Flavor Changing Neutral Currents Transition of the Σ Q to Nucleon in Full QCD and Heavy Quark Effective Theory
aa r X i v : . [ h e p - ph ] M a y Flavor Changing Neutral Currents Transition of the Σ Q to Nucleon in Full QCD andHeavy Quark Effective Theory K. Azizi , † , M. Bayar , ‡ , M. T. Zeyrek , ∗ , † Physics Division, Faculty of Arts and Sciences, Do˘gu¸s University, Acıbadem-Kadık¨oy,34722 Istanbul, Turkey ‡ Department of Physics, Kocaeli University, 41380 Izmit, Turkey ∗ Physics Department, Middle East Technical University, 06531, Ankara, Turkey [email protected] [email protected] [email protected] The loop level flavor changing neutral currents transitions of the Σ b → n l + l − and Σ c → p l + l − are investigated in full QCD and heavy quark effective theory in the light cone QCD sum rulesapproach. Using the most general form of the interpolating current for Σ Q , Q = b or c , as membersof the recently discovered sextet heavy baryons with spin 1/2 and containing one heavy quark, thetransition form factors are calculated using two sets of input parameters entering the nucleon distri-bution amplitudes. The obtained results are used to estimate the decay rates of the correspondingtransitions. Since such type transitions occurred at loop level in the standard model, they can beconsidered as good candidates to search for the new physics effects beyond the SM. PACS numbers: 11.55.Hx, 13.30.-a, 14.20.Mr, 14.20.Lq, 12.39.Hg
I. INTRODUCTION
The Σ b → n l + l − and Σ c → p l + l − are governed by flavor changing neutral currents (FCNC) transitions of b → d and c → u , respectively. These transitions are described via electroweak penguin and weak box diagrams in thestandard model (SM) and they are sensitive to new physics contributing to penguin operators. Looking for SUSYparticles [1], light dark matter [2] and also probable fourth generation of the quarks is possible by investigating suchloop level transitions. This transitions are also good framework to reliable determination of the V tb , V td , V cb , and V bu as members of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, CP and T violations and polarization asymmetries.The Σ b,c as members of the spin 1/2 sextet heavy baryons containing a single heavy bottom or charm quark areconsidered by their most general interpolating currents which generalize the Ioffe current for these baryons. In therecent years, important experimental progresses has been made in the spectroscopy of the heavy baryons containingheavy b or c quark [3–10]. Having the heavy quark makes these states be experimentally narrow, so their isolationand detection are easy comparing with the light baryons. Experimentally, investigation of the semileptonic decays ofthe heavy baryons, may be considered at large hadron collider (LHC) in the future, hence theoretical calculations ofthe decay properties can play crucial role in this respect.In our two recent works, we analyzed the tree level semileptonic decays of Σ b to proton [11] and Λ b (Λ c ) → p ( n ) lν [12] in light cone QCD sum rules. In full theory, these tree level transitions in the SM are described via six formfactors (for details and more about the works devoted to the semileptonic decays of the heavy baryons using differentphenomenological methods see [11, 12] and references therein). In the present work, considering the long and shortdistance effects, we calculate the 12 form factors entering the semileptonic loop level Σ b → n l + l − and Σ c → p l + l − transitions using the light cone QCD sum rules in full theory as well as heavy quark effective theory (HQET). Theshort distance effects are calculated using the perturbation theory and long distance contributions are expanded interms of the nucleon distribution amplitudes (DA’s) with increasing twists near the light cone, x ≃
0. We usethe value of the eight independent parameters entering to the nucleon DA’s from two different sources: predictedusing a simple model in which the deviation from the asymptotic DAs is taken to be 1/3 of that suggested by theQCD sum rule estimates [13] and obtained via lattice QCD [14–16]. Using the obtained form factors, we predict thecorresponding transition rates. Investigation of these decays can also give essential information about the internalstructure of Σ b,c baryons as well as the nucleon DA’s.The layout of the paper is as follows: in section II, we introduce the theoretical framework to calculate the formfactors in light cone QCD sum rules method in full theory. The HQET relations among the form factors are alsointroduced in this section. Section III is devoted to the numerical analysis of the form factors and their extrapolationin terms of the transferred momentum squired, q , their HQET limit and our predictions for the decay rates obtainedin two different sets of parameters entering the nucleon distribution amplitudes. II. LIGHT CONE QCD SUM RULES FOR TRANSITION FORM FACTORS
