Light cone QCD sum rules for the $g_{Ξ_QΞ_Qπ}$ coupling constant

0.31 MB·English

by  Su Houng Lee

Light cone QCD sum rules for the g coupling constant
ΞQΞQπ
Su Houng Lee1∗, A. Ozpineci2∗∗, Y. Sarac3∗∗∗,
1 Institute of Physics and Applied Physics,
Yonsei University, 120-749, Seoul, Korea
∗ [e-mail]
2 Physics Department, Middle East Technical University,
06531, Ankara, Turkey
0 ∗∗ [e-mail]
1
0 3 Electrical and Electronics Engineering Department,
2
Atilim University, 06836 Ankara, Turkey
n ∗∗∗ [e-mail]
a
J
(Dated: January 17, 2010)

7
1 For theheavybaryonsΞc and Ξb thecoupling constantsgΞcΞcπ and gΞbΞbπ arecalculated in the
framework of light cone QCD sum rules. The most general form of the interpolating field of ΞQ is

] used in thecalculation.
h
p PACSnumbers: 11.55.Hx,13.30.-a,14.20.Lq,14.20.Mr
-
p
e
I. INTRODUCTION
h
[

Experimental progresses over the last few years provided exciting results in the heavy baryon sector. There have
1
v beenplentyofheavybaryonobservationsyieldingalargeamountofexperimentaldataoncharmandbottombaryons.
5 ObservationofBelle,BABAR,DELPHI,CLEO,CDF,DOetc[1–9]havemotivatedthetheoreticalinterestsonheavy
0 baryonscontainingcandbquarks. Themassspectroscopyofthesebaryonshavebeenstudiedusingvarioustheoretical
9 models[10–18]aswellasQCDsumrulesmethod[19–31]. TheirmassescalculatedviaQCDsumrulesinheavyquark
2 limit [22] and in the heavy quark effective theory[23–25].
.
1 Beside the mass spectrum there have been theoretical studies on the magnetic moments of these baryons utilizing
0 naivequarkmodel[32,33],quarkmodel[34,35],boundstateapproach[36],relativisticthree-quarkmodel[37],hyper
0 centralmodel[38],Chiralperturbationmodel[39],solitonmodel[40],skyrmionmodel[41],nonrelativisticconstituent
1
quark model [42], QCD sum rules in external magnetic fiels [43], and light cone QCD sum rules (LCQSR) method
:
v [44, 45].
i In the present work we make use of the (LCQSR) to calculate the coupling constant g . A similar work has
X ΞQΞQπ
been done in [46] for the coupling constant g . This paper is organized as follows. In section 2, we introduce
r ΣQΛQπ
a the interpolatingfieldfor ΞQ andgivethe detailsof (LCQSR)calculationsfor the coupling constant. Insection3 the
numerical analysis, discussion and conclusion are presented.
II. LIGHT CONE QCD SUM RULES FOR THE gΞQΞQπ COUPLING CONSTANT
To calculate the coupling constant g via the (LCQSR) one studies a suitably chosen correlation function of
ΞQΞQπ
the form
Π=i d4xeipx π(q) η (x)η¯ (0) 0 , (1)
h |T{ ΞQ ΞQ }| i
Z
where η (x) denotes the interpolating current of Ξ baryon and denotes the time ordering product. One can
ΞQ Q T
calculate this correlation function either phenomenologically, inserting a complete set of hadronic states into the
correlator to obtain a result containing hadronic parameters, or theoretically via the operator product expansion
(OPE) in deep Euclidean region p2 in terms of QCD parameters. Sum rules are obtained by matching these
→ ∞
two expressions after Borel transformations and the contribution of the higher states and continuum is subtracted.
2
The calculation of the phenomenological side is similar to the calculation in [46] and the details are presented for
completeness. To obtain the physical representation of the correlator a complete set of hadronic state having the
quantum number of Ξ baryon is inserted. Then the correlationfunction becomes
Q
0 η Ξ (p ) Ξ (p ) η¯ 0
Π = h | ΞQ | Q 2 i Ξ (p )π(q) Ξ (p ) h Q 1 | ΞQ | i +...,
p2 m2 h Q 2 | Q 1 i p2 m2
2− ΞQ 1− ΞQ
(2)
where p = p+q and p = p and ... represents the contribution of the higher states and continuum. The matrix
1 2
elements representing the coupling of the interpolating field to the baryon state under consideration are defined as
0 η Ξ (p,s) =λ u (p,s), (3)
h | ΞQ | Q i ΞQ ΞQ
where λ denotes the coupling strength and u is the spinor for the Ξ baryon. The coupling constant g is
ΞQ ΞQ Q ΞQΞQπ
defined by the matrix element in Eq. (2) which is given as
Ξ (p )π(q) Ξ (p ) = g u(p )iγ u(p ).
