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Meson mass formula in broken SUN symmetry and its applications
ANNALS
OF PHYSICS
Meson
101,
394412 (1976)
Mass Formula
in Broken
SUN Symmetry
and Its Applications*
H. HAYASHI Department of Physics, Faculty of Education, Shizuoka University, Shizuoka; Japan I. ISHIWATA Shizuoka Public Health Laboratory, Shizuoka, Japan s. IWAO’ Department of Physics, College of Liberal Arts, Kanazawa University, Kanazawa, Japan
M. SHAKO Data Processing Center, Kanazawa University, Kanazawa, Japan AND S. TAKESHITA Shizuoka Environmental Pollution Control & Research Center, Shizuoka, Japan Received March 24, 1976
Group theory of SU, and SU, symmetry is developed explicitly. Meson mass formula of Borchardt-Mathur-Gkubo type is generalized to broken SUN in a form suitable for computer calculation. The result is applied to SU, and SU6 cases, by assuming two possible assignments: (1) 3.95 or upsilon 5.97 GeV is the new lowest state of Jpc = l-member of SU, symmetry, and (2) 3.95 and 5.97 GeV states are new l--members of SU, and SU, , respectively. The derived wave functions and other consistency checks *do not exclude above two possibilities in the sense of currently accepted Okubo-ZweigIizuka rule. A further check of the validity of the model in future experiments is suggested from a close examination of the group theoretical structures studied in this paper and available information.
* A preliminary result of this paper has already been communicated. t Supported in part by the 1975 Scientific Research Fund of Ministry of Education. 394 Copyright AU rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
MESON MASS FORMULA IN BROKEN su,
395
1. INTRODUCTION
The so-called J/$ particle was discovered at first at BNL in pp collisions and at SLAC in efe- annihilation, independently [l]. Within a week the second resonance at 3.684 GeV (#‘) was discovered [l]. The quantum numbers of these states are now determined to be IG = O-, JPc = l--. The successive experimental studies discovered the third and fourth resonances at around 4 and 4.4 GeV, respectively. The former has fine structures: a small peak at 3.95 (width of about 30 MeV) and 4.1 GeV with a broad width but perhaps with a structure [2]. After a lot of speculations in the fall of 1974 and earlier 1975, the most common view is now that these may be bound states of a charm (c) and its antiparticle. The main ideas in the earlier period of these developments were summarized by Gaillard, Lee, and Rosner (GLR) [3] in the extended model of Gell-Mann, Oakes and Renner (GMOR) [4]. In order to explain the extremely narrow total widths of the first two states (J/# and $‘), it is commonly assumed that the so-called Okubo-Zweig-Iizuka (OZI) rule is in operation for ordinary meson decays and the pair of charmedmeson decay threshold is not open at these energies. The vector meson and ps-meson masses calculated in that thought verify the validity of the second point. Borchardt, Mathur, and Okubo (BMO) [5] estimated the wave functions of J/z,!+ $’ and associated neutral meson states. Their result shows the validness of the OZI rule. The smallness of the photonic decay width of J/t,h particle to the lightest ps-meson with cc structure (qJ was explained at that time by assuming the closeness of the masses of J/# and 7c [6]. Later on, the candidate of qc was discovered at about 2.8 GeV by DESY group [7]. Many people guided by the so-called asymptotic-free gauge theory [8] (with assumed infrared slavery) have calculated mass spectra of ortho- and paracharmonium states [9]. Such an estimate, though nonrelativistic in its approach, predicts additional p-wave cc bound states between 3.1 and 3.7 GeV. The candidates of these states (or possibly more states), so-called x’s have been observed experimentally [2]. The charmonium model predicts ortho-charmonium states at 4.1 and 4.4 GeV as dS, (n = 3 and 4). According to this idea an additional structure at around 4 GeV may be due to n3D, (n = I, 2,...), or something else. The concept of radial (and orbital) excitation picture involved in charmonium model is now believed to be correct at least for #‘23S, state. In addition to these the s-dependence of R = (e+e- + hadrons)/(e+e- + p+p-) and other experimental information at mid 1975 (before the discoveries of x’s) had lead Harari [lo] to propose a new model with additional flavors (b and t) by keeping the quark-lepton symmetry. That model is rephrased in terms of SU, symmetry [l 1, 121. The bottom and top quarks thus introduced may now be taken as a cause of new complexity of the hadron spectra. Intermediary shortness of
396
HAYASHI
ET AL.
the weak decay lifetimes (IO-12 to lo-14 set) [13] of charmed (perhaps bottomed and topped), or new flavored particles do not admit us to see their picturesque tracks at hand in the present limitation of experimental technique. The neutrino production of muon pair [14,15] and anomalous lepton production in e+e- annihilation [16] do not negate the higher symmetry, such as SU, , from a beauty of nature [I 11. In order to avoid right-handed current Suzuki [17] proposed SU, symmetry in the framework of minimal model of gauge theory of weak and electromagneitic interaction of Weinberg and Salam. All these ideas, though they stimulated by experimental discoveries, contain even number of flavors (Crermions Qi = 0). This is the reason why one of the authors (S.I.) [18] had tried broken SU,, (and chiral SU,, x SU,,) symmetry (2N = odd and even) as a generalization of GMOR and GLR. Supplementing the ps-meson vertex of the type (P,(p) 1 uj 1P&Y)) (j, k = 0, I,..., N2 - 1 for SUN. Here index 0 stands for unitary singlet meson) to the original GMOR and their successors’ works we can derive an equivalent mass formula to BMO and their generalizations. However, we shall not get further into ps-meson scalar vertex pointed out above in this paper. The main purpose of this paper is to develop thoroughly the SU, and SU, group theory and find out the generalized BMO formula in SUN. A complete way to slove the eigenvalue problem for a given SUN is discussed and applied to two examples: (1) 3.95 or recently observed upsilon 5.97 GeV state [19] is a new lowest Jpc = l-- member of SU, and (2) 3.95 and 5.97 GeV states are, respectively, new l-- members of SU, and SU, . As far as the mass formula concerns these are allowed two possibilities. Before getting into this problem we shall shortly comment on our knowledge of the applications of BMO formula in SU, . We examined systematically the meson states having various spins and parities by making use of BMO formula [20]. The main result of our analysis is that the charmed meson with the lowest mass could be axial vector mesons. They are found to be mD* = 1.85 and mF* = 1.88 GeV. Alterelli et al. [21] pointed out that anomalous lepton production in e+e- annihilation may be explained by assigning the lowest mass to the charmed vector meson. The present data do not negate about 20 % admixture of spin 1 charmed mesons [22]. Our mass formula analysis is not free from the ambiguity arising from both experimental and theoretical side. However, a general applicability of BMO formula for any spins and parities (for bosons) seems to be verified. In the last summer we discussed that the SU, symmetry is a good candidate if heavy lepton exists. However, according to our experience, it is better to work out first the SU, structure. Following this suggestion, H. Hayashi and his coworkers at Shizuoka worked out SU, and SU, group structures in a close analogy with the work of Amati et. al. [23]. Then, they reduced SU, BMO-like formula to three coupled nonlinear equations of three unknowns. They continued the similar proce-
391
MESON MASS FORMULA IN BROKEN su,
dure for SU, . Before getting final answer, one of us (S.I.) developed a generalization of GMOR and GLR approaches and then found a generalization of BMO formula in SU, symmetry. At the same time Shako completed a program for solving directly the secular equation in SU, without referring to the lengthy (analytic) expressions of coupled N - 2 (e.g., N = 6) nonlinear equations by computer, once a crude estimate of starting values of unknown parameters is given. The way to find out the starting values for a given sets of input masses was solved before [18]. As far as the meson mass formula concerns we now have a complete formula which contains a necessary dijk’s of SU, as numbers in an assumed approximation. For the sake of completeness and of additional purposes we shall list generators and constants diig’s and &‘s of SU, and SU, in Appendix. The cross check of some of these constants is made by making use of the results obtained from the generalized GMOR and GLR approaches. Applications of our formalism to the masses of N2 - N + 1 meson states of SU, , classifications of baryons [18], mass formula of the latters, strong decays of some of them [24], and magnetic moments of baryons have already been performed [25]. We shall comment in this paper on charmonium equivalent “bottomonium” and “toponium” decays. In Section II we shall discuss SU, group and the generalized BMO mass formula in that symmetry. Section III is devoted to the applications. In Section IV the consistency of the results obtained in the previous section is examined. In the last section we summarize our results.
