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Meson decuplets in the SU(4)×SU(4) scheme. A possible explanation of narrow e+ − e− resonance
Volume 55B, number 5
PHYSICS LETTERS
17 March 1975
M E S O N D E C U P L E T S IN T H E S U ( 4 ) × S U ( 4 ) SCHEME. A
POSSIBLE EXPLANATION
OF NARROW
e + -
e- RESONANCE
R. VERGARA CAFFARELLI Instituto di Fisica Dell'Universita' di Pisa, Italy, [nstituto Nazionale di Fisica Nucleate, Sezione di Pisa, Italy
Received 25 November 1974
The recent proposals [1 ] of unified theories of weak and electromagnetic interactions have raised considerable interest in SU(4) X SU(4) group as an approximate symmetry of the strong Hamiltonikn. The relevance of this group is also motived by an analysis of the GMOR Hamiltonian [2]. The most interesting features of the SU(4) X SU(4) scheme is in fact to provide a "unified" mass formula very well satisfied by the 0 - , 1- , and 2 + nonets. As we will see, the (4, 4) + (4, 4) representation of SU(4) X SU(4) allows to accomodate a decuplet of 0 - non charmed particles and consequently enable us to solve the r / - X 0 - E mixing problem. This information is crucial for the discussion of the decays of 0 - mesons [3]. SU(4) X SU(4) also provides a scheme for explaining the recently discovered narrow e + - e - resonance. As we will see, they can be identified with the missing tenth particle of the 1- decuplet * 1 The basic Hamiltonian H = H 0 + eOu 0 + e 8 u $ = HO0 + e u + eoU 0 + e g u 8 ,
(1)
where H00 is SU(4) × SU(4) invariant, is a refinement of the GMOR Hamiltonian in so far as one specifies the transformation properties o f H 0 under SU(4)X SU(4) .2 . The first step will be to discuss the 0 - decuplet. By using the Ward identities technique [4] we will discuss the physical consequences of the Hamiltonian (1), without making the arbitrary assumptions made in the literature (like SU(3) invariance of the vacuum, asymptotic symmetry of some SU(3)X SU(3) or SU(2)X SU(2) subgroup, the lepton-hadron symmetry, the neglect of some of the mixings etc.) [5]. The only approximation we will make is the pole dominance of the propagators at zero m o m e n t u m transfer and the approximate equality of the renormalization constants of the fields of the observed 0 - mesons. In this way we establish the following equations: (2)
M i / g ~ l c ~ k = ej ,
where M i j is the squared mass matrix ,3 g~ are tabulated by many autors [ 5 ] ' 4 , and h k --- (0 lUki0> are different '
I
, l SLAC data obtained with SPEAR; Brookhave data, MIT group; Frascati data, by the "y~,-2,MEA and BB groups. ,z In the literature dealing with SU(4) X SU(4) the above Hamiltonian is usually written in the form H = Hoe + eo,#o, + esu8 + e Is u 1~ where u o, = (x/~uo + u)/2; u is = (u o - x/3 u)/2; co' = (x/~ eo + e)/2; e Is = (co - x/~ e)/2, We prefer the form ( I ) because in this way the (4,~) + (~1,4) representation, defined by uo, ul, ...us, u9 .... u14, u, oo, ol .... us, 09, ... u14, o is a natural extension of the standard (3, 3) + ~ , 3) representation. In the following, for simplicity of notation we will also denote u and o by u16 and o16. • 3 Here and in the following we will speak of mass matrix even for the charmed fields, for which the identification with the inverse propagator is more doubtful. We want to stress however that for our discussion we do not need such a strict identification, but only that the inverse propagator is a lower bound for the mass. •4 For convenience of the reader we give the combination orgY's corresponding to our notation: f9,1o,o = .fl h12,0 = f13L14,o = 1/N/~; / 9 , 1 0 , 1 6 = f 1 1 , 1 2 , 1 6 = f 1 3 , 1 4 , 1 6 = -- 1/Vr2; f~/,o = fi,£ 16 = 0 f o ~ i = 1..... 8. di~],0 = ,4c2/3 6/t" fox i = 1..... 8; di,£o = 1/~/66i] for i = 9 ..... 15; di,/, 16 = 0 for i = 1, ..,, 8, d//,l 6 = l/Vr28 ~t"for i = 9 ..... 14; d15,16,16 = - ",/'3~481
17 March 1975
PHYSICS LETTERS
Volume 55B, number 5 from zero only for k = 0,8, The determination of the in the literature, being based of the physical implications. the exact eigenvectors of M.
