Tree Level Semileptonic Σ b to Nucleon Decay in Light Cone QCD Sum Rules
aa r X i v : . [ h e p - ph ] J u l Tree Level Semileptonic Σ b to Nucleon Decay in Light Cone QCD Sum Rules K. Azizi ∗ , M. Bayar † , A. Ozpineci ‡ , Y. Sarac § , Physics Department, Middle East Technical University,06531, Ankara, Turkey Department of Physics,Kocaeli University, 41380 Izmit, Turkey Electrical and Electronics Engineering Department,Atilim University, 06836 Ankara, Turkey (Dated: November 6, 2018)Using the most general form of the interpolating current of the heavy spin 1/2, Σ b baryon anddistribution amplitudes of the nucleon, the transition form factors of the semileptonic Σ b → Nlν decay are calculated in the framework of light cone QCD sum rules. It is obtained that the formfactors satisfy the heavy quark effective theory relations. The obtained results for the related formfactors are used to estimate the decay rate of this transition.
PACS numbers: 11.55.Hx, 13.30.-a, 14.20.Mr, 12.39.Hg ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] I. INTRODUCTION
The baryons containing a heavy quark have been at the focus of much theoretical attention, especially since thedevelopment of the heavy quark effective theory (HQET) and its application to the spectroscopy of these baryons.The heavy quark provides a window that permits us to see further under the skin of the non-perturbative QCDas compared the light baryons. These states are expected to be narrow, so that their isolation and detection arerelatively easy. Recently, experimental studies on the spectroscopy of these baryons have been accelerated and newheavy baryons have been discovered [1, 2, 3, 4, 5, 6, 7, 8]. The Σ b channels are expected to be very rich, so itwill be possible to check its semileptoic decays like its decay to the nucleon at LHC in the near future. There aremany works in literature which are devoted to the investigation of the mass and magnetic moments of the heavybaryons using different approaches. The masses of these baryons have been discussed within QCD sum rules in[9, 10, 11, 12, 13, 14, 15, 16, 17], in heavy quark effective theory (HQET) in [18, 19, 20, 21, 22, 23, 24] and usingdifferent quark models in [25, 26, 27, 28, 29, 30, 31, 32, 33]. The magnetic dipole moment of heavy spin 1/2 and 3/2baryons as well as the transition magnetic dipole and electric quadrupole moments of heavy spin 3/2 to heavy spin1/2 baryons have been calculated in the framework of different approaches ( see for example [9, 34, 35] and referencestherein). However, the semileptonic and nonleptonic decays of the heavy baryons have not been extensively discussedin the literature comparing their mass and electromagnetic properties. Transition form factors of the Λ b → Λ c andΛ c → Λ decays have been studied in three points QCD sum rules in [36], and then used in the study of the semileptonicdecays. The Λ b → pl ¯ ν transition has also been investigated using three point QCD sum rules within the framework ofheavy quark effective theory (HQET) in [37] and using SU(3) symmetry and HQET in [38]. Hyperfine mixing and thesemileptonic decays of double-heavy baryons in a quark model [39], strong decays of heavy baryons in Bethe-Salpeterformalism [40], strong decays of charmed baryons in heavy hadron Chiral perturbation theory [41] and semileptonicdecays of some heavy baryons containing single heavy quark in different quark models [42, 43, 44] are some otherworks related to the heavy baryon decays.In the present work, we calculate the form factors related to the semileptonic decay of the Σ b → N lν transitionin the framework of the light cone QCD sum rules using the nucleon distribution amplitudes. Here, N refers to twomembers of the octet baryons, namely neutron and proton. The parameters appearing in the nucleon distributionamplitudes have been calculated using various methods. In this work, for the values of these parameters, we use theresults of QCD sum rules approach [45] and also the results which are recently obtained from lattice QCD [46, 47, 48].Analyzing of such transitions can give essential information about the internal structure of the Σ b baryon as well asaccurate calculation of the nucleon wave functions. Since the spin of the heavy baryon carries information on thespin of the heavy quark, the study of such transitions might also lead us to study the spin effects in the heavy quarksector of the standard model.The outline of the paper is as follows: in section II, using