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Relativistic qqq spectra from Bethe-Salpeter premises
Volume 104B, number 1
PHYSICS LETTERS
13 August 1981
RELATIVISTIC qqq SPECTRA FROM BETHE-SALPETER PREMISES A.N. MITRA and I. SANTHANAN Department of Physics, University of Delhi, Delhi-l lO007, India Received 14 October 1980 Revised manuscript received 3 March 1981
A relativistic but three-dimensional form of the Bethe-Salpeter (BS) equation developed under the Instantaneous Approximation (ANM) for a qqq system, is solved in a closed manner under conditions of harmonic confinement. It is shown that the BS dynamics offers a built-in mechanism for explaining major symmetry-breaking effects, especially spinspin and spin-orbit splittings using as inputs (1) a 'reduced' spring constant ¢~ and (ii) the quark mass. The model provides a very good description of the (ud) baryon spectrum in terms of the same values of t~ and mud as employed for q~ systems as well.
The general consensus on qqq spectroscopy has so far been confined to the harmonic oscillator (HO) description, essentially summarised in the FKR treatment [1 ], though non-trivial variants have been proposed on the basis of spectroscopic information [2], An equally satisfactory dynamical theory of symmetrybreaking effects (spin-spin, spin-orbit, etc.) is still lacking, though there have been recent attempts [3,4] to explain these in terms of QCD corrections, such as the Fermi-Breit term. On the other hand, the inverse proportionality of such correction terms to the quark masses (mq) militates against their validity for light quark (uds) spectroscopy [6,7] while serving as a good approximation for heavy (c, b) quarks. We propose an alternative mechanism for generating spin- and momentum-dependent corrections to the qqq spectrum whose relativistic validity is not impaired by an m q 1 expansion, by a method developed recently [6]. The result for the qqq spectrum for equal-mass kinetics is expressible as FB(N ) = N + 3, where N = N~ = Nn = 2n + L is the total HO quantum number, and the function FB(M ) defined by FB(M) -- E l + if'2 + F Q C D '
ff'l = (M2 - m 2 ) / a B ,
3 f 2 = _ I ~BM_23,~2(_~ d ' S - ~ - ON - Q'N), may be regarded as the (mass) 2 of the baryon, corrected for several symmetry-breaking effects ( J - S ,
ON, Q~r and too2), and measured in units (GeV 2) of an FKR-like [ 1], but M-dependent, spring constant g2B . The J ' S term, (which, unlike L "S is a relativistically invariant concept), is the resultant of spin-spin and spin-orbit splittings; m02 = 9m 2 is the quark mass correction; ~)N, Q~¢ represent certain momentumdependent corrections to the effective BS hamiltonian, and are expressible in terms of the total quantum number N. 3'B is a slowly varying function of M, and FQC D is a (hopefully small) perturbative correction due to short-range one-gluon exchange effects [5], termed QCD for brevity. A very similar formula, including the treatment of unequal-mass kinematics, has been derived for the q~ system as well [8]. Before discussing the experimental test of the mass formula (1) we first outline the essential steps leading to (1), The basic dynamics is contained in the B e t h e Salpeter equation [6] for a qqq system under pairwise q - q interactions with a colour dependence X(1). ~~ X(2) = - ~ , and spin-dependence "t ~ .v(1)A.(2)~ y , j, reduced to the three-dimensional level under the instantaneous approximation [6]. For equal-mass kinematics, and in the small q~. approximation (i,] = 1,2, 3), this equation takes the form [6]
(l) D12D23D31~(PlP2P3) = ~ ( - ~ ) D 3 1 D 2 3 M 1 1 123
(2) X f d k x 2 ( 2 ~ ) - 3 ( q 12 tV [klz) T (12) ~(PtlP~2P3),
62 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
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PHYSICS LETTERS
where the energy p03 of the spectator quark q3 has been taken in proportion to its intrinsic mass [6], viz. ½ M, and with m 0 = 3mq, 2 _ M 2 2 ~ 4 (m 02 - M 2) + p 2 + 4 q 2 2 D 1 2 -_m 1 22 + 4qx2 The potential V has the form [6]
(3)
q~ interaction. On the other hand it is tempting to relate CO2q to co2-~ through a plausible ansatz, viz., only one pair of q~ interaction produces roughly the same (mass) 2 spacings (AM 2 ~ 1 GeV 2) in q~ spectra as three pairs of qq interaction do for the qqq spectra. Therefore we postulate [9] 36O2q =6o2~ = 2c~2mq ,
(qlglk) = 3 (27r)3CO2qV28(q _ k) + 4rrO~s(q-k)-2, and [6]
13 August 1981
(4)
T(12) =
4..2_ 2 --~lv/ * P 3 - - ( q 1 2 +k12) 2 + o(1),-(2)g- _ -- 8iS12"q12Xk12 i/ "il vt12 k12)j(q12-k12)l
- 2 M m o l ( q 2 2 - k 2 2 ) + 2i(a 1 - * 2 ) ' P 3 X (q12-k12).