II. SU, GROUP AND GENERALIZED
BMO
MASS FORMULA
SU, group is characterized by N by N traceless hermitian Ai (i = 1, 2,..., N2 - 1) satisfying semisimple Lie algebra:
[Ai 3&I = QfkA ,
N2 - 1 generators
(i,j, k = 1, 2 ,..., N2 - l),
(2.1)
where the structure constants &‘s are antisymmetric with respect to the odd permutations of i, j, and k. Explicit forms of generators in SU, and SU, are constructed and tabulated in Appendix. Introducing a matrix h, = (2/N)‘12 II, we find the anticommutators, satisfying {Xi 7 hi) = (4/N) aii + 2&dk
>
(i,j = 0, I,..., N2 - 1; k = I, 2,..., N* - I), (2.2)
where dijk’s are constants which are symmetric under the interchange of i, j and k. The nonvanishing values of the hjk’s and dijk’s for SU, and SU, are tabulated in
398
HAYASHI
ET AL.
additional tables, explicitly in the Appendix. Comparing these constants with known values for SU, and SU, we find that in our normalization the corresponding J;:jk’s and dijk’s take one and the same values in SU, (for N = 2, 3,..., N) if indices, ijk, coincide in that order. This fact gives us an interesting application. Hara [27] has shown that the decay amplitudes of J/4 particle to the known three and fourgs-mesons may be described in terms ofhia’s and dij,‘s (i, j, k = 1,2,..., 8). He also discussed relations among decay widths of final five and six ps-meson states to a certain extent. In his treatment the essential assumption is that the Jft,b is an SU, singlet with IG = O-, J pc = l--. In SU, (N = 4, 5,...) our additional flavors always appear in an isosinglet state. Thus, we can assume safely that the charmonium-like “ortho-bottomonium” I& and “ortho-toponium” #t in SU, and SU, , respectively, will have the quantum numbers ZG = O-, Jpc = I--. Our observation in the previous paragraph tells us that the similar statement to Hara can be made for these particles. Perhaps some irregularities relative to J/t,h decay will be observed due to the difference in Q values and to the dynamical effect inherent to each state. Let us turn our attention to SU, mass formula. Mass operator in SU, group may be defined as N
Hint = 1 anTn2--1,
(010= I>,
(2.3)
TZ=O
where oli (i = I,..., N - 3) represents relative strength of symmetry breaking to SU, . In terms of SUN tensor notation T,,m (m, n = I,2 ,..., N), Eq. (2.3) can be rewritten as N
Hint = c +Jn”, n=3
(x0 = 11,
(2.4)
where n-1
x,w3 = (l/3)
1 (6/k(k
( k=3
-
I))‘/”
mk-3 + (6n/(n -
I))“”
01,-~ , 1
(n = 4, 5 ,..., N; a0 = 1).
(2.5)
This relation was first derived by one of us (S.I.) [18] from quark mass terms in SUN (N = 5, 6, 7, 8) and then generalized. We have checked its validity in our present approach, specifically, in SU, and SU, . The parameters xi’s are also
related to quark masses: xi =
mQ+s - ml m, - ml '
(i = 1, 2 ,..., N - 3).
(2.6)
Here, we used a short hand notation: u = ql , s = q3 , c = q4, etc., for simplicity, by assuming m, = md .
MESON
MASS
FORMULA
IN BROKEN
399
su,
The meson mass formula in SU, may be given in the first order of symmetry breaking:
w3,i = A 5 &.2-1%3,
(ffo= 11,
(2.7)
n=3 (W),,
=
R,2.
These relations together with experimental masses m? (i = I,..., N - 1) form the eigenvalue problem to find the eigenvectors for meson states in SU, . From now on we shall confine ourselves to neutral vector mesons having the definite charge conjugation parity. We shall specify the meson states appearing in the secular equation by 1, 2,..., N - 1, for simplicity. In this definition, e.g., w, $ and J/t,b mesons in SU, correspond to numbers 1, 2, and 3, respectively. Let us write our eigenvalue problem in a form: (C)i,(~),
= mwi
(i,j = l)..., N - l),
>
w3)
where cij = cji )
(i # j).
(2.9)
Making use of these notations we find Cl, = ao2, c 1.n
=
ACL-Z,
(010= 1; n = 2, 3 )...) N - l),
c m,n = (2/n@ - l))‘/” Dci m _ 29
(m # n; m, n = 2 ,..., N - l),
C n.n = R2 + DC--(n - 1)(2/n(n + I>)‘/” 01,~~ + C,-,,,-,
= m2 - (N - 2) D(2/N(N
-
(2.10)
kz$+2 (2/W - 1))“” ~~1,
1))“” olNps.
Not all the parameters in the generalized formula two constraints:
are independent.
We obtain
N-l
MO2 = C mi2 - (N - 2) mo2 + (N - 1) D 5 (2/n(n - l))‘/” DL,,-~, i-l
(2.11)
n=3
and R2 = (1/3)(m,2 + 2m&) - D 5 (2/n(n - 1))112E,-~. n=4
(2.12)
400
HAYASHI ET AL.
The parameter D that appeared in Eqs. (2.7), (2.10)-(2.12) is a mere constant: (2.13)
D = (243/3)(112,2- w&,).
At this point let us denote the masses of the flavored mesons in SU, by (mN)k (k = 1, 2,..., N - 2). We find N-l
(mN)f = &f2 + $0
c (2/n(n - 1))“” 01,-a - (N - 2)(2/N(N
( n=3
- l))‘/” a,+,--3, 1
(2.14) N-l
(m,)2, = M2 + (l/2) -
(N
-
(
-(2k/(k
2)(2/N(N
+ I))“” 01~~~+
1
(2/n(n - 1))lj2 CY,-~
n=kf2
-
l))“2
+$-3
,
(a,, = 1; k = 2,..., N - 2).
1
We shall not consider a linear fit in this paper so we wrote all expressions in a quadratic form, but if one wishes he can use the formulas by merely putting linear input masses at the corresponding places. Summarizing this section in what follows after substitution of N - 1 inputs, the N - 2 parameters will be determined as a solution of the given secular equation of SUN meson mass formula. The corresponding eigenfunctions are determined simultaneously. The validity of that approach may be decided after solving the equation numerically and comparing the results with our present knowledge of fundamental particles. III.
APPLICATIONS
Let us summarize first several relevant statements before making applications: (I) According to the theoretical considerations, proved experimentally in SU, and SU, , N2 - N + 1 meson states of N2 ones in SU, belong to the pure (unmixed) irreducible representations of that group, (II) masses of ps-mesons of these pure states may be expressed in terms of mechanical [28] (free) quark masses appeared in GMOR, GLR, and their extended hamiltonians, (III) ps-meson mass formula for the pure states obtained in chiral SU, x SUN take one and the same expressions to those given by Eq. (2.14), checked explicitly in SU, and SU, by making use of &‘s given in the Tables in the Appendix, and (IV) the mass formulas for meson states apply universally for all meson states by merely changing a few of the parameters in the mass formulas, provided that they belong to the same irreducible representations of SUN . The last statement was assumed and tested in SU, case [5, 201. This means in general that we can predict all meson masses by changing inputs D and m, (or mKt)
MESON
MASS
FORMULA
IN
BROKEN
401
su,
and a single parameter A appropriately (see, previous section), once the remaining N - 3 parameters clli (i = l,..., N - 3) are determined, at least semiquantitatively. Based on the assumptions (I)-(IV) the meson mass formulas for N2 - N + 1 pure states in SU, was derived before and generalized to GLR forms [18]. Especially, the relations, Eqs. (24) and (25), in that reference were useful for determining the starting values of parameters 01~and 01~of our present approach. In the numberical work we analyze two possibilities: (1) 3.95 or upsilon at 5.97 GeV is a candidate of & and (2) 3.95 GeV and the upsilon are candidates of &, and & of the lowest J pc = 12- states of SU, and SU, , respectively. There is some inaccuracy for choosing a rather sensitive quantity m,, . We shall choose m, = 0.765 GeV throughout the paper. This value is determined from our previous SU, analysis. Except assumed masses for Z& and I,$ stated above, we assume that all other masses are up to date values. TABLE Result
of Analysis
I in SU, Approach
Parameters Case
A(GeV”)
a2(GeV2)
ii;i,Z(GeVZ)
a1
(3 (ii)
-0.1692 -0.1692
5.2724 9.1209
6.330 10.97
21.42 21.38
(9 States
Wavefunctions (l/l/Z)(u&+dd) 0.9964 0.7477 0.2704 0.3082
. 10-l . 10-l * 10-l
-0.7612 0.9966 0.2102 0.2309
(ii) States
* 10-l * 10-l * IO-’
(0 (ii)
* 10-l . 10-l
(Coefficients
ss -0.7603 0.9966 0.2214 0.2068 Predicted
Case
-0.2771 -0.2502 0.9959 0.8235
* 10-l
bb * 10-l * 10-l
-0.2678 -0.2334. -0.8370 0.9959
* 10-l
-0.2765 -0.2499 0.9987 0.3382
. 10-l * 10-l Masses
. 10-l 10-l * 10-l
of)
bb
CE * IO-’
31.42 81.47
of) CE
Wavefunctions (llz/Z)(un+dd) 0.9964 0.7483 0.2849 0.2849
(Coefficients
ss
%
* 10-l * 10-l
-0.2592 -0.2190 -0.3508 0.9988
9 10-l
- 10-l * 10-l - 10-l
in GeV
mDt
mF*
met
mf *
we
2.216 2.214
2.263 2.262
2.787 4.171
2.824 4.196
3.477 4.660
402
HAYASHI
ET AL.
We shall call case (i) and (ii), respectively, for rn,* = 3.95 and 5.97 GeV in SU, approach. The flavored mesons (bottomed ones) may be called G, H, and Z from low to high masses. The result of analysis for SU, is tabulated in Table I. We have given all the computer fitted parameters which were obtained by solving essentially the coupled nonlinear parameter equations. We did not avoid the duplication of paramters which will be used as a consistency check (see next section). We learn from the table that at the present stage it is hard to negate the assignment of 3.95 GeV state as the & in the sense of OZI rule. The point will be clarified by a careful study around 4 GeV. One sees that Hara-like relations (see previous section) will hold in some accuracy by comparing wave functions for & an &, . These tabulations also tell us the relative sign of transient coupling constants. We think that a more detailed study is necessary for the last point, but not considered further here. Now, we proceed to analyze case (2). For definiteness sake, we introduce notations for topped mesons 0, P, Q, and R for tii, tl, tE and t6, respectively. The result of our analysis is given Table II. Again, we cannot exclude the assignment considered here simply from the mass formula analysis alone. TABLE Result
of Analysis
II in SlJ,
Approach
Parameters
A(GeT)
m*(GeVz)
-0.1579
mOz(GeVz) 12.67
10.078
Wavefunctions States
21.42
(Coefficients
ss
(l/&z)(u&+dJ)
::
0.9960 0.7439 0.2950 0.2652
* 10-I ** 10-l 10-I
-0.7648 0.9964 0.2209 0.2064
vh
0.3054
* 10-l
0.2215
31.47
of)
bb
CE
to
.* 10-l
-0.2806. -0.2530 0.7951 0.9954
10-l - 10-l . 10-l
-0.2713 -0.2362 -0.8283 0.9945
* 10-l * 10-l - 10-l
* 10-l
0.3646
- 10-l
0.5312
* 10-l
* 10-l
Predicted
Masses
76.40
-0.2633 -0.2223 -0.3321 -0.5733
. 10-l - 10-l -* 10-I 10-l
0.9972
in GeV
mD*
mF’
mc*
mfft
mI*
m0*
mQ*
mp*
mR’
2.217
2.264
2.789
2.826
3.479
4.164
4.189
4.654
4.952
MESON
MASS
IV.