the nucleon distribution amplitudes and the most generalform of the interpolating currents for the Σ b baryon, we calculate the form factors entering to the semileptonic decayof the heavy Σ b baryon to nucleon in the framework of the light cone QCD sum rules. The heavy quark limit ofthe form factors and the relations between the form factors in this limit is also discussed in this section. Section IIIencompasses numerical analysis of the form factors, our predictions for the decay rate obtained in two different ways:first, using the DA’s obtained from QCD sum rules and second, the DA’s calculated in lattice QCD , and discussion. II. LIGHT CONE QCD SUM RULES FOR THE Σ b → N FORM FACTORS
This section is devoted to the calculation of form factors relevant for the Σ b → p and Σ − b → n transitions using thelight cone QCD sum rules approach. At quark level, these transitions are governed by the tree level b → u transition.Considering the SU(2) symmetry, the form factors of these two transitions are the same, so we will use the notationN instead of neutron and proton. The quark level transition is described by the effective Hamiltonian given by H eff = G F √ V ub ¯ uγ µ (1 − γ ) b ¯ lγ µ (1 − γ ) ν. (1)Hence, to study Σ b → N lν decay, one needs the matrix element h N | ¯ uγ µ (1 − γ ) b | Σ b i . To calculate this matrix element,following the general philosophy of QCD sum rules, we start by considering the correlation function,Π µ ( p, q ) = i Z d xe iqx h N ( p ) | T { J trµ ( x ) ¯ J Σ b (0) } | i , (2)where, J Σ b is interpolating currents of Σ b baryon, J trµ = ¯ uγ µ (1 − γ ) b is transition current and h N ( p ) | presents theproton sate. p denotes the proton momentum and q = ( p + q ) − p is the transferred momentum. To calculate theform factors, the following three steps will be applied: • The correlation function is calculated by saturating it with a tower of hadrons having the same quantum numberas the interpolating current, J Σ b called the phenomenological or physical side. • The correlation function is calculated in QCD or theoretical side via operator product expansion (OPE), wherethe short and long distance quark-gluon interactions are separated. The former is calculated using QCD per-turbation theory, whereas the latter are parameterized in terms of the light-cone distribution amplitudes of thenucleon. • The sum rules for form factors are calculated equating the two representation of the correlation function men-tioned above and applying Borel transformation to suppress the contribution of the higher states and continuum.To calculate the physical side, a complete set of hadronic state is inserted to the correlation function. Afterperforming integral over x, we obtainΠ µ ( p, q ) = X s h N ( p ) | J trµ ( x ) | Σ b ( p + q, s ) ih Σ b ( p + q, s ) | ¯ J Σ b (0) | i m b − ( p + q ) + ..., (3)where, the ... represents the contribution of the higher states and continuum. The matrix element h Σ b ( p + q, s ) | ¯ J Σ b (0) | i in (3) can be written as: h Σ b ( p + q, s ) | ¯ J Σ b (0) | i = λ Σ b ¯ u Σ b ( p + q, s ) , (4)where λ Σ b is residue of Σ b baryon. The transition matrix element, h N ( p ) | J trµ | Σ b ( p + q, s ) i is parameterized in termsof the form factors f i and g i as h N ( p ) | J trµ ( x ) | Σ b ( p + q ) i = ¯ N ( p ) (cid:2) γ µ f ( Q ) + iσ µν q ν f ( Q ) + q µ f ( Q ) + γ µ γ g ( Q ) + iσ µν γ q ν g ( Q )+ q µ γ g ( Q ) (cid:21) u Σ b ( p + q ) , (5)where Q = − q , and f i , and g i , are the form factors and N ( p ) and u Σ b ( p + q ) are the spinors of nucleon and Σ b ,respectively. Using Eqs. (3), (4) and ,(5) and summing over spins of the Σ b baryon using X s u Σ b ( p + q, s ) u Σ b ( p + q, s ) = p + q + m Σ b , (6)we obtain the following expressionΠ µ ( p, q ) = λ Σ b m b − ( p + q ) ¯ N ( p ) (cid:2) γ µ f ( Q ) + iσ µν q ν f ( Q + q µ f ( Q ) + γ µ γ g ( Q ) + iσ µν γ q ν g ( Q )+ q µ γ g ( Q ) (cid:21) ( p + q + m Σ b ) + · · · (7)Using ¯ N σ µν q ν u Σ b = ¯ N [( m N + m Σ b ) γ µ − (2 p + q ) µ ] u Σ b , (8)in Eq. (7), the final expression for the physical side of the correlation function is obtained asΠ λ ( p, q ) = λ Σ b m b − ( p + q ) ¯ N ( p ) (cid:20) f ( Q ) p µ + (cid:26) − f ( Q )( m N − m Σ b ) + f ( Q )( m N − m ) (cid:27) γ µ + (cid:26) f ( Q ) − f ( Q )( m N + m Σ b ) (cid:27) γ µ q + 2 f ( Q ) p µ q + (cid:26) f ( Q ) + f ( Q ) (cid:27) ( m N + m Σ b ) q µ + (cid:26) f ( Q ) + f ( Q ) (cid:27) q µ q − g ( Q ) p µ γ + (cid:26) g ( Q )( m N + m Σ b ) − g ( Q )( m N − m b ) (cid:27) γ µ γ − (cid:26) g ( Q ) − g ( Q )( m N − m Σ b ) (cid:27) γ µ qγ − g ( Q ) p µ qγ − (cid:26) g ( Q ) + g ( Q ) (cid:27) ( m N − m Σ b ) q µ γ − (cid:26) g ( Q ) + g ( Q ) (cid:27) q µ qγ (cid:21) + · · · (9)Among many structures appearing in Eq. (7), we chose the independent structures p µ , p µ q , q µ q , p µ γ , p µ qγ , and q µ qγ to evaluate the form factors f , f , f , g , g and g , respectively.On QCD side, to calculate the correlation function in deep Euclidean region where ( p + q ) ≪
