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Hyperfine interactions in negative parity baryons
Volume 72B, number 1
PHYSICS LETTERS
HYPERFINE
INTERACTIONS
IN NEGATIVE
5 December 1977
PARITY BARYONS*
Nathan ISGUR
Department of Physics, University of Toronto, Toronto, Canada amd Gabriel KARL
Department of Physics, University of Guelph, Guelph, Canada Received 11 October 1977 The hyperfine interaction between quarks suggested in chromodynamics can, without free parameters, explain the size and pattern of the splittings and the mixing angles observed experimentally in nonstrange, negative parity baryons. There is growing support for the idea [I] that gluonmediated quark-quark interactions inside hadrons are similar to photon-mediated interactions. De Rujula et al. [2] have shown very convincingly how the socalled Fermi contact term correlates the splitting of A-N, N-A and I~*-N in ground state baryons. More recently, Carlitz et al. [3] have shown that the same interaction accounts very well for the charge radius of the neutron* t. Apart from the Fermi contact term there is a second, so-called tensor term which is part of the hyperfine interaction. The neutron charge radius and the ground-state mass differences are not sensitive in leading order to the tensor forces which mix L = 2 components into the L = 0 ground state. Contact and tensor forces operate side-by-side, however, in negative parity baryons. We have found that the hyperfine interaction of chromodynamics can explain the pattern and size o f the splittings and the mixing angles of negative parity nonstrange baryons. The effective hyperfine interaction between two quarks 1 and 2 widely assumed in chromodynamics ([2], for a nice review, see [4]) simulates the magnetic dipole-magnetic dipole interaction of electrodynamics:
* Research supported by the National Research Council of Canada. , t Their considerations can be extended to the computation of the electric form factor of the neutron G~(q 2), which turns out to have the right size and to have one sign only (no nodes), in agreement with experiment.
/ / 1 2 = A {(8rr/3)S l "S 2 8 3 (p) + P-3(3S1" b S 2" b -S1"$2)} ,
(1)
where S 1 and S 2 are the spins o f the two quarks, 21/2 p -= r 1 - r 2 is a vector joining them, and A is an overall constant depending on quark masses and the interaction strength , 2 . As is well known in atomic physics [see e.g. [5]), the two terms with relative strength as displayed in eq. (1) are two parts of a single physical interaction: the static interaction o f two intrinsic magnetic dipoles. The second term (often called the 'tensor' part) averages to zero in orbital S states of the pair, and so is operative only when a pair has non-zero orbital angular momentum, while the first term (called the 'Fermi contact' term) is operative only when the pair has zero orbital angular momentum. Our discussion is independent of A except when dealing with the size o f splittings, and then we normalize A by the size o f the A-N mass difference. The lowest mass negative parity baryons are assigned [6] in the quark model to a seventyplet of orbital angular momentum L = 1. Our discussion relies on this assignment. There are three pairs of quarks in a baryon, and in the simplest models*3, which we assume, the ,2 The interaction (1) corresponds to the single gluon exchange part of the potential. The needed size of A is roughly in accord with a "color" coupling constant a s of order unity: A = as/(3 m2x/2), where m is the effective quark mass, for a pair of equal mass quarks separated by r12 = ~ . ,3 An example of this class of models is provided by three particles connected by harmonic forces. 109
Volume 72B, number 1
PHYSICS LETTERS
5 Decembez 1977
states are split into two groups: a pair of N*'s with JP = 1 / 2 - , 3 / 2 - and a second, higher set o f states consisting o f three N*'s with JP = 1 / 2 - , 3 / 2 - , 5 / 2 and two A*'s o f J P = 1 / 2 - , 3 / 2 - . Moreover, the size of the splitting
~800 -
t/00-
M(3/2) - M ( 1 / 2 ) : 2 A~r(ff~MIf3(p)I~M),
(3)
i
can be correlated with the size of the splitting between the A (1236) and the nucleon N (940). If we take harmonic oscillator wavefunctions (see e.g. [8]) we find:
,~oo- ~ q3
1500l
N 'I12-
N" 5 / 2
N'5/2
&'ll~-
A'3/2-
