Analysis of the vertexes $Ξ_Q^*Ξ'_Q V$, $Σ_Q^*Σ_Q V$ and radiative decays $Ξ_Q^*\to Ξ'_Q γ$, $Σ_Q^*\to Σ_Q γ$

0.25 MB·

by  Zhi-Gang Wang

Analysis of the vertexes Ξ Ξ V, Σ Σ V and radiative decays
∗Q ′Q ∗Q Q
Ξ Ξ γ, Σ Σ γ
∗Q → ′Q ∗Q → Q
Zhi-Gang Wang 1
Department of Physics, North China Electric Power University, Baoding 071003, P. R.
China
0
Abstract
1
0 ∗ ′ ∗
In this article, we study the vertexes Ξ Ξ V and Σ Σ V with the light-cone
2 Q Q Q Q
QCDsumrules,thenassumethevectormesondominanceoftheintermediateφ(1020),
n ∗ ′ ∗
ρ(770) and ω(782), and calculate the radiative decays Ξ Ξ γ and Σ Σ γ.
a Q → Q Q → Q
J
PACS numbers: 11.55.Hx, 12.40.Vv, 13.30.Ce, 14.20.Lq, 14.20.Mr
1
3 Key Words: Heavy baryons; Light-cone QCD sum rules

]
h 1 Introduction
p
-
p
The charm and bottom baryons which contain a heavy quark and two light quarks are
e
h particularlyinterestingforstudyingdynamicsofthelightquarksinthepresenceofaheavy
[ quark. They serve as an excellent ground for testing predictions of the quark models and

3 heavy quark symmetry [1, 2]. The three light quarks form an SU(3) flavor triplet 3, two
v light quarks can form diquarks of a symmetric sextet and an antisymmetric antitriplet,
2 i.e. 3 3= ¯3+6. For the S-wave charm baryons, the 1+ antitriplet states (Λ+, Ξ+,Ξ0),
1 × 2 c c c
1 and the 1+ and 3+ sextet states (Ω ,Σ ,Ξ ) and (Ω ,Σ ,Ξ ) have been well established;
2 2 2 c c ′c ∗c ∗c ∗c
whilethecorrespondingbottombaryonsarefarfromcomplete, onlytheΛ , Σ , Σ ,Ξ , Ω
. b b ∗b b b
0
have been observed [3]. Furthermore, several new excited charm baryon states have been
1
9 observed by the BaBar, Belle and CLEO Collaborations, such as Λc(2765)+, Λ+c (2880),
0 Λ+(2940), Σ+(2800), Ξ+(2980), Ξ+(3077), Ξ0(2980) , Ξ0(3077) [4, 5, 6].
c c c c c c
:
v In Ref.[7], we assume the charm mesons Ds0(2317) and Ds1(2460) with the spin-parity
i 0+ and 1+ respectively are the conventional cs¯ states, and calculate the strong coupling
X
constants D φD and D φD with the light-cone QCD sum rules, then take the
r h s∗ | s0i h s | s1i
a vector meson dominance of the intermediate φ(1020), study the radiative decays D
s0
→
D γ and D D γ. In Refs.[8, 9], we calculate the masses and the pole residues
ofs∗the 1+ hse1av→y barsyons Ω and the 3+ heavy baryons Ω with the QCD sum rules.
2 Q 2 ∗Q
Moreover, we study the vertexes Ω Ω φ with the light-cone QCD sum rules, then assume
∗Q Q
thevectormesondominanceoftheintermediateφ(1020),andcalculatetheradiativedecays
Ω Ω γ [10].
∗Q → Q
Inthisarticle,weextendourpreviousworkstostudythevertexesΞ Ξ V andΣ Σ V
∗Q ′Q ∗Q Q
with the light-cone QCD sum rules 2, then assume the vector meson dominance of the
intermediate φ(1020), ρ(770) and ω(782), and calculate the radiative decays Ξ Ξ γ
∗Q → ′Q
and Σ Σ γ to complete our works on radiative decays among the 1+ and 3+ sextet
∗Q → Q 2 2
states (Ω ,Σ ,Ξ ) and (Ω ,Σ ,Ξ ). In Ref.[11], Aliev et al study the radiative decays
Q Q ′Q ∗Q ∗Q ∗Q
1E-mail,[email protected].
2The results of the strong coupling constants among the nonet vector mesons, the octet baryons and
thedecuplet baryons will bepresented elsewhere.
1
Σ Σ γ, Ξ Ξ γ and Σ Λ γ with the light-cone QCD sum rules, where the
∗Q → Q ∗Q → Q ∗Q → Q
light-cone distribution amplitudes of the photon are used.