0, we need to knowthe explicit expression for the interpolating current of the Σ b baryon. It is chosen as J Σ b ( x ) = − √ ε abc (cid:20) (cid:26) u T a ( x ) Cb b ( x ) (cid:27) γ d c ( x ) − (cid:26) b T a ( x ) Cd b ( x ) (cid:27) γ u c ( x )+ β (cid:26) { u T a ( x ) Cγ b b ( x ) } d c ( x ) − { b T a ( x ) Cγ d b ( x ) } u c ( x ) (cid:27) (cid:21) , (10)where a, b, c are the color indices and C is the charge conjugation operator and β is an arbitrary parameter with β = − J trµ = ¯ uγ µ (1 − γ ) b and J Σ b and contractingout all quark pairs applying the Wick’s theorem, we obtainΠ µ = − i √ ǫ abc Z d xe iqx (h ( C ) ηλ ( γ ) γφ − ( C ) λφ ( γ ) γη i + β " ( Cγ ) ηλ ( I ) γφ − ( Cγ ) λφ ( I ) γη (1 + γ ) γ µ i σθ S Q ( − x ) λσ h N ( p ) | ¯ u aη (0)¯ u bθ ( x ) ¯ d cφ (0) | i , (11)where, S Q ( x ) is the heavy quark propagator which is represented as [49]: S Q ( x ) = S freeQ ( x ) − ig s Z d k (2 π ) e − ikx Z dv " k + m Q ( m Q − k ) G µν ( vx ) σ µν + 1 m Q − k vx µ G µν γ ν . (12)where S freeQ = m Q π K ( m Q √− x ) √− x − i m Q x π x K ( m Q p − x ) , (13)and K i are the Bessel functions. The terms proportional to the gluon strength tensor can give contribution to fourand five particle distribution functions but they are expected to be small [50, 51, 52] and for this reason, we willneglect these amplitudes in further analysis.For the calculation of Π µ in Eq. (11), the matrix element h N ( p ) | ǫ abc ¯ u aη (0)¯ u bθ ( x ) ¯ d cφ (0) | i is required. The nucleonwave function is given as [45, 50, 51, 52, 53]:4 h | ǫ abc u aα ( a x ) u bβ ( a x ) d cγ ( a x ) | N ( p ) i = S m N C αβ ( γ N ) γ + S m N C αβ (/ xγ N ) γ + P m N ( γ C ) αβ N γ + P m N ( γ C ) αβ (/ xN ) γ + ( V + x m N V M )(/ pC ) αβ ( γ N ) γ + V m N (/ pC ) αβ (/ xγ N ) γ + V m N ( γ µ C ) αβ ( γ µ γ N ) γ + V m N (/ xC ) αβ ( γ N ) γ + V m N ( γ µ C ) αβ ( iσ µν x ν γ N ) γ + V m N (/ xC ) αβ (/ xγ N ) γ + ( A + x m N A M )(/ pγ C ) αβ N γ + A m N (/ pγ C ) αβ (/ xN ) γ + A m N ( γ µ γ C ) αβ ( γ µ N ) γ + A m N (/ xγ C ) αβ N γ + A m N ( γ µ γ C ) αβ ( iσ µν x ν N ) γ + A m N (/ xγ C ) αβ (/ xN ) γ + ( T + x m N T M )( p ν iσ µν C ) αβ ( γ µ γ N ) γ + T m N ( x µ p ν iσ µν C ) αβ ( γ N ) γ + T m N ( σ µν C ) αβ ( σ µν γ N ) γ + T m N ( p ν σ µν C ) αβ ( σ µρ x ρ γ N ) γ + T m N ( x ν iσ µν C ) αβ ( γ µ γ N ) γ + T m N ( x µ p ν iσ µν C ) αβ (/ xγ N ) γ + T m N ( σ µν C ) αβ ( σ µν / xγ N ) γ + T m N ( x ν σ µν C ) αβ ( σ µρ x ρ γ N ) γ , (14) S = S px S = S − S TABLE I: Relations between the calligraphic functions and proton scalar DA’s. P = P px P = P − P TABLE II: Relations between the calligraphic functions and proton pseudo-scalar DA’s. where, the calligraphic functions, which are functions of the scalar product px and the parameters a i , i = 1 , , F ( a i px )= S i , P i , V i , A i , T i as: F ( a i px ) = Z dx dx dx δ ( x + x + x − e − ipx Σ i x i a i F ( x i ) . (15)Here x i with i = 1 , , x the expression for the correlation function in QCD or theoretical side is obtained. Equating the correspondingstructures from both representations of the correlation function and applying Borel transformation with respect to( p + q ) to suppress the contribution of the higher states and continuum, one can obtain sum rules for the form factors f , f , f , g , g and g . Finally, to subtract the contribution of the higher states and the continuum, quark-hadronduality is assumed.In heavy quark effective theory (HQET), the heavy quark symmetry reduces the number of independent formfactors to two namely, F and F [54, 55], i.e., h N ( p ) | ¯ u Γ b | Σ b ( p + q ) i = ¯ N ( p )[ F ( Q )+ vF ( Q )]Γ u Σ b ( p + q ) , (16)where, Γ is any Dirac structure and v = p + qm Σ b . Comparison between Eq. (16) with the general definition of the formfactors in Eq. (5) leads to the following relations among the form factors in HQET limit [56, 57] g = f = F + m N m Σ b F g = f = g = f = F m Σ b (17)Our calculations show that the deviation