Fig. I. Comparison between theory and experiment for nonstrange P-wave baryons. The unperturbed mass was chosen to be 1610 MeV. The shaded regions indicate the interval of masses quoted in the Baryon Table of the Particle Data Group, with the exception of the two star 3 - resonance at around 1740, which we have estimated. orbital angular momentum L = 1 resides in a single pair while the other two pairs have zero orbital angular momenta. Thus in one pair the tensor term is operative, while in the other two the contact force represents the hyperfine interaction. The negative parity baryons are as a result a good place to test both pieces of the hyperfine interaction. The nonstrange components o f the (70, 1 - } supermultiplet consist of five isodoublets - two of quark spin S = 1/2 and three of quark spin S = 3/2 - and two isoquartets of S = 1/2. We first discuss the splitting of these states induced by the contact term. To calculate a matrix element we must consider the interaction in all three pairs; by the overall symmetry o f the wavefunctions we can instead multiply a specific pair by a factor of three (see e.g. [7}). We find: (N;3 1,. JIHcontact IN,.3~ 1 ; j ) = z r A ( ~ M i 5 3 ( p ) l ~ M )
'
, (N; ~1,. J[HcontactlN,. ,y 1 ; J ) = --TrA(~MI53(p)[~j~M),
(A; ½ 1; JIHcontactlA; ~- 1; J) = ~rA(@)MI63(P)[ ~)M), (2) where ~;~M is the component o f the spatial wavefunction which is even under the transposition of particles 1 and 2 and has L = 1, L z = M. The odd component C o does not contribute to the contact term since it has to vanish when = 0. The expectation value is independent of M. The splitting pattern induced by this term is in qualitative agreement with experiment [2] : the 110
A-N = 4n'A(ffS0163(p)l~S0 ) ~ ~ = 4Ao~3~'-1/2 and ( ~ 1 1 8 3 ( P ) I ~ I > = a3~'-3/2 '
(4) (5)
where c~ is a harmonic oscillator parameter. From eqs. (3), (4) and (5): M(3/2) - M(1/2) = g' (A-N) -~ 150 MeV
(6)
in agreement with the experimental splitting, as evaluated for example from M [N*(1670)] - M [N*(1520)] -~ 1 6 7 0 - 1520 = 150 MeV. The main incorrect prediction, were we to stop at this point, is the expectation (see eq. (2)) that both low lying states N*(1520) and N*(1535) are pure quark spin states S = 1/2, in disagreement with experiment [9]. We now include the tensor force in our discussion as we must. The tensor force has S = 2, so that its matrix elements vanish between any two S = 1/2 states. However, the other matrix elements are nonzero. The direct computation of the matrix elements of the tensor force is somewhat lengthy; we found the following identity [10] useful in reducing the labour involved: ( 1 S J I p - 3 ( 3 S1 " PS 2 " D - S 1 "$2)11S'J}
= ( - - ) J - 1 - S ' ( 3 ( 2 S + 1)) 1/2 W(11SS'; 2 J ) X (1]}½x/'3p-3~+D+II1)(SII½,v/3S1-S2-IIS'),
(7)
where W is a Racah coefficient, and the last two factors are reduced matrix elements of the tensors whose +2 and - 2 components respectively are displayed. Using this identity and standard 70-plet wavefunctions we find for the total contribution of tensor forces in all three pairs: 35
35
(N; 1 ~ ~-IHtensorlN; 1 ~- -f) = 3 a / 4 , 3
3 3
_
(N; 1 ~ ~ IHtensor[N; 1 ~" ~) - - 3 a ,
(8 a) (Sb)
Volume 72B, number 1
PHYSICS LETTERS
(N;1}}[HtensorlN; 1 73 :1) = 15a/4, (N;1 }3~.lHtensor IN ,. 1 7' }) = -(3/4)(5/2) 3 1
1/2a ,
I l
(N; 1 -ff7 [Htensor[ N; 1 ~"7) = 15a/4,
(8c)
corresponding to
(8d)
0d= arctan (X,~/(14 + 2 x / ~ ) ) --~+6.3 ° ,
(8e)
to be compared with the empirical mixing, found by Hey, Lichtfield and Castunore from analysing decay data [9] *4
with
(9)
a -= ½ A ( ~ I Ip-3 (3 cos20o - I ) 1 ~ 1)=
-(4/15)Aot3n -l/2,
where we have assumed that only one pair has nonzero orbital angular momentum, and evaluated the 'radial' matrix element with harmonic oscillator wavefunctions as in eq. (5). Note that while the matrix elements of the contact term (2) depend only on the even (X) component of ~, the matrix elements of the tensor term depend on the odd (P) component. From eqs. (2), (5), (8) and (9) we obtain the full effect of the hyperfine interaction H in the nonstrange states, written below in units (Ac~3~r- I / 2 ) = 4 -1 (M a - MN) -~ 75 MeV. In the zS* sector we have from eq. (2) alone
1
.1
IN*(1520)> -~ -0.181}})
+0.981 77>, '3
(15)
corresponding to 0 a -~ +10 °. The eigenvalues in the J = 1/2 sector are: k+ 1/2 _--
2 _ 1 ( _ 1 -+x/5-) -~ + 0.618 ~ 45 MeV (16) --1.618 - ~ - 1 2 0 MeV,
with the state of low mass [identified with the N*(1530)] being IN*(1530)) = - s i n 0sl 7~71 )+ cos 0s177) l 31
11
= 0.526177) + 0.85117-¢),
(17)
corresponding to a mixing angle of
( A ' 7173 ~**hyp A~ * I 7 T3 ~, _*1
5 December 1977
1 --~ 75 MeV 0 s = -arctan½ ( x / 5 - 1) ~- - 3 1 . 7 ° ,
1
_
(A 771Httyp}A 7 7 ) - 1
~75MeV,
in the J = 5/2 N* sector from eqs. (2), (5), (8a) and (9)
to be compared with the empirical mixing, found very accurately by Hey, Lichtfield and Cashmore to be [9]:
N*37~-}Hh 5 yp} N 1 4 ~"* & 2 52-' = 1 - ?-=-~
IN*(1530))= 0.5317~-) + 0.851½½), 0 - ~ - 3 2 °. (18)
~
60MeV
(10)
in the J = 3/2 N* sector from eqs. (2), (5), (8b, d) and (9) the interaction is H h y p \,,,..~3) M*t 3 ,, ~'7
i~, , 177~ 3
\l/x/~-1
/\N
~-~]'
(ll)
while in the J = 1/2 N* sector from eqs. (2), (5), (8c,e) and (9) it is H,
--nyp
\
.il
N 77-
I
1
\M*131
-, ~ 7 -
"
It remains to diagonalize the J = 3/2 and 1/2 N* sectors to find the physical eigenstates. The eigenvalues in the J = 3/2 sector are: ~3/2 = 10_1(4+ ~ )
~ 1.835 -~ +135 MeV, --1.035
- ~ - 7 5 MeV,
(13)
with the eigenstate corresponding to the lower eigenvalue X (identifed with N*(1520)) being: IN*(1520)) = - s i n 0dl-~}) + cos 0 d [ ~ r ) 33
13
= - 0 . 1 1 1 ~ 7 ) + 0.99417~),
31
The agreement of the composition of states (14) and (17), predicted by the Hamiltonian (1), with the experimental determinations (15) and (18) respectively is very striking. The predictions (14) and (17) are independent of any choice of parameters such as the coupling strength A or choice of harmonic oscillator constant a. These mixings are brought about by the presence of tensor terms and their magnitude depends only on the relative size of contact and tensor terms which is given (apparently correctly!)in eq. (1). The pattern and size of the splittings predicte d by eqs. (10), (13) and (16) with the constant (Ac~37r-1/2) determined as in eq. (4) is also in agreement with experiment as illustrated in fig. 1, where the broad bands represent the limits within which the Particle Data Group feel the masses of the various resonances are likely to lie. The degree of agreement is surprisingly ...J ,4 Our conventions as to the relative sign between S = 1/2 and S = 3/2states are the same as those of Hey et al. [9]. Therefore, the comparison of relative signs is meaningful.
(14) 111
Volume 72B, number 1
PHYSICS LETTERS
good considering that the only parameter at our disposal, Aa3cr - 1/2, was determined from a splitting in an entirely different supermultiplet. Previous group-theoretical analyses of the spectrum of negative parity baryons [ 11 ] have ignored tensor terms b u t kept instead spin-orbit terms * s. These analyses had n o difficulty in fitting the mass spectrum (by appropriate choice of parameters) b u t failed to obtain any sizeable mixing as observed [9] in the J = I/2 sector. At the risk of a n n o y i n g the reader, we repeat that from the point of view of c h r o m o d y n a m i c s it is on the contrary m a n d a t o r y that the tensor and contact terms be treated together, as they have the same physical origin. Nevertheless in analogy with electrodynamics one generally expects spin-orbit coupling to be present as well .6 . Since we neglected spin-orbit coupling completely and obtained satisfactory agreement with experiment, it is relevant to ask how strong a spin-orbit coupling one could tolerate without destroying this agreement. To normalize our answer we take as standard the Breit Hamiltonian [2, 13]. For a 1/r potential b e t w e e n equal mass quarks this predicts a coupling (3 A / 2 O3)L 12" ($1 + $2) with the same A as in eq. (1). If spin-orbit coupling were present with this strength, it would shift the N * 5 / 2 state upward b y ~ 2 2 5 MeV relative to the unshifted A* states .7 . Since in reality the J = 5/2 N* and the A*'s are nearly degenerate the spin-orbit coupling if present at all can be n o stronger than about 10% o f the value predicted by a Breit Hamiltonian [12] *8 This conclusion is very similar to one reached b y Schnitzer [13} in this analysis of the P wave states of charmonium*9'~°. ,5 From a phenomenological point of view there is of course no reason at all to correlate the tensor and contact forces to each other. This is the main advantage of a dynamical prescription such as that suggested by chromodynamics. ,6 Recall, however, that while the hyperfine interaction is the interaction of two intrinsic colour magnets, the spin-orbit coupling arises from a distinct physical mechanism. ,7 An analysis including full-strength spin-orbit coupling (see [ 14 ]) clearly demonstrates that this option is untenable. ,8 Note that the contribution of the tensor term is not small when compared to the contact term, as claimed erroneously in [12]. For example in the S = 3•2, J = 1/2 state the tensor term exactly cancels the contribution of the contact term - leading to the zero total matrix element. ,9 Similar ideas about P wave states in charmonium were also discussed by K. Johnson in a lecture in December 1976.
