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Effect of quark-quark tensor and spin-spin force on the radiative decay of Δ-isobar
Volume 138B, number 1,2,3
PHYSICS LETTERS
12 April 1984
EFFECT OF QUARK-QUARK TENSOR AND SPIN-SPIN FORCE ON THE RADIATIVE DECAY OF A4SOBAR Jishnu DEY 1 and Mira DEY 1 Laboratoire de Physique Nueldaire, Universitd de Montreal, Montreal, Canada Received 4 April 1983 Revised manuscript received 9 December 1983
We use oscillator wavefunctions with Coulomb, finite-range spin-spin and harmonic confinement forces, stabilizing the nucleon and the isobar separately. Subsequent introduction of the Fermi-Breit tensor force leads to an E(2)/M(1) ratio in the A-N'r decay. The sign and magnitude of this ratio is interesting and it is shown that its measurement leads to an estimate of c~s in light hadrons.
The non-relativistic quark model is one of the phenomenological models to study low energy hadron physics [1]. Within this model the radiative decay of the delta isobar (A) to the nucleon (N) has been studied [2,3]. Transitions are very sensitive to parameters of the model. We reconsider this problem with an effective interquark interaction whose spin dependent and non central components are taken from one-gluon exchange. The parameters of the potentials are chosen to fit some of the ground state properties of N and A. The hamiltonian that we take has the form 3
,:
i =1
(m,+
"
i<]
7
'
(1)
where
aft)":
--2-' x i • ~xs.[-%/i,.isi
+ (87ras/3 m i m j) Si'Si(o3/Tr3/2) exp(--o2r2.) + r..3(3S. "r..S. "r../r 2. - S i ' S i ) l tl t q I 11 t]
(2)
where the Xi, m i and S i are the Gell-Mann colour matrices, masses and the spin matrices of the quarks respectively. Note the presence of the colour electric force (first term) in addition to the colour magnetic 1 On leave from Education Service Govt. of W. Bengal, India. 200
forces, namely the s p i n - s p i n finite range interaction and the tensor term. The colour electric force was not explicitly included in refs. [ 1 - 3 ] . Recently Liu and Ohta et al. [4] have reemphasized the importance of stability against variation in the oscillator size. To do this one has to introduce a finite range s p i n - s p i n interaction, otherwise the contact s p i n - s p i n force of the one gluon exchange potential leads to a collapse for the nucleon system [5]. This problem was not faced by Liu, who considered only colour electric forces or by Ohta et al. who included the Darwin term and worked in a restricted model space [4]. Since we want to include SU(6) breaking introduced by the s p i n - s p i n and tensor forces and also minimize with respect to the oscillator size parameter, we are forced to use a finite range s p i n - s p i n force. We use a volume normalised gaussian form with range parameter o. To maintain the short range nature of the s p i n - s p i n force in the Fermi-Breit potential we could use o = 6 fin -1 . This cr is larger than the one used in ref. [6] but corresponds to an effective gluon mass ~1.2 GeV in conformity with bag model estimates [7]. On the other hand, Blinder [8] showed that under some approximations one obtains a smeared F e r m i Breit s p i n - s p i n term with range o -1 ~ e 2 / m e. This was also discussed by Bhaduri et al. [5]. Blinder's prescription gives 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138B, number 1,2,3
PHYSICS LETTERS
2
o = mq/gO~ s
obtained the stability c o n d i t i o n w i t h o u t the tensor force and finally its c o n t r i b u t i o n has been calculated by re-diagonalising at the stable point. Our n o t a t i o n and configurations are identical with those o f Gershtein et al. [2] and we have followed their conventions and references in defining the E(2)/M(1) ratio in the decay A -+ NT. The helicity amplitudes A 3/2 and A1/2 are defined as
(3)
.
The values of % we have chosen yield o varying f r o m 4 . 5 6 - 2 . 0 3 fm -1 . We have used b o t h of the above prescriptions. Our results show a qualitative difference b e t w e e n the two choices o f o. We would like to stress that the nucleon and the isobar are ground states of different spin systems so that one should find out the sizes o f b o t h b y variation. Since the sizes are different, with the isobar larger than the nucleon, all the three forces, the C o u l o m b , the c o n f i n e m e n t and the s p i n - s p i n parts, contribute to the mass difference b e t w e e n them. Thus we can choose values o f the strong coupling constant in eq. (2) which are more reasonable with the QCD running coupling constant estimates for m o m e n t u m transfer Iql ~> 0.2 GeV/c. At q = 0.2 G e V / c one gets % ~ 1 for flavour n u m b e r N = 3. With a normalisation point a s = 0.2 at q = 3 GeV/c, we have as(q2 ) = 0.2/[1 +
12 April 1984
A~, = (A, M = 2,lHin t lyN, X = X~ - XN, K ) ,
(5)
and Hin t is the electromagnetic interaction as in ref. [2]. A~ is the transition matrix element b e t w e e n the N7 system with helicity X and 7-ray m o m e n t u m K (the nucleon m o m e n t u m , correspondingly, is - K ) and the A-isobar, having K-axis spin projection M = 2,. With these amplitudes the resonant multipoles in the reaction 7 N - A - T r N are defined as E(2) =
(1/2V/3)(A3/2 -
V/3Al/2),
0.2fin(q2/9)] , M ( 1 ) = - -~('v~-A3/2 + A 1 / 2 ) ,
f = (11 -
-}X)/47r.