The light-cone QCD sum rules carry out the operator product expansion near the
light-cone x2 0 instead of the short distance x 0, while the nonperturbative hadronic
≈ ≈
matrix elements are parameterized by the light-cone distribution amplitudes instead of
the vacuum condensates [12, 13, 14]. The nonperturbative parameters in the light-cone
distributionamplitudesarecalculatedwiththeconventionalQCDsumrulesandthevalues
areuniversal. Basedonthequark-hadronduality,wecanobtaincopiousinformationabout
the hadronic parameters at the phenomenological side [14, 15, 16]. The ρNN, ρΣΣ, ρΞΞ
and other strong coupling constants of the nonet vector mesons with the octet baryons
have been calculated using the light-cone QCD sum rules [17, 18, 19]. In Refs.[20, 21],
Aliev et al study the strong coupling constants of the pseudoscalar octet mesons with the
octet (decuplet) baryons comprehensively. In Refs.[22, 23], we study the strong decays
∆++ pπ, Σ Σπ and Σ Λπ using the light-cone QCD sum rules. Moreover,
∗ ∗
→ → →
the coupling constants of the vector mesons ρ and ω with the baryons are studied with
the external field QCD sum rules [24]. Recently, the strong coupling constants among the
light vector mesons and the heavy baryons are calculated with the light-cone QCD sum
rule in the leading order of heavy quark effective theory [25].
The article is arranged as follows: we derive the strong coupling constants g , g and
1 2
g of the vertexes B B V with the light-cone QCD sum rules in Sect.2; in Sect.3, we
3 Q∗ Q
present the numerical results and discussions; and Sect.4 is reserved for our conclusions.
2 The vertexes B B V with light-cone QCD sum rules
Q∗ Q
We parameterize the vertexes B B φ, B B ρ and B B ω with three tensor structures
Q∗ Q Q∗ Q Q∗ Q
dueto Lorentz invariance and introducethree strongcoupling constants g , g and g [26],
1 2 3
B (p+q)B (p)φ(q) = U(p+q)[g (q ǫ ǫ q)γ +g (P ǫq P qǫ )γ
h Q | Q∗ i 1 µ 6 − µ 6 5 2 · µ− · µ 5
+g (q ǫq q2ǫ )γ Uµ(p), (1)
3 µ µ 5
· −
1
B (p+q)B (p)ρ /ω(q) = U(p+q)[g (q ǫ (cid:3) ǫ q)γ +g (P ǫq P qǫ )γ
h Q | Q∗ 0 i √2 1 µ 6 − µ 6 5 2 · µ− · µ 5
+g (q ǫq q2ǫ )γ Uµ(p), (2)
3 µ µ 5
· −
(cid:3)
where the U(p) and U (p) are the Dirac spinors of the heavy baryon states B (Ξ , Σ )
µ Q ′Q Q
and B (Ξ , Σ ) respectively, the ǫ is the polarization vector of the mesons φ(1020),
Q∗ ∗Q ∗Q µ
ρ(770) and ω(782), and P = 2p+q.
2
In the following, we write down the two-point correlation functions Π (p,q),
µ
Πφ/ρ0(p,q) = i d4xe ipx 0T J(0)J¯ (x) φ/ρ (q) , (3)
µ − · h | µ | 0 i
Z
JΞ(x) = ǫijkqT(x)Cγ s (x)γ(cid:8) γµQ (x),(cid:9)
i µ j 5 k
JΣ(x) = ǫijkqT(x)Cγ q (x)γ γµQ (x),
i µ j′ 5 k
JΞ(x) = ǫijkqT(x)Cγ s (x)Q (x),
µ i µ j k
JΣ(x) = ǫijkqT(x)Cγ q (x)Q (x), (4)
µ i µ j′ k
2
where Q = c,b and q,q = u,d, the i,j,k are color indexes, the Ioffe type heavy baryon
′
currents J(x) (JΞ(x), JΣ(x)) and J (x) (JΞ(x), JΣ(x)) interpolate the 1+ baryon states
µ µ µ 2
Ξ , Σ and the 3+ baryon states Ξ , Σ , respectively, the external vector states φ(1020)
′Q Q 2 ∗Q ∗Q
and ρ(770) have the four momentum q with q2 = M2 . The quark constituents of the
µ φ/ρ
vector mesons ρ and ω are 1 u¯u d¯d and 1 u¯u + d¯d respectively, the isospin
0 √2 | i−| i √2 | i | i
triplet meson ρ and isospin singlet meson ω have approximately degenerate masses. We
0 (cid:0) (cid:1) (cid:0) (cid:1)
assume that the vector mesons ρ and ω have similar light-cone distribution amplitudes,
0
and obtain the corresponding strong coupling constants by symmetry considerations, as
the ω-meson light-cone distribution amplitudes have not been explored yet, see Appendix
A for detailed discussions.
