Radiative Decays Involving $f_0(980)$ and $a_0(980)$ and Mixing Between Low and High Mass Scalar Mesons

0.19 MB·English

by  T. Teshima

Radiative Decays Involving f (980) and a (980)
0 0
and
Mixing Between Low and High Mass Scalar Mesons
T. Teshima, I. Kitamura, and N. Morisita
∗
Department of Applied Physics, Chubu University, Kasugai 487-8501, Japan
(Dated: February 2, 2008)
5
0
Abstract
0
2
n Weanalyzetheexperimentaldataforφ→ f0(980)γ, φ→ a0(980)γ, f0(980) → γγ anda0(980) →
a
γγ decaywidthsinaframeworkwheref (980)anda (980)areassumedtobemainlyqqq¯q¯lowmass
J 0 0

0
scalar mesons and mixed with qq¯high mass scalar mesons. Applied the vector meson dominance
1

1 model (VDM), these decay amplitudes are expressed by coupling parameters B describing the
v
3 S(qqq¯q¯scalarmeson)-V(vector meson)-V(vector meson)couplingandB′ describingtheS′(qq¯scalar
7
0 meson)-V-V coupling. Adopting the magnitudes for B and B as 2.8GeV 1 and 12GeV 1,
′ − −
1 ∼ ∼
0 respectively, the mixing angle between a (980) and a (1450) as 9 , and the mixing parameter
0 0 ◦
5 ∼
0 λ causing the mixing between I = 0 qqq¯q¯ state and qq¯ state as 0.24GeV2 , we can interpret
/ 01 ∼
h
p these experimental data, consistently.
-
p
e
PACS numbers: 12.15.Ff, 13.15.+g,14.60.Pq
h
:
v
i
X
r
a
∗
Electronic address: [email protected]
1
I. INTRODUCTION
From the recent experimental and theoretical analyses for the f (600) and κ(900), these
0
scalar mesons are considered as the light scalar nonet together with the f (980) and a (980)
0 0
[1]. From the status that though the masses of f (980) and a (980) are degenerate, the
0 0
f (980) state has the strangeness flavor rich character, many authors suggest this nonet as
0
non qq¯structure, e. g. KK¯ molecule [2], qqq¯q¯state [3]. Onthis standpoint, many literatures
assume that the higher mass scalar mesons, f (1370), a (1450), K (1430), f (1500) and
0 0 0∗ 0
f (1710) would be traditional qq¯nonet and glueball state.
0
In order to confirm the structure of the light scalar mesons, that is whether the light
scalar mesons are constituted of qq¯ or qqq¯q¯, the radiative decays of the φ meson to the
scalar mesons f (980) and a (980), φ f γ and φ a γ, have long been analyzed by
0 0 0 0
→ →
assuming the model, in which the decay φ f (a )γ proceeds through the charged K loop,
0 0
→
φ K+K f (a )γ [4]. Almost literature suggests that f (980) and a (980) mesons
− 0 0 0 0
→ →
¯
contain significant qqq¯q¯content, specifically being (uu¯ bb)ss¯.
±
The two γ decays of the f (980) and a (980) mesons are also analyzed by using the
0 0
various models, the linear sigma model [5], vector meson dominance model (VDM) [6] and
the quark-hadron duality idea [7]. Literature [5] suggested that the f (980) meson is mostly
0
composed of ss¯ component under the picture of the nonet scalar for light scalar mesons
f (600),κ(900),a (980) and f (980). Literature [6] studied the a (980) γγ and f (980)
0 0 0 0 0
→ →
γγ decays and further φ f (980)γ and φ a (980)γ decays comprehensively assuming
0 0
→ →
the VDM and the qqq¯q¯structure for light scalar mesons. It could explain the experimental
results for radiative decays except for the φ f (980)γ decay. Literature [7] considered the
0
→
nonet scalar for light scalar mesons and assumed that these were composed of qq¯ or qqq¯q¯
states. The author analyzed the two γ decays considering the mixing between qqq¯q¯and qq¯,
latter of which is the dominant structure of the higher mass scalar mesons .
We analyzed the mixing between the light scalar nonet, f (600), κ(900), a (980), f (980)
0 0 0
assumed as qqq¯q¯ states dominantly and high mass scalar nonet + glueball, f (1370),
0
a (1450), K (1430), f (1500), f (1710) assumed as L = 1 qq¯ states dominantly + glue-
0 0∗ 0 0
ball state, in our previous work [8]. The estimated mixing is very strong because of the
fact that the high mass scalar meson masses are very high compared with the masses sup-
posed from the L = 1 qq¯ 1++ and 2++ meson masses and relation m2(2++) m2(1++) =
−
2
2(m2(1++) m2(0++)) resulting from the L S force. Literature [9] took the similar conclu-
− ·
sion to ours in the mixing for I = 1 mesons and I = 1/2 mesons. That the mixing between
qqq¯q¯state and qq¯state is strong is recognized from the fact that the transition between qqq¯q¯
and qq¯states is caused by the OZI rule allowed diagram. In next work [10], we pursued this
problem analyzing the decay processes in which light scalar mesons and high mass scalar