(4)
(6)
yielding a ratio
Below [q[ = 0.2 G e V / c the % [eq. (3)] tends to explode since at 0.1 G e V / c it has the infrared pole. At Iql ~ 0.35 G e V / c a s comes out to be 0.5. We have considered some values of % b e t w e e n 0.5 and ~ 1 , and found the corresponding values o f r / t o fit roughly the mass difference Mzx - M N o f about 290 MeV. We have
E( 2 )/M(1 ) =-(1/~/3)(A3/2
x/~A1/2)/(~/f A3/2
+A1/2) • (7)
There is a factor o f - X / ~ b e t w e e n this and ref. [3] due
Table 1 Parameters of the hamiltonian, the oscillator sizes and the resulting E(2)/M(1) ratio. (a) Corresponds to short range spin force, (b) to Blinder's prescription [eq. (3)]. For Gershtein, set I and II and Isgur et al. there was no Coulomb force. %
(a) 0.5 (b) 0.5 (a) 0.87 (b) 0.87 (a) 1.12 (b) 1.12 Gershtein et al. [2] Set I 2 Set II 2 Isgur et al. [3] 2 (with a factor ( - x/~))
o (fm -1)
r/ Oscillator size (fm) (MeV fm -2) b N = 1/13 bLx = 1/a
6.0 4.56 6.0 2.61 6.0 2.03
520 600 160 340 60 400
0.38 0.38 0.42 0.42 0.36 0.40
0.45 0.43 0.58 0.48 0.69 0.44
~ ~ o~
135.7 135.7 135.7
0.62 0.62 0.62
0.62 0.62 0.62
Energy gained with tensor force (MeV)
E~-E N
E(2)/M(1)
E(2) (X 104)
M(1)
-1.1 5.3 -1.1 3.9 -1.0 19 -1.1 1.0 0.63 84.0 -1.1 -5.7
N
2x
2.6 2.9 4.2 6.0 9.1 10.0
5.5 6.4 6.2 13.6 5.2 12.7
276.7 274.5 289.9 274.1 297.4 297.7
5.9 -4.4 -19.5 -1.2 -53 6.5
-
290.0 290.0 290.0
33.1 55.3 42.0
(X 104)
-1.03 -32 -1.08 -51 -1.0 - 4 2
201
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PHYSICS LETTERS
presumably to different definition o f the E(2)/M(1) ,i 1 1 Isgur et al. [3] define M = -~X/ffA3/2 + -~A1/2 and E However, we stick to the defini= ~ A 3 / 2 - 7X/ffA1/2. a tions given in (6) and (7). The E(2)/M(1)ratio comes out to be rather small (table 1). In the absence o f tensor force the nucleon and the isobar are b o t h in S-states and there can be no E ( 2 ) transition. Since the D-state admixtures in the nucleon ( o f m i x e d spin s y m m e t r y 4 D m ) and the isobar (either s y m m e t r i c 4D s or m i x e d s y m m e t r i c 2D m) are small, the only i m p o r t a n t matrix elements are cross terms involving the lowest states 2Ss(N ) and ' 4Ss(A ). Other states are the 2S's(N ) and 4 Ss(A), also, k n o w n as the breathing m o d e states. These states are only nodal radial excitations o f the same s y m m e t r y as the lowest state, whence the name. In the nucleon the 2S m state is i m p o r t a n t and is mainly responsible for the negative squared charge radius o f the neutron. As has been pointed out b y Liu [ 4 ] , on stabilizing with respect to the oscillator parameter, the a d m i x t u r e o f the breathing m o d e practically disappears. This leads to different D-state admixtures, not so m u c h in the nucleon but in the isobar - which tilt the delicate bal-
12 April 1984
ance in the contribution f r o m these states to the E ( 2 ) transition strength, changing the sign of E(2)/M(1) (table 2), in most cases. Let us examine the short range spin force, o = 6 fm - 1 . The results are marked by (a) in the tables 1 , 2 and 3. We find that for various values o f c~s from 0.5 to 1 the E(2)/M(1) ratio goes on increasing. One cannot make c~s very m u c h larger than the m a x i m u m , c~s 1.12 as then even for a nominal or negligible conf i n e m e n t force M/, - M N is always greater than 300 MeV. It seems to be the upper limit in our m o d e l for o = 6. Results do not change very m u c h if o is changed to 6.5. As a s increases, the C o u l o m b force b e c o m e s steeper pushing d o w n the S-states. This results in a larger S m and c o n s e q u e n t l y S's a d m i x t u r e in the nucleon (table 3). This reduces M(1). This is because the main contributions to M ( 1 ) are A(4Ss) -- N(2Ss) and A(4Ss) - N(2Ss,). These two matrix elements are - 1 . 1 3 9 and 0.003 for as = 0.5 and - 1 . 0 9 and 0.471 for c~s = 1.12, respectively. But E ( 2 ) changes m u c h more, by a factor ~ 1 0 . A n o t h e r interesting feature for large o is the variation o f the size o f the oscillator for the nucleon. With large c~s the sharp attractive Coulomb and s p i n - s p i n force makes the size smaller, whereas for as small, the
,1 We are grateful to Professor G. Karl for confirming this.