Basing on the quark-hadron duality [15, 16], we can insert a complete set of interme-
diate hadronic states with the same quantum numbers as the current operators J(x) and
J (x) into the correlation functions Π (p,q) to obtain the hadronic representation. After
µ µ
isolating the ground state contributions from the pole terms of the heavy baryons Ξ , Σ
′Q Q
and Ξ , Σ , we get the following results,
∗Q ∗Q
0J(0)B (q+p) B (q+p)B (p)φ/ρ(q) B (p)J¯ (0)0
Πφ/ρ0(p,q) = h | | Q ih Q | Q∗ ih Q∗ | µ | i +
µ M2 (q+p)2 M2 p2 ···
BQ − BQ∗ −
= M2 (qλ+BpQh)λ2BQ∗M2 p2 igh1 MBQ +Mi BQ∗ 6ǫ6pγ5qµ
BQ − BQ∗ − n h i
h ih i
−g1 MBQ +MBQ∗ 6q6pγ5ǫµ+g26q6pγ5p·ǫqµ−g26q6pγ5q·pǫµ
g h i 1
2 qpγ q2ǫ g qpγ q2ǫ + 1/ + , (5)
5 µ 3 5 µ
− 2 6 6 − 6 6 ··· √2 ···
(cid:20) (cid:21)
o
where the following definitions have been used,
0J(0)B (p) = λ U(p,s),
h | | Q i BQ
h0|Jµ(0)|BQ∗(p)i = λBQ∗Uµ(p,s),
U(p,s)U(p,s) = p+M ,
6 BQ
s
X
γ γ 2p p p γ p γ
µ ν µ ν µ ν ν µ
Xs Uµ(p,s)Uν(p,s) = −(6p+MBQ∗) gµν − 3 − 3MB2Q∗ + 3M−BQ∗ ! , (6)
the factors 1 and 1 correspondto the correlation functions Πφ(p,q) and Πρ0(p,q) respec-
√2 µ µ
tively. The current J (x) couples not only to the spin-parity JP = 3+ states, but also
µ 2
to the spin-parity JP = 21− states. For a generic 12− resonance BQ∗, h0|Jµ(0)|BQ∗(p)i =
λ (γ 4pµ )U (p,s), whereλ is thepoleresidue, M is themass, andthespinorU (p,s)
sa∗tisµfie−s tMhe∗usu∗al Dirac equati∗on (p M )U (p) = 0.∗In this articlee, we choose tehe t∗ensor
∗
6 − ∗
structures ǫ pγ q , q pγ p ǫq and q pγ ǫ , the baryon state B has no contamination,
6 6 5 µ 6 6 5 · µ 6 6 5 µ Q∗
for example, we can study the contribution of the 21− baryon state BQ∗ to the correlation
e
e
3
φ
functions Π (p,q),
µ
0J(0)B (q+p) B (q+p)B (p)φ(q) B (p)J¯ (0)0
Πφ(p,q) = h | | Q ih Q | Q∗ ih Q∗ | µ | i +
µ M2 (q+p)2 [M2 p2] ···
BQ − e ∗ − e
= λBQλ∗M6qB2+Q6p−+(hqM+BQp)2 (cid:20)gV 6ǫ+iigTMǫBαQσα+βqMβ ∗(cid:21)γ5M6p+∗2−Mp∗2 (cid:20)γµ−4Mpµ∗(cid:21)
+
···
= f (γ,p,q,ǫ)γ +f (γ,p,q,ǫ)p + , (7)
1 µ 2 µ
···
where we introduce the strong coupling constants g and g to parameterize the vertexes
V T
B (q+p)B (p)φ(q) , the notations f and f are functions of γ , γ , ǫ , p and q , here
h Q | Q∗ i 1 2 α 5 α α α
we order the Dirac matrixes as ǫ,q,p, γ .