Form factors play essential role in analyzing the Σ b → n l + l − and Σ c → p l + l − transitions. At quark level, thesedecays proceed by loop b ( c ) → d ( u ) transition, which can be described by the following effective Hamiltonian: H eff = G F αV Q ′ Q V ∗ Q ′ q √ π (cid:26) C eff ¯ qγ µ (1 − γ ) Q ¯ lγ µ l + C ¯ qγ µ (1 − γ ) Q ¯ lγ µ γ l − m Q C q ¯ qiσ µν q ν (1 + γ ) Q ¯ lγ µ l (cid:27) , (1)where, Q ′ refers to the u, c, t for bottom case and d, s , b for charm case, respectively. The main contributions comefrom the heavy quarks, so we will consider Q ′ = t and Q ′ = b respectively for the Σ b → n l + l − and Σ c → p l + l − tran-sitions. The amplitude of the considered transitions can be obtained by sandwiching the above Hamiltonian betweenthe initial and final states. To proceed, we need to know the the matrix elements h N | J tr,Iµ | Σ Q i and h N | J tr,IIµ | Σ Q i ,where J tr,Iµ ( x ) = ¯ q ( x ) γ µ (1 − γ ) Q ( x ) and J tr,IIµ ( x ) = ¯ q ( x ) iσ µν q ν (1 + γ ) Q ( x ) are transition currents entering to theHamiltonian. From the general philosophy of the QCD sum rules, to obtain sum rules for the physical quantities westart considering the following correlation functions:Π Iµ ( p, q ) = i Z d xe iqx h N ( p ) | T { J tr,Iµ ( x ) ¯ J Σ Q (0) } | i , Π IIµ ( p, q ) = i Z d xe iqx h N ( p ) | T { J tr,IIµ ( x ) ¯ J Σ Q (0) } | i , (2)where, J Σ Q is interpolating currents of Σ b ( c ) baryon and p denotes the proton (neutron) momentum and q = ( p + q ) − p is the transferred momentum. The main idea in QCD sum rules is to calculate the aforementioned correlation functionsin two different ways: • In theoretical side, the time ordering product of the initial state and transition current is expanded in termsof nucleon distribution amplitudes having different twists via the operator product expansion (OPE) at deepEuclidean region. By OPE the short and large distance effects are separated. The short distance contributionis calculated using the perturbation theory, while the long distance phenomena are parameterized in terms ofnucleon DA’s. • From phenomenological or physical side, they are calculated in terms of the hadronic parameters via saturatingthem with a tower of hadrons with the same quantum numbers as the interpolating currents.To get the sum rules for the physical quantities, the two above representations of the correlation functions areequated through the dispersion relation. To suppress the contribution of the higher states and continuum and isolatethe ground state, the Borel transformation as well as continuum subtraction through quark-hadron duality assumptionare applied to both sides of the sum rules expressions.The first task is to calculate the aforementioned correlation function from QCD side in deep Euclidean region where( p + q ) ≪
0. To proceed, the explicit expression of the interpolating field of the Σ Q baryon is needed. Considering thequantum numbers, the most general form of interpolating current creating the Σ Q from the vacuum can be writtenas J Σ b ( x ) = − √ ε abc (cid:20) (cid:26) q T a ( x ) CQ b ( x ) (cid:27) γ q c ( x ) − (cid:26) Q T a ( x ) Cq b ( x ) (cid:27) γ q c ( x )+ β (cid:26) { q T a ( x ) Cγ Q b ( x ) } q c ( x ) − { Q T a ( x ) Cγ q b ( x ) } q c ( x ) (cid:27) (cid:21) , (3)where, C is the charge conjugation operator and β is an arbitrary parameter with β = − q and q are the u and d quarks, respectively and a, b, c are the color indices. Using the transition currents,and J Σ Q and contracting out all quark pairs via the Wick’s theorem, we obtain the following representations of thecorrelation functions in QCD side:Π Iµ = − i √ ǫ abc Z d xe iqx ( h ( C ) βη ( γ ) ρφ − ( C ) φβ ( γ ) ρη i + β " ( Cγ ) βη ( I ) ρφ − ( Cγ ) φβ ( I ) ρη γ µ (1 − γ ) i σθ ) S Q ( − x ) βσ h N ( p ) | ¯ d aη (0) ¯ d bθ ( x )¯ u cφ (0) | i , (4)Π IIµ = − i √ ǫ abc Z d xe iqx ( h ( C ) βη ( γ ) ρφ − ( C ) φβ ( γ ) ρη i + β " ( Cγ ) βη ( I ) ρφ − ( Cγ ) φβ ( I ) ρη iσ µν q ν (1 + γ ) i σθ ) S Q ( − x ) βσ h N ( p ) | ¯ d aη (0) ¯ d bθ ( x )¯ u cφ (0) | i , (5)where, S Q ( x ) is the heavy quark propagator and its expression is given as [17]: 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 . (6)where S freeQ = m Q π K ( m Q √− x ) √− x − i m Q x π x K ( m Q p − x ) , (7)and K i are the Bessel functions. When doing calculations, we neglect the terms proportional to the gluon fieldstrength tensor because they are contributed mainly to the four and five particle distribution functions and expectedto be very small in our case [18–20]. The matrix element h N ( p ) | ǫ abc ¯ d aη (0) ¯ d bθ ( x )¯ u cφ (0) | i appearing in Eqs. (4,5)denotes the nucleon wave function, which is given in terms of some calligraphic functions [13, 18–21]:4 h | ǫ abc d aα ( a x ) d bβ ( a x ) u 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 . (8)The calligraphic functions have not definite twists but they can be expressed in terms of the nucleon distributionamplitudes (DA’s) with definite and increasing twists by the help of the scalar product px and the parameters a i , i = 1 , ,
3. The explicit expressions for scalar, pseudo-scalar, vector, axial vector and tensor DA’s for nucleons aregiven in Tables I, II, III, IV and V, respectively. S = S px S = S − S TABLE I: Relations between the calligraphic functions and nucleon scalar DA’s.