Table 2 Comparison of dominant E(2) transition matrix elements, both for short range spin force (a) and Blinder's prescription (b).
(la) (lb) (2a) (2b) (3)
a s = 0.5 %=0.5 % = 1.12 C~s=l.12 Gershtein
r~ = 520 ~ =600 ~ =60 ~=400 et al. [2]
o = 6.0 o=4.56 a=6.0 a=2.03
/x(4Ss) - N(4Dm)
A(4Ds) - N(2Ss)
A(2Dm) -- N(2Ss)
-0.633 × -0.653x -0.839× -1.428× -0.712 ×
0.045 × 0.046× 0.039× 0.119× 0.209 X
0.543 X 0.573× 0.533× 1.380× 0.869 X
10 -2 10 -2 10 -2 10 -2 10 -2
10 - 2
10 -2 10 -2 10 -2 10 -2
10 -2 10 -2 10 -2 10 -2 10 -2
Table 3 Comparison of wave functions. Parameters as in table 2. Nucleon
(la) (lb) (2a) (2b) (3)
202
Isobar
2 Ss
2 Ss
2Sm
4 Dm
4 Ss
4 S~
4Ds
2Din
0.9889 0.9918 0.9488 0.9932 0.9496
0.0727 0.0604 0.1100 0.0152 -0.2353
-0.1231 -0.1048 -0.2878 -0.0716 0.2029
-0.0402 -0.0414 -0.0701 -0.0904 -0.0423
0.9978 0.9977 0.9951 0.9874 0.9727
-0.0015 0.0016 -0.0033 -0.0055 0.2006
-0.0531 -0.0551 -0.0804 -0.1251 -0.0967
0.0385 0.0403 0.0572 0.0964 0.0649
Volmne 138B, number 1,2,3
PHYSICS LETTERS
strong confinement necessary for keeping M a - M N fixed, does the same job. It seems there is an optimum value o f b = (h/mqW) 1/2 ~ 0.4 fm at c~s ~ 0.8. This leads to a proton charge radius o f " 0 . 5 fm which is rather small. The quark mass mq was taken to be 300 MeV. If we neglect the Sm state as in Ohta et al. [4] our energies are higher by 5 0 - 7 0 MeV. This shows that the s p i n - s p i n term breaks SU(6) symmetry quite severely producing a mixed symmetric Sm state of amplitude of around "-0.20. As shown by lsgur et al. [9] this reproduces the ratio of the squared charged radius of the neutron to the proton almost exactly. Let us now concentrate on the spin force range given by eq. (3). For small % the range is not too long, for example, c~s = 0.5 gives o = 4.56 fin -1 and this increase in the range (1/o) is compensated with an increase in the confinement. The results are nearly identical. More interesting is to compare the large c~s region where the range is rather long (for c~s = 1.12, o = 2.03 fm 1) and the role of the range is more apparent. For larger range the s p i n - s p i n force is weaker and a larger confinement is necessary (r/changes from 60 to 400 MeV fm - 2 ) to keep M a - M N fixed. With this weaker spin force the isobar size is closer to the nucleon size, with a larger effective tensor force yielding a bigger D-state admixture. In the nucleon the weaker spin force leads to a smaller Sm which in turn is responsible for a smaller S's state and a larger D m state admixture. The M(1) does not change much. The E(2) however, depends on a delicate cancellation of the matrix elements as shown in table 2. Its sign is determined mainly from relative increase in the N(4D m) and A (2D m ) states. The increase in the 2 Dm and 4 D s state in A overcomes the rise in the N(4Dm)contribution producing a change in the sign of the E(2)/M(1) ratio. It is still one order of magnitude smaller than the values given in refs. [2,3]. For the long range s p i n - s p i n force the ratio of the proton to neutron squared charge radii is not reproduced, a factor of two discrepancy compared to the short range force. But this may not be an adequate ground for ruling out such forces. The experiments predict the E(2)/M(1) ratio to be less than 3% [5] whereas the maximum in our table 1 is ~1%. As the experimental accuracy increases, in the near future, one will be able to make more definite prediction about the magnitude of this ratio as well as
12 April 1984