(5) In the "quadratic" approximation, and with the neglect of the Coulomb term in (4), eq. (2) reduces to
(13)
thus expressing the qqq spectra in terms of the q?t parametrization. With this assumption, the parameters ofeq. (1) are
UZB =6O3TB(moM)I/2,
"I,2 = 1+ 9 o32/Mm0 .
(14)
Further, the "proper" wave function ~ which incorporates probability conservation is given in terms of ~, eq. (6), by q~({,q) = ~(~,,q) exp [_9 ({2 +q2)/Mmq]
[_~(m 2 _ M 2) + ~(~2 + n 2 ) l ~(g ,n) = (~/32)-3/2 ex p [__~ (~2 +II2)/3B2] ,
3 2 ,---1 [4~ ( ~ 2 + r l 2 ) - 4 J . S + 9 + - ~ Q1 B* --~COqqm _
+ i QB - 6Mmo1 (~'V¢ + II"Vn)] ~(~,q) ,
(6)
where the basis set ~, r7 is defined by Pl = --tl'
--
1
--
P2,3 - 5Sl + l x / ~
'
(7)
and QB = ~ (4 q12V12 2 2 2 2 + 8ql 2 "V12 + 6 -P3V23 ) , 123
(8)
Q'B = - 2 ( l + m 2 M - 2 ) ~ - ~ ( q 2 2 + q 2 3 + a, 2P2 + a1 P3)V23' 2 2 123 (9) Except for the QB-terms' eq. (6) admits an exact solution. We indicate below a procedure for extracting the diagonal parts of QB, (~B in the (~, r/) basis in the form (N= N~ = N~) ~)B + ~)]3 ~ 6QN + 6Q~v,
(10)
QN _2 (N 2 + 3 N - 3)
(11)
, = Q'N
½( 2 N + 3) 2 -= S4Q N ,
~1 (1 + m2M -2) [7~ Q N - - I9( N + $ )3 2 + ~9l .
(12)
With these substitutions in (6), its solution is expressible as FB(M ) = N + 3, where FB(M ) is given by eq. (1), and the QCD contribution arising from the Coulomb term of eq. (4) has been added perturbatively. The quantities f2 B and TB are function of 6Oqq which, in principle, is a different parameter from 6Oq~ which would appear in eq. (4) for the corresponding
27/323, 2 = £2B .
(15)
Finally, for the reduction of ~)B, ~)B, we first express + + these in terms of the ladder operators au, au, ani, ani and drop the 4-step ladders a 4, a +4 which contribute only via second-order transitions with large energy denominators. This leaves us with operators of the types +2 2 2 +2 + + (16) a~i at/, anian] , auania~janj , and their hermitian conjugates. Next we make the following replacements separately for ~ and ~ operators (N - ai ai). _
+
+
.
+
ai(aia] )a] ~ a i ( 1 N + 1)alia; , +
+
+
1
a i (ai a/)aj -+ a i (~N)Si]a / .
(17) (18)
This amounts to a sort of "averaging" process [8] for the aia7 and a+a/operators consistently with their basic commutation relations. The results of these manipulations are the ON and Q~v terms listed in eqs. (10), (11).
To compare eq. (1) with the data [10] it is convenient to compute the function FB(M ) in terms of experimental masses, for specififed values of cJ and mud preferably the same (o3 = 0.15, mud = 0.28) as employed for q~ physics [8]. Such a common parametrization for qq and qqq spectroscopy should hopefully constitute a sufficiently wide framework for a 63
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PHYSICS LETTERS
comprehensive test of our BS formalism for harmonic confinement. Table 1 gives a limited comparison (for u, d quarks only) for non-strange 56 and 70 states, and fig. 1 exhibits their main sequence Regge resonances as a twin device for bringing out certain characteristic features of BS dynamics. In this regard it is good to keep in mind the intentions behind, as well as expectations from, the F B-function which for an "ideal" situation must be fully degenerate for all members of a given L-supermultiplet (both flavour- and spin-wise) as well as exhibit the unit spacing rule for successive supermultiplets. On the other hand since this quantity is being determined from the experimental masses, any overambitious expectation in this regard needs to be tempered with a more realistic appraisal not only of the obvious limitations of formula (1) in relation to the dimensions of its expected theoretical coverage, but
13 August 1981
also the equally obvious hazards of attaching too much literal significance to the officially listed masses of various states, without regard to their mass dispersions which are also given therein. To illustrate, the low-J states for a given L are usually listed with a high enough mass so that they appear nearly degenerate with their high-J partners, giving rise to the hasty impression of a small s p i n - o r b i t term, had it not also been for their mass-spreads which it would be perfectly legitimate to take cognizance of for purposes of a more "global" assessment in terms of the F B-function. To pursue the "apparent degeneracy" argument a step further, note, e.g., that D15(1670) and F15 (1688) look almost degenerate, but any serious attempt to search for a theoretical reason would bring back memories of parity doublets [11], a theory abandoned long ago since the so-called Carlitz-Kislinger cuts came into existence [ 12].