Each distribution amplitude F ( a i px )= S i , P i , V i , A i , T i can be expressed as: F ( a i px ) = Z dx dx dx δ ( x + x + x − e ipx Σ i x i a i F ( x i ) . (9) P = P px P = P − P TABLE II: Relations between the calligraphic functions and nucleon pseudo-scalar DA’s. 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 nucleon vector DA’s. where, x i with i = 1 , Iµ ( p, q ) = X s h N ( p ) | J tr,Iµ (0) | Σ Q ( p + q, s ) ih Σ Q ( p + q, s ) | ¯ J Σ Q (0) | i m Q − ( p + q ) + ..., (10)Π IIµ ( p, q ) = X s h N ( p ) | J tr,IIµ (0) | Σ Q ( p + q, s ) ih Σ Q ( p + q, s ) | ¯ J Σ Q (0) | i m Q − ( p + q ) + ..., (11)where, the ... denotes the contribution of the higher states and continuum. The baryonic to the vacuum matrixelement of the interpolating current, i.e., h Σ Q ( p + q, s ) | ¯ J Σ Q (0) | i can be parameterized in terms of the residue, λ Σ Q as: h Σ Q ( p + q, s ) | ¯ J Σ Q (0) | i = λ Σ Q ¯ u Σ Q ( p + q, s ) . (12)To proceed, we also need to know the transition matrix elements, h N ( p ) | J tr,Iµ | Σ Q ( p + q, s ) i and h N ( p ) | J tr,IIµ | Σ Q ( p + q, s ) i . In full theory, they are parameterized in terms of 12 transition form factors, f i , g i , f Ti and g Ti with i = 1 → h N ( p ) | J tr,Iµ ( x ) | Σ Q ( 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 Q ( p + q ) , (13)and h N ( p ) | J tr,IIµ ( x ) | Σ Q ( p + q ) i = ¯ N ( p ) (cid:2) γ µ f T ( Q ) + iσ µν q ν f T ( Q ) + q µ f T ( Q ) + γ µ γ g T ( Q ) + iσ µν γ q ν g T ( Q )+ q µ γ g T ( Q ) (cid:21) u Σ Q ( p + q ) , (14)where Q = − q . Here, N ( p ) and u Σ Q ( p + q ) are the spinors of nucleon and Σ Q , respectively. Using Eqs. (10), (11),(12), (13) and (14) and performing summation over spins of the Σ Q baryon using X s u Σ Q ( p + q, s ) u Σ Q ( p + q, s ) = p + q + m Σ Q , (15) 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 nucleon axial vector DA’s. 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 nucleon tensor DA’s. we obtain the following expressionsΠ Iµ ( p, q ) = λ Σ Q m Q − ( 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 Q ) + · · · (16)andΠ IIµ ( p, q ) = λ Σ Q m Q − ( p + q ) ¯ N ( p ) (cid:2) γ µ f T ( Q ) + iσ µν q ν f T ( Q ) + q µ f T ( Q ) + γ µ γ g T ( Q ) + iσ µν γ q ν g T ( Q )+ q µ γ g T ( Q ) (cid:21) ( p + q + m Σ Q ) + · · · (17)Using the relation ¯ N σ µν q ν u Σ Q = i ¯ N [( m N + m Σ Q ) γ µ − (2 p + q ) µ ] u Σ Q , (18)in Eqs. (16) and Eqs. (17), we attain the final expressions for the physical side of the correlation functions:Π Iλ ( p, q ) = λ Σ Q m Q − ( p + q ) ¯ N ( p ) (cid:20) f ( Q ) p µ + (cid:26) − f ( Q )( m N − m Σ Q ) + f ( Q )( m N − m Q ) (cid:27) γ µ + (cid:26) f ( Q ) − f ( Q )( m N + m Σ Q ) (cid:27) γ µ q + 2 f ( Q ) p µ q + (cid:26) f ( Q ) + f ( Q ) (cid:27) ( m N + m Σ Q ) q µ + (cid:26) f ( Q ) + f ( Q ) (cid:27) q µ q + 2 g ( Q ) p µ γ − (cid:26) g ( Q )( m N + m Σ Q ) − g ( Q )( m N − m Q ) (cid:27) γ µ γ + (cid:26) g ( Q ) − g ( Q )( m N − m Σ Q ) (cid:27) γ µ qγ + 2 g ( Q ) p µ qγ + (cid:26) g ( Q ) + g ( Q ) (cid:27) ( m N − m Σ Q ) q µ γ + (cid:26) g ( Q ) + g ( Q ) (cid:27) q µ qγ (cid:21) + · · · (19)andΠ IIλ ( p, q ) = λ Σ Q m Q − ( p + q ) ¯ N ( p ) (cid:20) f T ( Q ) p µ + (cid:26) − f T ( Q )( m N − m Σ Q ) + f T ( Q )( m N − m Q ) (cid:27) γ µ + (cid:26) f T ( Q ) − f T ( Q )( m N + m Σ Q ) (cid:27) γ µ q + 2 f T ( Q ) p µ q + (cid:26) f T ( Q ) + f T ( Q ) (cid:27) ( m N + m Σ Q ) q µ + (cid:26) f T ( Q ) + f T ( Q ) (cid:27) q µ q − g T ( Q ) p µ γ + (cid:26) g T ( Q )( m N + m Σ Q ) − g T ( Q )( m N − m Q ) (cid:27) γ µ γ − (cid:26) g T ( Q ) − g T ( Q )( m N − m Σ Q ) (cid:27) γ µ qγ − g T ( Q ) p µ qγ − (cid:26) g T ( Q ) + g T ( Q ) (cid:27) ( m N − m Σ Q ) q µ γ − (cid:26) g T ( Q ) + g T ( Q ) (cid:27) q µ qγ (cid:21) + · · · (20)In order to calculate the form factors or their combinations, f , f , f + f , g , g and g + g , we will choose theindependent structures p µ , p µ q , q µ q , p µ γ , p µ qγ , and q µ qγ from Eq. (19), respectively. The same structures arechosen to calculate the form factors or their combinations labeled by T in the second correlation function in Eq. (20).Having computed both sides of the correlation functions, it is time to obtain the sum rules for the related formfactors. Equating the