meson decays to two pseudoscalar mesons, and get the result that the mixing angle between
I = 1 a (980) and a (1450) is 10 .
0 0 ◦
∼
In the present work, we wil analyze the φ(1020) a (980)γ, φ(1020) f (980)γ,
0 0
→ →
a (980) γγ and f (980) γγ decays assuming that the light scalar mesons have the qqq¯q¯
0 0
→ →
component and qq¯ component, and we will reveal the mixing ratio of these components.
We analyze this problem using the vector dominance model, wherein we can treat these
radiative decay processes comprehensively.
II. MIXING BETWEEN LOW AND HIGH MASS SCALAR MESONS
In this section, we briefly review the mixing among the low mass scalar, high mass scalar
and glueball discussed in our previous work [8, 10]. The qqq¯q¯ scalar SU(3) nonet Sb are
a
represented by the quark triplet q and anti-quark triplet q¯a as [3], [9]
a
Sa ǫacdq q ǫ q¯eq¯f (1)
b ∼ c d bef
and have the following flavor configuration:
s¯d¯us, 1 (s¯d¯ds s¯u¯us), s¯u¯ds a+, a0, a
√2 − ⇐⇒ 0 0 −0
s¯d¯ud, s¯u¯ud, u¯d¯us, u¯d¯ds κ+, κ0, κ0, κ
−
⇐⇒
1 (s¯d¯ds+s¯u¯us) f f (980)
√2 ⇐⇒ NS ∼ 0
¯
u¯dud f f (600)
NN 0
⇐⇒ ∼
We use the notation f for 1 (s¯d¯ds+s¯u¯us) and f for u¯d¯ud in this paper, but we used
NS √2 NN
f and f for 1 (s¯d¯ds+s¯u¯us) and u¯d¯ud, respectively in our previous literature [8, 10]. The
N S √2
high mass scalar mesons S a are the ordinary SU(3) nonet
b′
S a q¯aq .
b′ ∼ b
3
qqq¯q¯ qq¯
Fig.1 OZI rule allowed graph for qqq¯q¯and qq¯states transition
A. Inter-mixing between I = 1,1/2 low and high mass scalar mesons
The mixing between qqq¯q¯and qq¯states, for which we call ”inter-mixing”, may be large,
because the transition between qqq¯q¯and qq¯states is caused by the OZI rule allowed diagram
shown in fig. 1. This transition is represented as
Lint = −λ01ǫabcǫdefNadNb′eδcf = λ01[a+0a′−0 +a−0a′+0 +a00a′00 +κ+K0∗−
+κ K + +κ0K 0 +κ¯+K¯ √2f f f f √2f f ]. (2)
− 0∗ 0∗ 0∗− − N N′ − S N′ − N S′
The inter-mixing parameter λ represents the strength of the inter-mixing and can be
01
considered as rather large.
We first consider the I = 1 a (980) and a (1450) mixing. Representing the before mixing
0 0
qqq¯q¯state by a (980) and qq¯by a (1450) and mixing angle as θ , physical after mixing state
0 0 a
a (980) and a (1450) are written as follows;
0 0
a (980) = cosθ a (980) sinθ a (1450)
0 a 0 a 0
− (3)
a (1450) = sinθ a (980)+cosθ a (1450)
0 a 0 a 0
and the mixing matrix is represented as
m2 λa
 a0(980) 01 , (4)
λa m2
 01 a0(1450) 
where m and m are the before mixing masses for a (980) and a (1450) states.
a0(980) a0(1450) 0 0
For the values of m and m , in the first our work [8], we adopted the values
a0(980) a0(1450)
m = 1271 31MeV, m = 1236 20MeV,
a0(980) ± a0(1450) ±
estimated from the relation m2(2++) m2(1++) = 2(m2(1++) m2(0++)) resulting from the
− −
L S force. Diagonalising the mass matrix in Eq. (4) and taking the eigenvalues of masses
·
m = 984.8 1.4MeV, m = 1474 19MeV, (5)
a0(980) ± a0(1450) ±
4
we can get the result
λa = 0.600 0.028GeV2, θ = 47.1 3.5 . (6)
01 ± a ± ◦
Similarly, we estimated the strength λK and mixing angle θ for I = 1/2 κ(900) and
01 K
K (1430) mixing case. Using the mass values before mixing and after mixing,
0∗
m = 1047 62MeV, m = 1307 11MeV,
κ(900) ± K0∗(1430) ± (7)
mκ(900) = 900±70MeV, mK0∗(1430) = 1412±6MeV,
we get the results
λK = 0.507 84GeV2, θ = 29.5 15.5 . (8)
01 ± K ± ◦
In our next work [10], we estimated the mixing angle θ and θ analyzing the a (980),
a K 0
a (1450) and K (1430) decay processes to two pseudoscalar meson, and get the results,
0 0∗
θ = θ = (9 4) . (9)
a K ◦
±
The value of λ and mass values before mixing of a (980), a (1450) and κ(900), K (1430)
01 0 0 0∗
for this mixing angle are estimated as follows;
λa = λK = 0.19+0.07GeV2,
01 01 0.09
−
m = 1.00+0.02GeV, m = 1.46 0.01GeV,
a0(980) 0.01 a0(1450) ±
−
m = 0.92 0.01GeV, m = 1.40 0.01GeV. (10)
K0(900) ± K0∗(1430) ±
B. Inter-mixing between I = 0 low and high mass scalar mesons
Among the I = 0,L = 1 qq¯ scalar mesons, there are the intra-mixing weaker than the
inter-mixing, caused from the transition between themselves represented by the OZI rule
suppression graph shown in Fig. 2, and furthermore the mixing between the qq¯ scalar
meson and the glueball caused from the transition represented by the graph shown in Fig.
3. Thus, the mass matrix for these I = 0,L = 1 qq¯scalar mesons and glueball is represented
as
m2 +2λ √2λ √2λ
 N′ 1 1 G 
√2λ 2m2 +λ λ , (11)
 1 S′ 1 G 
 
