Radiative Decays of the Heavy Flavored Baryons in Light Cone QCD Sum Rules

Transcription

1 arxiv:91.76v1 hep-ph 31 Dec 8 Radiative Decays of the Heavy Flavored Baryons in Light Cone QCD Sum Rules T. M. Aliev, K. Azizi, A. Ozpineci Physics Department, Middle East Technical University, 6531, Ankara, Turkey Astract The transition magnetic dipole and electric quadrupole moments of the radiative decays of the sextet heavy flavored spin 3 to the heavy spin 1 aryons are calculated within the light cone QCD sum rules approach. Using the otained results, the decay rate for these transitions are also computed and compared with the existing predictions of the other approaches. PACS: Hx, 13.4.Em, 13.4.Gp, 13.3.Ce, 14..Lq, 14..Mr Permanent address: Institute of Physics, Baku, Azeraijan taliev@metu.edu.tr e14634@metu.edu.tr ozpineci@metu.edu.tr

2 1 Introduction In the last years, considerale experimental progress has een made in the identification of new ottom aryon states. The CDF Collaoration 1 reported the first oservation of the Σ ± and Σ ±. After this discovery, the DO and CDF 3 Collaorations announced the oservation of the Ξ state (for a review see 4). The BaBar Collaoration discovered the Ω c state 5 and oserved the Ω c state in B decays 6. Lastly, the BaBar and BELLE Collaorations oserved Ξ c state 7. Heavy aryons containing c and quarks have een suject of the intensive theoretical studies (see 8 and references therein). A theoretical study of experimental results can give essential information aout the structure of these aryons. In this sense, study of the electromagnetic properties of the heavy aryons receives special attention. One of the main static electromagnetic quantities of the aryons is their magnetic moments. The magnetic moments of the heavy aryons have een discussed within different approaches in the literature (see 9 and references therein). In the present work, we investigate the electromagnetic decays of the ground state heavy aryons containing single heavy quark with total angular momentum J = 3 to the heavy aryons with J = 1 in the framework of the light cone QCD sum rules method. Note that, some of the considered decays have een studied within heavy hadron chiral perturation theory 1, heavy quark and chiral symmetries 11, 1, in the relativistic quark model 13 and light cone QCD sum rules at leading order in HQET in 14. Here, we also emphasize that the radiative decays of the light decuplet aryons to the octet aryons have een studied in the framework of the light cone QCD sum rules in 15. The work is organized as follows. In section, the light cone QCD sum 1

3 rules for the form factors descriing electromagnetic transition of the heavy J = 3, to the heavy aryons with J = 1 are otained. Section 3 encompasses the numerical analysis of the transition magnetic dipole and electric quadrupole moments as well as the radiative decay rates. A comparison of our results for the total decay width with the existing predictions of the other approaches is also presented in section 3. Light cone QCD sum rules for the electromagnetic form factors of the heavy flavored aryons We start this section with a few remarks aout the classification of the heavy aryons. Heavy aryons with single heavy quark elong to either SU(3) antisymmetric 3 F or symmetric 6 F flavor representations. For the aryons containing single heavy quark, in the m Q limit, the angular momentum of the light quarks is a good quantum numer. The spin of the light diquark is either S = 1 for 6 F or S = for 3 F. The ground state will have angular momentum l =. Therefore, the spin of the ground state is 1/ for 3 F representing the Λ Q and Ξ Q aryons, while it can e oth 3/ or 1/ for 6 F, corresponding to Σ Q, Σ Q, Ξ Q, Ξ Q, Ω Q and Ω Q states, where indicates spin 3/ states. After this remark, let us calculate the electromagnetic transition form factors of the heavy aryons. For this aim, consider the following correlation function, which is the main tool of the light cone QCD sum rules: T µ (p, q) = i d 4 xe ipx T {η(x) η µ ()} γ, (1) where η and η µ are the generic interpolating quark currents of the heavy flavored aryons with J = 1/ and 3/, respectively and γ means the external

4 electromagnetic field. In this work we will discuss the following electromagnetic transitions where Q = c or quark. Σ Q Σ Q γ, Ξ Q Ξ Q γ, Σ Q Λ Qγ, In QCD sum rules approach, this correlation function is calculated in two different ways: from one side, it is calculated in terms of the quarks and gluons interacting in QCD vacuum. In the phenomenological side, on the other hand, it is saturated y a tower of hadrons with the same quantum numers as the interpolating currents. The physical quantities are determined y matching these two different representations of the correlation function. The hadronic representation of the correlation function can e otained inserting the complete set of states with the same quantum numers as the interpolating currents. η (p) 1(p + q) η µ T µ (p, q) = (p) 1(p + q) p m γ +..., (3) (p + q) m 1 where 1(p+q) and (p) denote heavy spin 3/ and 1/ states and m 1 and m represent their masses, respectively and q is the photon momentum. In the aove equation, the dots correspond to the contriutions of the higher states and continuum. For the calculation of the phenomenological part, it follows from Eq. (3) that we need to know the matrix elements of the interpolating currents etween the vacuum and aryon states. They are defined as () 1(p + q, s) η µ () = λ 1 ū µ (p + q, s), η() (p, s ) = λ u(p, s ), (4) 3