coefficients of the corresponding structures from both sides of the correlation functions throughthe dispersion relations and applying Borel transformation with respect to ( p + q ) to suppress the contribution ofthe higher states and continuum, one can obtain sum rules for the form factors f , f , f , g , g , g , f T , f T , f T , g T , g T and g T . In heavy quark effective theory (HQET), where m Q → ∞ , the number of independent form factorsis reduced to two, namely, F and F . In this limit, the transition matrix element can be parameterized in terms ofthese two form factors in the following way [22, 23]: h N ( p ) | ¯ d Γ b | Σ Q ( p + q ) i = ¯ N ( p )[ F ( Q )+ vF ( Q )]Γ u Σ Q ( p + q ) , (21)where, Γ is any Dirac matrices and v = p + qm Σ Q . Here we should mention that the above relation is exact for Λ-likebaryons, where the light degrees of freedom are spinless. For the Σ like baryons this relation cannot hold exactly andhas to be replaced by a more complicated relation. In the present work, we will use the above approximate relationfor the considered transitions. Comparing this matrix element and our definitions of the form factors in Eqs. (13)and (14), we get the following relations among the form factors in HQET limit (see also [24, 25]) f = g = f T = g T = F + m N m Λ b F f = g = f = g = F m Σ Q f T = g T = F m Σ Q q f T = − F m Σ Q ( m Σ Q − m N ) g T = F m Σ Q ( m Σ Q + m N ) (22)Looking at the above relations, we see that it is possible to write all form factors in terms of f and f , so we willpresent the explicit expressions for these two form factors in the Appendix and give extrapolation of the other formfactors in finite mass as well as HQET in terms of q in the numerical analysis section.The expressions of the sum rules for form factors show that we need to know also the residue λ Σ Q . This residue isdetermined in [26]: − λ Q e − m Q /M B = Z s m Q e − sM B ρ ( s ) ds + e − m QM B Γ , (23)where, ρ ( s ) = ( < dd > + < uu > ) ( β − π ( m m Q (6 ψ − ψ − ψ ) + 3 m Q (2 ψ − ψ − ψ + 2 ψ ) ) + m Q π [5 + β (2 + 5 β )][12 ψ − ψ + 2 ψ − ψ + ψ − ln ( sm Q )] , (24)and Γ = ( β − < dd >< uu > (cid:20) m Q m M B + m M B − . (25)Here, ψ nm = ( s − m Q ) n s m ( m Q ) n − m are some dimensionless functions. III. NUMERICAL RESULTS
This section deals with the numerical analysis of the form factors as well as the total decay rate of the loop levelΣ b −→ nℓ + ℓ − and Σ c −→ pℓ + ℓ − transitions in both full theory and HQET limit. In obtaining numerical values,we use the following inputs for masses and quark condensates: h ¯ uu i (1 GeV ) = h ¯ dd i (1 GeV ) = − (0 . GeV , m n = 0 . GeV , m p = 0 . GeV , m b = 4 . GeV , m c = 1 . GeV , m Σ b = 5 . GeV , m Σ c = 2 . GeV and m (1 GeV ) = (0 . ± . GeV [27]. From the sum rules expressions for the form factors, it is clear that the nucleonDA’s (see Appendix) are the main input parameters. These DA’s contain eight independent parameters, namely, f N , λ , λ , V d , A u , f d , f u and f d . All of these parameters have been calculated in the framework of the lightcone QCD sum rules [13] and most of them are now available in lattice QCD [14–16] (see Table VI). Here, we shouldstress that in [13] those parameters are obtained both as QCD sum rules and assymptotic sets, but to improve theagreement with experimental data on nucleon form factors, a set of parameters is obtained using a simple model inwhich the deviation from the asymptotic DAs is taken to be 1/3 of that suggested by the QCD sum rule estimates(see [13]). We will use this set of parameters in this paper and refer it as set1 (see Table VI). In the following, wealso will denote the lattice QCD input parameters by set2. set1 [13] set2 or Lattice QCD [14–16] f N (5 . ± . × − GeV (3 . ± . ± . × − GeV λ − (2 . ± . × − GeV ( − . ± . ± . × − GeV λ (5 . ± . × − GeV (7 . ± . ± . × − GeV V d .
30 0 . ± . ± . A u .
13 0 . ± . ± . f d . − f u . − f d . − TABLE VI: The values of the 8 independent parameters entering the nucleon DA’s. The first errors in lattice values arestatistical and the second errors correspond to the uncertainty due to the Chiral extrapolation and renormalization. For lasttree parameters, the values are not available in lattice and we will use the set1 values for both sets of data.