h Q 2 | Q 1 i ΞQΞQπ 2 5 1
(4)
Using the Eqs. (3) and(4)in Eq. (2) one obtains the phenomenologicalside of the correlator as
g λ 2
Π = i ΞQΞQπ| ΞQ| p qγ m qγ . (5)
(p2 m2 )(p2 m2 ) −6 6 5− ΞQ 6 5
1− ΞQ 2− ΞQ
(cid:2) (cid:3)
The coefficient of any one of the structures p qγ or qγ can be used. In this work, we will work with the structure
5 5
6 6 6
qγ .
5
6
For the calculationofthe QCD side ofthe correlationfunction which is obtained via (OPE)one needs to know the
explicit expression of the interpolating field of Ξ which is given in the following form:
Q
1
η = ǫabc uTCs γ5Q +β uTCγ5s Q + uTCQ γ5s +β uTCγ5Q s (6)
ΞQ √2 "(cid:18) a b(cid:19) c (cid:18) a b(cid:19) c (cid:18) a b(cid:19) c (cid:18) a b(cid:19) c#
where Q represents the heavy quarks c or b, β is an arbitrary parameter with β = 1 corresponding to the Ioffe
−
current, C is the charge conjugation operator and a,b,c are the color indices. After inserting the interpolating fields
into Eq. (1) and carrying out the contractions, the following expression is obtained:
Π = iǫabcǫa′b′c′ d4xeipx π(q) γ Scb′CSTaa′CSbc′γ γ Scb′CSTaa′CSbc′γ
−2 h |(− 5 Q u s 5− 5 s u Q 5
Z
+Tr[CSTaa′CSbb′]γ Scc′γ +Tr[CSTaa′CSbb′]γ Scc′γ +β γ Scb′γ CSTaa′CSbc′
u s 5 Q 5 u Q 5 s 5 "− 5 Q 5 u s
γ Scb′γ CSTaa′CSbc′ Scb′CSTaa′Cγ Sbc′γ Scb′CSTaa′Cγ Sbc′γ
− 5 s 5 u Q − Q u 5 s 5− s u 5 Q 5
+Tr[γ CSTaa′CSbb′]γ Scc′ +Tr[γ CSTaa′CSbb′]γ Scc′ +Tr[CSTaa′Cγ Sbb′]Scc′γ
5 u s 5 Q 5 u Q 5 s u 5 s Q 5
+Tr[CSTaa′Cγ Sbb′]Scc′γ +β2 Scb′γ CSTaa′Cγ Sbc′ Scb′γ CSTaa′Cγ Sbc′
u 5 Q s 5# "− Q 5 u 5 s − s 5 u 5 Q
+Tr[γ CSTaa′Cγ Sbb′]Scc′ +Tr[γ CSTaa′Cγ Sbb′]Scc′ 0 . (7)
5 u 5 s Q 5 u 5 Q s #)| i
Note that this is a schematicalrepresentation. The pion can be emitted from any one of the u or d quarks and hence
both contribution should be summed. To obtain the contribution of pion emission from any one of the quarks, its
propagator is replaced by : qa(0)qb(x) :. To proceed with the calculation the heavy and light quark propagators are
3
needed. In this work, the following propagators are used: [47]
d4k 1 k+m 1
S (x) = Sfree(x) ig e−ikx dv 6 Q Gµν(vx)σ + vx Gµνγ ,
Q Q − sZ (2π)4 Z0 "(m2Q−k2)2 µν m2Q−k2 µ ν#
q¯q x2
S (x) = Sfree(x) h i m2 q¯q
q q − 12 − 192 0h i
1 x i
ig du 6 G (ux)σ uxµG (ux)γν . (8)
− s 16π2x2 µν µν − µν 4π2x2
Z0 (cid:20) (cid:21)
where the free light and heavy quark propagators in Eq. (8) are given in x representation as
i x
Sfree = 6 ,
q 2π2x4
m2 K (m √ x2) m2 x
Sfree = Q 1 Q − i Q 6 K (m x2),
Q 4π2 √ x2 − 4π2x2 2 Q −
− p
(9)
where K are the Bessel functions. As seen from Eq. (7) as well as the propagators the matrix elements of the form
i
π(q)q¯(x )Γ q(x )0 arealsoneeded. HereΓ representsanymemberoftheDiracbasisi.e. 1,γ ,σ /√2,iγ γ ,γ .