5 where λ 1 and λ are the residues of the heavy aryons, u µ (p, s) is the Rarita- Schwinger spinor and s and s are the polarizations of the spin 3/ and 1/ states, respectively. The electromagnetic vertex (p) 1(p + q) γ of the spin 3/ to spin 1/ transition is parameterized in terms of the three form factors in the following way 16, 17 (p) 1(p + q) γ = eū(p, s ) { G 1 (q µ ε ε µ q) + G (Pε)q µ (Pq)ε µ γ 5 + G 3 (qε)q µ q ε µ γ 5 }u µ (p + q), where P = p+(p+q) and ε µ is the photon polarization vector. Since for the considered decays the photon is real, the terms proportional to G 3 are exactly zero, and for analysis of these decays, we need to know the values of the form factors G 1 (q ) and G (q ) only at q =. From the experimental point of view, more convenient form factors are magnetic dipole G M, electric quadrupole G E and Coulom quadrupole G C which are linear cominations of the form factors G 1 and G ( see 15). At q =, these relations are G M = (3m 1 + m ) G 1 m + (m 1 m )G m 1 3, G1 m G E = (m 1 m ) + G m 1 3. (6) In order to otain the explicit expressions of the correlation function from the phenomenological side, we also perform summation over spins of the spin 3/ particles using u µ (p, s)ū ν (p, s) = ( p + m ) m { g µν γ µγ ν p µp ν 3m p µγ ν p ν γ µ }. (7) 3m s In principle, the phenomenological part of the correlator can e otained with the help of the Eqs. (3-7). As noted in 9, 15, at this point two prolems 4 (5)

6 appear: 1) all Lorentz structures are not independent, ) not only spin 3/, ut spin 1/ states also give contriutions to the correlation function. In other words the matrix element of the current η µ etween vacuum and spin 1/ states is nonzero. In general, this matrix element can e written in the following way η µ () B(p, s = 1/) = (Ap µ + Bγ µ )u(p, s = 1/). (8) Imposing the condition γ µ η µ =, one can immediately otain that B = A 4 m. In order to remove the contriution of the spin 1/ states and deal with only independent structures in the correlation function, we will follow 9, 15 and remove those contriutions y ordering the Dirac matrices in a specific way. For this aim, we choose the ordering for Dirac matrices as ε q pγ µ. Using this ordering for the correlator, we get 1 1 T µ = eλ 1 λ p m (p + q) m 1 ε µ (pq) (εp)q µ { G 1 m 1 G m 1 m + G (p + q) + G 1 G (m 1 m ) p + m G q G q p}γ 5 + q µ ε ε µ q { G 1 (p + m 1 m ) G 1 (m 1 + m ) p } γ 5 +G 1 ε(pq) q(εp)q µ γ 5 G 1 ε q {m + p}q µ γ 5 other structures with γ µ at the end or which are proportional to (p + q) µ. For determination of the form factors G 1 and G, we need two invariant structures. We otained that among all structures, the est convergence comes from the structures ε pγ 5 q µ and q pγ 5 (εp)q µ for G 1 and G, respectively. To 5 (9)

7 get the sum rules expression for the form factors G 1 and G, we will choose the same structures also from the QCD side and match the corresponding coefficients. We also would like to note that, the correlation function receives contriutions from contact terms. But, the contact terms do not give contriutions to the chosen structures (for a detailed discussion see e. g. 15, 18 ). On QCD side, the correlation function can e calculated using the operator product expansion. For this aim, we need the explicit expressions of the interpolating currents of the heavy aryons with the angular momentums J = 3/ and J = 1/. The interpolating currents for the spin J = 3/ aryons are written in such a way that the light quarks should enter the expression of currents in symmetric way and the condition γ µ η µ = should e satisfied. The general form of the currents for spin J = 3/ aryons satisfying oth aforementioned conditions can e written as 9 { } η µ = Aǫ ac (q1 at Cγ µ q)q c + (q at Cγ µ Q )q1 c + (Q at Cγ µ q1)q c, (1) where C is the charge conjugation operator and a, and c are color indices. The value of normalization factor A and quark fields q 1 and q for corresponding heavy aryons is given in Tale 1 (see 9). The general form of the interpolating currents for the heavy spin 1/ aryons can e written in the following form: η ΣQ = 1 ǫ ac {(q at 1 CQ )γ 5 q c + β(qat 1 Cγ 5Q )q c (Q at Cq )γ 5q c 1 + β(qat Cγ 5 q )qc 1 }, η ΞQ,Λ Q = 1 6 ǫ ac {(q at 1 Cq )γ 5Q c + β(q at 1 Cγ 5q )Qc + (q at 1 CQ )γ 5 q c + β(q at 1 Cγ 5Q )q c + (QaT Cq )γ 5q c 1 + β(qat Cγ 5 q )qc 1}, (11) 6

8 Heavy spin 3 aryons A q 1 q Σ +(++) (c) 1/ 3 u u /3 u d Σ (+) (c) Σ () (c) 1/ 3 d d Ξ (+) (c) /3 s u Ξ () (c) /3 s d Ω () (c) 1/ 3 s s Tale 1: The value of normalization factor A and quark fields q 1 and q for the corresponding heavy spin 3/ aryons. where β is an aritrary parameter and β = 1 corresponds to the Ioffe current. The quark fields q 1 and q for the corresponding heavy spin 1/ aryons are as presented in Tale. Heavy spin 1 aryons q 1 q Σ +(++) (c) u u Σ (+) (c) u d Σ () (c) d d Ξ (+) (c) u s Ξ () (c) d s Λ (+) (c) u d Tale : The quark fields q 1 and q for the corresponding heavy spin 1/ aryons. After performing all contractions of the quark fields in Eq. (1), we get the following expression for the correlation functions responsile for the Σ Σ γ and Ξ Ξ γ transitions in terms of the light and heavy quark 7