FORMULA
IN
CONSISTENCY
BROKEN
403
su,
CHECK
Let us rewrite Eqs. (2.11) and (2.12) in the forms: N-l
n1,2 = R,2 - (N - 1) D 5 (2/n@ - l))l/?
a&3
)
(4.1)
n=3
(1/3)(~,2 + 2&e)
= m2 + D ;
n=4
(2/n(n - >>"2in...3
.
(4.2)
The purpose of this section is to find out consistency of these conditions in referring to the left-hand side and right-hand side by making use of inputs and tabulations in Tables I and II. It would be most convenient to list the results in tabular form. Moreover, we have recalculated SU, and SU, cases by our present approach. The result for SU, completely coincides with our previous answer[20]. Including those cases the comparison stated above is listed in Table III. One may find how each fitting machinery is going on. TABLE
III
Consistency Check
SK Case case (i)
case (ii)
sue
10.646
25.078
45.117
60.134
10.060
24.022
45.113
60.133
(GeV2)
SU,
lhs Eq. (4.1)
1.0672
rhs Eq. (4.1)
1.0672
lhs Eq. (4.2)
0.7258
0.7258
0.7258
0.7258
0.7258
rhs Eq. (4.2)
0.7258
0.7261
0.7256
0.7257
0.7262
V.
SUMMARY
AND
DISCUSSION
Meson mass formula is derived in a form suitable to apply to unitary N2 members of SUN in the first order of symmetry breaking. A partial proof of the general expression may be given by making use of constants dijk’s of SU, and SU, group summarized in Appendix. The first step of the generali-
404
HAYASHI
ET
TABLE
IV
Generators
A,=
i . . .
1 . . . ,
. . . . .
. . . . .
. . . . .
A,=
Ax,,=
&=
A,e=
&=
A,,=
i . . .
: . . .
: . . .
: . . . .
: . . . . : ;-I’. . . . . . . . . . . . .
.
.---I.
:
:
:
:
:
i .
. .
. .
. .
. .
. . . .
. . . .
. . .
. . . . 1 .
.
.
1 .
. .
. .
. . .
. . .
. . .
. . .
1 . .
i
:
:
:
: .
.
.
.
.
.
.
.
.--I
.
.
.
.
.
.
.
.
.
.
.
I
.
.
.
.
. . . . .
. . . . .
. . . . 1
. . . 1 .
: . .
of SU,
A,==
.-i. i . . .
.
. .
A,=
. .--I. . . . i.... . , . . . .
. .
. .
: . .
. . 1 . .
1
.
.
.
. .
: . .
: . . . . . . .
. . 1 . . Ar=
AL.
X8 = l/z/J
A,,=
A,, =
A,,=
Azo=
A,,=
. . . .
. . . .
. . . . .
. . . . .
A,=
. &=
: . .
. . 1 , -2 . . . .
.
.
.
: .
: 1 .
: . .
. 1 . . .
.
.
. .
. .
. .
1 1 . -i . . I . . . . .
. . .
. . . . i
. . . . .
. . . . .
.---I . . . . . . . .
. .
. .
. .
. .
. .
. . .
. . .
. . 1
. . .
1 . .
.
.
: . .
: . .
. . . . .
. . . . . . .-I 2 .
1 .-1. . . .
A,=
&=
Ala = l/1/6
h,B=
As4 = l/10’/%
.
.
. * .
. . .
. . . . .
. 1 . . .
. . . . . 1
.
::::: 1 . . .
. .
. .
. .
. . . . .
. . . I .
. . . -2. . . . . . .
.
1 . . . .
. 1 . . .
.
.
.
i ..
:
. . . . .
. . . . 1
. . . . .
. . . . .
. 1 . . .
1
. 1 . . .
.
. .
. .
i .-
:
: . .
i . .
. . .
: 3 . . .
4
MESON
MASS
FORMULA
IN
TABLE
BROKEN
405
su,
V
New Generators in SU, . AZ&=
: . .
.
.
.
.
1
: . .
: . .
: . .
: . .
: . .
1 .
.
.
.
.
.
xzB=
A,,=
. A,,=
A,,=
: . .
. . . . .
. . . . .
.
. . . . . 1
. . . . . .
.
.
. : . . .
. . i . .
.
.
.
.
.
: . . .
: . . .
: . . .
: : . . .--I z .
A,,=
A,, = l/lW
. .
. .
. .
.-z . .
: . .
. . .
. . .
. . .
. . .
. . .
i
.
.
.
.
.
.
.
.
:::::i . . . . . . .
.
.
.
.
.
.
.
: . . .
: : . . . . 1 .
: . . .
: . . .
1 . . . .
.
.
.
.
.
.
.
.
. . . . 1 .
. . .
. . .
1 . . .
1 . . .
1 . . z
1 . . .
1 . . .
.
.
.
.
.
.
.
.
.
.
.
: . . .
: . . .
: . . .
: . . z
: : .--I . . . .
: . . .
: . . .
: . . .
: . . .
: . .
. .
1 . . . . .
i . . . .
: 1 . . .
:
:
i . .
: : 1 . .- 5
A,,=
A,, =
:
.&=
-i . . .
1 1 .
406
HAYASHI
ET AL.
TABLE VI Nonzero Elements of& and &
1
2
3
4
1
2
2 4 5 9 10 16 17 4 5 9 10 16 17 4 6 9 11 16 18 5 9 10 16 17
3 7 6 12 11 19 18 6 7 11 12 18 19 5 7 10 12 17 19 8 14 13 21 20
1 1 1 4 5 9 10 16 17 2 2 2 4 5 9
8 15 24 6 7 11 12 18 19 8 15 24 7 6 12
5
6
7
8
9 10 3
4
in SU,
13 14 20 21 8 14 13 21 20 13 14 20 21 10 12 14 17 19 21 15 23 22 22 23
11
10 16 17 7 11 12 18 19 11 12 18 19 9 11 13 16 18 20 10 16 17 16 17 9 10 11 12 16 17 18 19 4 4 4 9 10 16 17
9 10 11 12 16 17 18 19 8 15 24 13 14 20 21
6
9
12 13
14 15
16 18 20 22
7
8
12 18 19 18 19 14 20 21 20 21 16 18 20 22 17 19 21 23
15 23 22 22 23 15 23 22 22 23 17 19 21 23 24 24 24 24
11 12 18 19 7 7 7 11 12 18 19 8 8 8 9
13 14 20 21 8 15 24 14 13 21 20 8 15 24 9
21.\/8 4 -12 + 4 2/h + -fr 3 1 1,2%h 1/2x& 1/2d/;6 -3/21/a 5/2(1O)l/” 5/2(10)‘/* 5/2(1O)l/* 5/2( 1oy
Table continued
MESON
MASS
FORMULA
TABLE
2
1.
10 16 17
11 19 18
3 3 3 4 5 6 7
8 15 24 4 5 6 7
l/d/5 lid/a l/lo’12
8
21
21
-l/d5
9
9 9 16 17
15 24 22 23
--1/d/6
10 10 16 17
15 24 23 22
-lid6 l/lo112
11 11 18 19
15 24 22 23
-l/d/6 l/lO’/2
12 12 18 19
15 24 23 22
-l/d6 l/lO’lZ -$,
13 13 20 21
15 24 22 23
--l/d6 l/lO1’a
14 14 20 21
15 24 23 22
-l/d6 l/lo’/2
3
10
11
12
13
14
595/101/z-5
9 10 16 17 6 6 6
14 13 21 20 8 15 24
15
15
--2/1/G
15 16 17 18
24 16 17 18
l/101/2 l/266 1/2-\/i; li2A
19 20 21 22
19 20 21 22
1/2-\/a 1/2x& l/21/6 -3/2v%
16 17 18
23 16 17 18
23 24 24 24
--3/2x& -3/2(10)‘/2 -3/2(lO)l/” -33/2(10)‘/*
19 20 21 22
19 20 21 22
24 24 24 24
-3/2(lO)‘/2 -3/2(1O)l/” -3/2(lO)1/2 -3/2(10)‘/*
23 24
23 24
24 24
-3/2(lO)l/” -3/10’12
6
4 15
1/10”2 + 4
4.
.;.
3 +
+
a Q
-a +
407
su,v
VI (continued)
8 15 24
4
-+ -.
BROKEN
5 5 5
5
-*
4 d
IN
-l/24’; 116 l/lo’12
10 11 12
10 11 12
l/263 1/2d/J l/21/5
-$ -p
13 14 16
13 14 16
-l/b 1/q/B l/lo’/2
17 18 19 20
17 18 19 20
-l/d/3 -l/d/3 1/2d/3 1/2d/3 1/2d\/3 1:24/3 -l/1/3
-4
8
408
HAYASHI
ET AL.