Table 1. The non-strange qqq spectrum under harmonic confinement. For notations see text and PDG tames [10]. State
Resonance
M2
fZB
F2
FB(M)
56-even+
Pit -=N(938) P~3 -= A (1236) P'~1(1470) p~3(1690 ) N2 ~- F15 (1690) 2~2 ~ F37(1950 ) F35 (1890) P~l (1850-1910) a) N4 ~ Hi9 (2220) 2x4 ~. H3,11(2420) A6(2880) 2x8 (3350)
0.88 1.53 2.16 2.86 2.86 3.80 3.50 3.42-3.65 4.93 5.86 8.29 11.22
0.82 0.94 1.02 1.09 1.09 1.17 1.15 1.14-1.16 1.25 1.30 1.41 1.52
0.82 0.13 1.52 0.94 1.06 0.54 0.76 0.90-0.84 1.53 1.04 1.52 1.95
1.14 1.05 3.11 3.02 3.10 3.23 3.28 3.42-3.50 5.00 5.04 6.93 8.93
(70+, N = 2)
dPi'1(1700) qP~'3 F17(1950_1990 )
2.89 3.28 3.80-3.96
1.09 1.12 1.17-1.18
1.09 0.81 0.53-0.51
3.16 3.14 3.20-3.30
(70-odd-)
D'I3(1530) S'~1(1520) Dis(1685) D~'3(1670-1700) Si'l(1620 -1650) D33(1680 ) S~1(1640) D35 (1940 -1960) G 17(2120 -2150 ) G19(2200) I1,11(2580_2650 )
2.34 2.31 2.84 2.79 -2.89 2.62-2.72 2.82 2.69 3.76-3.84 4.49 -4.62 4.84 6.66-7.02
1.04 1.04 1.09 1.08 - 1.09 1.07-1.08 1.09 1.08 1.16-1.17 1.22 - 1.23 1.24 1.34-1.36
0.69 0.81 0.28 0.38 -0.36 0.51-0.49 0.55 0.68 1.49 -1.47 1.11 - 1.08 0.76 1.59-154
2.41 2.40 2.38 2.35 - 2.41 2.36 2.41 2.58 2.59 4.19-4.22 4.27 -4.33 4.23 6.10-6.22
a) However, a (70, 0+) assignment gives for P~l (1910), F(M) = 3.31 in much closer agreement with F(M) values for F37(1950) and other N = 2 members. 64
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9
8
7
t, -ms u_ 4
3
~
.
x -
~
N2, t
states
2
1
O
I
I
2
~
I
I
4
s
I
6
I
7
N ~
Fig. 1. Principal (J = L + S) Regge recurrences of FB-values for (56, even+) and (70,odd-) states. The vertical bars "1" for (70, odd-) states represent the extent of mass spread indicated in table 1. Subject to such words of caution, table 1 reveals several interesting features which are exhibited in terms of separate listings of (56, even + ) and (70, o d d - ) states, together with an intermediate category of a few " g o o d " states of ( 7 0 , N = 2) as well. It is also convenient to break up the cases in terms of (1) stretched (J --L + S) states which show very little deviation from the expected behaviour of FB(M), and (ii) unstretched ( J < L + S) states which seem to show modest deviations. To assess the latter, a reasonable mass spread consistent with the PDG tables [10] has been included, while for the stretched category a single mass value (usually the listed one) seems to suffice. (1) For the 56 states the F B values leave practically little to be desired. Among the notable success of the J" S term, in which the background of the QN terms has also played a vital numerical role, are the near equality o f f B for (NL, AL) all the way up to L = 4, despite huge differences in theirM 2 values. The same t tt is also true of the radially excited pair ( P l l , P33)(2) The more explicit role of the QN-terms is manifest in the remarkable closeness o f F B for the lowt tt mass radial states ( P l l , P33) to their high-mass L = 2 partners F15 and F37.