from the relations g = f and g = f = g = f are negligible in the case ofHQET limit. However, when we consider finite mass, the violation is (10 − / for Q > Q < f and f in the Appendix–A. However, we will give the extrapolation of allform factors in finite mass in terms of Q in the numerical analysis section.From the explicit expressions of the form factors, it is clear that we need to know the expression for the residue ofthe Σ b baryon. The residue λ Σ b is determined from sum rule and its expression is given in [58] as: − λ b e − m b /M = Z s m b e − sM ρ ( s ) ds + e − m bM Γ , (18)with ρ ( s ) = ( < dd > + < uu > ) ( β − π ( m m b (6 ψ − ψ − ψ ) + 3 m b (2 ψ − ψ − ψ + 2 ψ ) ) + m b π [5 + β (2 + 5 β )][12 ψ − ψ + 2 ψ − ψ + ψ − ln ( sm b )] , (19) V = V px V = V − V − V V = V px V = − V + V + V + 2 V px V = V − V px ) V = − V + V + V + V + V − V TABLE III: Relations between the calligraphic functions and proton vector DA’s. A = A px A = − A + A − A A = A px A = − A − A − A + 2 A px A = A − A px ) A = A − A + A + A − A + A TABLE IV: Relations between the calligraphic functions and proton axial vector DA’s.
Γ = ( β − < dd >< uu > (cid:20) m b m M + m M − , (20)where, s is continuum threshold, M is the Borel mass parameter and ψ nm = ( s − m b ) n s m ( m b ) n − m are some dimensionlessfunctions. III. NUMERICAL RESULTS
This section is devoted to the numerical analysis for the form factors and total decay rate for Σ b −→ N ℓν transition.Some input parameters used in the analysis of the sum rules for the form factors are h ¯ uu i (1 GeV ) = h ¯ dd i (1 GeV ) = − (0 . GeV , m N = 0 . GeV , m b = 4 . GeV , m Σ b = 5 . GeV , and m (1 GeV ) = (0 . ± . GeV [59]. Thenucleon DA’s are the main input parameters, whose explicit expressions can be found in [45]. These DA’s contain8 independent parameters f N , λ , λ , V d , A u , f d , f u and f d . These parameters have been calculated also in [45]within the light cone QCD sum rules. Recently, most of these parameters have been calculated in the framework ofthe lattice QCD [46, 47, 48]. We will use these two sets of data from QCD sum rules and lattice QCD and for eachparameter which have not been calculated in lattice, we will use the values from QCD sum rules prediction. Theseparameters are given in Table VI.The sum rules for form factors also contain 3 auxiliary parameters namely, continuum threshold s , Borel massparameter M and general parameter β entering to the general current of the Σ b baryon. These are not physicalquantities, hence the form factors should be independent of them. Therefore, we look for working regions such thatin these regions our results are practically independent of these mathematical objects. The continuum threshold, s is not completely arbitrary and it is related to the energy of the exited states. Our numerical analysis for formfactors show that the results are weakly depend on s in the interval, ( m Σ b + 0 . ≤ s ≤ ( m Σ b + 0 . . In order toobtain the working region for β , we plot the form factors with respect to cosθ in the interval − ≤ cosθ ≤ −∞ ≤ β ≤ ∞ , where β = tanθ and look for a region at which the dependency is weak. The commonworking region for β is obtained to be − . ≤ cosθ ≤ .
6. The Ioffe current which corresponds to cosθ = − .
71 isout of this region. The similar results have been obtained in [35]. The lower limit on Borel mass squared, M isdetermined from condition that the contribution of higher states and continuum to the correlation function shouldbe enough small, i.e., the contribution of the highest term with power 1 /M is less than, say, 20–25% of the highestpower of M . The upper limit of this parameter is acquired from the condition that series of the light cone expansionwith increasing twist should be convergent. Generally, this means that the higher states, higher twists and continuumcontributions to the correlation function should be less than 40–50% of the total value. Our numerical analysis showthat both conditions are satisfied in the region 15 GeV ≤ M B ≤ GeV , which we will use in numerical analysis. T = T px T = T + T − T T = T px T = T − T − T px T = − T + T + 2 T px ) T = 2 T − T − T + 2 T + 2 T + 2 T px T = T − T px ) T = − T + T + T − T + 2 T + 2 T TABLE V: Relations between the calligraphic functions and proton tensor DA’s.QCD sum rules [45] Lattice QCD [46, 47, 48] f N (5 . ± . × − GeV (3 . ± . ± . × − GeV λ − (2 . ± . × − GeV ( − . ± . ± . × − GeV λ (5 . ± . × − GeV (7 . ± . ± . × − GeV V d . ± .