5
6 6 6
In the foellowing, we briefly outline the operator product expansion for the correlation
functionsΠ (p,q)inperturbativeQCD.Thecalculations areperformedatthelargespace-
µ
like momentum regions (q+p)2 0 and p2 0, which correspond to the small light-cone
≪ ≪
distance x2 0 required by the validity of the operator product expansion approach. We
≈
write down the ”full” propagator of a massive quark in the presence of the quark and
gluon condensates firstly [12, 16],
iδ x δ m δ iδ δ x2
ij ij s ij ij ij
S (x) = 6 s¯s + m s¯s x s¯g σGs
ij 2π2x4 − 4π2x2 − 12h i 48 sh i6 − 192 h s i
iδ x2 i 1
+ ij m s¯g σGs x dvGij (vx)[(1 v)xσµν +vσµν x]+ ,
1152 sh s i6 − 16π2x2 µν − 6 6 ···
Z0
αβ
Sij(x) = i d4ke ikx δij gsGij σαβ(6k+mQ)+(6k+mQ)σαβ
Q (2π)4 − · k m − 4 (k2 m2)2
Z (6 − Q − Q
π2 α GG k2+m k
s Q
+ δ m 6 + , (8)
3 h π i ij Q(k2 m2)4 ···
− Q )
where s¯g σGs = s¯g σ Gαβs and αsGG = αsGαβGαβ (the corresponding full propa-
h s i h s αβ i h π i h π i
gators U (x) and D (x) of the quarks u and d respectively can be obtained with a simple
ij ij
replacement), then contract the quark fields in the correlation functions Π (p,q) with
µ
Wick theorem, and obtain the following results:
ΠΞ∗QΞQφ(p,q) = iǫijkǫi′j′k′ d4xe ipx
µ − ·
Z
γ5γαSQkk′(−x)Tr γαh0|sj(0)s¯j′(x)|φ(q)iγµCU/DiTi′(−x)C , (9)
(cid:2) (cid:3)
ΠΞ∗QΞQρ(p,q) = iǫijkǫi′j′k′ d4xe ipx
µ − ·
Z
γ5γαSQkk′(−x)Tr γαSjj′(−x)γµCh0|qi(0)q¯i′(x)|ρ(q)iTC , (10)
(cid:2) (cid:3)
4
ΠΣ∗QΣQρ(p,q) = Aiǫijkǫi′j′k′ d4xe ipx
µ − ·
Z
̟γ5γαSQkk′(−x)Tr γαh0|uj(0)u¯j′(x)|ρ(q)iγµCUiTi′(−x)C
n+γ5γαSQkk′(−x)Tr γ(cid:2)αUjj′(−x)γµCh0|ui(0)u¯i′(x)|ρ(q)iTC(cid:3) ,(11)
o
(cid:2) (cid:3)
here we take isospin limit for the quarks u and d, the symmetry factor A= 1, ̟ = 1 for
−
thechannelsΣ (Qud)Σ(Qud)ρ, andA= 2and̟ = 1forthechannelsΣ (Quu)Σ(Quu)ρ
∗ ∗
±
and Σ (Qdd)Σ(Qdd)ρ respectively.
∗
Performing the Fierz re-ordering to extract the contributions from the two-particle
and three-particle vector meson light-cone distribution amplitudes respectively, then sub-
stituting the full q and Q quark propagators into the correlation functions in Eqs.(9-11)
and completing the integral in the coordinate space, finally integrating over the variable
k, we can obtain the correlation functions Π (p,q) at the level of quark-gluon degree of
µ
freedom. In calculation, the two-particle and three-particle vector meson light-cone dis-
tribution amplitudes have been used [27, 28, 29, 30]. The parameters in the light-cone
distribution amplitudes are scale dependent and are estimated with the QCD sum rules
[29, 30]. In this article, the energy scale µ is chosen to be µ = 1GeV.
TakingdoubleBoreltransformwithrespecttothevariablesQ2 = p2 andQ2 = (p+
1 − 2 −
q)2 respectively, then subtracting the contributions from the high resonances and contin-
uum states by introducing the threshold parameter s0 (i.e. M2n → Γ[1n] 0s0dssn−1e−Ms2),
finally we can obtain 30 sum rules for the strong coupling constants g , g and G3 =
1 2
R
M +M g M2 g2 +g respectively, the explicit expressions are presented
− BQ ∗BQ 1− φ/ρ0 2 3
in(cid:16)the appendix(cid:17)A3.
(cid:0) (cid:1)
3 Numerical result and discussion
The masses of the established hadrons are taken from the Particle Data Group M =
φ
1.019455GeV, Mρ = 0.77549GeV, Mω = 0.78265GeV, MΞ∗c = 2.6459GeV, MΣ∗c++ =
2.5184GeV, MΣ∗c+ = 2.5175GeV, MΣ∗c0 = 2.5180GeV, MΣ∗b+ = 5.8290GeV, MΣ∗b− =
5.8364GeV, MΞ′c+ = 2.5756GeV, MΞ′c0 = 2.5779GeV, MΣ+c+ = 2.45402GeV, MΣ+c =
2.4529GeV, MΣ0c = 2.45376GeV, MΣ+b = 5.8078GeV, and MΣ−b = 5.8152GeV [3]. In
calculation, we take the average values of the masses in each isospin multiplet and neglect
the small isospin splitting in the heavy baryon multiplet.