112
5 December 1977
In looking critically at our results it is n e x t relevant to ask whether they are sensitive to the use of the h a r m o n i c oscillator model. We stress that our conclusions do n o t depend on the use o f harmonic oscillator forces. The model was employed only to generate wavefunctions to c o m p u t e m a t r i x elements, and as shown b y Gromes and Stamatescu [ 12] h a r m o n i c oscillator wavefunctions are a good approximation to the eigenfunctions of low-lying states of a system b o u n d by Coulomb-plus-linear potentials. We conclude that the QCD inspired hyperfine interaction (1) is d o m i n a n t in determining the splittings and mixing angles in P-wave baryons. It remains to be seen whether such effects can be rigorously derived in Q u a n t u m Chromodynamics. We thank F.E. Close and M. Kugler for conversations which helped us into this line of thought, and the Directors of the Les Houches S u m m e r School (1976) for their splendid physical facilities which helped initiate our work.
References [1} M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1006; H. Fritzsch, M. GeU-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365. [2] A. De Ru]ula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147; see also: T. De Grand, R.L. Jaffe, K. Johnson and I. Kiskis, Phys. Rev. D12 (1975) 2060. [3] R.D. Carlitz, S.D. Ellis and R. Savit, Phys. Lett. 68B (1976) 443; see also: N. Isgur, Lectures at the XVIIth Cracow School of Theoretical Physics, to be published in Acta Phys. Polon. [4] .I.D, Jackson, Lectures on the new particles, Proc. 1976 Summer Institute on Particle Physics, Stanford Linear Accelerator Center, Report No. 198 (1976); see also: J. Pumplin, W. Repko and A. Sato, Phys. Rev. Lett. 35 (1975) 1538; H.J. Schnitzer, Phys. Rev. Lett. 35 (1975) 1540; Phys. , l o The splitting between the S wave states qJ and ~c is determined by the contact term in mesons. It is amusing to note note that if one takes the "experimentally determined" q~ wavefunctions at zero relative separation (which rise roughly in proportion to the reduced mass of the q~ pair), then the large q~-~cmass difference can be understood in terms of the o-~r spfitting even though the contact term is inversely proportional to the product of the quark masses.
Volume 72B, number 1
[5] [6} [7] [8] [9]
PHYSICS LETTERS
Rev. D13 (1976) 74; R, Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 445. H.A. Bethe and E.E. Salpeter, in: Handbuch der Physik, vol. X X X V (Springer, Berlin, 1957) p. 267. R.H. Dalitz, Proc. Summer School Les Houches (1965). L.A. Copley, G. Karl and E. Obryk, Nucl. Phys. B13 (1969) 303. G. Karl and E. Obryk, Nucl. Phys. B8 (1968) 609. A.J.G. Hey, P.J. Litchfield and R.J. Cashmore, Nucl. Phys. B95 (1975) 516; see also: D. Faiman and D.E. Plane, Nucl. Phys. B50 (1972) 379.
5 December 1977
[10] D,M. Brink and G.R. Satchler, Angular momentum (Oxford University Press, Oxford, 1962). [11] D.R. Divgi and O.W. Greenberg, Phys. Rev. 175 (1968) 2024; D. Horgan, Nucl. Phys. B71 (1974) 514; For a recent review see: R.R. Horgan, in: Proc. Topical Conf. on Baryon Resonances, eds. R.T. Ross and D.H, Saxon, Rutherford Laboratory S.R,C. Chilton, Didcot, UK (1976) p. 434. [12[ D. Gromes and I.O. Stamatescu, Nucl. Phys. B112 (1976) 213. [131 H.J. Schnitzer, Phys. Lett. 65B (1976) 239. [14] W. Celmaster, Phys. Rev. D15 (1977) 1391.
