange forces which incorporate correct bag boundary conditions on gluon and pion fields. Calculation with these forces is very similar to working with constituent quark models. We have shown that the bag model contains all features necessary to explain experimental findings on spin-orbit forces. The main contributions to the suppression of spin-orbit forces are found to originate from confinement of the gluon fields. Deviations from the 1/r behaviour of the gluon propagator at larger distances, which here is simulated by the bag boundary conditions on color electric and magnetic fields, affect mainly the spin-orbit terms, at least in the space of N* and A* states considered here. The spin-orbit forces coming from Thomas precession of the relativistically moving quarks then cancel to a large extent the spin-orbit forces from one-gluon exchange in the odd-parity nucleons and add to them in the isobar states. Forces from the exchange of pions outside the bag show the same features. In the odd-parity states the spin-exchange and tensor forces pieces operate in even quantitatively the same manner as their gluon counterparts. As to the spin-orbit forces, they again are quite negligible in the nucleon states and contribute like the gluon exchange to the odd-parity J = ½ and J = 3 isobar splitting. These features explain the "mystery of the missing spin-orbit force" without commitment to special values for parameters, which are the color coupling constant, the pion decay constant and the bag radius. The similarity to constituent-quark-model calculations allows a direct comparison. Constituent quark models using a Fermi-Breit interaction without spinorbit terms give roughly the same results apart from the A* split. This is quite surprising since the bag-model approach is extremely relativistic and incorporates important effects from confinement. Therefore one is lead to believe that the agreement is just accidental. On the other hand, the remarkable success of the
H.R. Fiebig, B. Schwesinger / Splitting mechanisms
371
c o n s t i t u e n t q u a r k m o d e l in fitting a vast a m o u n t of b a r y o n p r o p e r t i e s s e e m s to c o n t r a d i c t an a c c i d e n t a l a g r e e m e n t . A s t h e t h e o r i e s st an d now, t h e b a g - m o d e l a p p r o a c h n o t only has t h e i n d i s p u t a b l e asset of b e i n g m o r e f u n d a m e n t a l , it also is c a p a b l e of e x p l a i n i n g t h e e x p e r i m e n t a l findings c o n c e r n i n g s p i n - o r b i t forces, w h i ch in c o n s t i t u e n t q u a r k m o d e l s a p p e a r to b e a mystery. W e are g r a t e f u l to G . E . B r o w n for i n t e r e s t i n g us in t h e p r o b l e m . T h e a u t h o r s also a c k n o w l e d g e h e l p f u l discussions with J. D u r s o an d t h a n k M. H a r v e y f o r c h e c k i n g phases.
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