1 i 2 i α αβ 5 α 5
h | | i { }
In terms of the pion light cone distribution amplitudes the matrix elements π(q)q¯(x )Γ q(x )0 are given explicitly
1 i 2
h | | i
as [48, 49]
1 1
π(p)q¯(x)γ γ q(0)0 = if p dueiu¯px ϕ (u)+ m2x2A(u)
h | µ 5 | i − π µ π 16 π
Z0 (cid:18) (cid:19)
i x 1
f m2 µ dueiu¯pxB(u),
− 2 π πpx
Z0
1
π(p)q¯(x)iγ q(0)0 = µ dueiu¯pxϕ (u),
5 π P
h | | i
Z0
i 1
π(p)q¯(x)σ γ q(0)0 = µ 1 µ˜2 (p x p x ) dueiu¯pxϕ (u),
h | αβ 5 | i 6 π − π α β − β α σ
Z0
(cid:0) (cid:1) 1
π(p)q¯(x)σ γ g G (vx)q(0)0 = iµ p p g (p x +p x )
µν 5 s αβ π α µ νβ ν β β ν
h | | i − px
(cid:20) (cid:18) (cid:19)
1
p p g (p x +p x )
α ν µβ µ β β µ
− − px
(cid:18) (cid:19)
1
p p g (p x +p x )
β µ να ν α α ν
− − px
(cid:18) (cid:19)
1
+ p p g (p x +p x )
β ν µα µ α α µ
− px
(cid:18) (cid:19)(cid:21)
αei(αq¯+vαg)px (α ),
i
× D T
Z
1
π(p)q¯(x)γ γ g G (vx)q(0)0 = p (p x p x ) f m2 αei(αq¯+vαg)px (α )
h | µ 5 s αβ | i µ α β − β α px π π D Ak i
Z
1
+ p g (p x +p x )
β µα µ α α µ
− px
(cid:20) (cid:18) (cid:19)
1
p g (p x +p x ) f m2
− α µβ − px µ β β µ π π
(cid:18) (cid:19)(cid:21)
αei(αq¯+vαg)px (α ),
⊥ i
× D A
Z
4
1
π(p)q¯(x)γ ig G (vx)q(0)0 = p (p x p x ) f m2 αei(αq¯+vαg)px (α )
h | µ s αβ | i µ α β − β α px π π D Vk i
Z
1
+ p g (p x +p x )
β µα µ α α µ
− px
(cid:20) (cid:18) (cid:19)
1
p g (p x +p x ) f m2
− α µβ − px µ β β µ π π
(cid:18) (cid:19)(cid:21)
αei(αq¯+vαg)px (α ), (10)
⊥ i
× D V
Z
where µ =f m2π , µ˜ = mu+md, α=dα dα dα δ(1 α α α ) and the ϕ (u), A(u), B(u), ϕ (u), ϕ (u),
π πmu+md π mπ D q¯ q g − q¯− q− g π P σ
(α ), (α ), (α ), (α ) and (α ) are functions of definite twist and their expressions will be given in the
i ⊥ i k i ⊥ i k i
T A A V V
numerical analysis section.
Withtheseinputs,the correlationfunctioncanbecalculatedintermsofquark-gluondegreesoffreedom. Tomatch
the two representation, their spectral representation is used. The contributions of the higher states and continuum
are subtracted using quark-hadron duality. Furthermore, to eliminate the unknown polynomials in the spectral
representation and suppress the contribution of higher states and continuum, Borel transformation is applied with
respect to p2 and (p+q)2. Finally, the sum rules is obtained from the integral:
e−mMΞ2QmΞQ|λΞQ|2gΞQΞQπ = Zms2Q0 eM−s2ρ(s)ds+e−Mm22QΓ, (11)
where the explicit expressions of ρ(s) and Γ are given in Appendix A.