03 0 . ± . ± . A u . ± .
15 0 . ± . ± . f d . ± . − f u . ± . − f d . ± . − TABLE VI: The values of independent parameters entering to the nucleon DA’s. The first errors in lattice values are statisticaland the second errors represent the uncertainty due to the chiral extrapolation and renormalization.
Considering the above requirements, we obtained that the form factors obey the following extrapolations in terms of q : f i ( q )[ g i ( q )] = a (1 − q m fit ) + b (1 − q m fit ) , (21)The values of the parameters a, b and m fit are given in Tables VII and VIII related to the QCD sum rules andlattice QCD input parameters, respectively. These parameterizations show that increasing in the value of q leads toincreasing in the absolute value of the form factors and they have no pole inside the physical region. The values of m fit presents the pole outside the allowed region of q and related to this and accordance to mesons, one can calculatethe coupling constant g Σ b Σ ∗ b N , where, Σ ∗ b can be considered as the exited state of Σ b baryon. For detailed analysisin this respect see [60, 61, 62]. Note that, as we work near the light cone, x ≃
0, from the considered correlationfunction it is clear that our predictions at low q are not reliable and we need the above parameterization to extendthe results to full physical region. As an example, to show how the actual sum rules results, and the parameterizationfit to each other, we present the dependency of f (both actual sum rule result and fit parameterization) on q forQCD sum rules input parameters and at fixed values of auxiliary parameters in Fig. 1.The values of form factors at q = 0 is also obtained as presented in Table IX. Our next task is to calculate thetotal decay rate of Σ b −→ pℓν transition in the whole physical region, i.e., m l ≤ q ≤ ( m Σ b − m N ) . The decay widthfor such transition is given by the following expression [63, 64]Γ(Σ b → P lν l ) = G F π m b | V bu | Z m l dq (1 − m l /q ) p (Σ − q )(∆ − q ) N ( q ) (22) (GeV )-2-1.5-1-0.50 f Sum Rule ResultFit Function
FIG. 1: The dependency of f (both actual sum rule result and fit parameterization) on q for QCD sum rules input parametersat M = 25 GeV , s = 6 . GeV and β = 5. a b m fit f f f -0.09 -0.02 4.92 g g -0.02 0.015 5.96 g -0.02 -0.009 5.65TABLE VII: Parameters appearing in the fit function for QCD sum rules set of data. where N ( q ) = F ( q )(∆ (4 q − m l ) + 2Σ ∆ (1 + 2 m l /q ) − (Σ + 2 q )(2 q + m l ))+ F ( q )(∆ − q )(2Σ + q )(2 q + m l ) /m b + 3 F ( q ) m l (Σ − q ) q /m b + 6 F ( q ) F ( q )(∆ − q )(2 q + m l )Σ /m Σ b − F ( q ) F ( q ) m l (Σ − q )∆ /m Σ b + G ( q )(Σ (4 q − m l ) + 2Σ ∆ (1 + 2 m l /q ) − (∆ + 2 q )(2 q + m l ))+ G ( q )(Σ − q )(2∆ + q )(2 q + m l ) /m b + 3 G ( q ) m l (∆ − q ) q /m b − G ( q ) G ( q )(Σ − q )(2 q + m l )∆ /m Σ b + 6 G ( q ) G ( q ) m l (∆ − q )Σ /m Σ b . (23)Where F ( q ) = f ( q ), F ( q ) = m Σ b f ( q ), F ( q ) = m Σ b f ( q ), G ( q ) = g ( q ), G ( q ) = m Σ b g ( q ), G ( q ) = m Σ b g ( q ), Σ = m Σ b + m p and ∆ = m Σ b − m p . G F = 1 . × − GeV − is the Fermi coupling constant, and m l is the leptonic (electron, muon or tau) mass. For the corresponding CKM matrix element V ub = (4 . ± .
30) 10 − is used [65]. Our final results for total decay rates are given in Table X. From this Table, we see that the obtainedresults for the decay rates are in the same order of magnitudes for two sets of input parameters. The central valuesof the decay rate for e and µ obtained using the lattice QCD input parameters are about 2 times greater thanthat of the QCD sum rules input parameters while, for τ case the result obtained by sum rules input parameters isabout 1.5 time larger than the prediction acquired using the lattice input parameters. However, when we considerthe uncertainties, results obtained using both sets of input parameters coincide for all leptons. Here, we shouldstress that as we mentioned before, the Λ b → pl ¯ ν decay has been studied in three point QCD sum rules and HQET a b m fit f f f -0.06 -0.015 4.93 g g -0.03 -0.002 5.97 g -0.028 -0.009 5.95TABLE VIII: Parameters appearing in the fit function for lattice QCD set of data.For QCD sum rules input parameters For lattice QCD input parameters f (0) 0 . ± .