sign. If it turns out to be negative it may imply that the spin--spin force is long range and not short range. Qualitatively, the E(2) is proportional to the transition matrix element r 2 and the M ( 1 ) t o the magnetic moment. Since our model predicts reasonable magnetic moment for the nucleon (in fact slightly larger than experiment, about 3 nuclear magnetons for protons and - 2 for neutrons) but gives small charge radius, we are probably calculating a lower limit to E(2)/M(1). Corrections to the charge radius may be of pionic origin, or the new m o m e n t u m dependent effects in the non relativistic quark model, discussed by Hayne and Isgur [10]. The position of the excited D-states affects the Dstate admixture which in turn plays a decisive role in the ElM ratio. If we identify the experimental 1920 MeV level as the isobar 4D s state we could determine which of the various forces in table 1 is more acceptable. In table 4 we list 4D s energy and the corresponding E/M. Large o values are generally low in energy compared to small o. Parameters (2a) and (3a) may even be said to be acceptable. Both yield positive ratio. If we arbitrarily adjust the diagonal matrix elements of 2hco states to yield 4D s energy we find comparable ElM only for sets (2a) and (3a). 'This arbitrary lowering although not justified, is done, in the same spirit as Forsyth and Cutkowsky [11] where an ad hoc term in the hamiltonian was introduced to lower the 5 6 ( 1 - ) 3hco state. Table 4 Isobar 4Ds energy and the corresponding E/M a) Parameters
4Ds energy (MeV)
E/M X 104
(la) H' (lb) H' (2a) t1' (2b) H' (3a) H' (3b) H'
2440 1948 2618 1940 2015 1920 2320 1930 1780 1912 2482 1937
5.3 -17.5 3.9 -23.8 19.2 14.1 10 -23.8 84 83 5.7 56.2
a) In the sets denoted by H' the 4D s energy is adjusted to ~1920-1950 MeV by adding an arbitrary constant to the 2~w levels. 203
Volume 138B, number 1,2,3
PHYSICS LETTERS
On the other hand, one could argue that all the observed excited states may not be due to single particle excitations. One could invoke symmetry and fit the spectrum as coming from the two boson state, q q q ( q q - ) 2 [12]. A small admixture of 2 boson states will be present in ground states of N and A. However, the single particle transition operator will not connect the 2 boson excited states to zero boson ground states described b y qqq. Neglecting the admixture mentioned above our observation regarding the single particle transition ratio E/M would still remain valid. We would like to comment on the possible implication of our calculation on the derivation of the NN interaction from q u a r k - q u a r k force. Since the value of a s needed to fit the A--N mass difference in between half to a quarter of what is conventionally used [4], the value of the repulsion in the NN interaction may correspondingly decrease. The size difference of the nucleon and the isobar should be included in the AN and A/~ channels in the resonating group method. We are grateful to Professor R.K. Bhaduri who suggested the problem and was in constant touch with its progress throughout. It is a pleasure to thank
204
12 April 1984
Professor Gabriel Karl for an illuminating discussion and Professor J. LeTourneux, Y. Nogami and Dr. A. Suzuki for encouragement, clarification and help. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References [1] N. lsgur and G. Karl, Phys. Rev. D19 (1979) 2653;D20 (1979) 1191. [2] S.S. Gershtein and D.V. Dzhikiya, Sov. J. Nucl. Phys. 34 (1981) 870. [3] N. Isgur, G. Karl and R. Koniuk, Phys. Rev. D25 (1982) 2394. [4] K.F. Liu, Phys. Lett. l14B (1982) 222; S. Ohta, M. Oka, A. Arima and K. Yazaki, Phys. Lett. l19B (1982) 35. [5] R.K. Bhaduri, L.E. Cohler and Y. Nogami, Phys. Rev. Lett. 44 (1980) 1369. [6] J. Weinstein and N. lsgur, Phys. Rev. D27 (1983) 588. [7] K. Kapusta, Phys. Rev. D23 (1981) 2444. [8] S.M. Blinder, J. Mol. Spec. 5 (1980) 17. [9] N. Isgur, G. Karl and R. Koniuk, Phys. Rev. Lett. 19 (1978) 1269; 45 (1980) 1738(E). [10] C. Hayne and N. Isgur, Phys. Rev. D25 (1982) 1944. [11] C.P. Forsyth and R.E. Cutkowsky, Z. Phys. C18 (1983) 219. [12] H. Toki, J. Dey and M. Dey, Phys. Lett. 133B (1983) 20.