The explicit expressions for the form factors also show their dependency to three auxiliary mathematical objects,namely, continuum threshold s , Borel mass parameter M B and general parameter β entering to the most generalform of the interpolating current of the initial state. The form factors as physical quantities should be independentof these parameters, hence we need to look for working regions for them. The working region for Borel mass squaredis determined as follows: the upper limit of M B is chosen demanding that the series of the light cone expansionwith increasing twist should be convergent. The lower limit is determined from condition that the higher states andcontinuum contributions constitute a small fraction of total dispersion integral. Both conditions are satisfied in theregions 15 GeV ≤ M B ≤ GeV and 4 GeV ≤ M B ≤ GeV for bottom and charm cases, respectively. Thevalue of the continuum threshold s is not completely arbitrary and it is correlated to the first exited state withquantum numbers of the initial particle interpolating current. Our numerical calculations show that the form factorsweakly depend on the continuum threshold in the interval, ( m Σ Q + 0 . ≤ s ≤ ( m Σ Q + 0 . . To obtain the workingregion for β at which the form factors are practically independent of it, we look for the variation of the form factorswith respect to cosθ in the interval − ≤ cosθ ≤ −∞ ≤ β ≤ ∞ , where β = tanθ . Asa result, the interval − . ≤ cosθ ≤ . β for both charm and bottom cases. In this interval, thedependency on this parameter is weak.The next step is to discuss the behaviour of the form factors in terms of the q . The sum rules predictions for theform factors are not reliable in the whole physical region. To be able to extend the results for the form factors tothe whole physical region, we look for a parametrization of the form factors such that in the reliable region whichis approximately 1 GeV below the perturbative cut, the original form factors and their fit parametrization coincideeach other. Our numerical results lead to the following extrapolation for the form factors in terms of q : f i ( q )[ g i ( q )] = a (1 − q m fit ) + b (1 − q m fit ) , (26)where the fit parameters a, b and m fit in full theory and HQET limit are given in Tables VII, VIII, IX and X using twosets for the independent parameters. These Tables, show poles of the form factors outside the allowed physical region.Therefore, the form factors are analytic in the full physical interval. In principle, we can use fit parametrization eitherwith single pole or double poles. However, when we combine them the accuracy of the fitting becomes very high,specially when the pole is the same for two parts. We could start from f i ( q ) = a − q /m + b − q /m fit , however for allform factors m fit gets too close to m , so the fit becomes numerically unstable. In such a case, it is appropriate toexpand the above relation to first order in m fit − m , which gives the Eq. (26) used to extrapolate the factors overthe whole range of q . For the same situation in B −→ D mesonic transition see for instance [28, 29]. The valuesof form factors at q = 0 are presented in Tables XI and XII in both full theory and HQET for bottom and charmcases, respectively. In extraction of the values of form factors at q = 0, the mean values of the form factors obtainedfrom the quoted ranges for the auxiliary parameters have been considered. When we look at these Tables, we see thatalthough the values for the eight independent input parameters for two sets are close to each other but the resultsfor the central values of some form factors differ in two sets, considerably. The numerical results show that the resultof sum rules are very sensitive to these parameters specially f N , λ and A u . Within the errors, the quoted valuesbecome close to each other for both sets. The numerical analysis depicts also that all form factors approximatelysatisfy the HQET limit relations in Eq. (22) within the errors for both sets of input parameters and bottom case, Q = b at q = 0. However for the charm case, Q = c although some of the relations are satisfied but most of themare violated at q = 0. This is an expected result since the m c → ∞ limit is not as reasonable as the m b → ∞ .Our next task is to calculate the total decay rate of the FCNC Σ b −→ pℓ + ℓ − and Σ c −→ nℓ + ℓ − transitions in thefull allowed physical region, namely, 4 m l ≤ q ≤ ( m Σ b,c − m p,n ) . To derive the expression for the decay rate, wewill make the following assumptions (see also [30]): the CLEO predicts the value R = F F = − . ± . ± .
08 forthe ratio of the form factors of Λ c → Λ ee ν at HQET limit [31]. This result shows that | F | < | F | and consideringEq. (22), the form factors f , g , f , f T and g T are expected to be large comparing to the other form factors sincethey are proportional to the F . Moreover, it is clear from the considered Hamiltonian as well as the definition of thetransition matrix elements in terms of the form factors that the form factors labeled by T are related to the Wilsoncoefficient C which is about one order of magnitude smaller than the other coefficients entered to the Hamiltonian,i.e., C and C , hence their effects expected to be small. As a result of the above procedure, the following results fordecay width describing such transitions is obtained [30]: d Γ ds (cid:0) Σ Q → N l + l − (cid:1) = G F α em | V Q ′ Q V ∗ Q ′ q | π m Q p φ ( s ) s − m l q ¯ f R Σ Q ( s ) , (27)where R Σ Q ( s ) = Γ ( s ) + Γ ( s ) + Γ ( s ) (28) set1 set2a b m fit a b m fit f − .
16 0 .
29 5 .
70 0 .
027 0 .
044 5 . f . − .
02 5 .
96 0 . − .
024 7 . f . − .
024 6 . − . − .
003 6 . g − .
21 0 .
33 5 . − .
13 0 .
20 5 . g . − .
02 5 . − .
01 0 .
003 5 . g . − .
02 5 .
87 0 . − .
023 7 . f T − .
06 0 .
023 5 . − . − .
018 5 . f T − .
16 0 .
29 6 .
47 0 . − .
017 5 . f T − .
18 0 .
25 8 .
81 0 . − .
023 5 . g T − .
15 0 .
14 5 . − . − .
028 5 . g T − .
20 0 .
31 5 .
24 0 .
026 0 .
04 4 . g T . − .
25 5 .
76 0 . − .
18 5 . b → nℓ + ℓ − .set1 set2a b m fit a b m fit f .
08 0 .
097 1 . − .
12 0 .