113
PHYSICS LETTERS
HYPERFINE
INTERACTIONS
IN NEGATIVE
5 December 1977
PARITY BARYONS*
Nathan ISGUR
Department of Physics, University of Toronto, Toronto, Canada amd Gabriel KARL
Department of Physics, University of Guelph, Guelph, Canada Received 11 October 1977 The hyperfine interaction between quarks suggested in chromodynamics can, without free parameters, explain the size and pattern of the splittings and the mixing angles observed experimentally in nonstrange, negative parity baryons. There is growing support for the idea [I] that gluonmediated quark-quark interactions inside hadrons are similar to photon-mediated interactions. De Rujula et al. [2] have shown very convincingly how the socalled Fermi contact term correlates the splitting of A-N, N-A and I~*-N in ground state baryons. More recently, Carlitz et al. [3] have shown that the same interaction accounts very well for the charge radius of the neutron* t. Apart from the Fermi contact term there is a second, so-called tensor term which is part of the hyperfine interaction. The neutron charge radius and the ground-state mass differences are not sensitive in leading order to the tensor forces which mix L = 2 components into the L = 0 ground state. Contact and tensor forces operate side-by-side, however, in negative parity baryons. We have found that the hyperfine interaction of chromodynamics can explain the pattern and size o f the splittings and the mixing angles of negative parity nonstrange baryons. The effective hyperfine interaction between two quarks 1 and 2 widely assumed in chromodynamics ([2], for a nice review, see [4]) simulates the magnetic dipole-magnetic dipole interaction of electrodynamics:
* Research supported by the National Research Council of Canada. , t Their considerations can be extended to the computation of the electric form factor of the neutron G~(q 2), which turns out to have the right size and to have one sign only (no nodes), in agreement with experiment.
/ / 1 2 = A {(8rr/3)S l "S 2 8 3 (p) + P-3(3S1" b S 2" b -S1"$2)} ,
(1)
where S 1 and S 2 are the spins o f the two quarks, 21/2 p -= r 1 - r 2 is a vector joining them, and A is an overall constant depending on quark masses and the interaction strength , 2 . As is well known in atomic physics [see e.g. [5]), the two terms with relative strength as displayed in eq. (1) are two parts of a single physical interaction: the static interaction o f two intrinsic magnetic dipoles. The second term (often called the 'tensor' part) averages to zero in orbital S states of the pair, and so is operative only when a pair has non-zero orbital angular momentum, while the first term (called the 'Fermi contact' term) is operative only when the pair has zero orbital angular momentum. Our discussion is independent of A except when dealing with the size o f splittings, and then we normalize A by the size o f the A-N mass difference. The lowest mass negative parity baryons are assigned [6] in the quark model to a seventyplet of orbital angular momentum L = 1. Our discussion relies on this assignment. There are three pairs of quarks in a baryon, and in the simplest models*3, which we assume, the ,2 The interaction (1) corresponds to the single gluon exchange part of the potential. The needed size of A is roughly in accord with a "color" coupling constant a s of order unity: A = as/(3 m2x/2), where m is the effective quark mass, for a pair of equal mass quarks separated by r12 = ~ . ,3 An example of this class of models is provided by three particles connected by harmonic forces. 109
Volume 72B, number 1
PHYSICS LETTERS
5 Decembez 1977
states are split into two groups: a pair of N*'s with JP = 1 / 2 - , 3 / 2 - and a second, higher set o f states consisting o f three N*'s with JP = 1 / 2 - , 3 / 2 - , 5 / 2 and two A*'s o f J P = 1 / 2 - , 3 / 2 - . Moreover, the size of the splitting
~800 -
t/00-
M(3/2) - M ( 1 / 2 ) : 2 A~r(ff~MIf3(p)I~M),
(3)
i
can be correlated with the size of the splitting between the A (1236) and the nucleon N (940). If we take harmonic oscillator wavefunctions (see e.g. [8]) we find:
,~oo- ~ q3
1500l
N 'I12-
N" 5 / 2
N'5/2
&'ll~-
A'3/2-