To obtain a prediction for g , the residue λ is also needed. The residue can be calculated using mass sum
ΞQΞQπ ΞQ
rules and is given as:
−λ2ΞQe−m2ΞQ/M2 = Zms2Q0 eM−s2ρ1(s)ds+e−Mm22QΓ1, (12)
where explicit expressions of ρ (s) and Γ are given in Appendix B. From Eq. (12), the mass can be obtained by
1 1
differentiation with respect to M2 as
ms02 eM−s2s ρ1(s)ds+M4dMd2(e−Mm22QΓ1)
m = Q . (13)
ΞQ R ms02 eM−s2ρ1(s)ds+e−Mm22QΓ1
Q
R
III. NUMERICAL ANALYSIS
Inthissectionthenumericalanalysisforthecouplingconstantg ispresented. Therequiredinputparameters
ΞQΞQπ
are given as: u¯u (1 GeV) = (0.243)3 GeV3, ß(1 GeV) = 0.8 u¯u (1 GeV), m = 4.7 GeV, m = 1.23 GeV,
b c
m =5.792 GheV,im =2.470−GeV, and m2(1 GeV)=(0.8 0.2)hGeiV2 [50], f =0.131[48, 50], m =0.135GeV
Ξb Ξc 0 ± π π
. We also need the π-meson wave functions for the coupling constant calculation, whose explicit forms are presented
as [48, 49]
3
φ (u) = 6uu¯ 1+aπC (2u 1)+aπC2(2u 1) ,
π 1 1 − 2 2 −
(cid:16) 1 (cid:17)
(α ) = 360η α α α2 1+w (7α 3) ,
T i 3 q¯ q g 32 g−
(cid:18) (cid:19)
5 1 1
φ (u) = 1+ 30η C2(2u 1)
P 3− 2µ2 2 −
(cid:18) π(cid:19)
27 1 81 1 1
+ 3η w aπ C2(2u 1),
− 3 3− 20µ2 − 10µ2 2 4 −
(cid:18) π π (cid:19)
1 7 3 3
φ (u) = 6uu¯ 1+ 5η η w µ2 µ2aπ C2(2u 1) ,
σ 3− 2 3 3− 20 π− 5 π 2 2 −
(cid:20) (cid:18) (cid:19) (cid:21)
(α ) = 120α α α (v +v (3α 1)),
k i q q¯ g 00 10 g
V −
(α ) = 120α α α (0+a (α α )),
k i q q¯ g 10 q q¯
A −
5
3
(α ) = 30α2 h (1 α )+h (α (1 α ) 6α α )+h (α (1 α ) (α2+α2)) ,
V⊥ i − g 00 − g 01 g − g − q q¯ 10 g − g − 2 q¯ q
(cid:20) (cid:21)
1
(α ) = 30α2(α α ) h +h α + h (5α 3) ,
A⊥ i g q¯− q 00 01 g 2 10 g−
(cid:20) (cid:21)
B(u) = g (u) φ (u),
π π
−
1 1 1
g (u) = g C2(2u 1)+g C2(2u 1)+g C2(2u 1),
π 0 0 − 2 2 − 4 4 −
16 24 20 1 1 7 10 3
A(u) = 6uu¯ + aπ +20η + η + + η w η C2(2u 1)
15 35 2 3 9 4 −15 16 − 27 3 3− 27 4 2 −
(cid:20) (cid:18) (cid:19)
11 4 3
+ aπ η w C2(2u 1)
−210 2 − 135 3 3 4 −
(cid:18) (cid:19) (cid:21)
18
+ aπ +21η w 2u3(10 15u+6u2)lnu
− 5 2 4 4 −
(cid:18) (cid:19)
+ 2u¯3(10 15u¯+6u¯2)(cid:2)lnu¯+uu¯(2+13uu¯) , (14)
−
where Ck(x) are the Gegenbauer polynomials, (cid:3)
n
1
h = v = η ,
00 00 4
−3
21 9
a = η w aπ,
10 8 4 4− 20 2
21
v = η w ,
10 4 4
8
7 3
h = η w aπ,
01 4 4 4− 20 2
7 3
h = η w + aπ,
10 4 4 4 20 2
g = 1,
0
18 20
g = 1+ aπ +60η + η ,
2 7 2 3 3 4
9
g = aπ 6η w . (15)
4 −28 2 − 3 3
The constants in the Eqs. (14) and (15) are calculated at the renormalization scale µ =1 GeV2 and are given as
aπ =0, aπ =0.44, η =0.015, η =10, w = 3 and w =0.2.
1 2 3 4 3 − 4
Looking at the result of LCQCD sum rules calculation for the coupling constant g one encounters three
ΞQΞQπ
auxiliary parameters. These parameters are the Borel mass M2, the continuum threshold s and the arbitrary
0
parameter β and there should be no dependency of a physical quantity, such as the coupling constant for our case,
on them. Therefore at this stage a working region of these auxiliary parameters should be determined. In order to
determine the upper and lower bound of M we use the requirements that the continuum contribution be less than
that of the ground state, and the highest power of 1/M2 be less than 300/ of the highest power of M. The former
0
(latter)isusedtodetermineupper(lower)boundofM2. Todeterminethevalueofcontinuumthresholds weusethe
0
two-pointcorrelationfunctionfromwhichweobtainthemasssumrules. InFig.1andFig.2,weplotthe dependence
of our prediction on m to the Borel parameter M2, and cosθ, where β = cosθ, respectively. For these plots, the
Ξc
continuumthresholdischosentobes =8Gev2 ands =9Gev2. Forthesevaluesofthecontinuumthresholdwesee
0 0
fromthesefiguresthatourpredictionsareinagreementwithexperimentalresultsandthatthereisnodependenceon
the auxiliary parameters. In Figs. 3, 4, we carry out the same analysis for m and find the continuum threshold to
Ξb
be s =36 Gev2 and s =37 Gev2. In Fig. 4, we also observe that our prediction is stable with respect to variations
0 0
of θ for the region 0.5<cosθ <0.5 which corresponds to β >1.7.