9 propagators T Σ Σ µ = i 3 ǫ ac ǫ a c d 4 xe ipx γ(q) { γ 5 S c c d TrS γ µ S a a u + γ 5 S c c u TrS γ 5 S c a d γ µs a u + γ 5 S c a u γ µ S a d γ µs a a c S S c d βs c u γ µs a d γ 5 S a c T Ξ Ξ µ = i 3 ǫ acǫ a c + βtrγ 5 S aa u + γ 5 S c d γ µs a u βsd ca γ µ S a u + βs c a u γ µs a Sa c u γ 5 S c γ µs Sd cc γ 5 S c u γ µs a d + βs c d γ µ S a βtrγ 5 S aa S a c γ 5 S a c u γ µ S d Su cc γ 5 S c d, (1) d 4 xe ipx γ(q) {γ 5 S c c s TrS γ µ S a a u γ 5 S c c TrS a s γ µ S a u + βtrγ 5S a u γ µs a s S c c βtrγ 5 S a a u γ µs S c c s + γ 5 S c c u TrS s γ µs a a βtrγ 5 S a a γ µ S s S c c u γ 5S c u γ µs a s + γ 5 S c a γ 5 S c s γ µs a + βs c a s where S = CS T C. γ µ S s S a c u γ 5 S c S a c u + βs c a γ µ S a u γ 5S c βs c βs c u γ µs a s γ 5 S a c γ µ S a a u S c s + γ 5 S c a s γ µ S s γ 5 S a c u βsc s γ µs a + βs c a u γ µs a γ 5 S a c u γ µ S a u γ µ S a a u + γ 5 S c a u γ µs a S c γ 5 S c s S c s S a c γ 5 S c s β, (13) The correlation functions for all other possile transitions can e otained y the following replacements T Σ Σ µ = T Σ Σ µ (u d), T Σ + Σ + µ = T Σ Σ µ (d u), T Σ + c Σ + c µ = T Σ Σ µ ( c), T Σ c Σ c Σ µ = T Σ µ ( c), T Σ ++ c Σ ++ c Σ + µ = T Σ + µ ( c), T Ξ Ξ µ = T Ξ Ξ µ (u d), T Ξ c Ξ c Ξ µ = T Ξ µ ( c), 8

10 T Ξ + c Ξ + c µ = T Ξ Ξ µ ( c), T Σ Λ µ = T Ξ Ξ µ (s d), T Σ + c Λ + c µ = T Σ Λ µ ( c). (14) The correlators in Eqs. (1, 13) get three different contriutions: 1) Perturative contriutions, ) Mixed contriutions, i.e., the photon is radiated from short distance and at least one of the quarks forms a condensate. 3) Non-perturative contriutions, i.e., when photon is radiated at long distances. This contriution is descried y the matrix element γ(q) q(x 1 )Γq(x ) which is parameterized in terms of photon distriution amplitudes with definite twists. The results of the contriutions when the photon interacts with the quarks perturatively is otained y replacing the propagator of the quark that emits the photon y { Sαβ a } a d 4 ys free (x y) AS free (y). (15) αβ The free light and heavy quark propagators are defined as: S free q = i x π x 4 S free Q m q 4π x, K 1 (m Q x ) 4π x = m Q i m Q x 4π x K (m Q x ), (16) where K i are Bessel functions. The non-perturative contriutions to the correlation function can e easily otained from Eq. (1) replacing one of the light quark propagators that emits a photon y S a αβ 1 4 qa Γ j q (Γ j ) αβ, (17) 9

11 { where Γ is the full set of Dirac matrices Γ j = 1, γ 5, γ α, iγ 5 γ α, σ αβ / } and sum over index j is implied. Remaining quark propagators are full propagators involving the perturative as well as the non-perturative contriutions. The light cone expansion of the light and heavy quark propagators in the presence of an external field is done in 19. The operators qgq, qggq and qq qq, where G is the gluon field strength tensor give contriutions to the propagators and in, it was shown that terms with two gluons as well as four quarks operators give negligile small contriutions, so we neglect them. Taking into account this fact, the expressions for the heavy and light quark propagators are written as: S Q (x) = S free Q (x) ig s + 1 m Q kvx µg µν γ ν d 4 k (π) 4e ikx, S q (x) = Sq free (x) m q 4π x qq 1 1 x ig s du i m q 3π G µνσ µν 1 k + mq dv (m Q (vx)σ k ) Gµν µν ( 1 i m ) ( q 4 x x 19 m qq 1 i m ) q 6 x 16π x G µν(ux)σ µν ux µ G µν (ux)γ ν i 4π x ( ( ) ) x Λ ln + γ E, (18) 4 where Λ is the scale parameter and we choose it at factorization scale i.e., Λ = (.5 GeV 1 GeV ) (see 9, 1). As we already noted, for the calculation of the non-perturative contriutions, the matrix elements γ(q) qγ i q are needed. These matrix elements are calculated in terms of the photon distriution amplitudes (DA s) as follows. γ(q) q(x)σ µν q() = ie q qq(ε µ q ν ε ν q µ ) i (qx) e q qq 1 ( ) ( εx εx x ν ε µ q µ x µ ε ν q ν qx qx 1 due iūqx ( χϕ γ (u) + x ) 1 due iūqx h γ (u) ) 16 A(u)