13 August 1981
(3) The (70 + , N = 2) states also seem to exhibit highly degenerate F B-values, not only in remarkable conformity with the patterns (1) and (2) but also in good numerical overlap with their 56-partners. This last feature is particularly welcome in view of the highly controversial status of these otherwise four-star states [13]. (4) The expected A F B = 2 spacing rule for successive Regge excitations is fulfilled with a shortfall of 5% all the way to L = 8, while simultaneously respecting the (NL, AL) degeneracy where data are available (L = 0, 2, 4). Note that the ~ ~ x/M variation implies an asymptotic variation M 3/2 ~ N (reminiscent of a linear potential in an NR context), in preference to the M 2 ~ N law characterizing the HO model [1]. The actual F Bplots against N for the stretched (NL, AL) states are shown in fig. 1. (5) The (70, odd ) states which are listed separately show much the same precise quantitative features for the stretched states, revealing the roles of the J - S as well as the QN-terms (e.g. D13 versus D15 ; G17 versus G/~9_ versus D35). Especially interesting is the natural - 3 - state which fits D35 but on which there seems to have been some controversy [13]. (6) The A F B = 2 spacing rule is independently in evidence for the 70-sequence (L = 1,3, 5), again with a 5% shortfall in parallel with the 56-case, as seen from fig. I. However, a small 5 6 - 7 0 breaking is visible from a 10% upward shift of the 70-line above the 56-line. t! tt (7) For the S = 3/2 unstretehedJ-states (S12 , D13 , tt P31) there is a tendency twoards increased F B in terms of the listed mass values, a disease which is only partly cured after taking account of their allowed variations. A similar problem occurs for the D33 and $31 too. Yet it is interesting to note a near equality of the F B-values for these complementary states (I = 1/2, S = 3/2 versus I = 3/2, S = 1/2). This figure stands at a 10% higher value than the nearly degenerate stretched-J case. Together with the P3'l case, these 10% higher values represent about the "worst" cases in the whole table, yet modest compared with the average 6M 2 spreads for L = 1 and L = 2. All other cases (of higher J ) have fared far better not merely on average, but individually. In fig. 1, it is futile to display the corresponding M 2 versus N plots, because of their huge scatter, which is best exhibited in the table itself. The QCD effect has been calculated perturbatively with the same method as employed for q~ mesons [8], the smallness OfFQc D provides an afortiori justification for its perturbative 65
Volume 104B, number 1
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treatment and, unlike ref. [3] it is not also being "missed" in the context o f the otherwise good agreement with only the confining interaction. To conclude, we have proposed an explicit form o f the qqq spectrum in terms of BS dynamics for harmonic confinement, using a common reduced spring constant c~ for mesons and baryons, under the plausible assumption (12). Our model contrasts with ref. [4] (which would prefer w/3-~ for the qqq system). It also contrasts with the QCD-dominant mechanism [3] employed in an NR framework whose validity has been questioned recently [14]. The present framework being relativistic, is hopefully free from the NR objections for light quarks. The data have given good and unequivocal support for the QN and J ' S terms except for certain low-J problems (which might suggest a slightly weaker s p i n - o r b i t effect). In particular the F(M)representation seems to be so much superior to the usualN2-display for revealing the symmetry-breaking effects, that its use could perhaps be profitably recommended on wider grounds than the strict premises of this model. Corresponding work on strange baryons is in progress.
66
13 August 1981
References [1 ] R.P. Feynman et al., Phys. Rev. D3 (1971) 2706; referred to as FKR. [2] See e.g., A.N. Mitra, Phys. Rev. D l l (1975) 3270. [3] See, e.g., N. Isgur and G. Karl, Phys. Rev. D18 (1978) 4187. [4] D.P. Stanley and D. Robson, Phys. Rev. Lett. 45 (1980) 235. [5 ] A. de Rfijula et al., Phys. Rev. D12 (1975) 147. [6] A.N. Mitra, Delhi Univ. preprint (March 1980); Z. Phys. C (1980), to be published. [7] R.K. Bhaduri et al., Phys. Rev. Lett. 44 (1980) 1369. [8] A.N. Mitra and I. Santhanam, Z. Phys. C, to be published. [9] Cf.: V.A. Novikov et al., Phys. Rep. 41C (1978) 1; see also: G.C. Joshi and R. Anderson, Phys. Rev. D20 (1979) 736. [10] Particle Data Group, Phys. Lett. 75B (1978); Rev. Mod. Phys. 52 (1980) No. 2. [11] S.W. McDoweU, Phys. Rev. 116 (1959) 774; V.N. Gribov, Sov. Phys. JETP 16 (1963) 1080. [12] R. Carlitz and M. Kislinger, Phys. Rev. Lett. 24 (1980) 186. [13] See, e.g., R.R. Horgan, CERN-TH-2916 (1980); also for relevant references. [14] A.H.G. Hey, at: Baryon -80 Conf. (Toronto, 1980).