Fig. I. Comparison between theory and experiment for nonstrange P-wave baryons. The unperturbed mass was chosen to be 1610 MeV. The shaded regions indicate the interval of masses quoted in the Baryon Table of the Particle Data Group, with the exception of the two star 3 - resonance at around 1740, which we have estimated. orbital angular momentum L = 1 resides in a single pair while the other two pairs have zero orbital angular momenta. Thus in one pair the tensor term is operative, while in the other two the contact force represents the hyperfine interaction. The negative parity baryons are as a result a good place to test both pieces of the hyperfine interaction. The nonstrange components o f the (70, 1 - } supermultiplet consist of five isodoublets - two of quark spin S = 1/2 and three of quark spin S = 3/2 - and two isoquartets of S = 1/2. We first discuss the splitting of these states induced by the contact term. To calculate a matrix element we must consider the interaction in all three pairs; by the overall symmetry o f the wavefunctions we can instead multiply a specific pair by a factor of three (see e.g. [7}). We find: (N;3 1,. JIHcontact IN,.3~ 1 ; j ) = z r A ( ~ M i 5 3 ( p ) l ~ M )
'
, (N; ~1,. J[HcontactlN,. ,y 1 ; J ) = --TrA(~MI53(p)[~j~M),
(A; ½ 1; JIHcontactlA; ~- 1; J) = ~rA(@)MI63(P)[ ~)M), (2) where ~;~M is the component o f the spatial wavefunction which is even under the transposition of particles 1 and 2 and has L = 1, L z = M. The odd component C o does not contribute to the contact term since it has to vanish when = 0. The expectation value is independent of M. The splitting pattern induced by this term is in qualitative agreement with experiment [2] : the 110
A-N = 4n'A(ffS0163(p)l~S0 ) ~ ~ = 4Ao~3~'-1/2 and ( ~ 1 1 8 3 ( P ) I ~ I > = a3~'-3/2 '
(4) (5)
where c~ is a harmonic oscillator parameter. From eqs. (3), (4) and (5): M(3/2) - M(1/2) = g' (A-N) -~ 150 MeV
(6)
in agreement with the experimental splitting, as evaluated for example from M [N*(1670)] - M [N*(1520)] -~ 1 6 7 0 - 1520 = 150 MeV. The main incorrect prediction, were we to stop at this point, is the expectation (see eq. (2)) that both low lying states N*(1520) and N*(1535) are pure quark spin states S = 1/2, in disagreement with experiment [9]. We now include the tensor force in our discussion as we must. The tensor force has S = 2, so that its matrix elements vanish between any two S = 1/2 states. However, the other matrix elements are nonzero. The direct computation of the matrix elements of the tensor force is somewhat lengthy; we found the following identity [10] useful in reducing the labour involved: ( 1 S J I p - 3 ( 3 S1 " PS 2 " D - S 1 "$2)11S'J}
= ( - - ) J - 1 - S ' ( 3 ( 2 S + 1)) 1/2 W(11SS'; 2 J ) X (1]}½x/'3p-3~+D+II1)(SII½,v/3S1-S2-IIS'),
(7)
where W is a Racah coefficient, and the last two factors are reduced matrix elements of the tensors whose +2 and - 2 components respectively are displayed. Using this identity and standard 70-plet wavefunctions we find for the total contribution of tensor forces in all three pairs: 35
35
(N; 1 ~ ~-IHtensorlN; 1 ~- -f) = 3 a / 4 , 3
3 3
_
(N; 1 ~ ~ IHtensor[N; 1 ~" ~) - - 3 a ,
(8 a) (Sb)
Volume 72B, number 1
PHYSICS LETTERS
(N;1}}[HtensorlN; 1 73 :1) = 15a/4, (N;1 }3~.lHtensor IN ,. 1 7' }) = -(3/4)(5/2) 3 1
1/2a ,
I l
(N; 1 -ff7 [Htensor[ N; 1 ~"7) = 15a/4,
(8c)
corresponding to
(8d)
0d= arctan (X,~/(14 + 2 x / ~ ) ) --~+6.3 ° ,
(8e)
to be compared with the empirical mixing, found by Hey, Lichtfield and Castunore from analysing decay data [9] *4
with
(9)
a -= ½ A ( ~ I Ip-3 (3 cos20o - I ) 1 ~ 1)=
-(4/15)Aot3n -l/2,
where we have assumed that only one pair has nonzero orbital angular momentum, and evaluated the 'radial' matrix element with harmonic oscillator wavefunctions as in eq. (5). Note that while the matrix elements of the contact term (2) depend only on the even (X) component of ~, the matrix elements of the tensor term depend on the odd (P) component. From eqs. (2), (5), (8) and (9) we obtain the full effect of the hyperfine interaction H in the nonstrange states, written below in units (Ac~3~r- I / 2 ) = 4 -1 (M a - MN) -~ 75 MeV. In the zS* sector we have from eq. (2) alone
1
.1
IN*(1520)> -~ -0.181}})
+0.981 77>, '3
(15)
corresponding to 0 a -~ +10 °. The eigenvalues in the J = 1/2 sector are: k+ 1/2 _--
2 _ 1 ( _ 1 -+x/5-) -~ + 0.618 ~ 45 MeV (16) --1.618 - ~ - 1 2 0 MeV,
with the state of low mass [identified with the N*(1530)] being IN*(1530)) = - s i n 0sl 7~71 )+ cos 0s177) l 31
11
= 0.526177) + 0.85117-¢),
(17)
corresponding to a mixing angle of
( A ' 7173 ~**hyp A~ * I 7 T3 ~, _*1
5 December 1977
1 --~ 75 MeV 0 s = -arctan½ ( x / 5 - 1) ~- - 3 1 . 7 ° ,
1
_
(A 771Httyp}A 7 7 ) - 1
~75MeV,
in the J = 5/2 N* sector from eqs. (2), (5), (8a) and (9)
to be compared with the empirical mixing, found very accurately by Hey, Lichtfield and Cashmore to be [9]:
N*37~-}Hh 5 yp} N 1 4 ~"* & 2 52-' = 1 - ?-=-~