18 1 . f − . − .
056 1 . − . − .
029 1 . f − .
025 0 .
012 1 .
61 0 . − .
047 1 . g − .
015 0 .
31 1 . − .
038 0 .
21 1 . g − . − .
12 1 .
55 0 . − .
14 1 . g − . − .
13 1 . − . − .
14 1 . f T − .
23 0 .
19 1 .
52 0 . − .
097 1 . f T .
066 0 .
067 1 .
63 0 .
12 0 .
13 1 . f T .
15 0 .
006 1 .
56 0 .
21 0 .
032 1 . g T − .
45 0 .
29 1 . − .
17 0 .
09 1 . g T .
009 0 .
08 1 . − .
026 0 .
14 1 . g T − . − .
11 1 . − . − .
16 1 . c → pℓ + ℓ − . set1 set2a b m fit a b m fit f − .
22 0 . .
96 0 .
037 0 .
06 5 . f . − .
024 5 .
13 0 . − .
028 5 . f . − .
02 5 . − . − .
002 5 . g − .
029 0 .
19 5 . − .
18 0 .
28 5 . g . − .
02 5 . − .
01 0 .
003 5 . g . − .
018 4 .
87 0 . − .
021 5 . f T − .
065 0 .
025 5 . − .
03 0 .
019 5 . f T − .
24 0 .
43 5 .
04 0 . − .
026 5 . f T − .
19 0 .
27 5 .
13 0 . − .
025 5 . g T − .
14 0 .
13 5 . − . − .
011 5 . g T − .
03 0 .
17 5 .
58 0 .
04 0 .
06 5 . g T . − .
27 5 .
16 0 . − .
17 4 . b → nℓ + ℓ − .set1 set2a b m fit a b m fit f .
02 0 .
13 1 . − .
17 0 .
25 1 . f − . − .
077 1 . − . − .
04 1 . f − .
034 0 .
017 1 .
73 0 . − .
065 1 . g − .
021 0 .
43 1 . − .
053 0 .
29 1 . g − . − .
12 1 . − . − .
14 1 . g − .
26 0 .
10 1 . − . − .
13 1 . f T − .
097 0 .
05 1 .
58 0 . − . . f T .
099 0 . .
69 0 .
18 0 .
196 1 . f T .
14 0 .
009 1 .
51 0 . − .
048 1 . g T − . .
26 1 . − .
15 0 .
08 1 . g T .
014 0 .
12 1 . − .
039 0 .
21 1 . g T − . − . . − . − .
15 1 . c → pℓ + ℓ − . Full Theory HQETset1 set2 set1 set2 f (0) 0 . ± .
04 0 . ± .
02 0 . ± .
05 0 . ± . f (0) − . ± . − . ± . − . ± . − . ± . f (0) − . ± . − . ± . − . ± . − . ± . g (0) 0 . ± .
03 0 . ± .
02 0 . ± .
04 0 . ± . g (0) − . ± . − . ± . − . ± . − . ± . g (0) − . ± . − . ± . − . ± . − . ± . f T (0) − . ± . − . ± . − . ± . − . ± . f T (0) 0 . ± .
04 0 . ± .
020 0 . ± .
05 0 . ± . f T (0) 0 . ± .
02 0 . ± .
020 0 . ± .
03 0 . ± . g T (0) − . ± . − . ± . − . ± . − . ± . g T (0) 0 . ± .
03 0 . ± .
021 0 . ± .
04 0 . ± . g T (0) − . ± . − . ± . − . ± . − . ± . q = 0 for Σ b → nℓ + ℓ − .Full Theory HQETset1 set2 set1 set2 f (0) 0 . ± .
05 0 . ± .
02 0 . ± .
05 0 . ± . f (0) − . ± . − . ± . − . ± . − . ± . f (0) − . ± . − . ± . − . ± . − . ± . g (0) 0 . ± .
09 0 . ± .
06 0 . ± .
12 0 . ± . g (0) − . ± . − . ± . − . ± . − . ± . g (0) − . ± . − . ± . − . ± . − . ± . f T (0) − . ± . − . ± . − . ± . − . ± . f T (0) 0 . ± .
04 0 . ± .
07 0 . ± .
06 0 . ± . f T (0) 0 . ± .
05 0 . ± .
07 0 . ± .
04 0 . ± . g T (0) − . ± . − . ± . − . ± . − . ± . g T (0) 0 . ± .
03 0 . ± .
03 0 . ± .
