Analysis of the strong coupling constant $G_{D_{s}^{*}D_{s}\phi}$ and the decay width of $D_{s}^{*}\rightarrow D_{s}\gamma$ with QCD sum rules

0.9 MB·English

by  G. L. Yu

Analysis of the strong coupling constant G and the decay width of
Ds∗Dsφ
D∗ D γ with QCD sum rules
s → s
Guo-Liang Yu1,∗ Zhen-Yu Li2, and Zhi-Gang Wang1†
1 Department of Mathematics and Physics, North China Electric power university,
Baoding 071003, People’s Republic of China
2 School of Physics and Electronic Science, Guizhou Normal College,
5 Guiyang 550018, People’s Republic of China
1
0 (Dated: February 15, 2015)
2
n In this article, we calculate the form factors and the coupling constant of the vertex Ds∗Dsφ
a using the three-point QCD sum rules. We consider the contributions of the vacuum condensates
J
up to dimension 7 in the operator product expansion(OPE). And all possible off-shell cases are
8
1 considered,φ,Ds andDs∗,resultinginthreedifferentformfactors. Thenwefittheformfactorsinto

] analyticalfunctionsandextrapolatethemintotime-likeregions,whichgivingthecouplingconstant
h
for the process. Our analysis indicates that the coupling constant for this vertex is G =
p Ds∗Dsφ
- 4.12±0.70GeV−1. Theresultsof thiswork are veryusefulin theotherphenomenological analysis.
p
e Asanapplication, wecalculatethecouplingconstantforthedecaychannelDs∗ →Dsγ andanalyze
h
[ the width of this decay with the assumption of the vector meson dominance of the intermediate

1 φ(1020). Ourfinalresult about thedecay width of this decay channelis Γ=0.59±0.15keV.
v
8
PACSnumbers: 13.25.Ft;14.40.Lb
9
6
1
0 1 Introduction
.
2
In relativistic heavy ion collisions J/ψ suppression has been recognized as an important tool to
0
5
identify the possiblephasetransitionto quark-gluonplasma[1]. ThedissociationofJ/ψ inthe quark-
1
: gluonplasmadueto colorscreeningcanleadtoareductionofitsproduction. Peopleusallyexplained
v
i this phenomenon as a process of the J/ψ absorption by π, ρ or φ mesons in a meson-exchange
X
r model[2]. And we can calculate the the absorption cross sections based on the interractions among
a
the quarkonia and mesons, where the hadronic coupling constants are basic input parameters. A
detailed knowledge of the hadronic coupling constants is of great importance in understanding the
effectsofheavyquarkoniumabsorptionsinhadronicmatter. Besides,thehadroniccouplingconstants
about the heavy-light mesons can also help us understanding the final-state interacions in the heavy
quarkoniumdecays[3]. Furthermore,someexoticmesonshavebeendetectedinrecentyears[4],which
are beyond the usual quark-modeldescription as qq pairs. And people interpret them as quark-gluon
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
hybrids (qqg), tetraquark states (qqqq), molecular states of two ordinary mesons, glueballs, states
with exotic quantum numbers and many others[5]. The form factors and coupling constants play an
important role in understanding the nature of these exotic mesons.
However, the strong coupling constant used in the above questions can not be explained by per-
turbative theories, because the associate interactions lie in the low energy region. It is fortunate that
the QCDSR approach can help us to solve the difficulty. The QCDSR is one of the most powerful
non-perturbativemethods, whichis alsoindependent ofmodelparameters. Inrecentyears,numerous
research articles have been reported about the precise determination of the strong form factors and
coupling constants via QCDSR, light-cone QCDSR or lattice calculation[6–8]. And many strong cou-
pling constants have been determined by different groups, for example, D∗D K, Ds∗DK, B∗B Υ,
s c c
B∗B ψ, B∗BK, J/ψD∗D , J/ψD D , J/ψD∗D∗, D∗D η′[6,9,10]. Inthis work,we usethe QCDSR
c c s s s s s s s s s
formalism to obtain the coupling constant of the meson vertice D∗D φ, where the contributions of
s s
the vacuum condensates up to dimension 7 in the OPE are considered. The results of this work are
very useful in these phenomenological analysis mentioned above.
It is indicated by the BaBar collaboration that Γ(D∗) < 1.9MeV and Γ(Ds∗→Dsγ) 0.94[11].
s ΓTotal ≈
However, the exact value of the decay width have yet not been determined. A more exact result can
help us understanding the nature of the meson and testing the validity of the theoretical model. As
an application, we also give an analysis about the decay D∗ D γ in the end of this paper, where
s → s
the electromagnetic coupling constant GDs∗Dsγ will be used. This coupling constant can be easily
obtained, when we set Q2 =0 in the analytical function of coupling constant GDs∗Dsφ(Q2) in Sec.III.
Theoutline ofthispaperisasfollows. InSec.II,westudy theD∗D φverticesusingthe three-point
s s
QCDSR. In order to reduce the uncertainties of the result, we calculate the three-point correlation
functions: one with the vector meson φ off-shell, another with the pseudoscalar meson D off-shell,
s
and a third one with the vector meson D∗ off-shell. Besides of the perturbative contribution, we also
s
considerthe contributionof qq , qgσ.Gq , g2G2 , f3G3 , qq 2 and qq GG atOPEside. InSec.
h i h i h i h i h i h ih i
III, we present the numerical results and discussions, and Sec IV is reserved for our conclusions.
2 QCD sum rules for the D∗D φ
s s
In this work, the D∗D φ is a vector-pseudoscalar-vector(VPV) vertex. With each meson off-shell,
s s
we write down the three-point correlationfunctions:
Πφ (p,p′)=i2 d4xd4yeip′.x+i(p−p′).y 0T J (x)j (y)J†(0) 0 (1)
µν | { 5 µ ν } |
Z
ΠDs(p,p′)=i2 d4xd4yeip′.x+i(p−p′).y(cid:10)0T J (x)J (y)j†(0) 0(cid:11) (2)
µν | { ν 5 µ } |
Z
ΠDs∗(p,p′)=i2 d4xd4yeip′.x+i(p−p′).y(cid:10)0T j (x)J (y)J†(0) 0(cid:11) (3)
µν | { µ ν 5 } |
Z D E
3
where T is the time orderedproduct and J†(x), J (x) and j (x) are the interpolating currents of the
ν 5 µ
mesons D∗, D and φ respectively:
s s
J†(x)=s¯(x)γ c(x) (4)
ν ν
J (x)=c¯(x)iγ s(x) (5)
5 5
j (x)=s¯(x)γ s(x) (6)
µ µ
According to the QCDSR, these correlation functions can be calculated in two different ways: using
hadrondegreesoffreedom,calledthephenomenologicalside,orusingquarkdegreesoffreedom,called
the OPE side. In the following we will obtain the sum rule according to above formulations.
2.1 The phenomenological side
We insert a complete set of intermediate hadronic states with the same quantum numbers as the
current operators Jν†(x), J5(x) and jµ(x) into the correlation functions Πφµν(p,p′), ΠDµνs(p,p′) and
ΠDs∗(p,p′) to obtainthe phenomenologicalrepresentations. After isolating the ground-statecontribu-
µν
tions, we get the following functions for the mesons φ, D and D∗ off-shell cases.
s s
CG(φ) (q2)pαp′βε
Πphen(φ) = − Ds∗Dsφ µναβ +h.r. (7)
µν (p2+m2 )(q2+m2)(p′2+m2 )
Ds∗ φ Ds
CG(Ds) (q2)pαp′βε
Πphen(Ds) = − Ds∗Dsφ µναβ +h.r. (8)
µν (p2+m2)(q2+m2 )(p′2+m2 )
φ Ds Ds∗
Πphen(Ds∗) = −CG(DDs∗s∗D)sφ(q2)pαp′βεµναβ +h.r. (9)
µν (p2+m2 )(q2+m2 )(p′2+m2)
Ds Ds∗ φ
where C = fDsm2DsfDs∗mDs∗fφmφ and h.r. stand for the contributions of higher resonances and contin-
(ms+mc)
uum states of each meaon. And in the derivation, we have used the following effective Lagrangian £
and definitions for the decay constants fDs∗, fDs and fφ:
£=GDs∗Dsφεαβλτ(∂αDs∗+βDs−∂λ+∂αDs∗−βDs+∂λ)φτ (10)
h0|Jν(0)|Ds∗(p)i=fDs∗mDs∗ζµ (11)
0J (0)D (p′) =f m2 /(m +m ) (12)
h | 5 | s i Ds Ds s c
0j (0)φ(q) =f m ξ (13)
µ φ φ µ
h | | i
where ζ and ξ are the polarization vectors. From Eqs.(7) (9), we can see that there is only one
µ µ
∼
tensor structure to work within the formalism of the QCDSR.
2.2 The OPE side
4
Now, we briefly outline the operator product expansion for the correlation functions Eqs.(1) (3)
∼
Firstly, we contract the quark fields in the correlation functions with Wich’s theorem.
Π(φ) = i2 d4xd4yeip′x+i(p−p′)ytr iγ smn(x y)γ snk(y 0)γ ckm(0 x) (14)
µν − { 5 − µ − ν − }
Z
Π(Ds) = i2 d4xd4yeip′x+i(p−p′)ytr γ cmn(x y)iγ snk(y 0)γ skm(0 x) (15)
µν − { ν − 5 − µ − }
Z
Π(Ds∗) = i2 d4xd4yeip′x+i(p−p′)ytr γ smn(x y)γ cnk(y 0)iγ skm(0 x) (16)
µν µ ν 5
− { − − − }
Z
Then, we replace the c and s quark propagators cij(x) and sij(x) with the corresponding full
propagators[12],
iδ x/ δ m δ ss iδ x/m ss δ x2 sg σGs iδ x2x/m sg σGs
ij ij s ij ij s ij s ij s s
S (x) = h i + h i h i + h i
ij 2π2x4 − 4π2x4 − 12 48 − 192 1152
igsGaαβtaij(x/σαβ +σαβx/) iδijx2x/gs2hssi2 δijx4hssi gs2GG hsjσµνsiiσµν
− 32π2x2 − 7776 − 27648 − 8
(cid:10) (cid:11)
s γµs γ
j i µ
h i + , (17)
− 4 ···
C (x) = i d4ke−ik.x δij gsGnαβtnij σαβ(k/+mc)+(k/+mc)σαβ
ij (2π)4 k m − 4 (k2 m2)2
Z (cid:26) − c − c
g D Gn tn(fλβα+fλαβ) g2(tatb) Ga Gb (fαβµν +fαµβν +fαµνβ)
+ s α βλ ij s ij αβ µν + ,(18)
3(k2 m2)4 − 4(k2 m2)5 ···
− c − c (cid:27)
fλαβ =(k/+m )γλ(k/+m )γα(k/+m )γβ(k/+m ) (19)
c c c c
fαβµν =(k/+m )γα(k/+m )γβ(k/+m )γµ(k/+m )γν(k/+m ) (20)
c c c c c
where g2GG = g2Gn Gnαβ ,tn = λn,theλn istheGell-Mannmatrix,D =∂ ig Gntn,andthe
h s i h s αβ i 2 α α− s α
i,j are color indices. Then we compute the integrals both in the coordinate and momentum spaces,
and obtain the correlationfunctions. Finally, the correlationfunctions can be divided into two parts:
ΠOPE(M) =Πpert(M)+Πnon−pert(M) (21)
µν µν µν
where M is the off-shell meson(M =φ,D ,D∗). Using dispersionrelations,the perturbative term for
s s
a given meson M off-shell can be written in the following form:
1 ∞ ∞ ρpert(M)(s,u,q2)
Πpert(M)(p,p′)= µν dsdu (22)
µν −4π2 (s p2)(u p′2)
Z0 Z0 − −
andthequantitiess=p2,u=p′2andq =p p′. WeputallquarklinesonmassshellusingCutkosky’s
−
rules(Fig.1 (a) and (b)) and obtain the spectral density ρpert(M)(s,u,q2)
µν
5
(a) (b)
FIG. 1: The perturbative contributions for φ, D and D∗ off-shell. The Dashed lines denote the
s s
Cutkosky cuts.
3 q2(s+u q2+2m2 2m2)
ρpert(φ)(s,u,q2) = (m m ) − s− c m ε pαp′β (23)
µν −√λ c− s λ(s,u,q2) − s µναβ
(cid:20) (cid:21)
3 (u q2)(s+u q2) 2s(u+m2 m2)
ρpert(Ds)(s,u,q2) = (m m ) − − − c − s m ε pαp′β
µν −√λ c− s λ(s,u,q2) − c µναβ
(cid:20) (cid:21)
(24)
ρpert(Ds∗)(s,u,q2) = 3 (m m )u(s+u−q2)−2u(m2c −m2s+u−q2) m ε pαp′β
µν −√λ c− s λ(s,u,q2) − s µναβ
(cid:20) (cid:21)
(25)
whereλ(s,u,q2 =(s+u q2))2 4su. Astothenon-perturbativecontributions,wetakeintoaccount
− −
the contribution of ss , sgσ.Gs , g2G2 , f3G3 , ss 2 and ss GG , which are showed explicitly
h i h i h i h i h i h ih i
inFigs2and3. ItshouldbenoticedthatastheconsequenceoftheuseofthedoubleBoreltransform,
the φ off-shellcase has only the contributions of g2G2 and f3G3 (Fig.2). Full expressionsfor these
h i h i
contributions of Figs 2 and 3 for φ, D and D∗ off-shell cases can be found in Appendix A,B and C,
s s
where the following representations will be used:
( 1)a+b+cπ2i ∞
Nabc = − dτ(τ +1)a+b+c−4τ1−b−c
m1m2m3 Γ(a)Γ(b)Γ(c)(M2)b(M2)c(M2)a−2
1 2 Z0
1 Q2 (τ +1)m2 (τ +1)m2 (τ +1)m2
exp 1 2 3 (26)
−τ M2+M2 − M2 − τM2 − τM2
(cid:26) 1 2 1 2 (cid:27)
( 1)a+b+cπ2i ∞
Iabc = − dτ(τ +1)a+b+c−5τ1−b−c
m1m2m3 Γ(a)Γ(b)Γ(c)(M2)b(M2)c+1(M2)a−3
1 2 Z0
1 Q2 (τ +1)m2 (τ +1)m2 (τ +1)m2
exp 1 2 3 (27)
−τ M2+M2 − M2 − τM2 − τM2
(cid:26) 1 2 1 2 (cid:27)
6
( 1)a+b+cπ2i ∞
Iabc = − dτ(τ +1)a+b+c−5τ1−b−c
m1m2m3 Γ(a)Γ(b)Γ(c)(M2)b+1(M2)c(M2)a−3
1 2 Z0
e exp 1 Q2 (τ +1)m21 (τ +1)m22 (τ +1)m23 (28)
−τ M2+M2 − M2 − τM2 − τM2
(cid:26) 1 2 1 2 (cid:27)
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
FIG. 2: Contributions of the condensate parts g2G2 and f3G3 for φ off-shell case
h i h i
2.3 The Coupling Constant and the Meson decay
We makethe changeofvariablesp2 P2,p′2 P′2 andq2 Q2 andperformadoubleBorel
→− →− →−
transform[13] to the physical as well as the OPE sides, which involves the transformation: P2 M2
→ 1
and P′2 M2, where M and M are the Borel parameters. Then, we equate the phenomenological
→ 2 1 2
and OPE sides, invoking the quark-hadronduality from which the sum rule is obtained.
Inordertoeliminatetheh.r.termsfromthephenomenologicalsideinEqs.(7) (9),twocontinuum
∼
thresholdparameterss andu intheOPEsideareintroduced. Theseparametersfulfillthefollowing
0 0
relations: m2 <s <m′2 and m2 <u <m′2, where m and m are the masses of the incoming and
i 0 i o 0 o i o
out-coming mesons respectively and m′ is the mass of the first excited state of these mesons. After
these performaions, the form factors can be written as:
7
(a) (b) (c) (d) (e)
(f) (g) (h) (i) (j)
(k) (l) (m) (n) (o)
(p) (q) (r) (s) (t)
(u) (v) (w) (x) (y)
(z) (aa) (bb) (cc)
FIG. 3: Contributions of the non-perturbative parts for D (D∗) off-shell case
s s
Gφ (Q2) = −4π2M112M22 ss10 uu10ρpert(φ)(s,u,Q2)e−Ms12e−Mu22dsdu+BB Πnon−pert(φ) , (29)
Ds∗Dsφ R R (Q2+mC2φ)M12M22e−m2Ds∗/M12e−m2Ds/M22 (cid:2) (cid:3)
8
GDs (Q2) = −4π2M112M22 ss10 uu10ρpert(Ds)(s,u,Q2)e−Ms12e−Mu22dsdu+BB Πnon−pert(Ds) ,(30)
Ds∗Dsφ R R (Q2+m2DCs)M12M22e−m2φ/M12e−m2Ds∗/M22 (cid:2) (cid:3)
GDs∗ (Q2) = −4π2M112M22 ss10 uu10ρpert(Ds∗)(s,u,Q2)e−Ms12e−Mu22dsdu+BB Πnon−pert(Ds∗) ,(31)
Ds∗Dsφ R R (Q2+m2DCs∗)M12M22e−m2Ds/M12e−m2φ/M22 (cid:2) (cid:3)
where BB[ ] stands for the double Borel transform. Now, we can calculate the form factors in
the space-like region according to these above equations. However, in order to obtain the coupling
constants, it is necessary to extrapolate these results into physical regions(Q2 <0), which is realized
by fit the form factors into suitable analytical functions. It is indicated that we should get the
same values for the coupling constants Gφ , GDs and GDs∗ [14], when we take Q2 = m2,
Ds∗Dsφ Ds∗Dsφ Ds∗Dsφ − φ
Q2 = m2 and Q2 = m2 separately. This above procedure is used to minimize the uncertainties
− Ds − Ds∗
in the calculation of the coupling constant, which will be quite clear in the following section.
Withthe assumptionofthe vectormesondominance(φ(1020)),the radiativedecaysD∗ D γ can
s → s
be described by the following electromagnetic lagrangian £′,
£′ = eQ sγ sAµ (32)
s µ
−
where the A , Q are the electromagnetic field and the charge number. From the lagrangian £′, we
µ s
can obtain the decay amplitude[15],
D (p)γ(q,ε)D∗(p′,ξ)
h s | s i
i
= γ(q,ε)φ(q,η) D (p)φ(q,η)D∗(p′,ξ)
h | iq2 m2 h s | s i
− φ
i
= D (p)φ(q,η)D∗(p′,ξ) f m eQ ( i)ε∗ηµ
h s | s iq2 m2 φ φ s − µ
− φ
i
= GDs∗Dsγǫαβλτp′αqλξβητ∗q2 m2fφmφeQs(−i)ε∗µηµ (33)
− φ
The parametersGDs∗Dsγ and fφ are the coupling constantand the weak decay constant, respectively.
p′ and q are the four momenta of the D and γ. ηµ, ε∗ and ξ are the polarizationvectors of the φ,
α λ s µ β
γ and Ds∗, respectively. The strongcoupling constantGDs∗Dsγ canbe relatedto the effective coupling
constant in the heavy quark effective Lagrangianby Eq.(10) in this paper.
3 Results and Discussions
Presentsectionisdevotedtothenumericalanalysisofthesumrulesforthecouplingconstants. The
decayconstantsandhadronicparametersusedinthisworkaretakenasf =0.229 0.003[16], f =
φ ± Ds
0.257±0.006[16],fDs∗ =0.301±0.013[16],mφ =1.019±0.020GeV[16],mDs =1.968±0.00032GeV[16],
mD∗ =2.112 0.0005GeV[16]. Thevacuumcondensatesaretakentobethestandardvalues<ss>=
s ±
(0.8 0.1) (0.24 0.01GeV)3[13, 17], < sg σGs >= m2 < ss >[13, 17], m2 = (0.8 0.1)GeV2,
− ± × ± s 0 0 ±
9
< g2GG >= (0.022 0.004)GeV4[18], < f3G3 >= (8.8 5.5)GeV2 < g2GG >[18]. And we also
s ± ± s
take the masses of quark m = (1.275 0.025)GeV, m = 0.095 0.005GeV from the Particle Data
c s
± ±
Group[16]. The continuum parameters, s and u in Eqs.(29) (31), are defined as s = (m + )2
0 0 0 i i
∼ △
and u = (m + )2, where the quantities and are determined imposing the most stable
0 o o i o
△ △ △
Borel window. In order to include the pole and to exclude the h.r. contributions for the cases of φ,
Ds and Ds∗ mesons off-shell, the values for △φ, △Ds and △Ds∗ can not be far from the experimental
value of the distance between the pole and the first excited state[13]. In addtion, the results of
the form factors in Eqs.(29) (31) should also not depend on the Borel parameters M2 and M2.
∼ 1 2
Therefore, we have to work in a region where the approximations made are supposedly acceptable
and where the results depend only moderately on the Borel variables[13]. Using the Borel region
5.0 M2 7.0GeV2 and5.0 M2 7.0GeV2 (Q2 =3.0GeV2 forφ off-shell), 6.0 M2 8.0GeV2
≤ 1 ≤ ≤ 2 ≤ ≤ 1 ≤
and 6.0 M2 8.0GeV2 (Q2 = 1.0GeV2 for D and D∗ off-shell) we found a good stability with
≤ 2 ≤ s s
△φ =△Ds =△Ds∗ =0.5GeV(Fig.4).
Fromthefigure,wecanseethatthevaluesareratherstalbewithvariationsoftheBorelparameters,
it is reliable to extract the form factors. Besides of the pertubative term, we can also see that ss
h i
give a considerable contribution for D and D∗ off-shell cases(Fig.4 (c) (f)). And the contributions
s s ∼
of the other condensate terms are small(< 1%). To the case of φ off-shell, condensate parts g2G2
h i
and f3G3 make up 1% 2% of the total contributions. It should be noticed that although these
h i ∼
condensatesterms, allexceptfor the perturbative termand ss , give smallcontributionsto the form
h i
factors,they haveasignificantinfluence onthe followinganalyticalfunctions(Eqs.(34) (36)), which
∼
are obtained by numerical fitting. Thus, these condensates contributions should not be neglected in
the calculation.
The formfactorsGφ , GDs andGDs∗ areshownexplicitly inFig.5andarefittedintothe
Ds∗Dsφ Ds∗Dsφ Ds∗Dsφ
folowing analytical functions:
Gφ =Aexp( BQ2), (34)
Ds∗Dsφ −
C
GDs = exp( EQ2), (35)
Ds∗Dsφ 1+DQ2 −
GDs∗ = C′ exp( E′Q2), (36)
Ds∗Dsφ 1+D′Q2 −
where
A=2.964 0.089GeV−1, B =0.1621 0.0077GeV−2,
± ±
C =2.755 0.008,D = 0.1944 0.0186,E =0.256 0.0265,
± − ± ±
C′ =2.825 0.012GeV−1, D′ = 0.1855 0.0171GeV−2, E′ =0.2593 0.0257,
± − ± ±
Considering uncertainties of all the input parameters, such as quark and mesons masses, decay
10
2 2
1.5 1.5
A A
B B
f22G(Q=3GeV)fDs*Ds 0.15 CH f22G(Q=3GeV)fD*Dss 0.15 CH
0 0
−0.5 −0.5
5.0 5.5 6.0 6.5 7.0 5.0 5.5 6.0 6.5 7.0
M12(GeV2) M22(GeV2)
(a) (b)
5 5
A A
4.5 4.5
B B
4 C 4 C
D D
3.5 3.5
E E
2GeV) 3 FG 2GeV) 3 FG
2Q=1 2.5 H 2Q=1 2.5 H
( 2 ( 2
DsG*fDsDs 1.5 DsG*fDsDs 1.5
1 1
0.5 0.5
0 0
−0.5 −0.5
6.0 6.5 7.0 7.5 8.0 6.0 6.5 7.0 7.5 8.0
M12(GeV2) M22(GeV2)
(c) (d)
5 5
4.5 A 4.5 AA
B BB
4 C 4 CC
3.5 D 3.5 DD
E EE
22Q=1GeV) 2.35 FGH 22Q=1GeV) 2.35 FGHFGH
( 2 ( 2
*DsG*fDsDs 1.5 *DsG*fDsDs 1.5
1 1
0.5 0.5
0 0
−0.5 −0.5
6.0 6.5 7.0 7.5 8.0 6.0 6.5 7.0 7.5 8.0
M12(GeV2) M22(GeV2)
(e) (f)
FIG. 4: The contributions of different condensate terms in the OPE with variations of the Borel
parameters M2 and M2 for φ((a),(b)), D ((c),(d)) and D∗((e),(f)) off-shell, where A-H denote the
1 2 s s
perturbative term, g2G2 , f3G3 , ss , sgσ.Gs , ss 2, ss GG and Total contributions.
h i h i h i h i h i h ih i