IN*(1530))= 0.5317~-) + 0.851½½), 0 - ~ - 3 2 °. (18)
~
60MeV
(10)
in the J = 3/2 N* sector from eqs. (2), (5), (8b, d) and (9) the interaction is H h y p \,,,..~3) M*t 3 ,, ~'7
i~, , 177~ 3
\l/x/~-1
/\N
~-~]'
(ll)
while in the J = 1/2 N* sector from eqs. (2), (5), (8c,e) and (9) it is H,
--nyp
\
.il
N 77-
I
1
\M*131
-, ~ 7 -
"
It remains to diagonalize the J = 3/2 and 1/2 N* sectors to find the physical eigenstates. The eigenvalues in the J = 3/2 sector are: ~3/2 = 10_1(4+ ~ )
~ 1.835 -~ +135 MeV, --1.035
- ~ - 7 5 MeV,
(13)
with the eigenstate corresponding to the lower eigenvalue X (identifed with N*(1520)) being: IN*(1520)) = - s i n 0dl-~}) + cos 0 d [ ~ r ) 33
13
= - 0 . 1 1 1 ~ 7 ) + 0.99417~),
31
The agreement of the composition of states (14) and (17), predicted by the Hamiltonian (1), with the experimental determinations (15) and (18) respectively is very striking. The predictions (14) and (17) are independent of any choice of parameters such as the coupling strength A or choice of harmonic oscillator constant a. These mixings are brought about by the presence of tensor terms and their magnitude depends only on the relative size of contact and tensor terms which is given (apparently correctly!)in eq. (1). The pattern and size of the splittings predicte d by eqs. (10), (13) and (16) with the constant (Ac~37r-1/2) determined as in eq. (4) is also in agreement with experiment as illustrated in fig. 1, where the broad bands represent the limits within which the Particle Data Group feel the masses of the various resonances are likely to lie. The degree of agreement is surprisingly ...J ,4 Our conventions as to the relative sign between S = 1/2 and S = 3/2states are the same as those of Hey et al. [9]. Therefore, the comparison of relative signs is meaningful.
(14) 111
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good considering that the only parameter at our disposal, Aa3cr - 1/2, was determined from a splitting in an entirely different supermultiplet. Previous group-theoretical analyses of the spectrum of negative parity baryons [ 11 ] have ignored tensor terms b u t kept instead spin-orbit terms * s. These analyses had n o difficulty in fitting the mass spectrum (by appropriate choice of parameters) b u t failed to obtain any sizeable mixing as observed [9] in the J = I/2 sector. At the risk of a n n o y i n g the reader, we repeat that from the point of view of c h r o m o d y n a m i c s it is on the contrary m a n d a t o r y that the tensor and contact terms be treated together, as they have the same physical origin. Nevertheless in analogy with electrodynamics one generally expects spin-orbit coupling to be present as well .6 . Since we neglected spin-orbit coupling completely and obtained satisfactory agreement with experiment, it is relevant to ask how strong a spin-orbit coupling one could tolerate without destroying this agreement. To normalize our answer we take as standard the Breit Hamiltonian [2, 13]. For a 1/r potential b e t w e e n equal mass quarks this predicts a coupling (3 A / 2 O3)L 12" ($1 + $2) with the same A as in eq. (1). If spin-orbit coupling were present with this strength, it would shift the N * 5 / 2 state upward b y ~ 2 2 5 MeV relative to the unshifted A* states .7 . Since in reality the J = 5/2 N* and the A*'s are nearly degenerate the spin-orbit coupling if present at all can be n o stronger than about 10% o f the value predicted by a Breit Hamiltonian [12] *8 This conclusion is very similar to one reached b y Schnitzer [13} in this analysis of the P wave states of charmonium*9'~°. ,5 From a phenomenological point of view there is of course no reason at all to correlate the tensor and contact forces to each other. This is the main advantage of a dynamical prescription such as that suggested by chromodynamics. ,6 Recall, however, that while the hyperfine interaction is the interaction of two intrinsic colour magnets, the spin-orbit coupling arises from a distinct physical mechanism. ,7 An analysis including full-strength spin-orbit coupling (see [ 14 ]) clearly demonstrates that this option is untenable. ,8 Note that the contribution of the tensor term is not small when compared to the contact term, as claimed erroneously in [12]. For example in the S = 3•2, J = 1/2 state the tensor term exactly cancels the contribution of the contact term - leading to the zero total matrix element. ,9 Similar ideas about P wave states in charmonium were also discussed by K. Johnson in a lecture in December 1976.