05 0 . ± . g T (0) − . ± . − . ± . − . ± . − . ± . q = 0 for Σ c → pℓ + ℓ − . ( s ) = − √ rs (cid:20) − m Q ρ (cid:18) m l q (cid:19) ReC eff9 C ∗ + δ (cid:18)(cid:18) m l q (cid:19) (cid:12)(cid:12)(cid:12) C eff (cid:12)(cid:12)(cid:12) + (cid:18) − m l q (cid:19) | C | (cid:19)(cid:21) + (cid:20) − r (cid:18) m l q (cid:19) − t (cid:18) − m l q (cid:19) + 3 (1 + r ) t (cid:21) × (cid:20) (2 ˆ m Q ρ ) | C | + (cid:12)(cid:12)(cid:12) C eff (cid:12)(cid:12)(cid:12) + | C | (cid:21) +6 ˆ m l t (cid:20) (2 ˆ m Q ρ ) | C | + (cid:12)(cid:12)(cid:12) C eff (cid:12)(cid:12)(cid:12) − | C | (cid:21) , (29)Γ ( s ) = 6 √ r (1 − t ) (cid:26) (cid:18) m l q (cid:19) ˆ m Q ρ | C | + ρs (cid:20)(cid:18) m l q (cid:19) (cid:12)(cid:12)(cid:12) C eff (cid:12)(cid:12)(cid:12) + (cid:18) − m l q (cid:19) | C | (cid:21)(cid:27) +12 (cid:18) m l q (cid:19) ˆ m Q ( t − r ) (cid:0) sρ (cid:1) ReC eff9 C ∗ , (30)Γ ( s ) = 12 (cid:18) m l q (cid:19) ˆ m Q √ rsρ ReC eff9 C ∗ − (cid:20) t (cid:18) m l q (cid:19) + 4 r (cid:18) − m l q (cid:19) − r ) t (cid:21) × " m Q s | C | + sρ (cid:18)(cid:12)(cid:12)(cid:12) C eff (cid:12)(cid:12)(cid:12) + | C | (cid:19) − m l (2 r − (1 + r ) t ) "(cid:18) m Q s (cid:19) | C | + ρ (cid:16)(cid:12)(cid:12) C eff9 (cid:12)(cid:12) − | C | (cid:17) . (31)Here, G F = 1 . × − GeV − is the Fermi coupling constant, ¯ f = f + g , ρ = m Σ Q f + g f + g , δ = f − g f + g , s = q m Q ,ˆ m Q = m Q m Σ Q , ˆ m l = m l m Σ Q , r = m N m Q , t = m Q [ m Q + m N − q ] and m l is the lepton mass. For the Wilson coefficients,we use C = − . , C = 4 . , C = − .
669 [32]. Here we should mention that the Wilson coefficient C eff receives long distance contributions from J/ψ family, in addition to short distance contributions. In the presentwork, we do not take into account the long distance effects. The elements of the CKM matrix V tb = (0 . +0 . − . ), V td = (8 . ± . × − , V bc = (41 . ± . × − and V bu = (3 . ± .
36) 10 − have also been used [33].Using the formula for the decay rate the final results as shown in Table XIII are obtained. From this table, we see Σ b −→ ne + e − Σ b −→ nµ + µ − Σ b −→ nτ + τ − Σ c −→ pe + e − Σ c −→ pµ + µ − Full (set1) (4 . ± . × − (2 . ± . × − (1 . ± . × − (5 . ± . × − (9 . ± . × − Full (set2) (5 . ± . × − (2 . ± . × − (4 . ± . × − (1 . ± . × − (2 . ± . × − HQET(set1) (8 . ± . × − (4 . ± . × − (6 . ± . × − (7 . ± . × − (1 . ± . × − HQET(set2) (1 . ± . × − (5 . ± . × − (1 . ± . × − (2 . ± . × − (4 . ± . × − TABLE XIII: Values of the Γ(Σ b,c −→ n, p ℓ + ℓ − ) in GeV for different leptons and two sets of input parameters. that: a) The value of the decay rate decreases by increasing in the lepton mass. This is reasonable since the phasespace in for example τ case is smaller than that of the electron and µ cases. b) The order of magnitude on decayrate of bottom case shows the possibility of the experimental studies on the Σ b −→ n ℓ + ℓ − transition, specially the µ case, at large hadron collider (LHC) in the near future. The lifetime of the Σ b is not exactly known yet but ifwe consider its lifetime approximately the same order of the b-baryon admixture (Λ b , Ξ b , Σ b , Ω b ) lifetime, which is τ = (1 . +0 . . ) × − s [33], the branching fraction is obtained in 10 − order. Any measurements in this respectand comparison of the results with the predictions of the present work can give essential information about the natureof the Σ Q baryon, nucleon distribution amplitudes and search for the new physics beyond the standard model.3 IV. ACKNOWLEDGMENT