TABLE New
i
i
k
firn
Nonzero
Elements
VII of hjk and
dirk in
SUe
i
i
k
LL
i
j
k
f;ic
24
25 27
26 28
1/2(10)‘/2 l/2(10)1/”
29 31
30 32
1/2(10)‘/~ 1/2(10)1~~
____. 1
25 26
28 27
13
29 30
32 31
t 2
2
25 26
27 28
14
29 30
31 32
& i
3
25 27
26 28
15
25 27
26 28
l/2& l/2-\/6
25
33 26
34 35
4
25 26
30 29
29 31
30 32
l/21/6 -3/2&
27 29
28 30
35 35
3115112 3/151/a
5
25 26
27 28
16
25 26
34 33
-a
31 33
32 34
35 35
3/151/* 3/15’/$
27 28
30 29
17
25 26
33
ri
34
+
27 28
29 30
18
27 28
34 33
25 27 29
26 28 30
19
33
20
27 28 29
25 26
32 31
21
30 29
25 26
31 32
22
27 28
32 31
23
17
17 25
35 34
26 18
33 35
27 28 19 27
33 34 35 34
6 7 8
9 10 11 12
27 28
31 32
1
25 26
27 28
9
25 26
28 27
10
25 26 27 28
25 26 27 28
11
2 3
12
4
-2/101/e 3/d/151/2
7;. -+ L 2 + +
34 34 33
1 2
33
g
30 31
34 34
4 +
32 31
33 33
t
32
34
+
25 26
31
1
32
i
25 26
32 31
27 28 27 28
31
-$
32 32 31
if
-+
-$ 4
18
19
-4 +
- .&
Table continued
MESON
MASS
i
i
k
29 30
31 32
29 30
32 31
-$
15 25
35 25
l/151/2 1/2-\/a
26 27
26 27
l/2%& l/2-\/6
28 29 30 31 32 16 25 26
28 29 30 31 32 35 33 34
l/2-\/6 l/21/6 1/2v’/8 -3/2x& -3/2x& 1/151’2
2.5 26
29 30
13
25 26
30 29
14
27 28
29 30
15
27 28
30 29
8 21 25 26 21 28 29 30
35 21 25 26 27 28 29 30
1/15’/2 -l/d3 l/2-\/3 l/2%0 l/2-\/3 1/2d/3 -l/-\/3 -l/-\/S
24
27 28 29 30 31 32 33 34
27 28 29 30 31 32 33 34
l/2(10)‘/* l/2(10)*/* 1/2(10)‘12 1/2(1O)lj2 1/2(10)‘/2 l/2(10)1/* -2/10’/2 -2/10’12
25
25
35
-2/151/e
26
26
35
-2115w
27
27
35
-2/151J2
28
28
35
-2/151/2
29
29
35
-2/15’1=
30
30
35
-2/15’/2
31
31
35
-2/15112
32
32
35
-2/15l/=
33
33
35
-2/15112
34
34
35
-2/15112
35
35
3.5
-4/151/2
4
5
6 I
8
BROKEN
VII
k
AJr
IN
TABLE
i
i
FORMULA
16
409
su,
(continued)
Ljk
a iI
i
j
k
19 20
28 20
33 35
29 30
33 34
21 29
35 34
30 22
33 35
31 32 23 31 32 24 25 26
33 34 35 34 33 35 25 26
Q
3 -$
21
22
23
24
Air -$
l/151/2 g 3 1 p51ia 1.
4
1,‘151/z g 4 l/151/2 : 11151’2 l/Z(lO)~~~ 1/2(10)‘/2
410
HAYASHI
ET AL.
zation was performed by the method of GMOR and GLR which is quickly manageable and transparent in some aspect. A typical factor (2/n@ - 1))lP appeared many times in the generalized formula was found by this approach [18]. As we discussed this and BMO approaches are complementary from various points of view. The former approach, however, predicted the relation between quark and ps-meson masses. The light quark masses estimated by Leutwyler [29] are considered to be one of the clues to establish the quark dynamics. We do not discuss this important problem in relations to new flavors b and t in this paper. The mass formula is applied to SU, and SU, cases by assigning suitably the fourth and fifth neutral vector mesons in eigenstate of charge-conjugation number C = -1. From our analysis one sees that it is hard to decide either 3.95 GeV or upsilon state belongs to SU, . Hence the possibility to choose I,,&,as 3.95 and & as 5.97 upsilon state of SU, cannot be excluded in a similar application. A possibility to find out a goodness of above choices is suggested, i.e., by studying decay modes of these particles in relation to those of J/#. Of course, the direct check of the idea should be made by observing the flavored particles of respective symmetry as it should be done for charmed ones. Combining the analysis of the present paper and our previous SCJ, mass formula analysis we can make the following statement for the meson masses in other states, e.g., 0-, l+, 2+, etc. The masses of these states may be predicted by substituting input masses belonging to I = 1 (p-like) and &(strange) mesons of their own members, changing slightly a single parameter A, while keeping remaining parameters ai’s at the values found in the text. We shall leave many relevant problems in future.
APPENDIX
We shall summarize a group structure of SU, and SU, in this Appendix. The generators hi (i = l,..., 15) of SU, may simply be obtained by adding null line and null column elements to the fifth line and column, respectively, to those of Amati et al. [23]. Including all remaining generators & (k = 16,..., 24) the generators of SU, are listed in Table IV. The generalization procedure for generators discussed above applies also to that for SU, . We merely list the new generators of SU, in Table V. The structure constants &‘s and constants &‘s for SU, are tabulated in Table VI. By comparing these numbers with known values for SU, and SU, one sees that diib’s and&‘s take one and the same values if all the indices ijk are the same in that order. This makes our tabulation for new constants in SU, much simpler (see Table VII).
MESON MASS FORMULA IN BROKEN su,
411
REFERENCES 1. J. J. AUBERT, U. BECKER, P. J. BIGGS, J. BURGER, M. CHEN, G. EVERHART, P. GOLDHAGEN, J. LEONG, T. MCCOORRISTON, T. G. RHOADES, M. ROHDE, S. C. C. TING AND S. L. Wu, Phys. Rev. Left. 33 (1974), 1404; J.-E. AUGUSTIN, A. M. BOYARXI, M. BREIDENBACH, F. BULOS, J. T. DAKIN, G. J. FELDMAN, G. E. FIXHER, D. FRYBERGER, G. HANSON, B. JEANMARIE, R. R. LARSEN, V. LOATH, H. L. LYNCH, D. LYON, C. C. MOREHOUSE, J. M. PATERSON, M. L. PERL, B. RICHTER, P. RAPIDIS, R. F. SCH~ITTERS, W. M. TANENBAUM, F. VANNUCCI, G. S. ABRAMS, D. BRIGGS, W. CHINOWSKY, C. E. FRIEDBERG, G. GOLLX-IABER, R. J. HOLLEBEEK, J. A. KADYK, B. LULU, F. PIERRE, G. H. TRILLING, J. S. WHITAKER, J. Wrss AND J. E. ZI~SE, Phys. Rev. Left. 33 (1974), 1406; G. J. FELDMAN AND M. L. PERL, Phys. Rep. 19C (1975),
233. 2. R. F. SCHWITTERS, “Hadron Production at SPEAR,” Intern. Symposium on Lepton and Photon Interactions at Stanford, California, Aug., 1975, SLAC-PUB-1666, Nov., 1975; B. RICHTER, “Experiments at SPEAR,” A talk given at RIFP, Nov., 1975. 3. M. K. GAILLARD, B. W. LEE, AND J. L. ROSNER, Rev. Mod. Phys. 47 (1975), 277. 4. M. GELL-MANN, R. J. OAKES, AND B. RENNER, Phys. Rev. 175 (1968), 2195. 5. S. BORCHARDT, V. S. MATHUR, AND S. OKUBO, Phys. Rev. Lett. 34 (1975), 38; Phys. Rev. Dll (1975), 2572. 6. B. W. LEE AND C. QUIGG, “Some Considerations on TV, FERMILAB-74/110-THY Dec., 1974; R. F. DASHEN, I. J. M~ZINICH, B. W. LEE, AND C. QUIGG, “Coherent Production and Decay Modes of a Pseudoscalar Partner of the $(3105) Boson,” COO-2220-39 and FERMILAB-PUB/75/18/THY, Jan., 1975. 7. B. H. WIIK, “Recent Results from DORIS,” DESY 75/37, Oct., 1975; and A talk given at RIFP, Nov., 1975. 8. D. J. GROIN AND F. WILCZEK, Phys. Rev. Lett. 30 (1973), 1343; Phys. Rev. D8 (1973), 3633; H. D. POLITZER, Phys. Rev. Lett. 30 (1973), 1346. 9. T. APPELQUIST AND H. D. POL,ITZER, Phys. Rev. Lett. 34 (1975), 43; A. DE R~JULA AND S. L. GLASHOW, Phys. Rev. Lett. 34 (1975), 46; T. A~PELQUIST, A. DE R~XJLA, H. D. POLITZER AND S. L. GLASHOW, Phys. Rev. Lett. 34 (1975), 365; E. EICHTEN, K. GOTTFRIED, T. KINOSHITA, J. K~GUT, K. D. LANE, AND T. -M. YAN, Phys. Rev. Lett. 34 (1975), 369; B. J. HARRINGT~N, S. Y. PARK, AND A. YILDIZ, Phys. Rev. Lett. 34 (1975), 168. 10. H. HARARI, Ann. Phys. (N. Y.) 94 (1975), 391. 11. H. HARARI, “Theoretical Implications of the New Particles,” Intern. Symposium on Lepton and Photon Interactions at Stanford, California, Aug., 1975, WIS-75/40 Ph (1975), and “How Many Quarks ?,” A talk given at RIFP, Nov., 1975. 12. H. FRJXXH, M. GELL-MANN, AND P. MINKOVSKI, Phys. Lett. 59B (1975), 256. 13. K. Nru, E. MIKUMO, AND Y. MAEDA, Prog. Theor. Phys. 46 (1971), 1644; K. HOSHINO, S. KURAMATA, K. NIU, K. NIWA, S. TASAKA, E. MIKUMO AND Y. MAEDA, “On the X-particles observed in the Cosmic Ray Jet Showers,” HE 5-l l(l975) ; “X-particle Production in 205 GeV/c Proton Interactions,” HE 5-12 (1975); P. L. JAIN AND B. GIRARD, Phys. Rev. Lett. 34 (1975) 1238. 14. C. RUBBIA, “New Particle Production by Neutrinos,” EPS Intern. Conf. on High-Energy Physics, Palermo, June, 1975. 15. B. W. LEE, “Dimuon Events,” FERMILAB-PUB-Conf-75/78-THY, Sept., 1975. 16. G. J. FELDMAN, “New Particle Searches and Discoveries at SPEAR,” Intern. Symposium on Lepton and Photon Interactions at Stanford, California, Aug., 1975, SLAC-PUB-1662, Oct., 1975; M. L. PERL, “Anomalous Lepton Production,” Lectures on Electron-Positron Annihilation-Part II, SLAC-PUB-1592, June, 1975.