PHYSICS LETTERS
12 April 1984
EFFECT OF QUARK-QUARK TENSOR AND SPIN-SPIN FORCE ON THE RADIATIVE DECAY OF A4SOBAR Jishnu DEY 1 and Mira DEY 1 Laboratoire de Physique Nueldaire, Universitd de Montreal, Montreal, Canada Received 4 April 1983 Revised manuscript received 9 December 1983
We use oscillator wavefunctions with Coulomb, finite-range spin-spin and harmonic confinement forces, stabilizing the nucleon and the isobar separately. Subsequent introduction of the Fermi-Breit tensor force leads to an E(2)/M(1) ratio in the A-N'r decay. The sign and magnitude of this ratio is interesting and it is shown that its measurement leads to an estimate of c~s in light hadrons.
The non-relativistic quark model is one of the phenomenological models to study low energy hadron physics [1]. Within this model the radiative decay of the delta isobar (A) to the nucleon (N) has been studied [2,3]. Transitions are very sensitive to parameters of the model. We reconsider this problem with an effective interquark interaction whose spin dependent and non central components are taken from one-gluon exchange. The parameters of the potentials are chosen to fit some of the ground state properties of N and A. The hamiltonian that we take has the form 3
,:
i =1
(m,+
"
i<]
7
'
(1)
where
aft)":
--2-' x i • ~xs.[-%/i,.isi
+ (87ras/3 m i m j) Si'Si(o3/Tr3/2) exp(--o2r2.) + r..3(3S. "r..S. "r../r 2. - S i ' S i ) l tl t q I 11 t]
(2)
where the Xi, m i and S i are the Gell-Mann colour matrices, masses and the spin matrices of the quarks respectively. Note the presence of the colour electric force (first term) in addition to the colour magnetic 1 On leave from Education Service Govt. of W. Bengal, India. 200
forces, namely the s p i n - s p i n finite range interaction and the tensor term. The colour electric force was not explicitly included in refs. [ 1 - 3 ] . Recently Liu and Ohta et al. [4] have reemphasized the importance of stability against variation in the oscillator size. To do this one has to introduce a finite range s p i n - s p i n interaction, otherwise the contact s p i n - s p i n force of the one gluon exchange potential leads to a collapse for the nucleon system [5]. This problem was not faced by Liu, who considered only colour electric forces or by Ohta et al. who included the Darwin term and worked in a restricted model space [4]. Since we want to include SU(6) breaking introduced by the s p i n - s p i n and tensor forces and also minimize with respect to the oscillator size parameter, we are forced to use a finite range s p i n - s p i n force. We use a volume normalised gaussian form with range parameter o. To maintain the short range nature of the s p i n - s p i n force in the Fermi-Breit potential we could use o = 6 fin -1 . This cr is larger than the one used in ref. [6] but corresponds to an effective gluon mass ~1.2 GeV in conformity with bag model estimates [7]. On the other hand, Blinder [8] showed that under some approximations one obtains a smeared F e r m i Breit s p i n - s p i n term with range o -1 ~ e 2 / m e. This was also discussed by Bhaduri et al. [5]. Blinder's prescription gives 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138B, number 1,2,3
PHYSICS LETTERS
2
o = mq/gO~ s
obtained the stability c o n d i t i o n w i t h o u t the tensor force and finally its c o n t r i b u t i o n has been calculated by re-diagonalising at the stable point. Our n o t a t i o n and configurations are identical with those o f Gershtein et al. [2] and we have followed their conventions and references in defining the E(2)/M(1) ratio in the decay A -+ NT. The helicity amplitudes A 3/2 and A1/2 are defined as
(3)
.
The values of % we have chosen yield o varying f r o m 4 . 5 6 - 2 . 0 3 fm -1 . We have used b o t h of the above prescriptions. Our results show a qualitative difference b e t w e e n the two choices o f o. We would like to stress that the nucleon and the isobar are ground states of different spin systems so that one should find out the sizes o f b o t h b y variation. Since the sizes are different, with the isobar larger than the nucleon, all the three forces, the C o u l o m b , the c o n f i n e m e n t and the s p i n - s p i n parts, contribute to the mass difference b e t w e e n them. Thus we can choose values o f the strong coupling constant in eq. (2) which are more reasonable with the QCD running coupling constant estimates for m o m e n t u m transfer Iql ~> 0.2 GeV/c. At q = 0.2 G e V / c one gets % ~ 1 for flavour n u m b e r N = 3. With a normalisation point a s = 0.2 at q = 3 GeV/c, we have as(q2 ) = 0.2/[1 +
12 April 1984
A~, = (A, M = 2,lHin t lyN, X = X~ - XN, K ) ,
(5)
and Hin t is the electromagnetic interaction as in ref. [2]. A~ is the transition matrix element b e t w e e n the N7 system with helicity X and 7-ray m o m e n t u m K (the nucleon m o m e n t u m , correspondingly, is - K ) and the A-isobar, having K-axis spin projection M = 2,. With these amplitudes the resonant multipoles in the reaction 7 N - A - T r N are defined as E(2) =
(1/2V/3)(A3/2 -
V/3Al/2),
0.2fin(q2/9)] , M ( 1 ) = - -~('v~-A3/2 + A 1 / 2 ) ,
f = (11 -
-}X)/47r.