The parameters which determine the vector meson light-cone distribution amplitudes
are fφ = (0.215 ± 0.005)GeV, fφ⊥ = (0.186 ± 0.009)GeV, ak1 = 0.0, a⊥1 = 0.0, ak2 =
0.18±0.08, a⊥2 = 0.14±0.07, ζ3k = 0.024±0.008, λk3 = 0.0, ω3k = −0.045±0.015, κk3 = 0.0,
ω3k = 0.09 ± 0.03, λk3 = 0.0, κ⊥3 = 0.0, ω3⊥ = 0.20 ± 0.08, λ⊥3 = 0.0, ς4k = 0.00 ± 0.02,
ω4k = −0.02±0.01, ς4⊥ = −0.01 ±0.03, ς4⊥ = −0e.03 ±0.0e4, κk4 = 0.0, κ⊥4 = 0.0 for the
φ-meson; and fρ = (0.216±0.003)GeV, fρ⊥ = (0.165±0.009)GeV, ak1 = 0.0, a⊥1 = 0.0,
e e
ak2 = 0.15 ± 0.07, a⊥2 = 0.14 ± 0.06, ζ3k = 0.030 ± 0.010, λk3 = 0.0, ω3k = −0.09 ± 0.03,
3HerewepresentsometechnicaldetailsincalculatingthecorrelationefunctionsΠµΞe∗QΞQφ(p,q)toillustrate
5
κk3 = 0.0,ω3k = 0.15±0.05, λk3 = 0.0,κ⊥3 = 0.0,ω3⊥ = 0.55±0.25, λ⊥3 = 0.0,ς4k = 0.07±0.03,
ω4k = −0.03±0.01, ς4⊥ = −0.03±0.05, ς4⊥ = −0.08±0.05, κk4 = 0.0, and κ⊥4 = 0.0 for the
ρ-meson at the energy scale µ = 1GeV [29, 30].
e The QCD input parameters are takeen to be the standard values ms = (140 10)MeV,
±
m = (1.35 0.10)GeV, m = (4.7 0.1)GeV, q¯q = (0.24 0.01GeV)3, s¯s =
c b
± ± h i − ± h i
(0.8 0.2) q¯q , s¯g σGs = m2 s¯s , q¯g σGq = m2 q¯q , m2 = (0.8 0.2)GeV2, and
± h i h s i 0h i h s i 0h i 0 ±
αsGG = (0.33GeV)4 at the energy scale µ = 1GeV [15, 16, 31].
h π i
ThebottombaryonstatesΞ andΞ havenotbeenobservedyet, westudytheirmasses
∗b ′b
with the conventional QCD sum rules. The masses MBQ and MBQ∗ and pole residues λBQ
and λB∗ are determined by the following correlation functions,
Q
Π (p) = i d4xeipx 0T J (x)J¯ (0) 0 ,
µν · µ ν
h | | i
Z
(cid:8) (cid:9)
Π(p) = i d4xeipx 0T J(x)J¯(0) 0 , (13)
·
h | | i
Z
JΞ(x) = ǫijkuT(x)Cγ s (x(cid:8))γ γµQ (x(cid:9)),
i µ j 5 k
JΣ(x) = ǫijkuT(x)Cγ d (x)γ γµQ (x),
i µ j 5 k
JΞ(x) = ǫijkuT(x)Cγ s (x)Q (x),
µ i µ j k
JΣ(x) = ǫijkuT(x)Cγ d (x)Q (x). (14)
µ i µ j k
In Refs.[32, 33], the masses of the heavy baryon states containing one heavy quark are
studied using the QCD sum rules, the pole residues are not calculated. In this article,
we take the simple Ioffe type interpolating currents, which are constructed by considering
theprocedure,
Πµ(p,q) = iǫijk1ǫ2ij′k′ d4xe−ip·xh0|s¯(x)γλγ5s(0)|φ(q)iγ5γαSQkk′(−x)Tr γαγλγ5γµCUjTj′(−x)C +···
Z h i
= 33f2φπM6φγ5γα[ǫµqα−ǫαqµ] 1dug⊥(a)(1−u) dDxdDkei(k−p−uq)·xk2−6km2 x12 +···
Z0 Z Q
e
f M 1 1 Γ(ε)
= [6q6pγ5ǫµ−6ǫ6pγ5qµ] 1φ6π2φ dug⊥(a)(1−u) dtt (p+uq)2−m2 ε |ε→0 +···
Z0 Z0 Q
e
−→ [6q6pγ5ǫµ−6ǫ6pγ5qµ]f1φ6Mπ2φ 1dug⊥(a)(1−u) 1dtt(cid:2) e (cid:3)
Z0 Z0
e
M4 m2 +u(1−u)M2
exp − Q φ δ(u−u )+··· (takedouble Borel transform)
M2M2 M2 0
1 2 " #
e
−→ [6q6pγ5ǫµ−6ǫ6pγ5qµ]MM4E2M1(x2)fφMφg1⊥(6aπ)(21−u0) 1dtt
1 2 Z0
e
m2 +u (1−u )M2
exp − Q 0 0 φ +··· (subtract continuum contributions)
M2
" #
e
= λMΞ′Q2λMΞ2∗Q exp−MMΞ22∗Q − MMΞ22′Q g1 MΞ′Q +MΞ∗Q 6ǫ6pγ5qµ
1 2 1 2
n h i
 
−g1 MΞ′Q +MΞ∗Q 6q6pγ5ǫµ+g26q6pγ5p·ǫqµ−g26q6pγ5q·pǫµ+··· +··· . (12)
h i o
Fortechnical details about theBorel transform, one can consult theexcellent review [14].