16. mass matrix and its diagonalization is a non trivial problem and the existent solutions on drastic approximations, do not provide a sufficient basis for a detailed analysis As we will see, the whole situation will look completely different once one may have To this purpo_se we note that eq. (2) for the channels us, uo, u16, plus the trace equa= pnz1 + mf + M$,’ can be written in the form:
Iion 43,s + MO,O+ %6,16 hM=ct:,
(3)
where
A’f,
&SD(l-f)
(1+2f)/2&
-3a
Vf- wh-
(1 -f)216
0
Zfia(l
-(4f
-(2f-1)3/2fi
-f)
-1)ma
\
(4)
i’
(5)
&=rnzfn
&ml
-fm)
(4fm-
l)/&
-2+?a(l
-f)
z-
s+$!+]
(1f 2fm)/24
-3e
2t1-fm)f&
cl (2f-1)
maf4f-1)[E-s+4*]
i a.ndf =fK/f,,,fK ,” ?
= d@
\ s-1-E C
II
2d2a
(aho
The inverse propagators
for the charmed fields are:
(6) 2fm - 1+2e m2 77’
Mr313=“fi,14 = 2f-1+2a f
(7)
MS
(8)
=MS
999
10,lO
(9) exhibiting the interesting feature that scalars and pseudo-scalars get interchanged when e + -e, a + -a. The above eq. (3) can be used toobtain the values of e16 and x16 and the eigenvectors of the mass matrix M. 2fm - 1 - 2A,zA,zA,, e=+1/2 2f -
tl--(1+2A,lAm2)(1+2A,2A,,)(I+2A,JA,1)
(10)
1 - 2A, 1AmZAm3
‘= 2fm - 1 -2A,,A,zA,J The eigenvector
482
’
e ’
corresponding
(11) to the eigenvalue mi (mj, mk being the other eigenvahres and ml
I
Volume 55B, number 5
PHYSICS LETTERS
/ 1+2Am.Amk (a/'v/3)(Ami+ l) V ( A m / - ~ - Ami)
(llx/-6)(2Ami--l)
V(Am _ Ami)i'f4lmk'XAm~
17 March 1975
\ \ ,
(12)
)
- mi) ,/(1 + 2A m .A m .) ( l + 2A m .Am.)! ['Ami(e/a~mi) [V -(Am] t- Almi)(Amk2 Ami;/ A m.(ela
\(l/x/~) where
Ami= 1-2f( mi-m ]. \mi-1 ]
(13)
It is important to stress that no sum rule can obtained by using only the group properties of the Hamiltonian (1) unless one makes some ad hoc assumption, since the number of unknown parameters is the same as the number of equations. 2 e/a 1> m 2. This implies that at The first interesting consequence of eq. (I 1) is that if m 2 t> X 0 then m22 >~ ~ m~r least one of the charmed particles has mass lower than rn 3 , and therefore if one identifies m 1 , m 2, m 3 with r/, X 0, E respectively, one cannot avoid a charmed particle with mass lower than 1.416 MeV/c 2. The only way out of the above difficulty is to conclude that either X 0 or E does not belong to the decuplet and another particle o f sufficiently high mass must take its place. The choice of the new mass is however not arbitrary since the following inequality - ( 1 + 2Am1Am~) (1 + 2Am2Ams)(1 + 2AmsAm1) >~0 must be satisfied in order to keep el6 and ~k16 real. Furthermore in order to have high masses for the charmed particles el6 must be large and this implies
Am2Amx=n~ -
112 •
(11)
The exploitation of eq. (14) leads to interesting conclusions. Firstly, it can be satisfied by taking m 2 = X 0 and f ~ 1.43 or m 2 = E and f ~ 1.04. In both cases even if el6 is very large the ratio m2 e/a remains bounded by m 2, a value reached in the limit when the equality (14) holds strictly. In this limit all the charmed masses become equal to m 3 , and the only mixing is between o0 and v 8 * s. It is also interesting to note that Ares~ - 1 when f ~ 1, for large rn 3 values (/> 2GeWc2). This implies the following limit: [ 2 f - 1 - 2AmlAm2Am3] -* 1+2AmlAm2 ~ 0 *6 and of course X16 ~ 0. Alternatively, the same result follows from f = 1.5, because Am3 -~ - 2 and [ 2 f - 1 2AmlAm2Am3] ~ 2(1 - 2AmlAm2) .~ O. The second important consequence of eq. (14) is that completely clarifies the problem of nonet mass formulae, appeared in the literature. To this purpose one writes eq. (14) in the equivalent form:
(m2-m2)(m2-m2)=
2(2f-2f(2f-1)1) 2 + 1 (m2-m2)[
2m2+2m~ 4fm2+4(f-1)m212f1
"
(14)
Then, f o r / ' = 1, with m 1 = r/, m 2 = E one obtains the Glashow's formula [6] -g(m K (m2n-m2r)(m2-m2r)-2 2-
m 2 ) ( 2 m 2 + 2m2 - 4 m 2 ) ,
(15)
which may also be written in the form given by Schwinger i7] : 2
2
(m2_4m2--m2)(m2_4m3-m~r)=_~(m2K_m2)2 " 3