(4)
(6)
yielding a ratio
Below [q[ = 0.2 G e V / c the % [eq. (3)] tends to explode since at 0.1 G e V / c it has the infrared pole. At Iql ~ 0.35 G e V / c a s comes out to be 0.5. We have considered some values of % b e t w e e n 0.5 and ~ 1 , and found the corresponding values o f r / t o fit roughly the mass difference Mzx - M N o f about 290 MeV. We have
E( 2 )/M(1 ) =-(1/~/3)(A3/2
x/~A1/2)/(~/f A3/2
+A1/2) • (7)
There is a factor o f - X / ~ b e t w e e n this and ref. [3] due
Table 1 Parameters of the hamiltonian, the oscillator sizes and the resulting E(2)/M(1) ratio. (a) Corresponds to short range spin force, (b) to Blinder's prescription [eq. (3)]. For Gershtein, set I and II and Isgur et al. there was no Coulomb force. %
(a) 0.5 (b) 0.5 (a) 0.87 (b) 0.87 (a) 1.12 (b) 1.12 Gershtein et al. [2] Set I 2 Set II 2 Isgur et al. [3] 2 (with a factor ( - x/~))
o (fm -1)
r/ Oscillator size (fm) (MeV fm -2) b N = 1/13 bLx = 1/a
6.0 4.56 6.0 2.61 6.0 2.03
520 600 160 340 60 400
0.38 0.38 0.42 0.42 0.36 0.40
0.45 0.43 0.58 0.48 0.69 0.44
~ ~ o~
135.7 135.7 135.7
0.62 0.62 0.62
0.62 0.62 0.62
Energy gained with tensor force (MeV)
E~-E N
E(2)/M(1)
E(2) (X 104)
M(1)
-1.1 5.3 -1.1 3.9 -1.0 19 -1.1 1.0 0.63 84.0 -1.1 -5.7
N
2x
2.6 2.9 4.2 6.0 9.1 10.0
5.5 6.4 6.2 13.6 5.2 12.7
276.7 274.5 289.9 274.1 297.4 297.7
5.9 -4.4 -19.5 -1.2 -53 6.5
-
290.0 290.0 290.0
33.1 55.3 42.0
(X 104)
-1.03 -32 -1.08 -51 -1.0 - 4 2
201
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presumably to different definition o f the E(2)/M(1) ,i 1 1 Isgur et al. [3] define M = -~X/ffA3/2 + -~A1/2 and E However, we stick to the defini= ~ A 3 / 2 - 7X/ffA1/2. a tions given in (6) and (7). The E(2)/M(1)ratio comes out to be rather small (table 1). In the absence o f tensor force the nucleon and the isobar are b o t h in S-states and there can be no E ( 2 ) transition. Since the D-state admixtures in the nucleon ( o f m i x e d spin s y m m e t r y 4 D m ) and the isobar (either s y m m e t r i c 4D s or m i x e d s y m m e t r i c 2D m) are small, the only i m p o r t a n t matrix elements are cross terms involving the lowest states 2Ss(N ) and ' 4Ss(A ). Other states are the 2S's(N ) and 4 Ss(A), also, k n o w n as the breathing m o d e states. These states are only nodal radial excitations o f the same s y m m e t r y as the lowest state, whence the name. In the nucleon the 2S m state is i m p o r t a n t and is mainly responsible for the negative squared charge radius o f the neutron. As has been pointed out b y Liu [ 4 ] , on stabilizing with respect to the oscillator parameter, the a d m i x t u r e o f the breathing m o d e practically disappears. This leads to different D-state admixtures, not so m u c h in the nucleon but in the isobar - which tilt the delicate bal-
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ance in the contribution f r o m these states to the E ( 2 ) transition strength, changing the sign of E(2)/M(1) (table 2), in most cases. Let us examine the short range spin force, o = 6 fm - 1 . The results are marked by (a) in the tables 1 , 2 and 3. We find that for various values o f c~s from 0.5 to 1 the E(2)/M(1) ratio goes on increasing. One cannot make c~s very m u c h larger than the m a x i m u m , c~s 1.12 as then even for a nominal or negligible conf i n e m e n t force M/, - M N is always greater than 300 MeV. It seems to be the upper limit in our m o d e l for o = 6. Results do not change very m u c h if o is changed to 6.5. As a s increases, the C o u l o m b force b e c o m e s steeper pushing d o w n the S-states. This results in a larger S m and c o n s e q u e n t l y S's a d m i x t u r e in the nucleon (table 3). This reduces M(1). This is because the main contributions to M ( 1 ) are A(4Ss) -- N(2Ss) and A(4Ss) - N(2Ss,). These two matrix elements are - 1 . 1 3 9 and 0.003 for as = 0.5 and - 1 . 0 9 and 0.471 for c~s = 1.12, respectively. But E ( 2 ) changes m u c h more, by a factor ~ 1 0 . A n o t h e r interesting feature for large o is the variation o f the size o f the oscillator for the nucleon. With large c~s the sharp attractive Coulomb and s p i n - s p i n force makes the size smaller, whereas for as small, the
,1 We are grateful to Professor G. Karl for confirming this.