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In this Appendix, the explicit expressions for the form factors f and f for b case as well as the nucleon DA’s aregiven: 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) , (32) 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 n 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) , (33)where H ( x i ) = H ( x , x , − x − x ) ,s ( y, Q ) = (1 − y ) m N + (1 − y ) y Q + m b y . (34)The t = t ( s , Q ) is the solution of the equation s ( t , Q ) = s , i.e., t ( s , Q ) = m N − Q + p m N ( m b + Q ) + ( m N − Q − s ) − s m N . (35)Here, s is continuum threshold, M B is the Borel mass parameter. In calculations, the following short hand notationsfor the functions H ± i a , ± j b ,... = ± a H i ± b H j ... are used, and H i functions are written in terms of the DA’s in thefollowing way: 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 , , (36)where for each DA’s, we also have used X ± i, ± j,... = ± X i ± X j ... .7The explicit expressions for the nucleon DA’s is given as: V ( x i , µ ) = 120 x x x [ φ ( µ ) + φ +3 ( µ )(1 − x )] ,V ( x i , µ ) = 24 x x [ φ ( µ ) + φ +3 ( µ )(1 − x )] ,V ( x i , µ ) = 12 x { ψ ( µ )(1 − x ) + ψ − ( µ )[ x + x − x (1 − x )]+ ψ +4 ( µ )(1 − x − x x ) } ,V ( x i , µ ) = 3 { ψ ( µ )(1 − x ) + ψ − ( µ )[2 x x − x (1 − x )]+ ψ +5 ( µ )[1 − x − x + x )] } ,V ( x i , µ ) = 6 x [ φ ( µ ) + φ +5 ( µ )(1 − x )] ,V ( x i , µ ) = 2[ φ ( µ ) + φ +6 ( µ )(1 − x )] ,A ( x i , µ ) = 120 x x x φ − ( µ )( x − x ) ,A ( x i , µ ) = 24 x x φ − ( µ )( x − x ) ,A ( x i , µ ) = 12 x ( x − x ) { ( ψ ( µ ) + ψ +4 ( µ )) + ψ − ( µ )(1 − x ) } ,A ( x i , µ ) = 3( x − x ) {− ψ ( µ ) + ψ − ( µ ) x + ψ +5 ( µ )(1 − x ) } ,A ( x i , µ ) = 6 x ( x − x ) φ − ( µ ) A ( x i , µ ) = 2( x − x ) φ − ( µ ) ,T ( x i , µ ) = 120 x x x [ φ ( µ ) + 12 ( φ − − φ +3 )( µ )(1 − x )] ,T ( x i , µ ) = 24 x x [ ξ ( µ ) + ξ +4 ( µ )(1 − x )] ,T ( x i , µ ) = 6 x { ( ξ + φ + ψ )( µ )(1 − x ) + ( ξ − + φ − − ψ − )( µ )[ x + x − x (1 − x )]+( ξ +4 + φ +4 + ψ +4 )( µ )(1 − x − x x ) } ,T ( x i , µ ) = 32 { ( ξ + φ + ψ )( µ )(1 − x ) + ( ξ − + φ − − ψ − )( µ )[2 x x − x (1 − x )]+( ξ +5 + φ +5 + ψ +5 )( µ )(1 − x − x + x )) } ,T ( x i , µ ) = 6 x [ ξ ( µ ) + ξ +5 ( µ )(1 − x )] ,T ( x i , µ ) = 2[ φ ( µ ) + 12 ( φ − − φ +6 )( µ )(1 − x )] ,T ( x i , µ ) = 6 x { ( − ξ + φ + ψ )( µ )(1 − x ) + ( − ξ − + φ − − ψ − )( µ )[ x + x − x (1 − x )]+( − ξ +4 + φ +4 + ψ +4 )( µ )(1 − x − x x ) } ,T ( x i , µ ) = 32 { ( − ξ + φ + ψ )( µ )(1 − x ) + ( − ξ − + φ − − ψ − )( µ )[2 x x − x (1 − x )]+( − ξ +5 + φ +5 + ψ +5 )( µ )(1 − x − x + x )) } ,S ( x i , µ ) = 6 x ( x − x ) (cid:2) ( ξ + φ + ψ + ξ +4 + φ +4 + ψ +4 )( µ ) + ( ξ − + φ − − ψ − )( µ )(1 − x ) (cid:3) S ( x i , µ ) = 32 ( x − x ) (cid:2) − (cid:0) ψ + φ + ξ (cid:1) ( µ ) + (cid:0) ξ − + φ − − ψ (cid:1) ( µ ) x + (cid:0) ξ +5 + φ +5 + ψ (cid:1) ( µ )(1 − x ) (cid:3) P ( x i , µ ) = 6 x ( x − x ) (cid:2) ( ξ − φ − ψ + ξ +4 − φ +4 − ψ +4 )( µ ) + ( ξ − − φ − + ψ − )( µ )(1 − x ) (cid:3) P ( x i , µ ) = 32 ( x − x ) (cid:2)(cid:0) ψ + ψ − ξ (cid:1) ( µ ) + (cid:0) ξ − − φ − + ψ (cid:1) ( µ ) x + (cid:0) ξ +5 − φ +5 − ψ (cid:1) ( µ )(1 − x ) (cid:3) . (37)The following functions are encountered to the above amplitudes and they can be defined in terms of the eight8independent parameters, namely f N , λ , λ , V d , A u , f d , f d and f u : φ = φ = f N φ = φ = 12 ( λ + f N ) ξ = ξ = 16 λ ψ = ψ = 12 ( f N − λ ) φ − = 212 A u ,φ +3 = 72 (1 − V d ) ,φ − = 54 (cid:0) λ (1 − f d − f u ) + f N (2 A u − (cid:1) ,φ +4 = 14 (cid:0) λ (3 − f d ) − f N (10 V d − (cid:1) ,ψ − = − (cid:0) λ (2 − f d + f u ) + f N ( A u + 3 V d − (cid:1) ,ψ +4 = − (cid:0) λ ( − f d + 5 f u ) + f N (2 + 5 A u − V d ) (cid:1) ,ξ − = 516 λ (4 − f d ) ,ξ +4 = 116 λ (4 − f d ) ,φ − = 53 (cid:0) λ ( f d − f u ) + f N (2 A u − (cid:1) ,φ +5 = − (cid:0) λ (4 f d −
1) + f N (3 + 4 V d ) (cid:1) ,ψ − = 53 (cid:0) λ ( f d − f u ) + f N (2 − A u − V d ) (cid:1) ,ψ +5 = − (cid:0) λ ( − f d + 2 f u ) + f N (5 + 2 A u − V d ) (cid:1) ,ξ − = − λ f d ,ξ +5 = 536 λ (2 − f d ) ,φ − = 12 (cid:0) λ (1 − f d − f u ) + f N (1 + 4 A u ) (cid:1) ,φ +6 = − (cid:0) λ (1 − f d ) + f N (4 V d − (cid:1)(cid:1)