n"
On the other hand, for f = 1.5 with rn 1 = r/, m 2 = X 0 one obtains the Bjorken-Glashow formula , s In both cases m2 = Xo, m2 = E, the o 8 - o0 mixing angle is small. •~ In fact, in the range f ~ 1 and m 2 > X o we have A m2Am3 ~ 0 which implies ( 1 + 2A m2A m3 ) > 1, and for m a > E, ( 1 + 2AmlAm3) = 0 only for f , ~ 1.
483
Volume 55B, number 5
PHYSICS LETTERS
(m 2 - m2)(m2o _ m r)2 _.$ ( m K _ 2 _ m 22) (, 2 ,m , + 2m2^u -- 3 m 2 - m2") '
17 March 1975 (16)
which may also be written in the following form [6]" (m2
4m2-m2)(m2
°
4m2-m2)
3
3
2 2 =--g(mK-
m2)2 ~r- "
It is important to stress that all the above equations are very well satisfied experimentally. This shows that the condition of the high masses for the charmed particles is not only realizable but it gives rise to a sum rule (14) which strongly suggests f = 1; a choice motived by the following considerations: i) it implies that SU(3) is a good symmetry for the vacuum and therefore the classification o f particles into SU(3) multiplets, according to the original Gell-Mann idea, is justified, ii) the sum rule (15) is verified with exceptional precision by 1 - and 2 + mesons, iii) the positivity of the charmed masses is realized only if ia I > 1/2. In the limit f = 1, a = 1]2 all the scalar masses go to infinity and SU(4) become a good symmetry for the vacuum. If the vacuum is symmetric under SU(4), we can write the following mass formula, suggested by the octet enhancement of SU(3);
Mi j = MoSi j + (e16d16,i, j + eodo,i,j + esds,i,j)A ,
(17)
i = 1, ..., 15, and .7
MO,,O, = s-- M8, 8 - M15,15 .
(17')
Eqs. (17), ( 1 7 ' ) give rise to the same mass matrix for the channels 0, 8, 16, as obtained before (eq. (3)), with f = 1, a = 1/2, and therefore the eigenvectors have the same dependence on the masses as those given in eq. (13). Therefore, by exactly the same argument, the tenth particle o f the 1 - , 2 + multiplets (h016)) has negligible mixing with the ¢ - co and the f ' - f system, respectively. Thus, we can introduce only one "mixing angle"; for example, for the ¢ - co system we have: .4 w + 1 - 2X/~- (K*2 _ p 2 ) tan0 =xf2 2 A w _ 1 4 K . 2 _3602_/92 , By using the above equation, we have; 0 ( r / - E) ~ 7 °, 0 (~o- co) ~ 38 ° 0 ( f ' - f ) ~ 30°; we note that these values coincides with the SU(3) prediction ,8. We observe that eq. (14) is verified by t a k i n g f = 0.97 a n d f = 0.96 for 1 - and 2 + mesons respectively; it follows that only near these values el6 is real and we can obtain the SU(4) condition a ~ 1/2. By identifying the 3.105 MeV e + - e - resonance, with the tenth (~0c) particle of the 1 - decuplet one obtains e ~-271. This identification explains the narrow width of ~oc since i) the decay product of its ~016 component should contain at least a pair of charmed particles and this is forbidden, since the mass o f the charmed particles is too high for e ~- 271, ii) the decay through the 8th and 0th component is strongly depressed. With the above values of e we can determine the masses of the charmed particles of the 0 - , 1+, 2 + multiplets and predict the mass of the tenth particle of the 0 - and 2 + decuplet. O- ; 71c = 3.14 GeV/e 2 ;
Kcl/2 = 2.22 GeV/e 2 ;
Kc0 = 2.27 GeV/e 2
1- ;input
Klc/2 = 2.26 GeV/c2 ;
K 0 = 2.31 GeV/c 2
2+ : fc = 3.83 GeV/c 2
Klc/2 = 2.87 GeV/c 2 ;
K c0 = 2.92GeV/c 2
,7 Since the state 1~00,)is an SU(4) singlet eq. (17) cannot be generalized to i, ] = 0'. :~a By SU(3) prediction we mean the solution of the equation tan 20 = (4K .2 - 3~o2 - p2)/(4K*2 - 3to 2 -O ~) obtained from the Gell-Mann-Okubo formula. 484
Volume 55B, number 5
PHYSICS LETTERS
17 March 1975
I wish to thank prof. R. Gatto for a discussion about a preliminary version of this paper and for suggesting to apply the formalism with this identification of the e + - e - resonance at 3.105 GeV.