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17. M. SUZUKI, Phys. Rev. Lett. 35 (1975), 1553. 18. S. IWAO, “Particle Physics in Broken Higher Symmetry,” Ann. Sci. (Kanuzawu Univ.) 13 (1976), to appear. 19. D. C. HOM, L. M. LEDERMAN, H. P. PAAR, H. D. SNM)ER, J. M. WEISS, J. K. YOH, J. A. APPEL,
B. C. BROWN,
C. N. BROWN,
“Observation of High Mass Dilepton Left. 36 (1976), 1236.
W. R. INNES,
T. YAMANOUCHI
AND D.
M.
KAPLAN,
Pairs in Hadron Collisions at 400 GeV,” I’lys. Rev.
20. H. HAYASHI, T. ISHIWATA, S. IWAO, M. SHAKO, AND S. TAKESHITA, Prog. Theor. Phys. 55 (1976), 1912. 21. G. ALTERELU, N. CABIBBO, AND L. MAIANI, Phys. Rev. Lett. 35 (1975), 635. 22. S. IWAO AND M. SHAKO, Nuovo Cimento Lett., 16 (1976), 29. 23. D. AMAn, H. BACRY, J. NUYTS, AND J. PRENTKI, Nuovo Cimento 34 (1964), 1732. 24. S. IWAO, Nuovo Cimento Lett., 15 (1976), 569. 25. S. IWAO, Nuovo Cimento Lett. 15 (1976), 331; Prog. Theor. Phys. 55 (1976), 943. 26. H. HAYA~HI, T. ISHIWATA, S. IWAO, M. SHAKO, AND S. TAKESHITA, “Sub and Strong Interactions,” HPICK-024 (Feb., 1976). 27. Y. HARA, Phys. Lett. 60B (1975), 53. 28. M. GELL-MANN, “Elementary Particles,” Oppenheimer Lecture, Institute for Advanced Study, Oct., 1974. 29. H. L EUTWYLER, Phys. Lett. 48B (1974), 431.
OF PHYSICS
Meson
101,
394412 (1976)
Mass Formula
in Broken
SUN Symmetry
and Its Applications*
H. HAYASHI Department of Physics, Faculty of Education, Shizuoka University, Shizuoka; Japan I. ISHIWATA Shizuoka Public Health Laboratory, Shizuoka, Japan s. IWAO’ Department of Physics, College of Liberal Arts, Kanazawa University, Kanazawa, Japan
M. SHAKO Data Processing Center, Kanazawa University, Kanazawa, Japan AND S. TAKESHITA Shizuoka Environmental Pollution Control & Research Center, Shizuoka, Japan Received March 24, 1976
Group theory of SU, and SU, symmetry is developed explicitly. Meson mass formula of Borchardt-Mathur-Gkubo type is generalized to broken SUN in a form suitable for computer calculation. The result is applied to SU, and SU6 cases, by assuming two possible assignments: (1) 3.95 or upsilon 5.97 GeV is the new lowest state of Jpc = l-member of SU, symmetry, and (2) 3.95 and 5.97 GeV states are new l--members of SU, and SU, , respectively. The derived wave functions and other consistency checks *do not exclude above two possibilities in the sense of currently accepted Okubo-ZweigIizuka rule. A further check of the validity of the model in future experiments is suggested from a close examination of the group theoretical structures studied in this paper and available information.
* A preliminary result of this paper has already been communicated. t Supported in part by the 1975 Scientific Research Fund of Ministry of Education. 394 Copyright AU rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
MESON MASS FORMULA IN BROKEN su,
395
1. INTRODUCTION
The so-called J/$ particle was discovered at first at BNL in pp collisions and at SLAC in efe- annihilation, independently [l]. Within a week the second resonance at 3.684 GeV (#‘) was discovered [l]. The quantum numbers of these states are now determined to be IG = O-, JPc = l--. The successive experimental studies discovered the third and fourth resonances at around 4 and 4.4 GeV, respectively. The former has fine structures: a small peak at 3.95 (width of about 30 MeV) and 4.1 GeV with a broad width but perhaps with a structure [2]. After a lot of speculations in the fall of 1974 and earlier 1975, the most common view is now that these may be bound states of a charm (c) and its antiparticle. The main ideas in the earlier period of these developments were summarized by Gaillard, Lee, and Rosner (GLR) [3] in the extended model of Gell-Mann, Oakes and Renner (GMOR) [4]. In order to explain the extremely narrow total widths of the first two states (J/# and $‘), it is commonly assumed that the so-called Okubo-Zweig-Iizuka (OZI) rule is in operation for ordinary meson decays and the pair of charmedmeson decay threshold is not open at these energies. The vector meson and ps-meson masses calculated in that thought verify the validity of the second point. Borchardt, Mathur, and Okubo (BMO) [5] estimated the wave functions of J/z,!+ $’ and associated neutral meson states. Their result shows the validness of the OZI rule. The smallness of the photonic decay width of J/t,h particle to the lightest ps-meson with cc structure (qJ was explained at that time by assuming the closeness of the masses of J/# and 7c [6]. Later on, the candidate of qc was discovered at about 2.8 GeV by DESY group [7]. Many people guided by the so-called asymptotic-free gauge theory [8] (with assumed infrared slavery) have calculated mass spectra of ortho- and paracharmonium states [9]. Such an estimate, though nonrelativistic in its approach, predicts additional p-wave cc bound states between 3.1 and 3.7 GeV. The candidates of these states (or possibly more states), so-called x’s have been observed experimentally [2]. The charmonium model predicts ortho-charmonium states at 4.1 and 4.4 GeV as dS, (n = 3 and 4). According to this idea an additional structure at around 4 GeV may be due to n3D, (n = I, 2,...), or something else. The concept of radial (and orbital) excitation picture involved in charmonium model is now believed to be correct at least for #‘23S, state. In addition to these the s-dependence of R = (e+e- + hadrons)/(e+e- + p+p-) and other experimental information at mid 1975 (before the discoveries of x’s) had lead Harari [lo] to propose a new model with additional flavors (b and t) by keeping the quark-lepton symmetry. That model is rephrased in terms of SU, symmetry [l 1, 121. The bottom and top quarks thus introduced may now be taken as a cause of new complexity of the hadron spectra. Intermediary shortness of
396
HAYASHI
ET AL.