PHYSICS LETTERS
13 August 1981
RELATIVISTIC qqq SPECTRA FROM BETHE-SALPETER PREMISES A.N. MITRA and I. SANTHANAN Department of Physics, University of Delhi, Delhi-l lO007, India Received 14 October 1980 Revised manuscript received 3 March 1981
A relativistic but three-dimensional form of the Bethe-Salpeter (BS) equation developed under the Instantaneous Approximation (ANM) for a qqq system, is solved in a closed manner under conditions of harmonic confinement. It is shown that the BS dynamics offers a built-in mechanism for explaining major symmetry-breaking effects, especially spinspin and spin-orbit splittings using as inputs (1) a 'reduced' spring constant ¢~ and (ii) the quark mass. The model provides a very good description of the (ud) baryon spectrum in terms of the same values of t~ and mud as employed for q~ systems as well.
The general consensus on qqq spectroscopy has so far been confined to the harmonic oscillator (HO) description, essentially summarised in the FKR treatment [1 ], though non-trivial variants have been proposed on the basis of spectroscopic information [2], An equally satisfactory dynamical theory of symmetrybreaking effects (spin-spin, spin-orbit, etc.) is still lacking, though there have been recent attempts [3,4] to explain these in terms of QCD corrections, such as the Fermi-Breit term. On the other hand, the inverse proportionality of such correction terms to the quark masses (mq) militates against their validity for light quark (uds) spectroscopy [6,7] while serving as a good approximation for heavy (c, b) quarks. We propose an alternative mechanism for generating spin- and momentum-dependent corrections to the qqq spectrum whose relativistic validity is not impaired by an m q 1 expansion, by a method developed recently [6]. The result for the qqq spectrum for equal-mass kinetics is expressible as FB(N ) = N + 3, where N = N~ = Nn = 2n + L is the total HO quantum number, and the function FB(M ) defined by FB(M) -- E l + if'2 + F Q C D '
ff'l = (M2 - m 2 ) / a B ,
3 f 2 = _ I ~BM_23,~2(_~ d ' S - ~ - ON - Q'N), may be regarded as the (mass) 2 of the baryon, corrected for several symmetry-breaking effects ( J - S ,
ON, Q~r and too2), and measured in units (GeV 2) of an FKR-like [ 1], but M-dependent, spring constant g2B . The J ' S term, (which, unlike L "S is a relativistically invariant concept), is the resultant of spin-spin and spin-orbit splittings; m02 = 9m 2 is the quark mass correction; ~)N, Q~¢ represent certain momentumdependent corrections to the effective BS hamiltonian, and are expressible in terms of the total quantum number N. 3'B is a slowly varying function of M, and FQC D is a (hopefully small) perturbative correction due to short-range one-gluon exchange effects [5], termed QCD for brevity. A very similar formula, including the treatment of unequal-mass kinematics, has been derived for the q~ system as well [8]. Before discussing the experimental test of the mass formula (1) we first outline the essential steps leading to (1), The basic dynamics is contained in the B e t h e Salpeter equation [6] for a qqq system under pairwise q - q interactions with a colour dependence X(1). ~~ X(2) = - ~ , and spin-dependence "t ~ .v(1)A.(2)~ y , j, reduced to the three-dimensional level under the instantaneous approximation [6]. For equal-mass kinematics, and in the small q~. approximation (i,] = 1,2, 3), this equation takes the form [6]
(l) D12D23D31~(PlP2P3) = ~ ( - ~ ) D 3 1 D 2 3 M 1 1 123
(2) X f d k x 2 ( 2 ~ ) - 3 ( q 12 tV [klz) T (12) ~(PtlP~2P3),
62 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 104B, number 1
PHYSICS LETTERS
where the energy p03 of the spectator quark q3 has been taken in proportion to its intrinsic mass [6], viz. ½ M, and with m 0 = 3mq, 2 _ M 2 2 ~ 4 (m 02 - M 2) + p 2 + 4 q 2 2 D 1 2 -_m 1 22 + 4qx2 The potential V has the form [6]
(3)
q~ interaction. On the other hand it is tempting to relate CO2q to co2-~ through a plausible ansatz, viz., only one pair of q~ interaction produces roughly the same (mass) 2 spacings (AM 2 ~ 1 GeV 2) in q~ spectra as three pairs of qq interaction do for the qqq spectra. Therefore we postulate [9] 36O2q =6o2~ = 2c~2mq ,
(qlglk) = 3 (27r)3CO2qV28(q _ k) + 4rrO~s(q-k)-2, and [6]
13 August 1981
(4)
T(12) =
4..2_ 2 --~lv/ * P 3 - - ( q 1 2 +k12) 2 + o(1),-(2)g- _ -- 8iS12"q12Xk12 i/ "il vt12 k12)j(q12-k12)l
- 2 M m o l ( q 2 2 - k 2 2 ) + 2i(a 1 - * 2 ) ' P 3 X (q12-k12).