References [1] [2] [3] [4] [5]
S.L. Glashow, J. IUiopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. M. Gell-Mann, R.J. Oakes, and B. Renner; Phys. Rev. 175 (1968) 2195. F. Strocchi and R. Vergara-Caffarelli;Phys. Lett. 35B (1971) 595. G. Cicogna, F. Strocchi and R. Vergara-Caffarelli,Phys. Rev. D6 (1972) 301. Z. Maki, T. Maskawa and I. Umemura, Progr. Theor. Phys. 47 (1972) 1682; P. Dittner and S. Eliezer, Phys. Rev. D8 (1973) 1929; D.A. Dicus and V.S. Mathur, Phys. Rev. D9 (1974) 1903; S.C. Prasad, Phys. Rev. D9 (1974) 1917. [6] S.L. Glashow, Hadrons and their interactions, ed. A. Zichichi (Academic Press, New York, 1968). [7] J. Schwinger, Phys. Rev. Lett. 12 (1964) 237.
485
PHYSICS LETTERS
17 March 1975
M E S O N D E C U P L E T S IN T H E S U ( 4 ) × S U ( 4 ) SCHEME. A
POSSIBLE EXPLANATION
OF NARROW
e + -
e- RESONANCE
R. VERGARA CAFFARELLI Instituto di Fisica Dell'Universita' di Pisa, Italy, [nstituto Nazionale di Fisica Nucleate, Sezione di Pisa, Italy
Received 25 November 1974
The recent proposals [1 ] of unified theories of weak and electromagnetic interactions have raised considerable interest in SU(4) X SU(4) group as an approximate symmetry of the strong Hamiltonikn. The relevance of this group is also motived by an analysis of the GMOR Hamiltonian [2]. The most interesting features of the SU(4) X SU(4) scheme is in fact to provide a "unified" mass formula very well satisfied by the 0 - , 1- , and 2 + nonets. As we will see, the (4, 4) + (4, 4) representation of SU(4) X SU(4) allows to accomodate a decuplet of 0 - non charmed particles and consequently enable us to solve the r / - X 0 - E mixing problem. This information is crucial for the discussion of the decays of 0 - mesons [3]. SU(4) X SU(4) also provides a scheme for explaining the recently discovered narrow e + - e - resonance. As we will see, they can be identified with the missing tenth particle of the 1- decuplet * 1 The basic Hamiltonian H = H 0 + eOu 0 + e 8 u $ = HO0 + e u + eoU 0 + e g u 8 ,
(1)
where H00 is SU(4) × SU(4) invariant, is a refinement of the GMOR Hamiltonian in so far as one specifies the transformation properties o f H 0 under SU(4)X SU(4) .2 . The first step will be to discuss the 0 - decuplet. By using the Ward identities technique [4] we will discuss the physical consequences of the Hamiltonian (1), without making the arbitrary assumptions made in the literature (like SU(3) invariance of the vacuum, asymptotic symmetry of some SU(3)X SU(3) or SU(2)X SU(2) subgroup, the lepton-hadron symmetry, the neglect of some of the mixings etc.) [5]. The only approximation we will make is the pole dominance of the propagators at zero m o m e n t u m transfer and the approximate equality of the renormalization constants of the fields of the observed 0 - mesons. In this way we establish the following equations: (2)