the weak decay lifetimes (IO-12 to lo-14 set) [13] of charmed (perhaps bottomed and topped), or new flavored particles do not admit us to see their picturesque tracks at hand in the present limitation of experimental technique. The neutrino production of muon pair [14,15] and anomalous lepton production in e+e- annihilation [16] do not negate the higher symmetry, such as SU, , from a beauty of nature [I 11. In order to avoid right-handed current Suzuki [17] proposed SU, symmetry in the framework of minimal model of gauge theory of weak and electromagneitic interaction of Weinberg and Salam. All these ideas, though they stimulated by experimental discoveries, contain even number of flavors (Crermions Qi = 0). This is the reason why one of the authors (S.I.) [18] had tried broken SU,, (and chiral SU,, x SU,,) symmetry (2N = odd and even) as a generalization of GMOR and GLR. Supplementing the ps-meson vertex of the type (P,(p) 1 uj 1P&Y)) (j, k = 0, I,..., N2 - 1 for SUN. Here index 0 stands for unitary singlet meson) to the original GMOR and their successors’ works we can derive an equivalent mass formula to BMO and their generalizations. However, we shall not get further into ps-meson scalar vertex pointed out above in this paper. The main purpose of this paper is to develop thoroughly the SU, and SU, group theory and find out the generalized BMO formula in SUN. A complete way to slove the eigenvalue problem for a given SUN is discussed and applied to two examples: (1) 3.95 or recently observed upsilon 5.97 GeV state [19] is a new lowest Jpc = l-- member of SU, and (2) 3.95 and 5.97 GeV states are, respectively, new l-- members of SU, and SU, . As far as the mass formula concerns these are allowed two possibilities. Before getting into this problem we shall shortly comment on our knowledge of the applications of BMO formula in SU, . We examined systematically the meson states having various spins and parities by making use of BMO formula [20]. The main result of our analysis is that the charmed meson with the lowest mass could be axial vector mesons. They are found to be mD* = 1.85 and mF* = 1.88 GeV. Alterelli et al. [21] pointed out that anomalous lepton production in e+e- annihilation may be explained by assigning the lowest mass to the charmed vector meson. The present data do not negate about 20 % admixture of spin 1 charmed mesons [22]. Our mass formula analysis is not free from the ambiguity arising from both experimental and theoretical side. However, a general applicability of BMO formula for any spins and parities (for bosons) seems to be verified. In the last summer we discussed that the SU, symmetry is a good candidate if heavy lepton exists. However, according to our experience, it is better to work out first the SU, structure. Following this suggestion, H. Hayashi and his coworkers at Shizuoka worked out SU, and SU, group structures in a close analogy with the work of Amati et. al. [23]. Then, they reduced SU, BMO-like formula to three coupled nonlinear equations of three unknowns. They continued the similar proce-
391
MESON MASS FORMULA IN BROKEN su,
dure for SU, . Before getting final answer, one of us (S.I.) developed a generalization of GMOR and GLR approaches and then found a generalization of BMO formula in SU, symmetry. At the same time Shako completed a program for solving directly the secular equation in SU, without referring to the lengthy (analytic) expressions of coupled N - 2 (e.g., N = 6) nonlinear equations by computer, once a crude estimate of starting values of unknown parameters is given. The way to find out the starting values for a given sets of input masses was solved before [18]. As far as the meson mass formula concerns we now have a complete formula which contains a necessary dijk’s of SU, as numbers in an assumed approximation. For the sake of completeness and of additional purposes we shall list generators and constants diig’s and &‘s of SU, and SU, in Appendix. The cross check of some of these constants is made by making use of the results obtained from the generalized GMOR and GLR approaches. Applications of our formalism to the masses of N2 - N + 1 meson states of SU, , classifications of baryons [18], mass formula of the latters, strong decays of some of them [24], and magnetic moments of baryons have already been performed [25]. We shall comment in this paper on charmonium equivalent “bottomonium” and “toponium” decays. In Section II we shall discuss SU, group and the generalized BMO mass formula in that symmetry. Section III is devoted to the applications. In Section IV the consistency of the results obtained in the previous section is examined. In the last section we summarize our results.
II. SU, GROUP AND GENERALIZED
BMO
MASS FORMULA
SU, group is characterized by N by N traceless hermitian Ai (i = 1, 2,..., N2 - 1) satisfying semisimple Lie algebra:
[Ai 3&I = QfkA ,
N2 - 1 generators
(i,j, k = 1, 2 ,..., N2 - l),
(2.1)
where the structure constants &‘s are antisymmetric with respect to the odd permutations of i, j, and k. Explicit forms of generators in SU, and SU, are constructed and tabulated in Appendix. Introducing a matrix h, = (2/N)‘12 II, we find the anticommutators, satisfying {Xi 7 hi) = (4/N) aii + 2&dk
>
(i,j = 0, I,..., N2 - 1; k = I, 2,..., N* - I), (2.2)
where dijk’s are constants which are symmetric under the interchange of i, j and k. The nonvanishing values of the hjk’s and dijk’s for SU, and SU, are tabulated in
398
HAYASHI
ET AL.
additional tables, explicitly in the Appendix. Comparing these constants with known values for SU, and SU, we find that in our normalization the corresponding J;:jk’s and dijk’s take one and the same values in SU, (for N = 2, 3,..., N) if indices, ijk, coincide in that order. This fact gives us an interesting application. Hara [27] has shown that the decay amplitudes of J/4 particle to the known three and fourgs-mesons may be described in terms ofhia’s and dij,‘s (i, j, k = 1,2,..., 8). He also discussed relations among decay widths of final five and six ps-meson states to a certain extent. In his treatment the essential assumption is that the Jft,b is an SU, singlet with IG = O-, J pc = l--. In SU, (N = 4, 5,...) our additional flavors always appear in an isosinglet state. Thus, we can assume safely that the charmonium-like “ortho-bottomonium” I& and “ortho-toponium” #t in SU, and SU, , respectively, will have the quantum numbers ZG = O-, Jpc = I--. Our observation in the previous paragraph tells us that the similar statement to Hara can be made for these particles. Perhaps some irregularities relative to J/t,h decay will be observed due to the difference in Q values and to the dynamical effect inherent to each state. Let us turn our attention to SU, mass formula. Mass operator in SU, group may be defined as N
Hint = 1 anTn2--1,
(010= I>,
(2.3)
TZ=O
where oli (i = I,..., N - 3) represents relative strength of symmetry breaking to SU, . In terms of SUN tensor notation T,,m (m, n = I,2 ,..., N), Eq. (2.3) can be rewritten as N
Hint = c +Jn”, n=3
(x0 = 11,
(2.4)
where n-1
x,w3 = (l/3)
1 (6/k(k
( k=3
-
I))‘/”
mk-3 + (6n/(n -
I))“”
01,-~ , 1
(n = 4, 5 ,..., N; a0 = 1).
(2.5)
This relation was first derived by one of us (S.I.) [18] from quark mass terms in SUN (N = 5, 6, 7, 8) and then generalized. We have checked its validity in our present approach, specifically, in SU, and SU, . The parameters xi’s are also
related to quark masses: xi =
mQ+s - ml m, - ml '
(i = 1, 2 ,..., N - 3).
(2.6)
Here, we used a short hand notation: u = ql , s = q3 , c = q4, etc., for simplicity, by assuming m, = md .
MESON
MASS
FORMULA
IN BROKEN
399
su,
The meson mass formula in SU, may be given in the first order of symmetry breaking:
w3,i = A 5 &.2-1%3,
(ffo= 11,
(2.7)
n=3 (W),,
=
R,2.
These relations together with experimental masses m? (i = I,..., N - 1) form the eigenvalue problem to find the eigenvectors for meson states in SU, . From now on we shall confine ourselves to neutral vector mesons having the definite charge conjugation parity. We shall specify the meson states appearing in the secular equation by 1, 2,..., N - 1, for simplicity. In this definition, e.g., w, $ and J/t,b mesons in SU, correspond to numbers 1, 2, and 3, respectively. Let us write our eigenvalue problem in a form: (C)i,(~),
= mwi
(i,j = l)..., N - l),
>
w3)
where cij = cji )
(i # j).
(2.9)
Making use of these notations we find Cl, = ao2, c 1.n
=
ACL-Z,
(010= 1; n = 2, 3 )...) N - l),
c m,n = (2/n@ - l))‘/” Dci m _ 29
(m # n; m, n = 2 ,..., N - l),
C n.n = R2 + DC--(n - 1)(2/n(n + I>)‘/” 01,~~ + C,-,,,-,
= m2 - (N - 2) D(2/N(N
-
(2.10)
kz$+2 (2/W - 1))“” ~~1,
1))“” olNps.
Not all the parameters in the generalized formula two constraints:
are independent.
We obtain
N-l
MO2 = C mi2 - (N - 2) mo2 + (N - 1) D 5 (2/n(n - l))‘/” DL,,-~, i-l
(2.11)
n=3
and R2 = (1/3)(m,2 + 2m&) - D 5 (2/n(n - 1))112E,-~. n=4
(2.12)
400
HAYASHI ET AL.
The parameter D that appeared in Eqs. (2.7), (2.10)-(2.12) is a mere constant: (2.13)
D = (243/3)(112,2- w&,).
At this point let us denote the masses of the flavored mesons in SU, by (mN)k (k = 1, 2,..., N - 2). We find N-l
(mN)f = &f2 + $0
c (2/n(n - 1))“” 01,-a - (N - 2)(2/N(N
( n=3
- l))‘/” a,+,--3, 1
(2.14) N-l
(m,)2, = M2 + (l/2) -
(N
-
(
-(2k/(k
2)(2/N(N
+ I))“” 01~~~+
1
(2/n(n - 1))lj2 CY,-~
n=kf2
-
l))“2
+$-3
,
(a,, = 1; k = 2,..., N - 2).
1
We shall not consider a linear fit in this paper so we wrote all expressions in a quadratic form, but if one wishes he can use the formulas by merely putting linear input masses at the corresponding places. Summarizing this section in what follows after substitution of N - 1 inputs, the N - 2 parameters will be determined as a solution of the given secular equation of SUN meson mass formula. The corresponding eigenfunctions are determined simultaneously. The validity of that approach may be decided after solving the equation numerically and comparing the results with our present knowledge of fundamental particles. III.