 √2λ λ λ 
 G G GG 
5
g
qq¯ qq¯
g
Fig. 2. OZI rule suppression graph for qq¯ qq¯transition.
−
qq¯ gg gg gg
(a) (b)
Fig. 3. Transition graph between (a) qq¯and gg, and (b) gg and gg.
where m2 = m2 , m2 = 2m2 m2 , and λ is the transition strength among the I = 0, qq¯
N′ a′0 S′ K0′ − a′0 1
mesons, λ isthetransitionstrengthbetween qq¯andglueball gg andλ isthe pureglueball
G GG
mass square.
For the light I = 0 qqq¯q¯ scalar mesons, there are the intra-mixing caused from the
transition between themselves represented by the OZI rule suppression graph shown in Fig.
4, and the mass matrix for these I = 0 qqq¯q¯scalar meson is represented as
m2 +2λ √2λ
 NN 0 0 
, (12)
√2λ 2m2 +λ
 0 NS 0 
where m2 = 2m2 m2 , m2 = m2 , and λ represents the transition strength between
NN K0 − a0 NS a0 0
I = 0 qqq¯q¯mesons.
The inter- and intra-mixing among I = 0 low mass and high mass scalar mesons and
qqq¯q¯ qqq¯q¯
Fig. 4. OZI suppression graph for qqq¯q¯ qqq¯q¯transition.
−
6
glueball is expressed by the overall mixing mass matrix as
m2 +λ √2λ √2λ 0 0
 NN 0 0 01 
√2λ m2 +2λ λ √2λ 0
 0 NS 0 01 01 
 