12 ( ) εx 1 γ(q) q(x)γ µ q() = e q f 3γ ε µ q µ due iūqx ψ v (u) qx γ(q) q(x)γ µ γ 5 q() = e qf 3γ ǫ µναβ ε ν q α x β due iūqx ψ a (u) γ(q) q(x)g s G µν (vx)q() = ie q qq (ε µ q ν ε ν q µ ) Dα i e i(α q+vαg)qx S(α i ) γ(q) q(x)g s Gµν iγ 5 (vx)q() = ie q qq (ε µ q ν ε ν q µ ) Dα i e i(α q+vαg)qx S(αi ) γ(q) q(x)g s Gµν (vx)γ α γ 5 q() = e q f 3γ q α (ε µ q ν ε ν q µ ) Dα i e i(α q+vαg)qx A(α i ) γ(q) q(x)g s G µν (vx)iγ α q() = e q f 3γ q α (ε µ q ν ε ν q µ ) Dα i e i(α q+vαg)qx V(α i ) {( ) ( εx γ(q) q(x)σ αβ g s G µν (vx)q() = e q qq ε µ q µ g αν 1 ) qx qx (q αx ν + q ν x α ) q β ( ) ( εx ε µ q µ g βν 1 ) qx qx (q βx ν + q ν x β ) q α ( ) ( εx ε ν q ν g αµ 1 ) qx qx (q αx µ + q µ x α ) q β ( ) ( εx + ε ν q ν g βµ 1 ) q.x qx (q βx µ + q µ x β ) q α Dα i e i(α q+vαg)qx T 1 (α i ) ( ) ( εx + ε α q α g µβ 1 ) qx qx (q µx β + q β x µ ) q ν ( ) ( εx ε α q α g νβ 1 ) qx qx (q νx β + q β x ν ) q µ ( ) ( εx ε β q β g µα 1 ) qx qx (q µx α + q α x µ ) q ν ( ) ( εx + ε β q β g να 1 ) qx qx (q νx α + q α x ν ) q µ Dα i e i(α q+vαg)qx T (α i ) + 1 qx (q µx ν q ν x µ )(ε α q β ε β q α ) Dα i e i(α q+vαg)qx T 3 (α i ) + 1 } qx (q αx β q β x α )(ε µ q ν ε ν q µ ) Dα i e i(α q+vαg)qx T 4 (α i ), (19) where ϕ γ (u) is the leading twist, ψ v (u), ψ a (u), A and V are the twist 3 and h γ (u), A, T i (i = 1,, 3, 4) are the twist 4 photon DA s, respectively and χ 11

13 is the magnetic susceptiility of the quarks. The photon DA s is calculated in. The measure Dα i is defined as Dα i = dα q dα q dα g δ(1 α q α q α g ). () The coefficient of any structure in the expression of the correlation function can e written in the form T(q 1, q, Q) = e q1 T 1 (q 1, q, Q) + e q T 1(q 1, q, Q) + e Q T (q 1, q, Q), (1) where T 1, T 1 and T in the right side correspond to the radiation of the photon from the quarks q 1, q and Q, respectively. Starting from the expressions of the interpolating currents, one can easily otain that the functions T 1 and T 1 differ only y q 1 q exchange for Σ Q Σ Q transition. Using Eq. (14), in SU() symmetry limit, we otain T Σ +(++) Σ +(++) (c) (c) + T Σ () Σ () (c) (c) = T Σ (+) (c) Σ (+) (c). () Therefore, in the numerical analysis section, we have not presented the numerical results for T Σ () Σ () (c) (c). Now, using the aove equations, one can get the the correlation function from the QCD side. Separating the coefficients of the structures ε pγ 5 q µ and q pγ 5 (εp)q µ respectively for the form factors G 1 and G from oth QCD and phenomenological representations and equating them, we get sum rules for the form factors G 1 and G. To suppress the contriutions of the higher states and continuum, Borel transformations with respect to the variales p and (p + q) are applied. The explicit forms of the sum rules for the form factors G 1 and G can e as follows. For Σ G 1 G = m 1 = λ 1 λ (m 1 + m ) e 1 e λ 1 λ m 1 m M 1 M e 1 M 1 e m M Σ, we otain e q1 Π 1 + e q Π 1 (q 1 q ) + e Π 1 e q1 Π + e q Π (q 1 q ) + e Π, (3) 1

14 where q 1 = u, q = d. The functions Π i Π i can e written as: s Π i Π i = e s M ρ i (s)ρ m Q i (s)ds + e M Γ i Γ i, (4) m Q where, 3ρ1 (s) = q 1 q 1 q q β 3 χϕ γ(u ) + m q (1 + β) q 144m Q π (1 + ψ 11) βmq + q q 4π (ψ 1 ψ 1 ) + q 1 q 1 m Q 3(1 + β)(ψ 96π 1 ψ 1 )A(u ) ( + (ψ 1 ψ 1 ){(1 + 5β)η 1 η 4η 3 η 4 + 6η 5 + η 6 + 3η 7 η 8 β(η + η 4 η 5 η 6 7η 7 + η 8 )} 3(3 + β)( 1 + ψ + ψ 1 ψ 1 )(u 1)ζ 1 + {(1 + 3β)η 1 (3 + β)(η η 4 ) + (1 + 3β)η 7 }ln( s ) m Q + 3(1 + β)m Qχ{ψ 1 ψ + ψ 31 ln( s ) )}ϕ γ (u ) + + m4 Q (β 1) 4ψ 51π 4 3 5ψ 4 4ψ 5 + (4ψ 3 8ψ ψ 4 + 4ψ 5 )u + 6{ψ 1 ψ ln( s )}(1 + u ) m Q + f 3γm Q (β 1) 8{ (ψ 19π ψ 31 )(η 1 η ) + ψ 1 ln( s )(η m 3 + η 1 η )} Q + 6(ψ ψ 31 )ψ a (u ) + (3ψ ψ 3 + 3ψ 4 )(u 1)4ψ v (u ) dψa (u ) du 3ρ 1(s) = m ( q 1 q 1 + q q ) m Q 5(1 + 3β)(1 + ψ11 ) π m Q,