| |
The results for the coupling constant calculation are presented in Figs. 5, 6, 7 and 8. Figs. 5 and 7 depicts
respectively the dependence of the coupling constants g and g on M2 in the working region of M2. The
ΞcΞcπ ΞbΞbπ
results are given for two fixed values of β and two fixed values of s for each coupling constant. It follows from
0
the figures that the results are rather stable with respect to the variations of M2 in the given region of M2. The
dependenceofthecouplingconstantsoncosθarealsopresentedinFigs.6and8andwithrespecttothesefigureswhen
cosθ is in between 0.25 < cosθ < 0.25 the coupling constant g is practically independent of the unphysical
− ΞcΞcπ
parameterβ. Theintervalofβ thatgivescouplingconstantresultindependentofβ forg is 0.75<cosθ <0.75.
ΞbΞbπ −
6
As a result of our analysis we obtain the values of the coupling constants as
g =1.0 0.5, g =1.6 0.4.
ΞcΞcπ ± ΞbΞbπ ±
To summarize, in this work we present the results of the coupling constant for the coupling constants g and
ΞcΞcπ
g . To this end, we make use of the LCQSR approachwith the Ξ current applied in its most generalform. We
ΞbΞbπ Q
obtaintheappropriatevaluesofthethresholdparameterss fromthemasssumrulesand,usingthemandappropriate
0
intervals of Borel parameter and β, we attain the coupling constants g and g .
ΞcΞcπ ΞbΞbπ
IV. ACKNOWLEDGMENT
The authors would like to thank T. M. Aliev for his useful discussions. S. H. L. was supported by the Korean
ResearchFoundation KRF-2006-C00011. The work of A. O. and Y. S. have been supported in part by the European
Union (HadronPhysics2 project ”Study of strongly interacting matter”).
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8
Appendix A
f m2
ρ(s) = π Q 3m m (1+6β+β2)(2ψ ψ +ψ ) 4m (3 4β+β2)ψ
b Q 10 20 30 Q 10
−768√2π2" ( − − −
s
2[m (1+6β+β2)ψ 2m (3 4β+β2)]ln( ) +u2m28(1+7β+β2)ψ ϕ (u )
− Q 00− s − m2Q ) 0 π 32# π 0
µ s
π (β 1)(µ2 1) 2m2[( m +3βm +5m +3βm )ψ +m (1 3β)ln( )]
−384√2π2 − − " Q − Q Q s s 10 Q − m2Q