(5) In the "quadratic" approximation, and with the neglect of the Coulomb term in (4), eq. (2) reduces to
(13)
thus expressing the qqq spectra in terms of the q?t parametrization. With this assumption, the parameters ofeq. (1) are
UZB =6O3TB(moM)I/2,
"I,2 = 1+ 9 o32/Mm0 .
(14)
Further, the "proper" wave function ~ which incorporates probability conservation is given in terms of ~, eq. (6), by q~({,q) = ~(~,,q) exp [_9 ({2 +q2)/Mmq]
[_~(m 2 _ M 2) + ~(~2 + n 2 ) l ~(g ,n) = (~/32)-3/2 ex p [__~ (~2 +II2)/3B2] ,
3 2 ,---1 [4~ ( ~ 2 + r l 2 ) - 4 J . S + 9 + - ~ Q1 B* --~COqqm _
+ i QB - 6Mmo1 (~'V¢ + II"Vn)] ~(~,q) ,
(6)
where the basis set ~, r7 is defined by Pl = --tl'
--
1
--
P2,3 - 5Sl + l x / ~
'
(7)
and QB = ~ (4 q12V12 2 2 2 2 + 8ql 2 "V12 + 6 -P3V23 ) , 123
(8)
Q'B = - 2 ( l + m 2 M - 2 ) ~ - ~ ( q 2 2 + q 2 3 + a, 2P2 + a1 P3)V23' 2 2 123 (9) Except for the QB-terms' eq. (6) admits an exact solution. We indicate below a procedure for extracting the diagonal parts of QB, (~B in the (~, r/) basis in the form (N= N~ = N~) ~)B + ~)]3 ~ 6QN + 6Q~v,
(10)
QN _2 (N 2 + 3 N - 3)
(11)
, = Q'N
½( 2 N + 3) 2 -= S4Q N ,
~1 (1 + m2M -2) [7~ Q N - - I9( N + $ )3 2 + ~9l .
(12)
With these substitutions in (6), its solution is expressible as FB(M ) = N + 3, where FB(M ) is given by eq. (1), and the QCD contribution arising from the Coulomb term of eq. (4) has been added perturbatively. The quantities f2 B and TB are function of 6Oqq which, in principle, is a different parameter from 6Oq~ which would appear in eq. (4) for the corresponding
27/323, 2 = £2B .
(15)
Finally, for the reduction of ~)B, ~)B, we first express + + these in terms of the ladder operators au, au, ani, ani and drop the 4-step ladders a 4, a +4 which contribute only via second-order transitions with large energy denominators. This leaves us with operators of the types +2 2 2 +2 + + (16) a~i at/, anian] , auania~janj , and their hermitian conjugates. Next we make the following replacements separately for ~ and ~ operators (N - ai ai). _
+
+
.
+
ai(aia] )a] ~ a i ( 1 N + 1)alia; , +
+
+
1
a i (ai a/)aj -+ a i (~N)Si]a / .
(17) (18)
This amounts to a sort of "averaging" process [8] for the aia7 and a+a/operators consistently with their basic commutation relations. The results of these manipulations are the ON and Q~v terms listed in eqs. (10), (11).
To compare eq. (1) with the data [10] it is convenient to compute the function FB(M ) in terms of experimental masses, for specififed values of cJ and mud preferably the same (o3 = 0.15, mud = 0.28) as employed for q~ physics [8]. Such a common parametrization for qq and qqq spectroscopy should hopefully constitute a sufficiently wide framework for a 63
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comprehensive test of our BS formalism for harmonic confinement. Table 1 gives a limited comparison (for u, d quarks only) for non-strange 56 and 70 states, and fig. 1 exhibits their main sequence Regge resonances as a twin device for bringing out certain characteristic features of BS dynamics. In this regard it is good to keep in mind the intentions behind, as well as expectations from, the F B-function which for an "ideal" situation must be fully degenerate for all members of a given L-supermultiplet (both flavour- and spin-wise) as well as exhibit the unit spacing rule for successive supermultiplets. On the other hand since this quantity is being determined from the experimental masses, any overambitious expectation in this regard needs to be tempered with a more realistic appraisal not only of the obvious limitations of formula (1) in relation to the dimensions of its expected theoretical coverage, but
13 August 1981
also the equally obvious hazards of attaching too much literal significance to the officially listed masses of various states, without regard to their mass dispersions which are also given therein. To illustrate, the low-J states for a given L are usually listed with a high enough mass so that they appear nearly degenerate with their high-J partners, giving rise to the hasty impression of a small s p i n - o r b i t term, had it not also been for their mass-spreads which it would be perfectly legitimate to take cognizance of for purposes of a more "global" assessment in terms of the F B-function. To pursue the "apparent degeneracy" argument a step further, note, e.g., that D15(1670) and F15 (1688) look almost degenerate, but any serious attempt to search for a theoretical reason would bring back memories of parity doublets [11], a theory abandoned long ago since the so-called Carlitz-Kislinger cuts came into existence [ 12].