15 + ( q 1 q 1 + q q ) m Q(1 + 3β) ψ 8π 1 ψ 1 ln( s ) m Q + m4 Q (1 β) ( 3ψ 768π 4 1ψ ψ ψ ψ 41 ) 3(ψ 4 + 4ψ 5 ) + {(3ψ + 6ψ 3 18ψ 31 8ψ 3 ) + 3( 8ψ 41 + ψ 4 + 4ψ 5 )}u 3ρ (s) = q 1 q 1 ( + 1ψ 1 (7 + 5u ) 7(ψ 1 + u )ln( m Q s ) + 1{7 + ψ 1 ( 1 + u ) u }ln( s ) m Q 1 1m Q π, (5) 3(3 + β)( 1 + ψ 3 + ψ 1 + ψ )(u 1)ζ (1 + β) + (β 1)ψ 11 + (1 + 3β)(ψ 1 + ψ ) 3 + β ) ( 1 + ψ 1 + ψ )(ζ 9 ζ 5 ) + + m Q (β 1) (1ψ 18π ψ 3 + ψ ψ 4 )(u 1)u + f 3γ(β 1) 4{(1 ψ 96π )η 9 + ( 3 + 3ψ + ψ 1 ψ 1 )η 11 ( 1 + ψ + ψ 1 ψ 1 )η 1 + ( 1 + 3ψ ψ 3 )(u 1)ζ 11 } ( 1 + 3ψ ψ 3 )(u 1)ψ a (u ), 3ρ (s) = m Q (β 1) ( 1 + u 19π 4 )u {1ψ 1 6ψ + 4ψ ψ 3 + ψ ψ 4 1ln( s )} m Q, (6) βm 4 3Γ1 = m Q A(u ) < q 1 q 1 >< q q > + m Q 48M 6 43M {1η 4 η 3 η 4 + η 5 + β(η 1 + η η 3 + η 4 η 6 + η 8 ) + 3(u 1)ζ 1 βa(u )} 14

16 + 5β 54 χϕ 1 γ(u ) + 18M {(1 u )(β 5)ζ 1 + 9βm Qχϕ γ (u } βm Q A(u ) + < q 1 q 1 >< q q > + 1 1M 36 { 4η η 3 η 4 + η 5 + β(η 1 + η η 3 + η 4 η 6 + η 8 ) + 3(u 1)ζ 1 + 3βA(u )} + m βmq < q q > + { βm3 Q 16π 4M + (1 + β)m Q }f 4 16M 3γ ψ a (u ) β + < q q > 6 f 3γm Q ψ a (u ), (7) 3Γ 1 = m < q 1q 1 >< q q > βm Q 3M 4 + < q 1 q 1 >< q q > β, 3 3Γ = m < q 1 q 1 >< q q > 1 54M 4(u 1)(β 5)ζ m Q + 18M {3(u 6 1)ζ β(ζ 5 ζ 9 )} + < q 1 q 1 >< q q > 9M { 3(u 1)ζ + β(ζ 5 ζ 9 )}, (8) For Ξ G 1 G = Ξ, we have m 1 = λ 1 λ (m 1 + m ) e 1 e λ 1 λ m 1 m M 1 M e Γ 1 M 1 e =. (9) m M e q1 Θ 1 e q Θ 1 (q 1 q ) + e Θ 1 e q1 Θ e q Θ (q 1 q ) + e Θ, (3) where q 1 = u, q = s. The functions Θ i Θ i are also defined as: Θ i Θ i = s m Q e s M i(s) 15 i m Q (s)ds + e M i i, (31)

17 and 1 + β 1(s) = q 1 q 1 q q χϕ γ (u ) 9 (1 + β) + m q q 144m Q π (1 + ψ 11) βmq + q q 4π (ψ 1 ψ 1 ) + q 1 q 1 m Q 3(1 + 5β)(ψ 88π 1 ψ 1 )A(u ) ( + (ψ 1 ψ 1 ){( 1 + 7β)η 1 + η 4η 3 + η 4 + η 5 η 6 3η 7 + η 8 + β(5η 8η 3 + 5η 4 η 5 1η 6 3η 7 + 1η 8 )} + 3( 1 + β)( 1 + ψ + ψ 1 ψ 1 )(u 1)ζ 1 + ( 1 + β)(η 1 + η η 4 + η 7 )ln( s ) m Q 3(1 + 5β)m Qχ{ψ 1 ψ + ψ 31 ln( s ) )}ϕ γ (u ) + m Q + m4 Q (β 1) 6ψ 51π 4 4ψ 3 + 5ψ 4 + 4ψ 5 + (6ψ 4ψ 3 + 8ψ 41 5ψ 4 4ψ 5 )u 1{ψ 1 ln( s )}(1 + u ) f 3γm Q (β 1) 8{(ψ 576π ψ 31 )(η 1 η ) + ψ 1 ln( s )(3η m 3 η 1 + η )} Q + 18(ψ ψ 31 )ψ a (u ) + 3(3ψ ψ 3 + 3ψ 4 )(u 1)4ψ v (u ) dψa (u ) du 1 (s) = m ( q 1q 1 q q ) + ( q 1 q 1 q q ) m Q(β 1) 4π m Q 5(β 1)(1 + ψ11 ) 43π m Q ψ 1 ψ 1 ln( s ) m Q, (3) 16