e
[m (3β 1)(2ψ ψ ψ +2ψ )+m (5+3β)(ψ +ψ )]u m2 ϕ (u )
− Q − 10− 11− 12 21 s 11 12 0 π# σ 0
f m2
π π m m (1+6β+β2)ψ +m (3 4β+β2)ψ
Q Q 10 s 00
−256√2π2"− ( − )
f m4u
+2(1+7β+β2)(ψ +ψ )u2m2 A(u ) π Q 0[1+7β+β2) 6ψ 3ψ +ψ 2ψ
11 12 0 π# 0 − 384√2π2 " 10− 20 30− 41
6ψ ln( s ) ϕ′ (u ) µπm2Qu0(β 1)(µ2 1) m (3β 1)(2ψ ψ +ψ )
− 00 m2Q # π 0 − 768√2π2 − − " Q − 10− 20 31
+m (5+3β)(ψ ψ )+2m (1 3β)ln(es ) ϕ′ (u ) fπm2πm2Qu0(1+7β+β2)(ψ ψ )A′(u )
s 20− 31 Q − m2Q # σ 0 − 256√2π2 20− 31 0
f µ Λ2
π π m ( 1 4β+5β2)[m2( 1+3ψ ψ +ψ ) sψ ](η 2η )ln( )
−64√2π2mQ" s − − Q − 00− 01 10 − 00 1− 2 mQ
s
+m2µ m ( 1 4β+5β2)ψ (η 2η )ln( )+[m u (2+14β+2β2)(2η′ η′)
Q π( s − − 01 1− 2 Λ2 Q 0 4− 3
s
(5+14β+5β2)(2η′ η )+m (7 2β+7β2)η +m ( 5+4β+β2)(η 2η )]ln( )
− 8− 7 Q − 1 s − 1− 2 m2
Q
s m2
+m ( 1 4β+5β2)(η 2η ) γ (2ψ ψ +ψ ψ )+( 2ψ +ψ +ψ ψ )ln( − Q)
s − − 1− 2 E 00− 10 11− 21 − 00 01 10− 21 Λ2
m2(s m2) s
ψ ln( Q − Q ) [m4(1 4β+3β2)+m3m ( 5+2β+3β2)](2η′ η′)ln( )
− 00 Λ2s !)− Q − Q s − 6− 5 m2Q #
1
+ f m2µ 2m 3m u (1+β2)(ψ ψ )(2η′ η′)+m (1+β2)[(2η′ η′ 10η′ +5η′)
128√2π2" π π π( Q Q 0 20− 31 8− 7 Q 8− 7− 4 3
+14(1+β2)m η m (1 5β2)(η 2η )]ψ +m (1 5β2)(ψ ψ ψ )(η 2η )
Q 1 s 1 2 10 Q 00 01 21 1 2
− − − − − − − !
+3m u [m (ψ ψ ) 2m (ψ +ψ )]ζ+m u β2[3m (ψ ψ ) 2m (ψ +ψ )]ζ
Q 0 Q 20 31 s 10 21 Q 0 Q 20 31 s 10 21
− − − −
2u2m2(1+β2)[(12ψ 3ψ 3ψ +12ψ )η +( 10ψ +6ψ +6ψ 10ψ )η ]
− 0 π 10− 11− 12 21 1 − 10 11 12− 21 2 )
+2µ [(m 5m )+3β2(m +m )][(2η′ η′)ψ +2u m2(ψ +ψ )(2η η )]
π Q− s Q s 6− 5 10 0 π 10 21 6− 5 #
β
+ f m22m [7m u (2η′ η′ +η′ 2η′)+4m η
64√2π2" π π Q( q 0 4− 3 7− 8 Q 1
2m (η 2η )]ψ +2m (ψ ψ ψ )(η 2η )+m u [11m (ψ ψ )+4m (ψ +ψ )]ζ
s 1 2 10 s 00 01 21 1 2 Q 0 Q 20 31 s 10 21
− − − − − −
2u2m2[(6ψ +3ψ +3ψ +6ψ )η 14(ψ +ψ )η ] +2µ (2m m )[m2(2η′ η′)
− 0 π 10 11 12 21 1− 10 21 2 ) π( q− s Q 6− 5
s¯s
+2u m2(ψ +ψ )(2η η )] + h i 3f 2m (3 4β+β2) m (1+6β+β2) ϕ (u )
0 π 10 21 6− 5 )# 288√2" π( Q − − s ) π 0
( ) ( )
9
+µ u ( 5+2β+3β2)(µ2 1)ψ ϕ′ (u ) 6(1+γ )µ ( 5+2β+3β2)(2η′ η′)ψ