 √2λ λ m2 +2λ √2λ √2λ . (13)
 01 01 N′ 1 1 G 
 
 0 √2λ √2λ m2 +λ λ 
 01 1 S′ 1 G 
 
 0 0 √2λ λ λ 
G G GG
We use the values of λ = 0.19GeV2 tentatively, corresponding to the mixing angle θ =
01 a
θ = 9 estimated in analyses of the decay processes as shown in Eq. (10), and then
K ◦
diagonalize this 5 5 mass matrix. In this case, we input the values for m etc. as follows;
NN
×
mNN = 0.826GeV, mNS = 1.00GeV, mN′ = 1.46GeV, mS′ = 1.34GeV, (14)
predicted from the relations, m2 = 2m2 m2 , m2 = m2 and m2 = 2m2 m2 ,
NN K0 − a0 NS a0 S′ K0′ − a′0
m2 = m2 and the estimated values Eq. (10). Varying the parameters λ , λ , λ and
N′ a′0 0 1 G
λ , we get the best fit eigenvalues for the mass of f (600), f (980), f (1370), f (1500) and
GG 0 0 0 0
f (1710),
0
m = 0.77(0.80 0.40)GeV, m = 0.93(0.980 0.010)GeV,
f0(600) ± f0(980) ±
m = 1.37(1.350 0.150)GeV, m = 1.51(1.507 0.015)GeV, (15)
f0(1370) ± f0(1500) ±
m = 1.71(1.714 0.005)GeV,
f0(1710) ±
where the values in parenthesis are quoted from [11]. Best-fit values are obtained for the
values of λ etc.,
0
λ = 0.03GeV2,λ = 0.04GeV2,λ = 0.1GeV2,λ = 1.72GeV2, (16)
0 1 G GG
−
7
and at this time, mixing parameters are calculated as
f (600) f
 0   NN 
f (980) f
 0   NS 
   
f0(1370) = [Rf0(M)I] fN′ , (17)
   
f0(1500)  fS′ 
   
f (1710)  f 
0 G
0.949 0.250 0.185 0.047 0.013
 − − 
0.270 0.925 0.067 0.257 0.017
 − − 
 