18 + m3 Q m q (β 1) 6ψ 18π 4 1 ψ + ψ 31 {3 + ψ 1 }ln( s ), m Q (33) (s) = q 1 q 1 1 β 3( 1 + ψ 36m Q π 3 + ψ 1 + ψ )(u 1)ζ + ψ 11 (ζ 3 + ζ 5 ζ 7 ζ 9 ) + { 1 (ζ 9 ζ 5 ) + (ψ 1 + ψ )ζ (ζ 5 ζ 9 ) ζ 7 } + m Q (β 1) (1ψ 18π ψ 3 + ψ ψ 4 )(u 1)u f 3γ(β 1) 4{( 1 + ψ 88π + 4ψ 1 4ψ 1 )η 9 3( 1 + ψ + ψ 1 ψ 1 )η 11 + ( 1 + ψ ψ 1 + ψ 1 )η 1 + 3( 1 + 3ψ ψ 3 )(u 1)ζ 11 } 3( 1 + 3ψ ψ 3 )(u 1)ψ a (u ), (s) =, (34) (1 + β)m 4 1 = m Q A(u ) < q 1 q 1 >< q q > + m Q 144M 6 196M {1(1 β)(η 4 η 3 ) 3(1 + β)η 4 + ( + β)η 5 + (1 + β)(η 6 + η 7 η 8 ) + 3( + β)(u 1)ζ 1 5(1 + β) + (1 + β)a(u )} χϕ γ (u ) M {( 1 + u )(5 + 3β)ζ 1 3(1 + β)m Qχϕ γ (u } + < q 1 q 1 >< q q > (1 + β)m Q A(u ) M 18 {4(β 1)η + η 3 + 3η 4 (η 5 + η 6 + η 7 η 8 3ζ 1 ) 6u ζ 1 β(η 3 3η 4 + η 5 + 4(η 6 + η 7 η 8 ) + 3(u 1)ζ 1 ) 3(1 + β)a(u )} + m < q q > βmq 16π { βm3 Q 4M + (1 + β)m Q }f 4 16M 3γ ψ a (u ) 17

19 β + < q q > 6 f 3γm Q ψ a (u ), (35) 1 = m < q 5(β 1) m 3 Q m q 1q 1 >< q q > 43 M 6 (β 1)mQ m q < q 1 q 1 >< q q >, 36M m Qm q M 4 = m < q 1 1q 1 >< q q > 54M 4(u 1)(3β + 5)ζ + m Q 54M {3( + β)(u 6 1)ζ (1 + β)(ζ 5 ζ 9 )} + < q 1 q 1 >< q q > 7M {3( + β)(u 1)ζ (1 + β)(ζ 5 ζ 9 )}, (36) =. (37) Note that, in the aove equations, the terms proportional to m s are omitted ecause of their lengthy expressions, ut they have een taken into account in numerical calculations. The contriutions of the terms < G > are also calculated, ut their numerical values are very small and therefore for customary in the expressions these terms are also omitted. The functions entering Eqs. (5-37) are given as 1 η i = Dα i dvf i (α i )δ(α q + vα g u ), 1 η i = Dα i dvg i (α i )δ (α q + vα g u ), ζ i = ζ i = 1 h i (u)du (i = 1,, 11), u 1 Dα i d vh i (α i )θ(α q + vα g u ) (i = 3 1), 18

20 ψ nm = (s m Q ) n s m (m Q )n m, (38) and f 1 (α i ) = S(α i ), f (α i ) = S(α i ), f 3 (α i ) = v S(α i ), f 4 (α i ) = h 5 (α i ) = T (α i ), f 5 (α i ) = h 6 (α i ) = vt (α i ), f 6 (α i ) = h 8 (α i ) = vt 3 (α i ), f 7 (α i ) = h 9 (α i ) = T 4 (α i ), f 8 (α i ) = h 1 (α i ) = vt 4 (α i ), f 9 (α i ) = A(α i ), f 1 (α i ) = g 3 (α i ) = va(α i ), f 11 (α i ) = g (α i ) = vv(α i ), g 1 (α i ) = V(α i ), h 1 (u) = h γ (u), h (u) = (u u )h γ (u), h 3 (α i ) = T 1 (α i ), h 4 (α i ) = vt 1 (α i ), h 7 (α i ) = T 3 (α i ) and h 11 (u) = ψ v (u) are functions in terms of the photon distriution amplitudes. Note that in the aove equations, the Borel parameter M is defined as M = M 1 M and u M1 = M 1. Since the masses of the initial and final +M M1 +M aryons are very close to each other, we set M1 = M = M and u = 1/. From Eqs. (3) and (3) it is clear that for the calculation of the transition magnetic dipole and electric quadrupole moments, the expressions for the residues λ 1 and λ are needed. These residues are determined from two point sum rules. For the chosen interpolating currents, we get the following results for the residues λ 1 and λ (see also 9, 3): λ 1e m 1 M = A Π + Π (q 1 q ), (39) where Π = s m Q ds e {m s/m < q (m q1 6m Q )(ψ + ψ 1 1) 1q 1 > 19m Q π (ψ + ψ 1 ψ 1 1)m Q + (ψ 1)(3m q1 m q ) < q 1 q 1 > 3π m3 Q 8(3ψ 51π ψ 3 )m q1 + 3{(ψ 3 4ψ ψ 4 + ψ 5 )m Q 4ψ 4 m q1 } + 6(ψ 1 ψ )( 3 } m Q m q1 ) 6(4m q1 3m Q )ln( m Q s ) 19