π 0( − π − 00) σ 0 − E π − 6− 5 00#
+ hgs2G2i f −6m3Qmes(3 4β+β2)ln(s−m2Q)ψ + 1 [ m (1+6β+β2)
4608√2π2" π( M2 − Λ2 00 m2Q − Q
s m2
+2m (1 9γ )(3 4β+β2)]ψ +18m m (3 4β+β2)(ψ ψ 2ψ +2ψ ψ )ln( − Q)
s − E − 00 Q s − 00− 02− 10 21 22 Λ2
µ m (µ˜2 1) s m2 3 2
+4u2m2(1+7β+β2)ψ ϕ (u )+ π s π − ( 5+2β+3β2)ln( − Q)ψ [u2m2(
0 π 02) π 0 ( M2 − Λ2 10 0 π −M2 − m2Q
2m2 2m2 µ (β 1)(µ˜2 1)
Q)+ Q] π − π − m4 4m (3β 1)ψ 3m (5+3β)(56γ ψ ψ +ψ +ψ )
− M4 M4 − 6m6Q Q Q − 00− s E 00− 20 21 31
h
s m2
+12m (5+3β)(14ψ 14ψ 42ψ +30ψ +27ψ +18ψ +9ψ +24ψ +12ψ )ln( − Q)
s 00− 03− 10 20 21 22 23 31 32 Λ2
i
6u2m2 m2[m (3β 1)ψ +m (5+3β)( 61ψ +20ψ +20ψ +20ψ +122ψ +2ψ +5ψ +11ψ
− 0 π Q Q − 02 s − 00 01 02 03 10 12 13 14
h s m2
122ψ 61ψ 20ψ )]+6m (5+3β) [m2(39ψ +40ψ ) 40sψ ]ln( − Q)
− 21− 22− 23 s { Q 00 10 − 00 Λ2
s(s m2) 3m5 3m3 s m2
+m2ψ ln( − Q ) ϕ (u )+(3 4β+β2)f m2m ( Q Q)ln( − Q)
Q 04 m2QΛ2 } !) σ 0 − π π s( 2M8 − 4M6 Λ2
i
1 s m2
+18 [2ψ ψ ψ 3ψ +3ψ +ψ +2(ψ ψ 3ψ +3ψ +2ψ +ψ )ln( − Q)] A(u )
m3Q 00− 01− 02− 10 21 22 00− 03− 10 21 22 23 Λ2 ) 0
( 5+2β+3β2) m2 3 s m2
2(1+7β+β2)f u (ψ +ψ )ϕ′ (u )+ − m u µ (µ˜2 1)( Q + )ln( − Q)ψ
− π 0 10 21 π 0 ( M2 s 0 π π − M2 2 Λ2 00
1
+ (β 1)(µ˜2 1) (3β 1)m (ψ 3ψ 3ψ )+3m (5+3β)[(20γ 1)ψ +ψ 2ψ ψ +ψ ]
6m2Q − π − − Q 00− 10− 21 s E − 00 10− 12− 20 31
s m2
+12m (5+3β)(5ψ 5ψ 15ψ +30ψ +24ψ +12ψ +4ψ )ln( − Q) ϕ′ (u )
s 00− 03− 10 20 31 32 33 Λ2 !) σ 0
1 2m
+ (β 1)m m 3f m2(7β 13)(η 2η ) m µ (10+6β)(2η′ η′) Qµ2[m f (2β 14)(η 2η )
M4 − Q s( π π − 1− 2 − Q π 6− 5 − M2 π Q π − 1− 2
s m2 2
u (3β 9)ζ]+6u (5+3β)(2η η ) ln( − Q) (β 1) 3m m f µ2 (18 6β)u ζ[2ψ ψ
− 0 − 0 6− 5 ) Λ2 − m4Q − ( Q s π π − 0 00− 01
s m2
ψ 3ψ +3ψ +ψ +2(ψ ψ 3ψ +3ψ +2ψ +ψ )ln( − Q)]+(η 2η ) (1+5β)γ ψ
− 02− 10 21 22 00− 03− 10 21 22 23 Λ2 1− 2 E 00
+(15+5β)(ψ ψ ψ ) (ψ +3ψ 4ψ 36ψ +15ψ +15ψ +36ψ +20ψ +h4ψ
10 11 21 00 02 03 10 11 12 21 22 23
− − − − −
s m2
+β[5ψ +15ψ 20ψ 36ψ +3ψ +3ψ +36ψ +28ψ +20ψ ])ln( − Q)
00 02− 03− 10 11 12 21 22 23 Λ2 !