[R ] = 0.054 0.233 0.210 0.946 0.064 .
f0(M)I  − − 
 
0.150 0.161 0.932 0.160 0.239 
 − 
 
0.025 0.035 0.220 0.107 0.969 
The mixing parameters [R ] for f (600) and f (980) are similar to those obtained in
f0(M)I 0 0
our previous works [8], but the ones for f (1370), f (1500) and f (1710) are rather different
0 0 0
from them. This comes about from the smaller values of λ in the present analysis. Because
01
only the mixing parameters [R ] of f (980) and mixing angle θ between a (980) and
f0(M)I 0 a 0
f (1450) are necessary in this work, the deviation of the mixing parameters [R ] for
0 f0(M)I
f (1370), f (1500) and f (1710) from previous result does not get into trouble.
0 0 0
III. RADIATIVE DECAYS INVOLVING F (980) AND a (980)
0 0
A. Radiative decays involving f (980) and a (980) in VDM
0 0
In this work, we analyze the radiative decays involving f (980) and a (980); φ(1020)
0 0
→
f (980)γ, φ(1020) a (980)γ and f (980) γγ, f (980) γγ. We assume the vector
0 0 0 0
→ → →
meson dominance model (VDM) in this analysis, for the radiative processes involving pseu-
doscalar mesons, P γγ, V Pγ and P Vγ have been interpreted well in VDM [12].
→ → →
In VDM, V Sγ and S γγ processes are described by the diagram shown in Fig.5.
→ →
8
γ
γ
V
V V S
′
V
′
S
γ
(a) V Sγ decay (b) S γγ decay
→ →
Fig. 5. Diagram for V Sγ and S γγ decays in VDM
→ →
WeusethefollowinginteractionsforSVV,S VV andGVV couplingwithcouplingconstants
′
B, B and B , respectively,
′ ′′
L = Bεabcε SdVµνeV f +B S b Vµνc, V a +B G Vµνb, V a , (18)
I def a b µνc ′ a′ { b µνc} ′′ { a µνb}
where Vµν is the vector field strength ∂µVν ∂νVµ. These interactions are represented
−
graphically by the diagrams shown in fig. 6. Although interactions as Tr(SV )Tr(Vµν) and
µν
Tr(S V )Tr(Vµν) other than those represented by Eq. (18) may exist [9], these interac-
′ µν
tionsviolatetheOZI ruleandareconsidered tobesmallerthantheinteractionsinEq.(18).
V V
V
S G
S
′
V V V
Fig.6. SVV, S VV and GVV coupling
′
We define the coupling constants gSVV′ in the following expression,
1 1
LI = ga0K∗K∗√2Kµ∗ντ·a0K∗µν +ga′0K∗K∗√2Kµ∗ντ·a′0K∗µν +ga0ρωa0·ρµνωµν
+ ga′0ρωa′0·ρµνωµν +ga0ρφa0·ρµνφµν +ga′0ρφa′0·ρµνφµν
1 1
+ gK0∗K∗ρ(√2Kµ∗ντ·ρµνK0∗ +H.C.)+gκK∗ρ(√2Kµ∗ντ·ρµνκ+H.C.)
+ gκK∗ω(κKµ∗νωµν +H.C.)+gK0∗K∗ω(K0∗Kµ∗νωµν +H.C.)
+ gκK∗φ(κKµ∗νφµν +H.C.)+gK0∗K∗φ(K0∗Kµ∗νφµν +H.C.),
+ g f (M)ρ ρµν +g f (M)K K µν +g f (M)ω ωµν
f0(M)ρρ 0 µν· f0(M)K∗K∗ 0 µ∗ν ∗ f0(M)ωω 0 µν
+ g f (M)ω φµν +g f (M)φ φµν, (19)
f0(M)ωφ 0 µν f0(M)φφ 0 µν
9
where fields a represents the low mass I = 1 scalar mesons and a the high mass I = 1
0 ′0
scalar mesons. Then the coupling constants for I = 0 S, V mesons are, by using Eq. (18),
3
expressed as
g = 2Bcosθ ,
a0(980)ρφ − a
g = 2√2B sinθ ,
a0(980)ρω − ′ a
g = 2BR ,
f0(980)φω f0(980)NS
gf0(980)φφ = 2B′Rf0(980)S′ +2B′′Rf0(980)G,
gf0(980)ρρ = −BRf0(980)NN +√2B′Rf0(980)N′ +2B′′Rf0(980)G,
gf0(980)ωω = BRf0(980)NN +√2B′Rf0(980)N′ +2B′′Rf0(980)G. (20)
V γ coupling is defined as
−
em2
< 0 jem(0) V(p,ε) >= V ε (p), (21)
| µ | γ µ
V
where m and ε (p) are the mass and polarization vector of the vector meson, respectively.
V µ
4π α2 1
ThenΓ(V ll) = m ,wecangetthevalue = 0.201 0.002fromtheexperimental
→ 3 γ2 V γ ±
V ρ
data Γ(ρ ee) = 7.02 0.11keV [11].
→ ±
The decay amplitude and decay width for V Sγ are expressed as
→
2egSVV′
M(V Sγ) = (p kε ε p ε k ε ),
→ XV′ γV′ · V · γ − · γ · V
Γ(V Sγ) = 4α( gSVV′)2 k 3, (22)
→ 3 XV′ γV′ | γ|
where ε and ε are the polarization vector of the vector meson and the photon, respectively
V γ
and k is the photon momentum in the V rest frame. Only the ρ meson contributes to the
γ
intermediate V vector meson for the φ a (980)γ decay, and ω and φ mesons contribute
′ 0
→
to φ f (980)γ decay. Then the decay widths for the φ a (980)γ and φ f (980)γ
0 0 0
→ → →
decay are written as
4α
Γ(φ a (980)γ) = (g )2 k 3,
→ 0 3γ2 a0ρφ | γ|
ρ
4α g √2g
Γ(φ f (980)γ) = ( f0φω f0φφ)2 k 3, (23)
0 γ
→ 3γ2 3 − 3 | |
ρ
where we assumed the SU(3) symmetry for the V γ coupling,
−
m2 m2 m2 1 1 1 1 1 1
ρ : ω : φ = : : = : : .
γ γ γ ∼ γ γ γ √2 3√2 −3
ρ ω φ ρ ω φ
10