05 0 . ± . f (0) − . ± . − . ± . f (0) − . ± . − . ± . g (0) 0 . ± .
05 0 . ± . g (0) − . ± . − . ± . g (0) − . ± . − . ± . q = 0. in [37] and using SU(3) symmetry and HQET in [38]. Their predictions on the decay rate of the Λ b → pl ¯ ν are,1 . × − | V ub | GeV and 6 . × | V ub | s − , respectively. In order to have a sense of the order of decay rates, wecompare our average results presented in Table X with those predictions. Considering all results in the same unit, wesee that our average result is in the same order of magnitude with that of [37], but one order of magnitude is greaterthan the [38] prediction. For exact comparison the initial particles should be the same.In conclusion, using the most general form of the interpolating currents of the heavy spin 1/2, Σ b baryon anddistribution amplitudes of the nucleon, the transition form factors of the semileptonic Σ b → N lν were calculatedin the framework of the light cone QCD sum rules. Ignoring the negligible deviation, the form factors satisfied theHQET relations among the form factors. The obtained results for the related form factors were used to estimate thedecay rate of this transition for two different sets of independent parameters entering to expressions for the nucleondistribution amplitudes namely, QCD sum rules and lattice QCD input parameters. The obtained values for thedecay rate for these two sets of data are approximately consistent with each other. Further improvements would beachieved by determining the next leading order QCD corrections to the nucleon distribution amplitudes.
IV. ACKNOWLEDGMENT
The authors thank T. M. Aliev for his useful discussions. This work has been supported in part by the EuropeanUnion (HadronPhysics2 project “Study of strongly interacting matter”). K. A. thanks also TUBITAK, TurkishScientific and Technical Research Council, for their financial support provided under the project 108T502. [1] M. Mattson et al., (SELEX Collaboration), Phys. Rev. Lett. 89, 112001 (2002).[2] A. Ocherashvili et al., (SELEX Collaboration), Phys. Lett. B 628, 18 (2005).[3] D. Acosta et al., (CDF Collaboration), Phys. Rev. Lett. 96, 202001 (2006).[4] R. Chistov et al., (Belle Collaboration), Phys. Rev. Lett. 97, 162001 (2006).[5] B. Aubert et al., (BABAR Collaboration), Phys. Rev. Lett. 97, 232001 (2006); Phys. Rev. Lett. 99, 062001 (2007); Phys.Rev. D 77, 012002 (2008).[6] V. Abazov et al., (D0 Collaboration), Phys. Rev. Lett. 99, 052001 (2007); Phys. Rev. Lett. 101, 232002 (2008). 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In this section, we present the explicit expressions for the form factors f and f . f ( Q ) = 1 √ λ Σ b e m b /M B (cid:18) Z t dx Z − x dx e − s ( x ,Q ) /M B √ (cid:20) m b (cid:26) (1 + 3 β ) H ( x i ) − − β ) H ( x i ) − (3 + β ) H ( x i ) (cid:27) − m N x (cid:26) H , − , − , , − , ( x i ) + β H , , − , − , , , − ( x i ) (cid:27) (cid:21) + Z t dx Z − x dx Z x t dt e − s ( t ,Q ) /M B " − m N m b M B t √ − β ) x H ( x i ) − m N M B t √ (cid:26) m N x h ( − β ) H − , ( x i ) + 2 β H ( x i ) i + h m N x + m b { Q + s ( t , Q ) } ( − β ) x − m N m b ( − β )(2 + 3 x ) i H ( x i ) (cid:27) + m N M B t √ (cid:26) m N x h Q + s ( t , Q ) H , − ( x i ) + ( − β ) H ( x i ) − β H , ( x i ) i + m b ( − β ) h Q (1 + 3 x ) + s ( t , Q )(1 + x ) i H ( x i ) + m N m b h