APPLICATIONS
Let us summarize first several relevant statements before making applications: (I) According to the theoretical considerations, proved experimentally in SU, and SU, , N2 - N + 1 meson states of N2 ones in SU, belong to the pure (unmixed) irreducible representations of that group, (II) masses of ps-mesons of these pure states may be expressed in terms of mechanical [28] (free) quark masses appeared in GMOR, GLR, and their extended hamiltonians, (III) ps-meson mass formula for the pure states obtained in chiral SU, x SUN take one and the same expressions to those given by Eq. (2.14), checked explicitly in SU, and SU, by making use of &‘s given in the Tables in the Appendix, and (IV) the mass formulas for meson states apply universally for all meson states by merely changing a few of the parameters in the mass formulas, provided that they belong to the same irreducible representations of SUN . The last statement was assumed and tested in SU, case [5, 201. This means in general that we can predict all meson masses by changing inputs D and m, (or mKt)
MESON
MASS
FORMULA
IN
BROKEN
401
su,
and a single parameter A appropriately (see, previous section), once the remaining N - 3 parameters clli (i = l,..., N - 3) are determined, at least semiquantitatively. Based on the assumptions (I)-(IV) the meson mass formulas for N2 - N + 1 pure states in SU, was derived before and generalized to GLR forms [18]. Especially, the relations, Eqs. (24) and (25), in that reference were useful for determining the starting values of parameters 01~and 01~of our present approach. In the numberical work we analyze two possibilities: (1) 3.95 or upsilon at 5.97 GeV is a candidate of & and (2) 3.95 GeV and the upsilon are candidates of &, and & of the lowest J pc = 12- states of SU, and SU, , respectively. There is some inaccuracy for choosing a rather sensitive quantity m,, . We shall choose m, = 0.765 GeV throughout the paper. This value is determined from our previous SU, analysis. Except assumed masses for Z& and I,$ stated above, we assume that all other masses are up to date values. TABLE Result
of Analysis
I in SU, Approach
Parameters Case
A(GeV”)
a2(GeV2)
ii;i,Z(GeVZ)
a1
(3 (ii)
-0.1692 -0.1692
5.2724 9.1209
6.330 10.97
21.42 21.38
(9 States
Wavefunctions (l/l/Z)(u&+dd) 0.9964 0.7477 0.2704 0.3082
. 10-l . 10-l * 10-l
-0.7612 0.9966 0.2102 0.2309
(ii) States
* 10-l * 10-l * IO-’
(0 (ii)
* 10-l . 10-l
(Coefficients
ss -0.7603 0.9966 0.2214 0.2068 Predicted
Case
-0.2771 -0.2502 0.9959 0.8235
* 10-l
bb * 10-l * 10-l
-0.2678 -0.2334. -0.8370 0.9959
* 10-l
-0.2765 -0.2499 0.9987 0.3382
. 10-l * 10-l Masses
. 10-l 10-l * 10-l
of)
bb
CE * IO-’
31.42 81.47
of) CE
Wavefunctions (llz/Z)(un+dd) 0.9964 0.7483 0.2849 0.2849
(Coefficients
ss
%
* 10-l * 10-l
-0.2592 -0.2190 -0.3508 0.9988
9 10-l
- 10-l * 10-l - 10-l
in GeV
mDt
mF*
met
mf *
we
2.216 2.214
2.263 2.262
2.787 4.171
2.824 4.196
3.477 4.660
402
HAYASHI
ET AL.
We shall call case (i) and (ii), respectively, for rn,* = 3.95 and 5.97 GeV in SU, approach. The flavored mesons (bottomed ones) may be called G, H, and Z from low to high masses. The result of analysis for SU, is tabulated in Table I. We have given all the computer fitted parameters which were obtained by solving essentially the coupled nonlinear parameter equations. We did not avoid the duplication of paramters which will be used as a consistency check (see next section). We learn from the table that at the present stage it is hard to negate the assignment of 3.95 GeV state as the & in the sense of OZI rule. The point will be clarified by a careful study around 4 GeV. One sees that Hara-like relations (see previous section) will hold in some accuracy by comparing wave functions for & an &, . These tabulations also tell us the relative sign of transient coupling constants. We think that a more detailed study is necessary for the last point, but not considered further here. Now, we proceed to analyze case (2). For definiteness sake, we introduce notations for topped mesons 0, P, Q, and R for tii, tl, tE and t6, respectively. The result of our analysis is given Table II. Again, we cannot exclude the assignment considered here simply from the mass formula analysis alone. TABLE Result
of Analysis
II in SlJ,
Approach
Parameters
A(GeT)
m*(GeVz)
-0.1579
mOz(GeVz) 12.67
10.078
Wavefunctions States
21.42
(Coefficients
ss
(l/&z)(u&+dJ)
::
0.9960 0.7439 0.2950 0.2652
* 10-I ** 10-l 10-I
-0.7648 0.9964 0.2209 0.2064
vh
0.3054
* 10-l
0.2215
31.47
of)
bb
CE
to
.* 10-l
-0.2806. -0.2530 0.7951 0.9954
10-l - 10-l . 10-l
-0.2713 -0.2362 -0.8283 0.9945
* 10-l * 10-l - 10-l
* 10-l
0.3646
- 10-l
0.5312
* 10-l
* 10-l
Predicted
Masses
76.40
-0.2633 -0.2223 -0.3321 -0.5733
. 10-l - 10-l -* 10-I 10-l
0.9972
in GeV
mD*
mF’
mc*
mfft
mI*
m0*
mQ*
mp*
mR’
2.217
2.264
2.789
2.826
3.479
4.164
4.189
4.654
4.952
MESON
MASS
IV.
FORMULA
IN
CONSISTENCY
BROKEN
403
su,
CHECK
Let us rewrite Eqs. (2.11) and (2.12) in the forms: N-l
n1,2 = R,2 - (N - 1) D 5 (2/n@ - l))l/?
a&3
)
(4.1)
n=3
(1/3)(~,2 + 2&e)
= m2 + D ;
n=4
(2/n(n - >>"2in...3
.
(4.2)
The purpose of this section is to find out consistency of these conditions in referring to the left-hand side and right-hand side by making use of inputs and tabulations in Tables I and II. It would be most convenient to list the results in tabular form. Moreover, we have recalculated SU, and SU, cases by our present approach. The result for SU, completely coincides with our previous answer[20]. Including those cases the comparison stated above is listed in Table III. One may find how each fitting machinery is going on. TABLE
III
Consistency Check
SK Case case (i)
case (ii)
sue
10.646
25.078
45.117
60.134
10.060
24.022
45.113
60.133
(GeV2)
SU,
lhs Eq. (4.1)
1.0672
rhs Eq. (4.1)
1.0672
lhs Eq. (4.2)
0.7258
0.7258
0.7258
0.7258
0.7258
rhs Eq. (4.2)
0.7258
0.7261
0.7256
0.7257
0.7262
V.
SUMMARY
AND
DISCUSSION
Meson mass formula is derived in a form suitable to apply to unitary N2 members of SUN in the first order of symmetry breaking. A partial proof of the general expression may be given by making use of constants dijk’s of SU, and SU, group summarized in Appendix. The first step of the generali-
404
HAYASHI
ET
TABLE
IV
Generators
A,=
i . . .
1 . . . ,
. . . . .
. . . . .
. . . . .
A,=
Ax,,=
&=
A,e=
&=
A,,=
i . . .
: . . .
: . . .
: . . . .
: . . . . : ;-I’. . . . . . . . . . . . .
.
.---I.
:
:
:
:
:
i .
. .
. .
. .
. .
. . . .
. . . .
. . .
. . . . 1 .
.
.
1 .
. .
. .
. . .
. . .
. . .
. . .
1 . .
i
:
:
:
: .
.
.
.
.
.
.
.
.--I
.
.
.
.
.
.
.
.
.
.
.
I
.
.
.
.
. . . . .
. . . . .
. . . . 1
. . . 1 .
: . .
of SU,
A,==
.-i. i . . .
.
. .
A,=
. .--I. . . . i.... . , . . . .
. .
. .
: . .
. . 1 . .
1
.
.
.
. .
: . .
: . . . . . . .
. . 1 . . Ar=
AL.
X8 = l/z/J
A,,=
A,, =
A,,=
Azo=
A,,=
. . . .
. . . .
. . . . .
. . . . .
A,=
. &=
: . .
. . 1 , -2 . . . .
.
.
.
: .
: 1 .
: . .
. 1 . . .
.
.
. .
. .
. .
1 1 . -i . . I . . . . .
. . .
. . . . i
. . . . .
. . . . .
.---I . . . . . . . .
. .
. .
. .
. .
. .
. . .
. . .
. . 1
. . .
1 . .
.
.
: . .
: . .
. . . . .
. . . . . . .-I 2 .
1 .-1. . . .
A,=
&=
Ala = l/1/6
h,B=
As4 = l/10’/%
.
.
. * .
. . .
. . . . .
. 1 . . .
. . . . . 1
.
::::: 1 . . .
. .
. .
. .
. . . . .
. . . I .
. . . -2. . . . . . .
.
1 . . . .
. 1 . . .
.
.
.
i ..
:
. . . . .
. . . . 1
. . . . .
. . . . .
. 1 . . .
1
. 1 . . .
.
. .
. .
i .-
:
: . .
i . .
. . .
: 3 . . .
4
MESON
MASS
FORMULA
IN
TABLE
BROKEN
405
su,
V
New Generators in SU, . AZ&=
: . .
.
.
.
.
1
: . .
: . .
: . .
: . .
: . .
1 .
.
.
.
.
.
xzB=
A,,=
. A,,=
A,,=
: . .
. . . . .
. . . . .
.
. . . . . 1
. . . . . .
.
.
. : . . .
. . i . .
.
.
.
.
.
: . . .
: . . .
: . . .
: : . . .--I z .
A,,=
A,, = l/lW
. .
. .
. .
.-z . .
: . .
. . .
. . .
. . .
. . .
. . .
i
.
.
.
.
.
.
.
.
:::::i . . . . . . .
.
.
.
.
.
.
.
: . . .
: : . . . . 1 .
: . . .
: . . .
1 . . . .
.
.
.
.
.
.
.
.
. . . . 1 .
. . .
. . .
1 . . .
1 . . .
1 . . z
1 . . .
1 . . .
.
.
.
.
.
.
.
.
.
.
.
: . . .
: . . .
: . . .
: . . z
: : .--I . . . .
: . . .