6
the diquark theory and the heavy quark symmetry [34, 35]. We insert a complete set of
intermediate baryon states with thesame quantum numbers as the currentoperators J(x)
andJ intothecorrelationfunctionsΠ(p)andΠ (p)toobtainthehadronicrepresentation
µ µν
[15, 16]. After isolating the pole terms of the lowest states Ξ , Ξ , Σ and Σ , we obtain
∗Q ′Q ∗Q Q
the following results:
MB∗+p
Πµν(p) = λ2BQ∗ MB2Q∗Q−p6 2 [−gµν +···]+··· ,
M +p
Π(p) = λ2 BQ 6 + , (15)
BQM2 p2 ···
BQ −
we choose the tensor structures g , pg , 1 and p for analysis. After performing the
µν µν
6 6
standard procedure of the QCD sum rules, we obtain sixteen sum rules for the heavy
baryons states B and B ,
Q∗ Q
λ2ie−MMi22 = s0i dsρAi (s)e−Ms2 ,
Z∆i
λ2iMie−MMi22 = s0i dsρBi (s)e−Ms2 , (16)
Z∆i
where the i denote the channels Ξ , Ξ , Σ and Σ respectively; the s0 are the cor-
∗Q ′Q ∗Q Q i
responding continuum threshold parameters and the M2 is the Borel parameter. The
thresholds ∆ can be sorted into two sets, we introduce the qq¯, qs¯ to denote the light
i
quark constituents in the heavy baryon states to simplify the notations, ∆ = m2,
qq¯ Q
∆ = (m + m )2. The explicit expressions of the spectral densities ρA(s) and ρB(s)
qs¯ Q s i i
are given in the appendix B.
Differentiate the Eq.(16) with respect to 1 , then eliminate the pole residues λ , we
M2 i
can obtain the sixteen sum rules for the masses of the heavy baryon states B and B ,
Q∗ Q
s0 A/B s
M2 = ∆iidssρi (s)e−M2 . (17)
i s0 A/B s
R∆iidsρi (s)e−M2
R
In the conventional QCD sum rules [15, 16], there are two criteria (pole dominance
and convergence of the operator product expansion) for choosing the Borel parameter M2
and threshold parameter s . We impose the two criteria on the heavy baryon states to
0
choose theBorel parameter M2 andthresholdparameter s , thevalues areshown inTable
0
1. Finally we obtain the values of the masses and pole resides of the heavy baryon states
B and B , which are shown in Table 2.
Q∗ Q
From Table 2, we can see that the average values of the masses with the tensor struc-
turespg (p)andg (1)canreproducetheexperimentaldataapproximatelyfortheestab-
µν µν
6 6
lished heavy baryon states. So it is reasonable to take the average values MΞ∗ = 5.98GeV
b
and MΞ′ = 5.95GeV for the un-established bottom baryon states Ξ∗b and Ξ′b in numerical
b
analysis. The values of the pole residues from different tensor structures differ greatly
from each other in some channels, for example, Ξ , Σ , Ξ , Σ . In this article, we take the
′b b ′c c
average values and assume uniform uncertainties (about 20%) for the pole residues in all
7
M2(GeV2) s (GeV2)
0
Ξ 5.0 6.0 46.0 0.5
∗b − ±
Ξ 4.8 5.6 44.5 0.5
′b − ±
Σ 5.0 6.0 45.0 0.5
∗b − ±
Σ 4.8 5.6 43.5 0.5
b
− ±
Ξ 2.0 3.0 11.0 0.5
∗c − ±
Ξ 2.0 2.8 10.5 0.5
′c − ±
Σ 2.0 3.0 10.5 0.5
∗c − ±
Σ 1.9 2.7 10.0 0.5
c
− ±
Table 1: The Borel parameters M2 and threshold parameters s for the heavy baryon
0
states.