M i / g ~ l c ~ k = ej ,
where M i j is the squared mass matrix ,3 g~ are tabulated by many autors [ 5 ] ' 4 , and h k --- (0 lUki0> are different '
I
, l SLAC data obtained with SPEAR; Brookhave data, MIT group; Frascati data, by the "y~,-2,MEA and BB groups. ,z In the literature dealing with SU(4) X SU(4) the above Hamiltonian is usually written in the form H = Hoe + eo,#o, + esu8 + e Is u 1~ where u o, = (x/~uo + u)/2; u is = (u o - x/3 u)/2; co' = (x/~ eo + e)/2; e Is = (co - x/~ e)/2, We prefer the form ( I ) because in this way the (4,~) + (~1,4) representation, defined by uo, ul, ...us, u9 .... u14, u, oo, ol .... us, 09, ... u14, o is a natural extension of the standard (3, 3) + ~ , 3) representation. In the following, for simplicity of notation we will also denote u and o by u16 and o16. • 3 Here and in the following we will speak of mass matrix even for the charmed fields, for which the identification with the inverse propagator is more doubtful. We want to stress however that for our discussion we do not need such a strict identification, but only that the inverse propagator is a lower bound for the mass. •4 For convenience of the reader we give the combination orgY's corresponding to our notation: f9,1o,o = .fl h12,0 = f13L14,o = 1/N/~; / 9 , 1 0 , 1 6 = f 1 1 , 1 2 , 1 6 = f 1 3 , 1 4 , 1 6 = -- 1/Vr2; f~/,o = fi,£ 16 = 0 f o ~ i = 1..... 8. di~],0 = ,4c2/3 6/t" fox i = 1..... 8; di,£o = 1/~/66i] for i = 9 ..... 15; di,/, 16 = 0 for i = 1, ..,, 8, d//,l 6 = l/Vr28 ~t"for i = 9 ..... 14; d15,16,16 = - ",/'3~481
17 March 1975
PHYSICS LETTERS
Volume 55B, number 5 from zero only for k = 0,8, The determination of the in the literature, being based of the physical implications. the exact eigenvectors of M.
16. mass matrix and its diagonalization is a non trivial problem and the existent solutions on drastic approximations, do not provide a sufficient basis for a detailed analysis As we will see, the whole situation will look completely different once one may have To this purpo_se we note that eq. (2) for the channels us, uo, u16, plus the trace equa= pnz1 + mf + M$,’ can be written in the form:
Iion 43,s + MO,O+ %6,16 hM=ct:,
(3)
where
A’f,
&SD(l-f)
(1+2f)/2&
-3a
Vf- wh-
(1 -f)216
0
Zfia(l
-(4f
-(2f-1)3/2fi
-f)
-1)ma
\
(4)
i’
(5)
&=rnzfn
&ml
-fm)
(4fm-
l)/&
-2+?a(l
-f)
z-
s+$!+]
(1f 2fm)/24
-3e
2t1-fm)f&
cl (2f-1)
maf4f-1)[E-s+4*]
i a.ndf =fK/f,,,fK ,” ?