i
s m2
+µ m2(η′ 2η′) m (3β 1)ψ 18m (5+3β)[γ ψ (ψ ψ 2ψ +3ψ +3ψ +ψ )ln( − Q)]
π Q 5− 6 Q − 00− s E 00− 00− 02− 10 20 31 32 Λ2
h i
36m2m u (5+3β)(η 2η ) 2ψ ψ ψ 3ψ +3ψ +ψ +2(ψ ψ 3ψ +3ψ +2ψ
− π s 0 5− 6 00− 01− 02− 10 21 22 00− 03− 10 21 22
s m2 h
+ψ )ln( − Q) , (16)
23 Λ2
!)#
i
10
and
f m2 s¯s 3
Γ = π π m m γ M2( 1+4β+5β2)(η 2η )+ h i f m2m2m [m2(1+6β+β2)
32√2π2 Q s E − 1− 2 1728√2" π(M4 0 Q s Q
4u2m2(1+7β+β2)]+6m [m2(β2 1)+3m m (1+6β+β2)] 72m u2m2(1+7β+β2)
− 0 π Q 0 − Q s − s 0 π
m2
+ 0[9m3(3 4β+β2)+m2m ( 11+10β 11β2)+2m u2m2(13+46β+13β2)] ϕ (u )
M2 Q − Q s − − s 0 π ) π 0
+µ (µ˜2 1) 12( 5+2β+3β2)(m2 u2m2)+ 1 (β 1) m20m3Qms(3β 1)(m2 u2m )
π π − ( − Q− 0 π M2 − M4 − Q− 0 π
m2m
0 Q[m4(15+9β)+m2m (7β 5) m u2m2(15+3β) m u2m2(β 3)] 6m3m (3β 1)
− M2 Q Q s − − Q 0 π − s 0 π − − Q s −
m2m2(β 1)+m m2m (β 3)+2u2m2[m2(5+7β)+m m (9β 3)] ϕ (u )
− Q 0 − Q 0 s − 0 π 0 Q s − !) σ 0
+m2πfπ 3m20m4Qms[m2(1+6β+β2) 4u2m2(1+7β+β2)]+ 3[m m2(β2 1)+6m3(3 4β+β2)
M2 ( 4M6 Q − 0 π 2 Q 0 − Q −
m2
+12m u2m2(1+7β+β2)]+ Q [ 9m2m (1+6β+β2)+6m2m (5 6β+β2)m2m ( 11+10β 11β2)
s 0 π 2M2 − Q s 0 Q − 0 s − −
m2m2
+36m u2m2(1+7β+β2)]+ Q 0[ 9m3(3 4β+β2)+2m2m (1 32β+β2)
s 0 π 4M4 − Q − Q s −
m2
2m u2m2(7+4β+7β2)] A(u ) m2m u f (13+46β+13β2)+6 Q(1+7β+β2) ϕ′ (u )
− s 0 π ) 0 − 0 s 0 π( M2 ) π 0
m2m3m
+µ (µ˜2 1)u (β 1) m2(7+5β) 3m m (3β 1)+ 0 Q s(3β 1)
π π − 0 − (− 0 − Q s − 2M4 −
m2m m2 m2m4
0 Q[3m (5+3β)+m (β 3)] ϕ′ (u )+m u f m2 9(1+7β+β2)(1+ Q 0 Q)
− 2M2 Q s − ) σ 0 s 0 π π(− M2 − 6M6
4m2m2 3m2m2m f m2
+ 0 Q(7+4β+7β2) A′(u ) 0 Q s π π m2[(7 2β+7β2)η +u (3+22β+3β2)ζ]
M4 ) 0 − M6 ( Q − 1 0
3
u2m2[6(1+β+β2)η (5+14β+5β2)η ] + f m2 m m2[3( 5+4β+β2)(η 2η )
− 0 π 1− 2 ) M2( π π Q 0 − 1− 2
2u (β2 1)ζ]+6m2m [(7 2β+7β2)η +u (3+22β+3β2)ζ] 12m m u2m2[6(1+β+β2)η
− 0 − Q s − 1 0 − Q s 0 π 1
(5+14β+5β2)η ] +3m2m2µ ( 5+2β+3β2)(2η′ η′)
− 2 ! 0 Q π − 6− 5 )
3m2m2m2
0 Q π f 3m ( 5+4β+β2)(η 2η )+3m u (3 4β+β2)ζ+2m u (1+7β+β2)(2η′ η′)
− M4 ( π Q − 1− 2 Q 0 − s 0 4− 3
m (5+14β+5β2)η′ 2m (7 2β+7β2)η +m u (1+6β+β2)ζ 6u µ ( 5+2β+3β2)(2η η )
− s 7− s − 1 s 0 !− 0 π − 6− 5 )
+18m2 f 2m u (1+7β+β2)(2η′ η′) m u (5+14β+5β2)η′ +m u (3+22β+3β2)ζ
π( π s 0 4− 3 − s 0 7 s 0
+2m ( 5+4β+β2)(η 2η )+2m u (3 4β+β2)ζ 4u µ ( 5+2β+3β2)(2η η )
Q 1 2 Q 0 0 π 6 5
− − − !− − − )#
g2G2 18γ M2 m3 Λ2
+ h i m f (3 4β+β2) m + E Q[2+3ln( )] ϕ (u )
4608√2π2" s π − ( Q mQ − M2 m2Q ) π 0
4M2γ m2 Λ2 1
+µ (µ˜π2 1) m ( 5+2β+3β2) E + Q [2+3ln( )](m2 u2m2)+ [10m2 6u2m2
π − ( s − − m2Q 3M4 m2Q Q− 0 π 3M2 Q− 0 π
Λ2 (β 1)
3u2m2γ 6u2m2ln( )] + − m2[m (3β 1)+9m (5+3β)] u2m2[m (3β 1)
− 0 π E − 0 π m2Q ! 3m2Q Q Q − s − 0 π Q −