21 5m 3 + e m Q {m /M < q Q m q1 1q 1 >< q q > 1 + < q 1 q 1 >< q q > 1 m Qm q1 + m 1M < q 1 q 1 > + m Q(m Q + 5m q1 ) 5 144M 6 4M 4 4M 6mq 7m q1 19π }, (4) λ Σ Q (Ξ Q ) e m Σ Q (Ξ Q ) /M = s e s m Q M σ ΣQ (Ξ Q )(s)ds + e m Q M ω ΣQ (Ξ Q ), (41) with { σ ΣQ (s) = (< dd > + < uu >) (β 1) m (6ψ 64π 13ψ 6ψ 11 ) 4m Q } + 3m Q (ψ 1 ψ 11 ψ 1 + ψ 1 ) + m 4 Q 48π 45 + β( + 5β)1ψ 1 6ψ + ψ 3 4ψ 41 + ψ 4 1ln( s ), m Q (4) { (β 1) m σ ΞQ (s) = (< ss > + < uu >) 6(1 + β)ψ 19π (7 + 11β)ψ 4m Q } 6(1 + β)ψ 11 + (1 + 5β)m Q (ψ 1 ψ 11 ψ 1 + ψ 1 ) + m 4 Q 48π 45 + β( + 5β)1ψ 1 6ψ + ψ 3 4ψ 41 + ψ 4 1ln( s ), m Q (43) ω ΣQ = ω ΞQ = (β 1) 4 (β 1) 7 m Q m < dd >< uu > + m M 4 4M 1, m Q m < ss >< uu > ( β) M4, (44) + m 4M(5 + 3β) ( β) λ ΛQ = λ ΞQ (s d). (45)

22 In the expression for λ ΞQ also the terms proportional to the strange quark mass are taken into account in our numerical calculations, ut omitted in the aove formulas. 3 Numerical analysis This section is devoted to the numerical analysis for the magnetic dipole G M and electric quadrupole G E as well as the calculation of the decay rates for the considered radiative transitions. The input parameters used in the analysis of the sum rules are taken to e: ūu (1 GeV ) = dd (1 GeV ) = (.43) 3 GeV 3, ß(1 GeV ) =.8 ūu (1 GeV ), m (1 GeV ) = (.8±.) GeV 4, Λ = (.5 1) GeV and f 3γ =.39 GeV. The value of the magnetic susceptiility was otained in different papers as χ(1 GeV ) = 3.15 ±.3 GeV, χ(1 GeV ) = (.85 ±.5) GeV 5 and χ(1 GeV ) = 4.4 GeV 6. From sum rules for the magnetic dipole G M and electric quadrupole G E, it is clear that we also need to know the explicit form of the photon DA s : ( ) ϕ γ (u) = 6uū 1 + ϕ (µ)c 3 (u ū), ψ v (u) = 3 ( 3(u 1) 1 ) + 3 ( ) ( 15w V 64 γ 5wγ A 3 3(u 1) + 35(u 1) 4), ψ a (u) = ( 1 (u 1) )( 5(u 1) 1 ) ( wv γ 3 ) 16 wa γ, ( ) A(α i ) = 36α q α q αg 1 + wγ A 1 (7α g 3), V(α i ) = 54w V γ (α q α q )α q α q α g, h γ (u) = 1 ( 1 + κ +) C 1 (u ū), A(u) = 4u ū ( 3κ κ ) +8(ζ + 3ζ ) uū( + 13uū) 1

23 + u 3 (1 15u + 6u ) ln(u) + ū 3 (1 15ū + 6ū ) ln(ū), T 1 (α i ) = 1(3ζ + ζ + )(α q α q )α q α q α g, T (α i ) = 3αg (α q α q ) ( (κ κ + ) + (ζ 1 ζ 1 + )(1 α g) + ζ (3 4α g ) ), T 3 (α i ) = 1(3ζ ζ + )(α q α q )α q α q α g, T 4 (α i ) = 3αg (α q α q ) ( (κ + κ + ) + (ζ 1 + ζ 1 + )(1 α g) + ζ (3 4α g ) ), S(α i ) = 3αg {(κ + κ+ )(1 α g ) + (ζ 1 + ζ 1 + )(1 α g)(1 α g ) + ζ 3(α q α q ) α g (1 α g )}, S(α i ) = 3αg{(κ κ + )(1 α g ) + (ζ 1 ζ 1 + )(1 α g )(1 α g ) + ζ 3(α q α q ) α g (1 α g )}. (46) The constants entering the aove DA s are otained as ϕ (1 GeV ) =, wγ V = 3.8 ±1.8, wa γ =.1 ±1., κ =., κ+ =, ζ 1 =.4, ζ =.3, ζ 1 + = and ζ + =. The sum rules for the electromagnetic form factors also contain three auxiliary parameters: the Borel mass parameter M, the continuum threshold s and the aritrary parameter β entering the expression of the interpolating currents of the heavy spin 1/ aryons. The measurale physical quantities, i.e. the magnetic dipole and electric quadrupole moments, should e independent of them. Therefore, we look for regions for these auxiliary parameters such that in these regions the G M and G E are practically independent of them. The working region for M are found requiring that not only the contriutions of the higher states and continuum should e less than the ground state contriution, ut the highest power of 1/M e less than say 3 / of the highest power of M. These two conditions are oth satisfied in the region 15 GeV M 3 GeV and 6 GeV M 1 GeV for aryons containing and c-quark, respectively. The working regions for the continuum threshold s and the general parameter β are otained con-