pg /p (M) g /1 (M) pg /p (λ) g /1 (λ)
µν µν µν µν
6 6 6 6
Ξ 6.04 0.14 5.92 0.20 4.8 1.0 4.2 1.3
∗b ± ± ± ±
Ξ 6.04 0.10 5.85 0.15 9.0 1.8 6.0 1.3
′b ± ± ± ±
Σ 5.95 0.14 5.72 0.25 4.2 1.0 3.3 1.3
∗b ± ± ± ±
Σ 5.96 0.10 5.73 0.16 8.0 1.7 5.0 1.2
b
± ± ± ±
Ξ 2.60 0.20 2.68 0.18 3.1 0.8 3.1 0.7
∗c ± ± ± ±
Ξ 2.65 0.14 2.54 0.17 6.2 1.5 4.4 0.9
′c ± ± ± ±
Σ 2.50 0.20 2.59 0.19 2.7 0.7 2.7 0.7
∗c ± ± ± ±
Σ 2.54 0.15 2.42 0.20 5.4 1.4 3.6 1.0
c
± ± ± ±
Table 2: The masses and pole residues of the heavy baryon states from the sum rules
with different tensor structures. The masses M are in unit of GeV and the pole residues
λ are in unit of 10 2GeV3.
−
channels, the uncertainties originate from the parameters other than the Borel parame-
ter M2 are about 20%, we subtract the uncertainties originate from the Borel parameter
from the total uncertainties to avoid double counting. The values of the pole residues are
λΞ∗b = (4.5±0.8)×10−2GeV3,λΞ∗c = (3.1±0.5)×10−2GeV3,λΞ′b = (7.5±1.5)×10−2GeV3,
λΞ′ = (5.3 1.0) 10−2GeV3,λΣ∗ = (3.8 0.7) 10−2GeV3,λΣ∗ = (2.7 0.5) 10−2GeV3,
c ± × b ± × c ± ×
λ = (6.5 1.2) 10 2GeV3, and λ = (4.5 0.9) 10 2GeV3. The threshold pa-
Σb ± × − Σc ± × −
rameters are taken as s = (11.0 1.0)GeV2, (10.5 1.0)GeV2, (45.0 1.0)GeV2 and
0
± ± ±
(46.0 1.0)GeV2 in the channels Ξ (Ξ ), Σ (Σ ), Ξ (Ξ ) and Σ (Σ ) respectively; the
± ∗c ′c ∗c c ∗b ′b ∗b b
Borel parameters are taken as M2 = (2.0 3.0)GeV2 and (5.0 6.0)GeV2 in the charm
− −
and bottom channels respectively. Those parameters are determined by the two-point
QCD sum rules to avoid possible contaminations from the high resonances and continuum
states. In calculation, we observe that the values of the strong coupling constants g , g
1 2
and G3 are insensitive to threshold parameters s .
0
The main uncertainties originate from the parameters λBQ, λBQ∗ (as the strong cou-
pling constants g , g and G3 1 ) and m , the variations of those parameters can
1 2 ∝ λBQλBQ∗ Q
lead to relatively large changes for the numerical values, and almost saturate the total
8
uncertainties, i.e. the variations of the two hadronic parameters λBQ and λBQ∗ lead to an
uncertainty about 20% √2 = 28%, and the variations of the m lead to an uncertainty
Q
×
about (10 20)%, refiningthose parameters is of great importance. In the case of the sum
−
rules for the strong coupling constants g , the values are not stable enough with variations
2
of the Borel parameter, additional uncertainties are introduced, the total uncertainties are
very large, see Table 3. The contributions from the strong coupling constants g to the
2
radiative decay widths are very small comparing with the corresponding ones from the g ,
1
the predictions are insensitive to the Borel parameter. Although there are many param-
eters in the light-cone distributions amplitudes [29, 30], the uncertainties originate from
those parameters are rather small. In calculation, we neglect the contributions from the
high dimension vacuum condensates, such as f GaGbGc , q¯q αsGG , s¯s αsGG , etc.