: . . .
: . . .
: . . .
: . .
. .
1 . . . . .
i . . . .
: 1 . . .
:
:
i . .
: : 1 . .- 5
A,,=
A,, =
:
.&=
-i . . .
1 1 .
406
HAYASHI
ET AL.
TABLE VI Nonzero Elements of& and &
1
2
3
4
1
2
2 4 5 9 10 16 17 4 5 9 10 16 17 4 6 9 11 16 18 5 9 10 16 17
3 7 6 12 11 19 18 6 7 11 12 18 19 5 7 10 12 17 19 8 14 13 21 20
1 1 1 4 5 9 10 16 17 2 2 2 4 5 9
8 15 24 6 7 11 12 18 19 8 15 24 7 6 12
5
6
7
8
9 10 3
4
in SU,
13 14 20 21 8 14 13 21 20 13 14 20 21 10 12 14 17 19 21 15 23 22 22 23
11
10 16 17 7 11 12 18 19 11 12 18 19 9 11 13 16 18 20 10 16 17 16 17 9 10 11 12 16 17 18 19 4 4 4 9 10 16 17
9 10 11 12 16 17 18 19 8 15 24 13 14 20 21
6
9
12 13
14 15
16 18 20 22
7
8
12 18 19 18 19 14 20 21 20 21 16 18 20 22 17 19 21 23
15 23 22 22 23 15 23 22 22 23 17 19 21 23 24 24 24 24
11 12 18 19 7 7 7 11 12 18 19 8 8 8 9
13 14 20 21 8 15 24 14 13 21 20 8 15 24 9
21.\/8 4 -12 + 4 2/h + -fr 3 1 1,2%h 1/2x& 1/2d/;6 -3/21/a 5/2(1O)l/” 5/2(10)‘/* 5/2(1O)l/* 5/2( 1oy
Table continued
MESON
MASS
FORMULA
TABLE
2
1.
10 16 17
11 19 18
3 3 3 4 5 6 7
8 15 24 4 5 6 7
l/d/5 lid/a l/lo’12
8
21
21
-l/d5
9
9 9 16 17
15 24 22 23
--1/d/6
10 10 16 17
15 24 23 22
-lid6 l/lo112
11 11 18 19
15 24 22 23
-l/d/6 l/lO’/2
12 12 18 19
15 24 23 22
-l/d6 l/lO’lZ -$,
13 13 20 21
15 24 22 23
--l/d6 l/lO1’a
14 14 20 21
15 24 23 22
-l/d6 l/lo’/2
3
10
11
12
13
14
595/101/z-5
9 10 16 17 6 6 6
14 13 21 20 8 15 24
15
15
--2/1/G
15 16 17 18
24 16 17 18
l/101/2 l/266 1/2-\/i; li2A
19 20 21 22
19 20 21 22
1/2-\/a 1/2x& l/21/6 -3/2v%
16 17 18
23 16 17 18
23 24 24 24
--3/2x& -3/2(10)‘/2 -3/2(lO)l/” -33/2(10)‘/*
19 20 21 22
19 20 21 22
24 24 24 24
-3/2(lO)‘/2 -3/2(1O)l/” -3/2(lO)1/2 -3/2(10)‘/*
23 24
23 24
24 24
-3/2(lO)l/” -3/10’12
6
4 15
1/10”2 + 4
4.
.;.
3 +
+
a Q
-a +
407
su,v
VI (continued)
8 15 24
4
-+ -.
BROKEN
5 5 5
5
-*
4 d
IN
-l/24’; 116 l/lo’12
10 11 12
10 11 12
l/263 1/2d/J l/21/5
-$ -p
13 14 16
13 14 16
-l/b 1/q/B l/lo’/2
17 18 19 20
17 18 19 20
-l/d/3 -l/d/3 1/2d/3 1/2d/3 1/2d\/3 1:24/3 -l/1/3
-4
8
408
HAYASHI
ET AL.
TABLE New
i
i
k
firn
Nonzero
Elements
VII of hjk and
dirk in
SUe
i
i
k
LL
i
j
k
f;ic
24
25 27
26 28
1/2(10)‘/2 l/2(10)1/”
29 31
30 32
1/2(10)‘/~ 1/2(10)1~~
____. 1
25 26
28 27
13
29 30
32 31
t 2
2
25 26
27 28
14
29 30
31 32
& i
3
25 27
26 28
15
25 27
26 28
l/2& l/2-\/6
25
33 26
34 35
4
25 26
30 29
29 31
30 32
l/21/6 -3/2&
27 29
28 30
35 35
3115112 3/151/a
5
25 26
27 28
16
25 26
34 33
-a
31 33
32 34
35 35
3/151/* 3/15’/$
27 28
30 29
17
25 26
33
ri
34
+
27 28
29 30
18
27 28
34 33
25 27 29
26 28 30
19
33
20
27 28 29
25 26
32 31
21
30 29
25 26
31 32
22
27 28
32 31
23
17
17 25
35 34
26 18
33 35
27 28 19 27
33 34 35 34
6 7 8
9 10 11 12
27 28
31 32
1
25 26
27 28
9
25 26
28 27
10
25 26 27 28
25 26 27 28
11
2 3
12
4
-2/101/e 3/d/151/2
7;. -+ L 2 + +
34 34 33
1 2
33
g
30 31
34 34
4 +
32 31
33 33
t
32
34
+
25 26
31
1
32
i
25 26
32 31
27 28 27 28
31
-$
32 32 31
if
-+
-$ 4
18
19
-4 +
- .&
Table continued
MESON
MASS
i
i
k
29 30
31 32
29 30
32 31
-$
15 25
35 25
l/151/2 1/2-\/a
26 27
26 27
l/2%& l/2-\/6
28 29 30 31 32 16 25 26
28 29 30 31 32 35 33 34
l/2-\/6 l/21/6 1/2v’/8 -3/2x& -3/2x& 1/151’2
2.5 26
29 30
13
25 26
30 29
14
27 28
29 30
15
27 28
30 29
8 21 25 26 21 28 29 30
35 21 25 26 27 28 29 30
1/15’/2 -l/d3 l/2-\/3 l/2%0 l/2-\/3 1/2d/3 -l/-\/3 -l/-\/S
24
27 28 29 30 31 32 33 34
27 28 29 30 31 32 33 34
l/2(10)‘/* l/2(10)*/* 1/2(10)‘12 1/2(1O)lj2 1/2(10)‘/2 l/2(10)1/* -2/10’/2 -2/10’12
25
25
35
-2/151/e
26
26
35
-2115w
27
27
35
-2/151J2
28
28
35
-2/151/2
29
29
35
-2/15’1=
30
30
35
-2/15’/2
31
31
35
-2/15112
32
32
35
-2/15l/=
33
33
35
-2/15112
34
34
35
-2/15112
35
35
3.5
-4/151/2
4
5
6 I
8
BROKEN
VII
k
AJr
IN
TABLE
i
i
FORMULA
16
409
su,
(continued)
Ljk
a iI
i
j
k
19 20
28 20
33 35
29 30
33 34
21 29
35 34
30 22
33 35
31 32 23 31 32 24 25 26
33 34 35 34 33 35 25 26
Q
3 -$
21
22
23
24
Air -$
l/151/2 g 3 1 p51ia 1.
4
1,‘151/z g 4 l/151/2 : 11151’2 l/Z(lO)~~~ 1/2(10)‘/2
410
HAYASHI
ET AL.
zation was performed by the method of GMOR and GLR which is quickly manageable and transparent in some aspect. A typical factor (2/n@ - 1))lP appeared many times in the generalized formula was found by this approach [18]. As we discussed this and BMO approaches are complementary from various points of view. The former approach, however, predicted the relation between quark and ps-meson masses. The light quark masses estimated by Leutwyler [29] are considered to be one of the clues to establish the quark dynamics. We do not discuss this important problem in relations to new flavors b and t in this paper. The mass formula is applied to SU, and SU, cases by assigning suitably the fourth and fifth neutral vector mesons in eigenstate of charge-conjugation number C = -1. From our analysis one sees that it is hard to decide either 3.95 GeV or upsilon state belongs to SU, . Hence the possibility to choose I,,&,as 3.95 and & as 5.97 upsilon state of SU, cannot be excluded in a similar application. A possibility to find out a goodness of above choices is suggested, i.e., by studying decay modes of these particles in relation to those of J/#. Of course, the direct check of the idea should be made by observing the flavored particles of respective symmetry as it should be done for charmed ones. Combining the analysis of the present paper and our previous SCJ, mass formula analysis we can make the following statement for the meson masses in other states, e.g., 0-, l+, 2+, etc. The masses of these states may be predicted by substituting input masses belonging to I = 1 (p-like) and &(strange) mesons of their own members, changing slightly a single parameter A, while keeping remaining parameters ai’s at the values found in the text. We shall leave many relevant problems in future.
APPENDIX
We shall summarize a group structure of SU, and SU, in this Appendix. The generators hi (i = l,..., 15) of SU, may simply be obtained by adding null line and null column elements to the fifth line and column, respectively, to those of Amati et al. [23]. Including all remaining generators & (k = 16,..., 24) the generators of SU, are listed in Table IV. The generalization procedure for generators discussed above applies also to that for SU, . We merely list the new generators of SU, in Table V. The structure constants &‘s and constants &‘s for SU, are tabulated in Table VI. By comparing these numbers with known values for SU, and SU, one sees that diib’s and&‘s take one and the same values if all the indices ijk are the same in that order. This makes our tabulation for new constants in SU, much simpler (see Table VII).
MESON MASS FORMULA IN BROKEN su,
411
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