Table 1. The non-strange qqq spectrum under harmonic confinement. For notations see text and PDG tames [10]. State
Resonance
M2
fZB
F2
FB(M)
56-even+
Pit -=N(938) P~3 -= A (1236) P'~1(1470) p~3(1690 ) N2 ~- F15 (1690) 2~2 ~ F37(1950 ) F35 (1890) P~l (1850-1910) a) N4 ~ Hi9 (2220) 2x4 ~. H3,11(2420) A6(2880) 2x8 (3350)
0.88 1.53 2.16 2.86 2.86 3.80 3.50 3.42-3.65 4.93 5.86 8.29 11.22
0.82 0.94 1.02 1.09 1.09 1.17 1.15 1.14-1.16 1.25 1.30 1.41 1.52
0.82 0.13 1.52 0.94 1.06 0.54 0.76 0.90-0.84 1.53 1.04 1.52 1.95
1.14 1.05 3.11 3.02 3.10 3.23 3.28 3.42-3.50 5.00 5.04 6.93 8.93
(70+, N = 2)
dPi'1(1700) qP~'3 F17(1950_1990 )
2.89 3.28 3.80-3.96
1.09 1.12 1.17-1.18
1.09 0.81 0.53-0.51
3.16 3.14 3.20-3.30
(70-odd-)
D'I3(1530) S'~1(1520) Dis(1685) D~'3(1670-1700) Si'l(1620 -1650) D33(1680 ) S~1(1640) D35 (1940 -1960) G 17(2120 -2150 ) G19(2200) I1,11(2580_2650 )
2.34 2.31 2.84 2.79 -2.89 2.62-2.72 2.82 2.69 3.76-3.84 4.49 -4.62 4.84 6.66-7.02
1.04 1.04 1.09 1.08 - 1.09 1.07-1.08 1.09 1.08 1.16-1.17 1.22 - 1.23 1.24 1.34-1.36
0.69 0.81 0.28 0.38 -0.36 0.51-0.49 0.55 0.68 1.49 -1.47 1.11 - 1.08 0.76 1.59-154
2.41 2.40 2.38 2.35 - 2.41 2.36 2.41 2.58 2.59 4.19-4.22 4.27 -4.33 4.23 6.10-6.22
a) However, a (70, 0+) assignment gives for P~l (1910), F(M) = 3.31 in much closer agreement with F(M) values for F37(1950) and other N = 2 members. 64
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9
8
7
t, -ms u_ 4
3
~
.
x -
~
N2, t
states
2
1
O
I
I
2
~
I
I
4
s
I
6
I
7
N ~
Fig. 1. Principal (J = L + S) Regge recurrences of FB-values for (56, even+) and (70,odd-) states. The vertical bars "1" for (70, odd-) states represent the extent of mass spread indicated in table 1. Subject to such words of caution, table 1 reveals several interesting features which are exhibited in terms of separate listings of (56, even + ) and (70, o d d - ) states, together with an intermediate category of a few " g o o d " states of ( 7 0 , N = 2) as well. It is also convenient to break up the cases in terms of (1) stretched (J --L + S) states which show very little deviation from the expected behaviour of FB(M), and (ii) unstretched ( J < L + S) states which seem to show modest deviations. To assess the latter, a reasonable mass spread consistent with the PDG tables [10] has been included, while for the stretched category a single mass value (usually the listed one) seems to suffice. (1) For the 56 states the F B values leave practically little to be desired. Among the notable success of the J" S term, in which the background of the QN terms has also played a vital numerical role, are the near equality o f f B for (NL, AL) all the way up to L = 4, despite huge differences in theirM 2 values. The same t tt is also true of the radially excited pair ( P l l , P33)(2) The more explicit role of the QN-terms is manifest in the remarkable closeness o f F B for the lowt tt mass radial states ( P l l , P33) to their high-mass L = 2 partners F15 and F37.