h abc i h ih π i h ih π i
They are greatly suppressed by the large numerical denominators and additional inverse
powers of the Borel parameter 1 , and would not play any significant roles. Furthermore,
M2
we neglect some terms involving the light-cone distributions amplitudes f(u¯ ) and f(u¯ )
0 0
in case of the contributions from the terms f(u¯ ) are small, as
0
e
e e
f(u¯ ) f(u¯ )
0 0
40%, 10%. (18)
f(u¯ ) ≈ f(u¯ ) ≈
0 0
e
e e
Taking into account all the uncertainties of the revelent parameters, finally we obtain
thenumericalresultsofthestrongcouplingconstantsg ,g andG3,whichareshowninthe
1 2
2
Table3. Weestimatetheuncertaintiesδwiththeformulaδ = ∂f (x x¯ )2,
i ∂xi |xi=x¯i i − i
r
where the f denote strong coupling constants g , g and G3, the (cid:16)x de(cid:17)note the revelent
1 2 P i
parameters m , q¯q , s¯s , . In the numerical calculations, we take the approximation
Q
h i h i ···
2
∂f (x x¯ )2 [f(x¯ ∆x ) f(x¯ )]2 for simplicity. For the central values of the
∂xi i − i ≈ i± i − i
∗
s(cid:16)tron(cid:17)g coupling constants, g1Bb∗BbV (70 80)%, G3Bb∗BbV(MBc∗+MBc) (80 90)%, the
g1BcBcV ≈ − G3Bc∗BcV(MBb∗+MBb) ≈ −
heavy quark symmetry works rather well. Those strong coupling constants in the ver-
texes B V V are basic parameters in describing the interactions among the heavy baryon
Q∗ Q
states, once reasonable values are obtained, wecan usethem toperformphenomenological
analysis.
The radiative decays B B γ can be described by the following electromagnetic
Q∗ → Q
lagrangian ,
L
= eQ ¯bγ bAµ eQ c¯γ cAµ eQ s¯γ sAµ eQ u¯γ uAµ eQ d¯γ dAµ, (19)
b µ c µ s µ u µ d µ
L − − − − −
wherethe A is theelectromagnetic field. From thelagrangian , wecan obtain the decay
µ
L
amplitudeswiththeassumptionofthevectormesondominance,eT = B (p)γ(q) B (p+
h Q |L| Q∗
9
Vertexes g (GeV 1) g (GeV 2) G3
1 − 2 −
− −
Ξ +Ξ+φ, Ξ 0Ξ0φ 2.98+1.18 0.62+0.63 9.75+3.87
∗c ′c ∗c ′c 0.81 0.31 2.68
Ξ +Ξ+ρ,Ξ +Ξ+ω,Ξ 0Ξ0ω 3.54−+1.39 0.47−+0.46 13.61−+5.42
∗c ′c ∗c ′c ∗c ′c 0.97 0.24 3.76
Ξ 0Ξ0ρ (3.54−+1.39) (0.47−+0.46) (13.61−+5.42)
∗c ′c 0 − 0.97 − 0.24 − 3.76
Σ ++Σ++ρ ,Σ ++Σ++ω,Σ 0Σ0ω,Σ +Σ+ω 7.07+−3.09 0.97+−0.87 25.00+−10.54
∗c c 0 ∗c c ∗c c ∗c c 2.17 0.49 7.53
Σ +Σ+ρ 0− 0− 0−
∗c c 0
Σ 0Σ0ρ (7.07+3.09) (0.97+0.87) (25.00+10.54)
∗c c 0 − 2.17 − 0.49 − 7.53
Ξ∗b0Ξ′0bφ, Ξ∗b−Ξ′−b φ 2.25+−00..9639 0.11+−00..1161 20.14+−68..5142
Ξ∗b0Ξ′0bρ,Ξ∗b0Ξ′0bω,Ξ∗b−Ξ′−b ω 2.73−+01..1844 0.07−+00..1045 26.68+−81.12.504
Ξ∗b−Ξ′−b ρ −(2.73−+01..1844) −(0.07−+00..1045) −(26.68−+81.12.504)
Σ +Σ+ρ ,Σ +Σ+ω, Σ 0Σ0ω, Σ Σ ω 5.05+−2.17 0.13+−0.27 47.05+−19.72
∗b b 0 ∗b b ∗b b ∗b− −b 1.59 0.10 14.71
Σ 0Σ0ρ 0− 0− 0−
∗b b 0
Σ Σ ρ (5.05+2.17) (0.13+0.27) (47.05+19.72)
∗b− −b 0 − 1.59 − 0.10 − 14.71
− − −
Table 3: The values of the strong coupling constants g , g and G3.
1 2
Channels Γ (KeV)
Ξ + Ξ+γ 0.96+1.47
∗c → ′c 0.67
Ξ 0 Ξ0γ 1.26−+0.80
∗c → ′c 0.46
Σ ++ Σ++γ 6.36−+6.79
∗c → c 3.31
Σ + Σ+γ 0.40−+0.43
∗c → c 0.21
Σ 0 Σ0γ 1.58−+1.68
∗c → c 0.82
Ξ 0 Ξ0γ 0.047−+0.077
∗b → ′b 0.036
Ξ∗b− → Ξ′−b γ 0.066−+00..004257
Σ + Σ+γ 0.12+−0.13
∗b → b 0.06
Σ 0 Σ0γ 0.0076−+0.0079
∗b → b 0.0040
Σ Σ γ 0.030+−0.032
∗b− → −b 0.016
−
Table 4: The widths of the radiative decays B B γ.
Q∗ → Q
10