Table 2 Comparison of dominant E(2) transition matrix elements, both for short range spin force (a) and Blinder's prescription (b).
(la) (lb) (2a) (2b) (3)
a s = 0.5 %=0.5 % = 1.12 C~s=l.12 Gershtein
r~ = 520 ~ =600 ~ =60 ~=400 et al. [2]
o = 6.0 o=4.56 a=6.0 a=2.03
/x(4Ss) - N(4Dm)
A(4Ds) - N(2Ss)
A(2Dm) -- N(2Ss)
-0.633 × -0.653x -0.839× -1.428× -0.712 ×
0.045 × 0.046× 0.039× 0.119× 0.209 X
0.543 X 0.573× 0.533× 1.380× 0.869 X
10 -2 10 -2 10 -2 10 -2 10 -2
10 - 2
10 -2 10 -2 10 -2 10 -2
10 -2 10 -2 10 -2 10 -2 10 -2
Table 3 Comparison of wave functions. Parameters as in table 2. Nucleon
(la) (lb) (2a) (2b) (3)
202
Isobar
2 Ss
2 Ss
2Sm
4 Dm
4 Ss
4 S~
4Ds
2Din
0.9889 0.9918 0.9488 0.9932 0.9496
0.0727 0.0604 0.1100 0.0152 -0.2353
-0.1231 -0.1048 -0.2878 -0.0716 0.2029
-0.0402 -0.0414 -0.0701 -0.0904 -0.0423
0.9978 0.9977 0.9951 0.9874 0.9727
-0.0015 0.0016 -0.0033 -0.0055 0.2006
-0.0531 -0.0551 -0.0804 -0.1251 -0.0967
0.0385 0.0403 0.0572 0.0964 0.0649
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strong confinement necessary for keeping M a - M N fixed, does the same job. It seems there is an optimum value o f b = (h/mqW) 1/2 ~ 0.4 fm at c~s ~ 0.8. This leads to a proton charge radius o f " 0 . 5 fm which is rather small. The quark mass mq was taken to be 300 MeV. If we neglect the Sm state as in Ohta et al. [4] our energies are higher by 5 0 - 7 0 MeV. This shows that the s p i n - s p i n term breaks SU(6) symmetry quite severely producing a mixed symmetric Sm state of amplitude of around "-0.20. As shown by lsgur et al. [9] this reproduces the ratio of the squared charged radius of the neutron to the proton almost exactly. Let us now concentrate on the spin force range given by eq. (3). For small % the range is not too long, for example, c~s = 0.5 gives o = 4.56 fin -1 and this increase in the range (1/o) is compensated with an increase in the confinement. The results are nearly identical. More interesting is to compare the large c~s region where the range is rather long (for c~s = 1.12, o = 2.03 fm 1) and the role of the range is more apparent. For larger range the s p i n - s p i n force is weaker and a larger confinement is necessary (r/changes from 60 to 400 MeV fm - 2 ) to keep M a - M N fixed. With this weaker spin force the isobar size is closer to the nucleon size, with a larger effective tensor force yielding a bigger D-state admixture. In the nucleon the weaker spin force leads to a smaller Sm which in turn is responsible for a smaller S's state and a larger D m state admixture. The M(1) does not change much. The E(2) however, depends on a delicate cancellation of the matrix elements as shown in table 2. Its sign is determined mainly from relative increase in the N(4D m) and A (2D m ) states. The increase in the 2 Dm and 4 D s state in A overcomes the rise in the N(4Dm)contribution producing a change in the sign of the E(2)/M(1) ratio. It is still one order of magnitude smaller than the values given in refs. [2,3]. For the long range s p i n - s p i n force the ratio of the proton to neutron squared charge radii is not reproduced, a factor of two discrepancy compared to the short range force. But this may not be an adequate ground for ruling out such forces. The experiments predict the E(2)/M(1) ratio to be less than 3% [5] whereas the maximum in our table 1 is ~1%. As the experimental accuracy increases, in the near future, one will be able to make more definite prediction about the magnitude of this ratio as well as