= d@
\ s-1-E C
II
2d2a
(aho
The inverse propagators
for the charmed fields are:
(6) 2fm - 1+2e m2 77’
Mr313=“fi,14 = 2f-1+2a f
(7)
MS
(8)
=MS
999
10,lO
(9) exhibiting the interesting feature that scalars and pseudo-scalars get interchanged when e + -e, a + -a. The above eq. (3) can be used toobtain the values of e16 and x16 and the eigenvectors of the mass matrix M. 2fm - 1 - 2A,zA,zA,, e=+1/2 2f -
tl--(1+2A,lAm2)(1+2A,2A,,)(I+2A,JA,1)
(10)
1 - 2A, 1AmZAm3
‘= 2fm - 1 -2A,,A,zA,J The eigenvector
482
’
e ’
corresponding
(11) to the eigenvalue mi (mj, mk being the other eigenvahres and ml
I
Volume 55B, number 5
PHYSICS LETTERS
/ 1+2Am.Amk (a/'v/3)(Ami+ l) V ( A m / - ~ - Ami)
(llx/-6)(2Ami--l)
V(Am _ Ami)i'f4lmk'XAm~
17 March 1975
\ \ ,
(12)
)
- mi) ,/(1 + 2A m .A m .) ( l + 2A m .Am.)! ['Ami(e/a~mi) [V -(Am] t- Almi)(Amk2 Ami;/ A m.(ela
\(l/x/~) where
Ami= 1-2f( mi-m ]. \mi-1 ]
(13)
It is important to stress that no sum rule can obtained by using only the group properties of the Hamiltonian (1) unless one makes some ad hoc assumption, since the number of unknown parameters is the same as the number of equations. 2 e/a 1> m 2. This implies that at The first interesting consequence of eq. (I 1) is that if m 2 t> X 0 then m22 >~ ~ m~r least one of the charmed particles has mass lower than rn 3 , and therefore if one identifies m 1 , m 2, m 3 with r/, X 0, E respectively, one cannot avoid a charmed particle with mass lower than 1.416 MeV/c 2. The only way out of the above difficulty is to conclude that either X 0 or E does not belong to the decuplet and another particle o f sufficiently high mass must take its place. The choice of the new mass is however not arbitrary since the following inequality - ( 1 + 2Am1Am~) (1 + 2Am2Ams)(1 + 2AmsAm1) >~0 must be satisfied in order to keep el6 and ~k16 real. Furthermore in order to have high masses for the charmed particles el6 must be large and this implies
Am2Amx=n~ -
112 •
(11)
The exploitation of eq. (14) leads to interesting conclusions. Firstly, it can be satisfied by taking m 2 = X 0 and f ~ 1.43 or m 2 = E and f ~ 1.04. In both cases even if el6 is very large the ratio m2 e/a remains bounded by m 2, a value reached in the limit when the equality (14) holds strictly. In this limit all the charmed masses become equal to m 3 , and the only mixing is between o0 and v 8 * s. It is also interesting to note that Ares~ - 1 when f ~ 1, for large rn 3 values (/> 2GeWc2). This implies the following limit: [ 2 f - 1 - 2AmlAm2Am3] -* 1+2AmlAm2 ~ 0 *6 and of course X16 ~ 0. Alternatively, the same result follows from f = 1.5, because Am3 -~ - 2 and [ 2 f - 1 2AmlAm2Am3] ~ 2(1 - 2AmlAm2) .~ O. The second important consequence of eq. (14) is that completely clarifies the problem of nonet mass formulae, appeared in the literature. To this purpose one writes eq. (14) in the equivalent form:
(m2-m2)(m2-m2)=
2(2f-2f(2f-1)1) 2 + 1 (m2-m2)[
2m2+2m~ 4fm2+4(f-1)m212f1
"
(14)
Then, f o r / ' = 1, with m 1 = r/, m 2 = E one obtains the Glashow's formula [6] -g(m K (m2n-m2r)(m2-m2r)-2 2-
m 2 ) ( 2 m 2 + 2m2 - 4 m 2 ) ,
(15)
which may also be written in the form given by Schwinger i7] : 2
2
(m2_4m2--m2)(m2_4m3-m~r)=_~(m2K_m2)2 " 3
n"
On the other hand, for f = 1.5 with rn 1 = r/, m 2 = X 0 one obtains the Bjorken-Glashow formula , s In both cases m2 = Xo, m2 = E, the o 8 - o0 mixing angle is small. •~ In fact, in the range f ~ 1 and m 2 > X o we have A m2Am3 ~ 0 which implies ( 1 + 2A m2A m3 ) > 1, and for m a > E, ( 1 + 2AmlAm3) = 0 only for f , ~ 1.
483
Volume 55B, number 5
PHYSICS LETTERS
(m 2 - m2)(m2o _ m r)2 _.$ ( m K _ 2 _ m 22) (, 2 ,m , + 2m2^u -- 3 m 2 - m2") '
17 March 1975 (16)
which may also be written in the following form [6]" (m2
4m2-m2)(m2
°
4m2-m2)
3
3
2 2 =--g(mK-
m2)2 ~r- "
It is important to stress that all the above equations are very well satisfied experimentally. This shows that the condition of the high masses for the charmed particles is not only realizable but it gives rise to a sum rule (14) which strongly suggests f = 1; a choice motived by the following considerations: i) it implies that SU(3) is a good symmetry for the vacuum and therefore the classification o f particles into SU(3) multiplets, according to the original Gell-Mann idea, is justified, ii) the sum rule (15) is verified with exceptional precision by 1 - and 2 + mesons, iii) the positivity of the charmed masses is realized only if ia I > 1/2. In the limit f = 1, a = 1]2 all the scalar masses go to infinity and SU(4) become a good symmetry for the vacuum. If the vacuum is symmetric under SU(4), we can write the following mass formula, suggested by the octet enhancement of SU(3);
Mi j = MoSi j + (e16d16,i, j + eodo,i,j + esds,i,j)A ,
(17)
i = 1, ..., 15, and .7
MO,,O, = s-- M8, 8 - M15,15 .