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5 December 1977
In looking critically at our results it is n e x t relevant to ask whether they are sensitive to the use of the h a r m o n i c oscillator model. We stress that our conclusions do n o t depend on the use o f harmonic oscillator forces. The model was employed only to generate wavefunctions to c o m p u t e m a t r i x elements, and as shown b y Gromes and Stamatescu [ 12] h a r m o n i c oscillator wavefunctions are a good approximation to the eigenfunctions of low-lying states of a system b o u n d by Coulomb-plus-linear potentials. We conclude that the QCD inspired hyperfine interaction (1) is d o m i n a n t in determining the splittings and mixing angles in P-wave baryons. It remains to be seen whether such effects can be rigorously derived in Q u a n t u m Chromodynamics. We thank F.E. Close and M. Kugler for conversations which helped us into this line of thought, and the Directors of the Les Houches S u m m e r School (1976) for their splendid physical facilities which helped initiate our work.
References [1} M.Y. Han and Y. Nambu, Phys. Rev. 139B (1965) 1006; H. Fritzsch, M. GeU-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365. [2] A. De Ru]ula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147; see also: T. De Grand, R.L. Jaffe, K. Johnson and I. Kiskis, Phys. Rev. D12 (1975) 2060. [3] R.D. Carlitz, S.D. Ellis and R. Savit, Phys. Lett. 68B (1976) 443; see also: N. Isgur, Lectures at the XVIIth Cracow School of Theoretical Physics, to be published in Acta Phys. Polon. [4] .I.D, Jackson, Lectures on the new particles, Proc. 1976 Summer Institute on Particle Physics, Stanford Linear Accelerator Center, Report No. 198 (1976); see also: J. Pumplin, W. Repko and A. Sato, Phys. Rev. Lett. 35 (1975) 1538; H.J. Schnitzer, Phys. Rev. Lett. 35 (1975) 1540; Phys. , l o The splitting between the S wave states qJ and ~c is determined by the contact term in mesons. It is amusing to note note that if one takes the "experimentally determined" q~ wavefunctions at zero relative separation (which rise roughly in proportion to the reduced mass of the q~ pair), then the large q~-~cmass difference can be understood in terms of the o-~r spfitting even though the contact term is inversely proportional to the product of the quark masses.
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[5] [6} [7] [8] [9]
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Rev. D13 (1976) 74; R, Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 445. H.A. Bethe and E.E. Salpeter, in: Handbuch der Physik, vol. X X X V (Springer, Berlin, 1957) p. 267. R.H. Dalitz, Proc. Summer School Les Houches (1965). L.A. Copley, G. Karl and E. Obryk, Nucl. Phys. B13 (1969) 303. G. Karl and E. Obryk, Nucl. Phys. B8 (1968) 609. A.J.G. Hey, P.J. Litchfield and R.J. Cashmore, Nucl. Phys. B95 (1975) 516; see also: D. Faiman and D.E. Plane, Nucl. Phys. B50 (1972) 379.
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[10] D,M. Brink and G.R. Satchler, Angular momentum (Oxford University Press, Oxford, 1962). [11] D.R. Divgi and O.W. Greenberg, Phys. Rev. 175 (1968) 2024; D. Horgan, Nucl. Phys. B71 (1974) 514; For a recent review see: R.R. Horgan, in: Proc. Topical Conf. on Baryon Resonances, eds. R.T. Ross and D.H, Saxon, Rutherford Laboratory S.R,C. Chilton, Didcot, UK (1976) p. 434. [12[ D. Gromes and I.O. Stamatescu, Nucl. Phys. B112 (1976) 213. [131 H.J. Schnitzer, Phys. Lett. 65B (1976) 239. [14] W. Celmaster, Phys. Rev. D15 (1977) 1391.
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