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sign. If it turns out to be negative it may imply that the spin--spin force is long range and not short range. Qualitatively, the E(2) is proportional to the transition matrix element r 2 and the M ( 1 ) t o the magnetic moment. Since our model predicts reasonable magnetic moment for the nucleon (in fact slightly larger than experiment, about 3 nuclear magnetons for protons and - 2 for neutrons) but gives small charge radius, we are probably calculating a lower limit to E(2)/M(1). Corrections to the charge radius may be of pionic origin, or the new m o m e n t u m dependent effects in the non relativistic quark model, discussed by Hayne and Isgur [10]. The position of the excited D-states affects the Dstate admixture which in turn plays a decisive role in the ElM ratio. If we identify the experimental 1920 MeV level as the isobar 4D s state we could determine which of the various forces in table 1 is more acceptable. In table 4 we list 4D s energy and the corresponding E/M. Large o values are generally low in energy compared to small o. Parameters (2a) and (3a) may even be said to be acceptable. Both yield positive ratio. If we arbitrarily adjust the diagonal matrix elements of 2hco states to yield 4D s energy we find comparable ElM only for sets (2a) and (3a). 'This arbitrary lowering although not justified, is done, in the same spirit as Forsyth and Cutkowsky [11] where an ad hoc term in the hamiltonian was introduced to lower the 5 6 ( 1 - ) 3hco state. Table 4 Isobar 4Ds energy and the corresponding E/M a) Parameters
4Ds energy (MeV)
E/M X 104
(la) H' (lb) H' (2a) t1' (2b) H' (3a) H' (3b) H'
2440 1948 2618 1940 2015 1920 2320 1930 1780 1912 2482 1937
5.3 -17.5 3.9 -23.8 19.2 14.1 10 -23.8 84 83 5.7 56.2
a) In the sets denoted by H' the 4D s energy is adjusted to ~1920-1950 MeV by adding an arbitrary constant to the 2~w levels. 203
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On the other hand, one could argue that all the observed excited states may not be due to single particle excitations. One could invoke symmetry and fit the spectrum as coming from the two boson state, q q q ( q q - ) 2 [12]. A small admixture of 2 boson states will be present in ground states of N and A. However, the single particle transition operator will not connect the 2 boson excited states to zero boson ground states described b y qqq. Neglecting the admixture mentioned above our observation regarding the single particle transition ratio E/M would still remain valid. We would like to comment on the possible implication of our calculation on the derivation of the NN interaction from q u a r k - q u a r k force. Since the value of a s needed to fit the A--N mass difference in between half to a quarter of what is conventionally used [4], the value of the repulsion in the NN interaction may correspondingly decrease. The size difference of the nucleon and the isobar should be included in the AN and A/~ channels in the resonating group method. We are grateful to Professor R.K. Bhaduri who suggested the problem and was in constant touch with its progress throughout. It is a pleasure to thank
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Professor Gabriel Karl for an illuminating discussion and Professor J. LeTourneux, Y. Nogami and Dr. A. Suzuki for encouragement, clarification and help. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
References [1] N. lsgur and G. Karl, Phys. Rev. D19 (1979) 2653;D20 (1979) 1191. [2] S.S. Gershtein and D.V. Dzhikiya, Sov. J. Nucl. Phys. 34 (1981) 870. [3] N. Isgur, G. Karl and R. Koniuk, Phys. Rev. D25 (1982) 2394. [4] K.F. Liu, Phys. Lett. l14B (1982) 222; S. Ohta, M. Oka, A. Arima and K. Yazaki, Phys. Lett. l19B (1982) 35. [5] R.K. Bhaduri, L.E. Cohler and Y. Nogami, Phys. Rev. Lett. 44 (1980) 1369. [6] J. Weinstein and N. lsgur, Phys. Rev. D27 (1983) 588. [7] K. Kapusta, Phys. Rev. D23 (1981) 2444. [8] S.M. Blinder, J. Mol. Spec. 5 (1980) 17. [9] N. Isgur, G. Karl and R. Koniuk, Phys. Rev. Lett. 19 (1978) 1269; 45 (1980) 1738(E). [10] C. Hayne and N. Isgur, Phys. Rev. D25 (1982) 1944. [11] C.P. Forsyth and R.E. Cutkowsky, Z. Phys. C18 (1983) 219. [12] H. Toki, J. Dey and M. Dey, Phys. Lett. 133B (1983) 20.