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(3) The (70 + , N = 2) states also seem to exhibit highly degenerate F B-values, not only in remarkable conformity with the patterns (1) and (2) but also in good numerical overlap with their 56-partners. This last feature is particularly welcome in view of the highly controversial status of these otherwise four-star states [13]. (4) The expected A F B = 2 spacing rule for successive Regge excitations is fulfilled with a shortfall of 5% all the way to L = 8, while simultaneously respecting the (NL, AL) degeneracy where data are available (L = 0, 2, 4). Note that the ~ ~ x/M variation implies an asymptotic variation M 3/2 ~ N (reminiscent of a linear potential in an NR context), in preference to the M 2 ~ N law characterizing the HO model [1]. The actual F Bplots against N for the stretched (NL, AL) states are shown in fig. 1. (5) The (70, odd ) states which are listed separately show much the same precise quantitative features for the stretched states, revealing the roles of the J - S as well as the QN-terms (e.g. D13 versus D15 ; G17 versus G/~9_ versus D35). Especially interesting is the natural - 3 - state which fits D35 but on which there seems to have been some controversy [13]. (6) The A F B = 2 spacing rule is independently in evidence for the 70-sequence (L = 1,3, 5), again with a 5% shortfall in parallel with the 56-case, as seen from fig. I. However, a small 5 6 - 7 0 breaking is visible from a 10% upward shift of the 70-line above the 56-line. t! tt (7) For the S = 3/2 unstretehedJ-states (S12 , D13 , tt P31) there is a tendency twoards increased F B in terms of the listed mass values, a disease which is only partly cured after taking account of their allowed variations. A similar problem occurs for the D33 and $31 too. Yet it is interesting to note a near equality of the F B-values for these complementary states (I = 1/2, S = 3/2 versus I = 3/2, S = 1/2). This figure stands at a 10% higher value than the nearly degenerate stretched-J case. Together with the P3'l case, these 10% higher values represent about the "worst" cases in the whole table, yet modest compared with the average 6M 2 spreads for L = 1 and L = 2. All other cases (of higher J ) have fared far better not merely on average, but individually. In fig. 1, it is futile to display the corresponding M 2 versus N plots, because of their huge scatter, which is best exhibited in the table itself. The QCD effect has been calculated perturbatively with the same method as employed for q~ mesons [8], the smallness OfFQc D provides an afortiori justification for its perturbative 65
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treatment and, unlike ref. [3] it is not also being "missed" in the context o f the otherwise good agreement with only the confining interaction. To conclude, we have proposed an explicit form o f the qqq spectrum in terms of BS dynamics for harmonic confinement, using a common reduced spring constant c~ for mesons and baryons, under the plausible assumption (12). Our model contrasts with ref. [4] (which would prefer w/3-~ for the qqq system). It also contrasts with the QCD-dominant mechanism [3] employed in an NR framework whose validity has been questioned recently [14]. The present framework being relativistic, is hopefully free from the NR objections for light quarks. The data have given good and unequivocal support for the QN and J ' S terms except for certain low-J problems (which might suggest a slightly weaker s p i n - o r b i t effect). In particular the F(M)representation seems to be so much superior to the usualN2-display for revealing the symmetry-breaking effects, that its use could perhaps be profitably recommended on wider grounds than the strict premises of this model. Corresponding work on strange baryons is in progress.
66
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References [1 ] R.P. Feynman et al., Phys. Rev. D3 (1971) 2706; referred to as FKR. [2] See e.g., A.N. Mitra, Phys. Rev. D l l (1975) 3270. [3] See, e.g., N. Isgur and G. Karl, Phys. Rev. D18 (1978) 4187. [4] D.P. Stanley and D. Robson, Phys. Rev. Lett. 45 (1980) 235. [5 ] A. de Rfijula et al., Phys. Rev. D12 (1975) 147. [6] A.N. Mitra, Delhi Univ. preprint (March 1980); Z. Phys. C (1980), to be published. [7] R.K. Bhaduri et al., Phys. Rev. Lett. 44 (1980) 1369. [8] A.N. Mitra and I. Santhanam, Z. Phys. C, to be published. [9] Cf.: V.A. Novikov et al., Phys. Rep. 41C (1978) 1; see also: G.C. Joshi and R. Anderson, Phys. Rev. D20 (1979) 736. [10] Particle Data Group, Phys. Lett. 75B (1978); Rev. Mod. Phys. 52 (1980) No. 2. [11] S.W. McDoweU, Phys. Rev. 116 (1959) 774; V.N. Gribov, Sov. Phys. JETP 16 (1963) 1080. [12] R. Carlitz and M. Kislinger, Phys. Rev. Lett. 24 (1980) 186. [13] See, e.g., R.R. Horgan, CERN-TH-2916 (1980); also for relevant references. [14] A.H.G. Hey, at: Baryon -80 Conf. (Toronto, 1980).