x (1 + 3 β ) H ( x i )+2( − β ) H ( x i ) + (3 + β ) H ( x i ) − ( − β )(3 + x ) H ( x i ) i − m N h ( − β )(1 + x ) H , − ( x i ) − { β (1 + x ) H ( x i ) + (2 + 4 x ) H ( x i ) } i(cid:27) + m N M B √ (cid:26) − m b Q ( − β ) H ( x i )+ m N m b h ( − β ) H , − ( x i ) − (1 + 3 β ) H ( x i ) − (3 + β ) H ( x i ) i + m N h ( − t − x ) H ( x i )+( − β ) H , − ( x i ) − β H ( x i ) i + m N h Q ( − β )( − t − x ) H , − ( x i )+ Q ( − t + 6 x + 4 + 2 β ) H ( x i ) + 2 Q β (1 − t + x ) H ( x i ) i(cid:27) + m N M B t √ (cid:26) H , − , ( x i )+( − β ) H ( x i ) − H , − , ( x i ) (cid:27) + m N M B t √ (cid:26) [ Q + s ( t , Q )] h (3 + 25 β ) H ( x i ) + 2( − β ) H − , ( x i ) i − (5 + β ) H ( x i ) − m N h − β ) H − , ( x i ) − (11 + 3 β ) H ( x i ) + (5 + 67 β ) H ( x i ) i +2 x h ( − β ) H − , ( x i ) + β H ( x i ) i − m N m b h H , − , − , , , , − , ( x i ) + 4 x ( − β ) H ( x i )+ β H , − , − , , , , , − ( x i ) i(cid:27) + m N M B √ (cid:26) Q h H − , +12 , , − ( x i ) + β H , − , , − ( x i ) i +4 m N ( − β ) H ( x i ) + s ( t , Q ) h H , − ( x i ) + β H , − ( x i ) i + m N h β H , , − , − , , − , , − , , − , , ( x i ) + H − , − , , , − , , − , − , , − ( x i )+8( t − x ) H ( x i ) i(cid:27) + m N t √ (cid:26) − β ) H − , ( x i ) + (1 + 5 β ) H ( x i ) − (3 + β ) H ( x i ) (cid:27) + m N √ (cid:26) (1 + 21 β ) H ( x i ) − (3 + β ) H ( x i ) (cid:27) (cid:21) + Z t dx Z − x dx e − s /M B " m N t ( Q + m N t ) √ t − x ) (cid:26)h m N m b ( − β )( − t )( − t ) + 2 m N t { − t ) t } − m N t { Q ( − t ) + ( − t ) s ( s , Q ) }− m b ( − β ) t { Q ( − t ) + ( − t ) s ( s , Q ) } i H ( x i ) + m N t (cid:16) { m N ( − β )( − t ) − m N m b t (3 + β ) + ( − β ) t [ Q ( − t ) − s ( s , Q )] }H ( x i ) − h m N ( − β )( − t ) + m N m b t (1 + 3 β )+( − β ) t { Q ( − t ) − s ( s , Q ) } i H ( x i ) + 2 h − m N m b t ( − β ) + m N β (1 − t ) + βt ( Q (1 − t )+ s ( s , Q )) i H ( x i ) (cid:17) (cid:27) + m N ( Q + m N t ) √ t − x ) (cid:26)h m N m b ( − β )( − t )( − t )+2 m N t { − t ) t } − m N t { Q ( − t ) + ( − t ) s ( s , Q ) } − m b t ( − β ) { Q ( − t )+( − t ) s ( s , Q ) } i H ( x i ) + m N t h { m N ( − β )( − t ) − m N m b t (3 + β ) + ( − β ) t [ Q ( − t ) − s ( s , Q )] }H ( x i ) − { m N ( − β )( − t ) + m N m b t (1 + 3 β ) + ( − β ) t ( Q ( − t ) − s ( s , Q )) }H ( x i )+2 {− m N m b t ( − β ) + m N β (1 − t ) + βt [ Q (1 − t ) + s ( s , Q )] }H ( x i ) i(cid:27) m N ( Q + m N t )4 √ M B t (cid:26) m N ( t − x ) h m N m b ( − β )( − t )( − t ) + 2 m N t (2 + ( − t ) t )+ m b t ( − β ) { Q (1 − t ) + 2 M B t + (1 − t ) s ( s , Q ) } + 2 m N t { Q (2 − t ) } + 2 M B t +(2 − t ) s ( s , Q ) i H ( x i ) + t h m N M B H , − , − , ( x i ) + β H − , , − , ( x i ) i + 2 m N (1 − t ) H , ( x i )+ m N M B t H − , , , − ( x i ) + m N m b M B t H − , , , − , − , − , , − ( x i ) + M B Q t H , − , − , ( x i )+ m N βt H − , , ( x i ) + m N M B βt H , − , , − ( x i ) + m N m b M B βt H − , , , − , − , − , − , ( x i )+ M B Q βt H − , , − , ( x i ) + 2 m N t H − , ( x i ) + m N m b t H − , − , ( x i )+ m N M B t H − , − , , , − , , − , − , , − ( x i ) + 2 m N Q t H , − ( x i ) + M B Q t H − , , , − ( x i )+ m N βt H , − , − ( x i ) − m N m b βt H , , ( x i ) + m N M B βt H , , − , − , , − , , − , , − , , ( x i )+ m N Q βt H − , , ( x i ) + M B Q βt H , − , , − ( x i ) − Q t H , − ( x i ) + Q βt H , − , − ( x i )+ M B t s ( s , Q ) H , − , − , ( x i ) + M B βt s ( s , Q ) H − , , − , ( x i ) + 2 m N t s ( s , Q ) H , − ( x i )+ M B t s ( s , Q ) H , − ( x i ) + m N βt s ( s , Q ) H − , , ( x i ) + M B βt s ( s , Q ) H , − ( x i ) − m N x h { ( − β )( − t ) − m N m b t (3 + β ) + ( − β ) t [ − M B − Q (1 − t ) + s ( s , Q )] }H ( x i ) − h m N ( − β )( − t ) + m N m b t (1 + 3 β ) + ( − β ) t {− M B − Q (1 − t ) − s ( s , Q ) } i H ( x i )+2 h − m N m b t ( − β ) + m N β (1 − t ) + βt { M B + Q (1 − t ) + s ( s , Q ) } i H ( x i ) i(cid:27) (cid:19) , ( A. f ( Q ) = 1 √ λ Σ b e m b /M B (cid:18) Z t dx Z − x dx e − s ( x ,Q ) /M B √ x (cid:20) H , − , ( x i ) − β H , − , ( x i ) (cid:21) + Z t dx Z − x dx Z x t dt e − s ( t ,Q ) /M B " − m N M B t √ β ) x H ( x i ) + m N M B t √ (cid:26) m N m b x h (1 + 3 β ) H ( x i )+2( − β ) H ( x i ) + (3 + β ) H ( x i ) i + 2 h − m N m b x ( − β ) − { Q + s ( t , Q ) } (3 + β ) x + m N (3 + β +(5 + β ) x i H ( x i ) (cid:27) + m N M B t √ (cid:26) m N m b h (1 + 3 β ) H ( x i ) + 2( − β ) H ( x i ) + (3 + β ) H ( x i ) i +2 h m N m b (1 − β ) − s ( t , Q )(3 + β + x ) + m N (5 + β + x ) − Q (3 + β + (4 + β ) x ) i H ( x i ) (cid:27) + m N M B √ (cid:26)h m N − (4 + β ) Q − s ( t , Q ) i H ( x i ) (cid:27) + m N M B t √ (cid:26) − m b h H , , − , − ( x i ) + β H , , , ( x i ) i − β ) x H ( x i ) (cid:27) + m N M B t √ (cid:26) H − , − , , , − , − , , − ( x i ) + ( − β ) H ( x i ) − β H − , − , , , , − , , − , − , ( x i ) − H ( x i ) (cid:27) + m N √ M B H ( x i ) + Z t dx Z − x dx e − s /M B " m N t ( Q + m N t ) √ t − x ) (cid:26) − m N m b t h (1 + 3 β ) H ( x i ) + 2( − β ) H ( x i )+(3 + β ) H ( x i ) + 2( m N m b ( − β ) t + m N (3 + β − (5 + β ) t + t ) + t ( Q (3 + β ) − (4 + β ) t )+(3 + β − t ) s ( s , Q ) i H ( x i ) (cid:27) − m N ( Q + m N t ) √ t − x ) (cid:26) m N m b t h (1 + 3 β ) H ( x i ) + 2( − β ) H ( x i )+(3 + β ) H ( x i ) i + 2 h − m N m b ( − β ) t + Q t {− − β + (4 + β ) t } + m N {− β ( − t ) − ( − t ) t } + t ( − − β + t ) s ( s , Q ) i H ( x i ) (cid:27) + m N ( Q + m N t )4 √ M B t (cid:26) m N ( t − x ) h m N m b ( − β ) t + m N { β − (5 + β ) t + t } + t { M B (2 + β + 2 t ) + Q { β − (4 + β ) t } + (3 + β − t ) s ( s , Q ) }H ( x i ) − t h m b M B {H , , − ( x i ) + β H , , ( x i ) } + m N m b t {H , , − ( x i ) + β H , , ( x i ) } + m N M B t {H , , − , , − , , ( x i ) + β H − , − , , − , , − , , − ( x i ) } + 2 M B { m b ( − β )+ m N (1 + β ) t }H ( x i ) − m N m b x { (3 + β ) H ( x i ) + (1 + 3 β ) H ( x i ) + 2( − β ) H ( x i ) } i(cid:27) (cid:21) (cid:27) , ( A. H ( x i ) = H ( x , x , − x − x ) ,s ( y, Q ) = (1 − y ) m N + (1 − y ) y Q + m b y , ( A. t = t ( s , Q ) is the solution of the equation s ( t , Q ) = s , and is given as t ( s , Q ) = m N − Q − p − m N ( m b − Q ) + ( − m N + Q − s ) + s m N . ( A. H ± i a , ± j b ,... = ± a H i ± b H j ... , and H i are defined in terms of the distribution amplitudes as follows: H = S H = S , − H = P H = P , − H = V H = V , − , − H = V H = − V , − + V , H = V , − H = − V , − , − , − , − , H = A H = − A , − , H = A H = − A , − − A , H = A , − H = A , − , , , − , H = T H = T , − T H = T H = T , − − T H = − T , − + 2 T H = T , − , − , , , H = T , − H = − T , − , − , + 2 T , , ( A. X ± i, ± j,... = ± X i ± X j ......