(17')
Eqs. (17), ( 1 7 ' ) give rise to the same mass matrix for the channels 0, 8, 16, as obtained before (eq. (3)), with f = 1, a = 1/2, and therefore the eigenvectors have the same dependence on the masses as those given in eq. (13). Therefore, by exactly the same argument, the tenth particle o f the 1 - , 2 + multiplets (h016)) has negligible mixing with the ¢ - co and the f ' - f system, respectively. Thus, we can introduce only one "mixing angle"; for example, for the ¢ - co system we have: .4 w + 1 - 2X/~- (K*2 _ p 2 ) tan0 =xf2 2 A w _ 1 4 K . 2 _3602_/92 , By using the above equation, we have; 0 ( r / - E) ~ 7 °, 0 (~o- co) ~ 38 ° 0 ( f ' - f ) ~ 30°; we note that these values coincides with the SU(3) prediction ,8. We observe that eq. (14) is verified by t a k i n g f = 0.97 a n d f = 0.96 for 1 - and 2 + mesons respectively; it follows that only near these values el6 is real and we can obtain the SU(4) condition a ~ 1/2. By identifying the 3.105 MeV e + - e - resonance, with the tenth (~0c) particle of the 1 - decuplet one obtains e ~-271. This identification explains the narrow width of ~oc since i) the decay product of its ~016 component should contain at least a pair of charmed particles and this is forbidden, since the mass o f the charmed particles is too high for e ~- 271, ii) the decay through the 8th and 0th component is strongly depressed. With the above values of e we can determine the masses of the charmed particles of the 0 - , 1+, 2 + multiplets and predict the mass of the tenth particle of the 0 - and 2 + decuplet. O- ; 71c = 3.14 GeV/e 2 ;
Kcl/2 = 2.22 GeV/e 2 ;
Kc0 = 2.27 GeV/e 2
1- ;input
Klc/2 = 2.26 GeV/c2 ;
K 0 = 2.31 GeV/c 2
2+ : fc = 3.83 GeV/c 2
Klc/2 = 2.87 GeV/c 2 ;
K c0 = 2.92GeV/c 2
,7 Since the state 1~00,)is an SU(4) singlet eq. (17) cannot be generalized to i, ] = 0'. :~a By SU(3) prediction we mean the solution of the equation tan 20 = (4K .2 - 3~o2 - p2)/(4K*2 - 3to 2 -O ~) obtained from the Gell-Mann-Okubo formula. 484
Volume 55B, number 5
PHYSICS LETTERS
17 March 1975
I wish to thank prof. R. Gatto for a discussion about a preliminary version of this paper and for suggesting to apply the formalism with this identification of the e + - e - resonance at 3.105 GeV.
References [1] [2] [3] [4] [5]
S.L. Glashow, J. IUiopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. M. Gell-Mann, R.J. Oakes, and B. Renner; Phys. Rev. 175 (1968) 2195. F. Strocchi and R. Vergara-Caffarelli;Phys. Lett. 35B (1971) 595. G. Cicogna, F. Strocchi and R. Vergara-Caffarelli,Phys. Rev. D6 (1972) 301. Z. Maki, T. Maskawa and I. Umemura, Progr. Theor. Phys. 47 (1972) 1682; P. Dittner and S. Eliezer, Phys. Rev. D8 (1973) 1929; D.A. Dicus and V.S. Mathur, Phys. Rev. D9 (1974) 1903; S.C. Prasad, Phys. Rev. D9 (1974) 1917. [6] S.L. Glashow, Hadrons and their interactions, ed. A. Zichichi (Academic Press, New York, 1968). [7] J. Schwinger, Phys. Rev. Lett. 12 (1964) 237.
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