24 sidering the fact that the results of the physical quantities are approximately unchanged. We also would like to note that in Figs. 1-4, the asolute values are plotted since it is not possile to calculate the sign of the residue from the mass sum-rules. Hence, it is not possile to predict the signs of the G M or G E. The relative sign of the G M and G E can only e predicted using the QCD sum rules. The dependence of the magnetic dipole moment G M and the electric quadropole moment G E on cosθ, where β = tanθ at two fixed values of the continuum threshold s and Borel mass square M are depicted in Figs. 1-. All presented figures have two following common ehavior: a) They ecomes very large near to the end points, i.e, cosθ = ±1 and they have zero points at some finite values of cosθ. Note that similar ehavior was otained in analysis of the radiative decays of the light decuplet to octet aryons (see 15). Explanation of these properties is as follows. From the explicit forms of the interpolating currents, one can see that the correlation, hence, λ 1 λ (β)g M(E) as well as λ (β) is a linear function of the β. In general, zeros of these quantities do not coincide due to the fact that the OPE is truncated, i.e., the calculations are not exact. These points and any region etween them are not suitale regions for β and hence the suitale regions for β should e far from these regions. It should e noted that in many cases, the Ioffe current which corresponds to cosθ.71, is out of the working region of β. We also show the dependence of the magnetic dipole moment G M and the electric quadropole moment G E on M at two fixed values of the continuum threshold s and three values of the aritrary parameter β in Figs From all these figures, it follows that the sum rules for G M and G E exhiit very good staility with respect to the M in the working region. From these figures, we also see that the results depend on s weakly. We should 3

25 also stress that our results practically do not change considering three values of χ which we presented at the eginning of this section. Our final results on the asolute values of the magnetic dipole and electric quadrupole moments of the considered radiative transitions are presented in Tale 3. The quoted errors in Tale 3 are due to the uncertainties in m, variation of s, β and M as well as errors in the determination of the input parameters. Here, we would like to make the following remark. From Eq. (6), it follows that the G E is proportional to the mass difference m 1 m and these masses are close to each other. Therefore, reliaility of predictions for G E are questionale and one can consider them as order of magnitude. For this reason, we consider only the central values for G E in Tale 3. G M G E Σ Σ γ 1. ±.4.5 Σ Σ γ.1 ±.7.16 Σ + Σ + γ 4. ± Σ + c Σ + c γ 1. ±..14 Σ c Σ cγ.5 ±.1.3 Σ ++ c Σ ++ c γ.8 ±.8.3 Ξ Ξ γ 8.5 ± Ξ Ξ γ.9 ±.3.11 Ξ + c Ξ + c γ 4. ± Ξ c Ξ cγ.45 ±.15.7 Σ Λ γ 7.3 ±.8.75 Σ + c Λ + c γ 3.8 ± 1..6 Tale 3: The results for the magnetic dipole moment G M and electric quadrupole moment G E for the corresponding radiative decays in units of their natural magneton. At the end of this section, we would like to calculate the decay rate of the considered radiative transitions in terms of the multipole moments G E 4

26 and G M : Γ = 3 α (m 1 m )3 ( G 3 m 3 1m M + 3GE) (47) The results for the decay rates are given in Tale 4. In comparison, we also present the predictions of the constituent quark model in SU(N f ) O(3) symmetry 1, the relativistic three-quark model 13 and heavy quark effective theory (HQET) 14 in this Tale. Γ (present work) Γ 1 Γ 13 Γ 14 Σ Σ γ.8 ± Σ Σ γ.11 ± Σ + Σ + γ.46 ± Σ + c Σ + c γ.4 ± ±.4 - Σ c Σ cγ.8 ± Σ ++ c Σ ++ c γ.65 ± Ξ Ξ γ 135 ± Ξ Ξ γ 1.5 ± Ξ + c Ξ + c γ 5 ± 5-54 ± 3 - Ξ c Ξ cγ.66 ± ±.4 - Σ Λ γ 114 ± Σ + c Λ + c γ 13 ± ± 4 - Tale 4: The results for the decay rates of the corresponding radiative transitions in KeV. In summary, the transition magnetic dipole moment G M (q = ) and electric quadrupole moment G E (q = ) of the radiative decays of the sextet heavy flavored spin 3 to the heavy spin 1 aryons were calculated within the light cone QCD sum rules approach. Using the otained results for the electromagnetic moments G M and G E, the decay rate for these transitions were also calculated. The comparison of our results on these decay rates with the predictions of the other approaches is also presented. 5

27 4 Acknowledgment Two of the authors (K. A. and A. O.), would like to thank TUBITAK, Turkish Scientific and Research Council, for their partial financial support through the project numer 16T333 and also through the PhD scholarship program (K. A.). References 1 T. Aaltonen et. al, CDF Collaoration, Phys. Rev. Lett. 99 (7) 1. V. Aazov et. al, DO Collaoration, Phys. Rev. Lett. 99 (7) T. Aaltonen et. al, CDF Collaoration, Phys. Rev. Lett. 99 (7) 5. 4 R. Mizuk, Heavy Flavor Baryons, prepared for XXIII International Symposium on Lepton and Photon Interactions at High Energy (LP7), Aug 7, Daegu, Korea; arxiv:71.31 hep-ex. 5 B. Auert et. al, BaBar Collaoration, Phys. Rev. Lett. 97 (6) B. Auert et. al, BaBar Collaoration, Phys. Rev. Lett. 99 (7) B. Auert et. al, BaBar Collaoration, Phys. Rev. D 77 (8) 1; R. Christor et. al, BELLE Collaoration, Phys. Rev. Lett. 97 (6) W. Roerts, M.Pervin, Int. J. Mod. Phys. A (8)

28 9 T. M. Aliev, K. Azizi, A. Ozpineci, Nucl. Phys. B 88 (9) M. Banuls, A. Pich, I. Scimemi, Phys. Rev. D 61 () H. Y. Cheng, et. al, Phys. