g
QQcoupling constant via light cone QCD sum rules
K. Azizi,1,*M. Bayar,2,†A. Ozpineci,3,‡and Y. Sarac4,x
1Faculty of Arts and Sciences, Department of Physics, Dog˘us¸ University, Acibadem, Kadikoy, 34722, Istanbul, Turkey 2Department of Physics, Kocaeli University, 41380 Izmit, Turkey
3Physics Department, Middle East Technical University, 06531, Ankara, Turkey 4Electrical and Electronics Engineering Department, Atilim University, 06836, Ankara, Turkey
(Received 1 August 2010; published 11 October 2010)
Using the most general form of the interpolating currents, the coupling constants gbband gccare
calculated within the light cone QCD sum rules approach. It is found that gcc¼ 8:0 1:7 and
gbb¼ 11:0 2:1.
DOI:10.1103/PhysRevD.82.076004 PACS numbers: 11.55.Hx, 13.75.Gx
I. INTRODUCTION
Theoretical studies on heavy baryons containing a single heavy quark have gained pace recently as a result of the experimental progresses in the last few years (see [1] and references therein). The presence of a heavy b or c quark in these baryons makes them more attractive, theoretically. They can provide information on the structure of the QCD and its parameters as well as clues on new physics effects. For this reason, it is important to understand their proper-ties as precisely as possible. Couplings of these baryons to pions are one of the important properties, also because their pion cloud can significantly modify their properties.
In light of the experimental developments, the mass spec-trum of these baryons has been studied extensively via QCD sum rules method, both in the finite mass limit (see e.g. [2]) and heavy quark limit [3], in the heavy quark effective theory [4] and different phenomenological models (see for example [5–10]). Electromagnetic [11–17] and strong [18–21] interactions are two of other important character-istic of hadrons that can be used to probe their structures.
In the present work, the coupling of theQbaryons Q ¼
c or b to pion is calculated in the light cone QCD sum rules (LCQSR) framework. This problem has been studied using chiral perturbation theory in [21]. Note that the same framework has also been used to calculate the strong coupling constants of the light mesons with heavy baryons [18–20]. The layout of the paper is as follows: in the next section, we obtain sum rules for the corresponding cou-pling constants in the framework of LCQSR. Section III
contains our numerical analysis and discussion. II. LIGHT CONE QCD SUM RULES FOR
THE gQQCOUPLING CONSTANT
To study the coupling constant in LCQSR method, the following correlation function is studied:
¼ iZ d4xeipxhðqÞ j T f ud QðxÞ uu Qð0Þg j 0i; (1) where q1q2
Q ðxÞ is the interpolating current of q1q2 Q baryon
containing the quarks q1q2Q (qi¼ u, or d) and T stands
for the time ordered product. The correlator can be calcu-lated in terms of hadronic parameters inserting a complete set of hadronic states. This expression is called the phe-nomenological or physical side. It can also be calculated in theoretical side in terms of QCD parameters via operator product expansion (OPE) in deep Euclidean region, where p2 ! 1 and ðp þ qÞ2 ! 1. QCD sum rules for the
considered coupling constant is obtained matching both sides of the correlation function and applying Borel trans-formation in order to suppress contribution of the higher states and continuum.
First, we focus our attention to calculate the physical side of the correlation function. The phenomenological side of the sum rules for the heavy baryons is similar to the light baryons [22]. For this aim, we insert a complete set of hadronic state having the same quantum numbers of the interpolating currents into the correlator. As a result, we obtain ¼h0 j Q j Qðp2Þi p22 m2 Q hQðp2ÞðqÞ j Qðp1Þi hQðp1Þ j Q j 0i p21 m2 Q þ . . . ; (2) where p1¼ p þ q and p2¼ p and . . . denotes the con-tribution of the higher states and continuum. The matrix element creating baryon from the vacuum is parameterized as
h0 j Q j Qðp; sÞi ¼ QuQðp; sÞ; (3) where Q is the residue and uQ is the spinor for theQ
baryon. The remaining matrix element appearing in Eq. (2) defines the coupling constant gQQthrough
hQðp2ÞðqÞ j Qðp1Þi ¼ gQQuðp2Þi5uðp1Þ: (4)
*kazizi@dogus.edu.tr
†melahat.bayar@kocaeli.edu.tr ‡ozpineci@metu.edu.tr x
Using the Eqs. (3) and (4) in Eq. (2) we obtain the phenomenological side of the correlator as
¼ i gQQjQj 2 ðp2 1 m2QÞðp 2 2 m2QÞ ½qp5 mQq5: (5) In principle, one can choose any of the structures qp5and q5 existing in Eq. (5). Our calculations show that the structure qp5 leads to a more reliable result.
After obtaining the physical side, we proceed to calcu-late the QCD side of the correlation function. For this purpose, we use the interpolating currents in the following general form: ud Q ¼ 1p ffiffiffi2 abc½ðuT aCQbÞ5dcþ ðuTaC5QbÞdc ðQT aCdbÞ5uc ðQTaC5dbÞuc; uu Q ¼ 12 abc½ðuT aCQbÞ5ucþ ðuTaC5QbÞuc ðQT aCubÞ5uc ðQTaC5ubÞuc: (6)
In Eq. (6), the Q denotes the heavy quarks b or c, is an arbitrary parameter with ¼ 1 corresponding to the Ioffe current, C is the charge conjugation operator, and a, b, and c represent the color indices.
To proceed in our calculations in QCD side, we need also the propagators of the heavy and light quarks. They are given as [23]
SQðxÞ ¼ SfreeQ ðxÞ igsZ d 4k ð2Þ4eikx Z1 0 dv k þ m Q ðm2 Q k2Þ2 GðvxÞ þ 1 m2Q k2vxG ; SqðxÞ ¼ Sfreeq ðxÞ hqqi 12 x2 192m20hqqi igs Z1 0 du x 162x2 GðuxÞ uxGðuxÞ i 42x2 : (7)
The free light and heavy quark propagators in Eq. (7) are given in x representation as Sfreeq ¼ ix 22x4; SfreeQ ¼ m 2 Q 42 K1ðmQ ffiffiffiffiffiffiffiffiffi x2 p Þ ffiffiffiffiffiffiffiffiffi x2 p i m2Qx 42x2K2ðmQ ffiffiffiffiffiffiffiffiffi x2 p Þ; (8)
where Kiare the Bessel functions. Besides the propagators, the matrix elements of the form hðqÞjqðx1Þiqðx2Þj0i are also required. Hereirepresents any member of the Dirac
basis i.e., f1; ; =
ffiffiffi 2 p
; i5; 5g. The matrix elements
hðqÞjqðx1Þiqðx2Þj0i are parameterized in terms of the
pion light cone distribution amplitudes, and they are given explicitly as [24,25] hðpÞjqðxÞ5qð0Þj0i ¼ ifp Z1 0 due iupx ’ðuÞ þ 116m2x2AðuÞ i 2fm2 x px Z1 0 due iupxBðuÞ; hðpÞjqðxÞi5qð0Þj0i ¼ Z1 0 due iupx’ PðuÞ; hðpÞjqðxÞ 5qð0Þj0i ¼ i 6ð1 ~2Þðpx pxÞ Z1 0 due iupx’ ðuÞ; hðpÞjqðxÞ 5gsGðvxÞqð0Þj0i ¼ i pp g 1 pxðpxþ pxÞ pp g 1 pxðpxþ pxÞ pp g 1 pxðpxþ pxÞ þ pp g 1 pxðpxþ pxÞ Z DeiðqþvgÞpxT ð iÞ; hðpÞjqðxÞ5gsGðvxÞqð0Þj0i ¼ pðpx pxÞ 1 pxfm 2 Z DeiðqþvgÞpxAkð iÞ þp g 1 pxðpxþ pxÞ pg 1 pxðpxþ pxÞ fm2 Z DeiðqþvgÞpxA ?ðiÞ; hðpÞjqðxÞigsGðvxÞqð0Þj0i ¼ pðpx pxÞ 1 pxfm 2 Z DeiðqþvgÞpxV kðiÞ þ p g 1 pxðpxþ pxÞ p g 1 pxðpxþ pxÞ fm2 Z DeiðqþvgÞpxV ?ðiÞ; (9)
where ¼ f m2
muþmd; ~¼ muþmd
m , D ¼
dqdqdg ð1 q q gÞ and the ’ðuÞ, AðuÞ, BðuÞ, ’PðuÞ, ’ ðuÞ, T ðiÞ, A?ðiÞ, AkðiÞ, V?ðiÞ,
andVkðiÞ are functions of definite twist, and their
ex-pressions will be given in the numerical analysis section. Using these inputs, the QCD side of the correlator can be calculated in a straightforward fashion.
Having calculated the correlation function both in physi-cal and QCD sides, we equate the coefficients of the selected structure from both representations and apply Borel transformation with respect to variables p2 and
ðp þ qÞ2 to suppress the contribution of higher states and
continuum. As a result of these procedures, the QCD sum rules for coupling constant gQQ is obtained as
eðmQ=M2Þm QjQj 2g QQ ¼Zs0 m2Q eðs=M2Þðm2=4M2Þ ðsÞds þ eðm2Q=M2Þðm2=4M2Þ; (10) where the functions ðsÞ and are given as
ðsÞ ¼ 1 64pffiffiffi22 2ð1 Þ2m3 Qf 2c10c20þc31 2 ln s m2Q ’ðu0Þ 2ð1 2Þðc20c31Þ m2 Qð1 þ ~2Þ’ ðu0Þ þ 2mQð1 Þ2ðc10c21ÞAðu0Þfm2þ 2ð1 Þfð1 ÞmQf½2ðc10c21Þ ð3þ 1 2Þ ð1 2Þ ln s m2Q m2þ ð1 þ Þ m2Qð04 05Þðc10c20þc31 ln s m2Q þ 4ðc10c11c12þc21Þu0ð04 205Þm2m 2ð1 þ ÞmQln s m2Q ½ð1 Þð1 22Þfm2 þ ð1 þ ÞmQð04 205Þm þ 2ð1 þ Þf2ð1 þ Þðc10þc21ÞmQð3þ 1 2Þfm2 þ ð1 þ Þ½ðc10c20þc31Þm2Qð 0 4 205Þ þ 4ðc10c11c12þc21Þu0ð4 25Þm2mg þ huui 12p ð1 þ ffiffiffi2 2Þc00f’ðu0Þ; (11) ¼ huui 288p ð6mffiffiffi2 2Qþ 1Þð1 þ 2Þm20f’ðu0Þ huui 216pffiffiffi2M4½ð1 þ ~ 2 Þmð6M4mQð3 þ 2 þ 62Þ 3 2m20m3Qð3 þ 2 þ 62Þ þ M2m20mQð5 þ 4 þ 52ÞÞ’ ðu0Þ huui 1152pffiffiffi2M6ð1 Þ 2f m2½24M6 6m20m4Q þ 5M2m2 0m2Qþ 24M4m2QAðu0Þ þ huui 96pffiffiffi2M4ð1 þ 2Þð 3 1Þfm2ðm20m2Q 4M4Þ; (12) where j¼ Z Di Z1 0 dvfjðiÞ ðqþ ð1 vÞg u0Þ; 0j¼Z Di Z1 0 dvfjðiÞ 0ð qþ ð1 vÞg u0Þ; cnm¼ ðs m2 QÞn smðm2 QÞnm ; (13) and f1ðiÞ ¼ VkðiÞ, f2ðiÞ ¼ V?ðiÞ, f3ðiÞ ¼
AkðiÞ, f4ðiÞ ¼ T ðiÞ, f5ðiÞ ¼ vT ðiÞ are the pion
distribution amplitudes. Note that, in the above equations, the Borel parameter M2 is defined as M2 ¼ M21M22
M21þM22 and
u0¼ M21
M12þM22. Since the masses of the initial and final
baryons are the same, we can set M21 ¼ M22 and u0 ¼12. As it is clear from the expression of the coupling con-stant, we need also the residue Q, which is calculated using mass sum rules [20]:
2 Qe m2 Q=M2 ¼Zs0 m2Q es=M2 1ðsÞds þ em2Q=M2 1; (14) with 1ðsÞ ¼ ðh ddi þ huuiÞð 2 1Þ 642 m20 4mQ ð6c00 13c02 6c11Þ þ 3mQð2c10c11c12þ 2c21Þ þ m4Q 20484 5 þ ð2 þ 5Þ½12c10 6c20þ 2c30 4c41þc42 12ln s m2Q ; (15)
1 ¼ð 1Þ 2 24 h ddihuui m2 Qm20 2M4 þ m20 4M2 1 ; (16)
Note that, since only the square of the residue appears in Eq. (14), only the magnitude (not the sign) of Q can be determined from mass sum rules. This indeterminacy is also carried into the coupling constant calculation, and hence sum rules determine the coupling constant up to a sign.
III. NUMERICAL ANALYSIS
This section is devoted to the numerical analysis of the coupling constant gQQ. The numerical values of the required input parameters are given as hqqið1 GeVÞ ¼ ð0:243Þ3 GeV3, m
b¼ 4:7 GeV, mc ¼ 1:27 GeV,
mb ¼ 5:805 GeV,mc ¼ 2:455 GeV, m20ð1 GeVÞ ¼ ð0:8 0:2Þ GeV2 [26], f
¼ 0:131 [24,26], and m¼
0:135 GeV. In order to calculate the coupling constant, the -meson wave functions are needed, and explicit forms of the these wave functions are represented as [24,25]
ðuÞ ¼ 6u uð1 þ a 1C1ð2u 1Þ þ a2C3=22 ð2u 1ÞÞ; T ðiÞ ¼ 3603qq2g 1 þ w312ð7g 3Þ ; PðuÞ ¼ 1 þ 303 522 C1=22 ð2u 1Þ þ 33w3 27202 81102a2 C1=24 ð2u 1Þ; ðuÞ ¼ 6u u 1 þ53 123w3 7202 352a2 C3=22 ð2u 1Þ ; VkðiÞ ¼ 120qqgðv00þ v10ð3g 1ÞÞ; AkðiÞ ¼ 120qqgð0 þ a10ðq qÞÞ; V?ðiÞ ¼ 302g h00ð1 gÞ þ h01ðgð1 gÞ 6qqÞ þ h10ðgð1 gÞ 3 2ð2qþ 2qÞ ; A?ðiÞ ¼ 302gðq qÞ h00þ h01gþ 1 2h10ð5g 3Þ ; BðuÞ ¼ gðuÞ ðuÞ;
gðuÞ ¼ g0C1=20 ð2u 1Þ þ g2C1=22 ð2u 1Þ þ g4C1=24 ð2u 1Þ;
AðuÞ ¼ 6u u16 15þ 2435a2 þ 203þ 209 4þ 1 15þ 116 7273w3 10274 C3=22 ð2u 1Þ þ 11 210a2 41353w3 C3=24 ð2u 1Þ þ 18 5 a2 þ 214w4
½2u3ð10 15u þ 6u2Þ lnu
þ 2 u3ð10 15 u þ 6 u2Þ ln u þ u uð2 þ 13u uÞ; (17)
where Ck
nðxÞ are the Gegenbauer polynomials
h00¼ v00¼ 1
34; a10¼ 218 4w4 920a2; v10¼ 218 4w4; h01¼ 744w4 320a2;
h10¼ 7
44w4þ 320a2; g0¼ 1; g2 ¼ 1 þ187 a2 þ 603þ 203 4; g4 ¼ 928a2 63w3:
(18)
The constants in the Eqs. (17) and (18) were calculated at the renormalization scale ¼1 GeV2 using QCD sum rules [24,25,27] and are given as a
1 ¼ 0, a2 ¼ 0:44, 3 ¼
0:015, 4 ¼ 10, w3¼ 3, and w4¼ 0:2.
The sum rules for the coupling constants contain three auxiliary parameters, the Borel mass parameter M2, the continuum threshold s0, and the general parameter . These are not physical parameters, hence the sum rules for the coupling constants should be independent of them. Therefore, we should look for the working regions for these
parameters such that the dependency on these parameters is weak. At too large values of the Borel parameter M2, the suppression of the contribution of higher states and con-tinuum is reduced, increasing the error due to the quark-hadron duality approximation. The lower limit of the M2 can be obtained by requiring that the highest twist contri-bution, which is suppressed by a higher power of M2, should be small. Therefore, the higher states and continuum contributions should comprise a small percentage of the total dispersion integral. The continuum threshold s0 is not
completely arbitrary and is related to the energy of the first exited state. From the numerical analysis of the sum rules for the coupling constants, the continuum thresholds s0are obtained as 38 GeV2 s0 42 GeV2 and 9 GeV2 s0 11 GeV2 for the bb and cc vertexes,
re-spectively. In order to acquire the working region for parameter, we plot the gQQas a function ofcos in the interval 1 cos 1, which corresponds to 1 1, wheretan ¼ . From the Figs.1and2), which show the dependency of coupling constants on thecos, it is seen that for 0:5 cos 0:4 the dependence of the coupling constants on cos is weak. We present also the dependency of the coupling constants gcc and gbb on the Borel parameter M2 in Figs.3and4, respectively.
These Figures reveal that the coupling constants are quite stable in the chosen Borel mass region. From these Figures, the numerical values of the coupling constants gccand
gbbcan be extracted as follows:
gcc¼ 8:0 1:7; gbb¼ 11:0 2:1: (19) The errors appearing in our predictions are due to the uncertainties in the input as well as the auxiliary parameters.
At the end of this section, let us compare our results with the existing prediction on the gþ0þ coupling constant [22] for the light baryon. In light case the s quark and in
FIG. 1. The dependence of the coupling constant gcc on
cos for the value of Borel parameter M2¼ 7 GeV2 and the
different values of continuum threshold s0.
FIG. 2. The dependence of the coupling constant gbb on
cos for the value of Borel parameter M2¼ 30 GeV2 and the
different values of continuum threshold s0.
FIG. 4. The dependence of the coupling constant gbbon the
Borel parameter M2for the value of arbitrary parameter ¼ 3 and the different values of the continuum threshold s0. FIG. 3. The dependence of the coupling constant gccon the
Borel parameter M2for the value of arbitrary parameter ¼ 3 and the different values of the continuum threshold s0. gQQCOUPLING CONSTANT VIA LIGHT. . . PHYSICAL REVIEW D 82, 076004 (2010)
our case (baryons with a single heavy quark) the c or b quark is the spectator quark. These quarks do not partici-pate in the considered strong interactions, so one expects these coupling constants (heavy and light cases) have values close to each other. In [22], it was estimated that gþ0þ¼ 9 2. The same transition for the light baryon was also studied in [28,29]. In [28], the ratio gþ0þ=gNN’ 0:8 was obtained, which leads to the value of the coupling constant gþ0þ’ 11:9, with the experimental value of gNN¼ 14:9 [30]. The value of the
same coupling constant was given in [29] as gþ0þ¼ 11:9 0:4. Comparison of our results with that of [22,28,29] supports the expectation that the spectator quark does not effect the pionic coupling, significantly.
ACKNOWLEDGMENTS
The work has been supported in part by the European Union (HadronPhysics2 project ‘‘Study of strongly inter-acting matter’’).
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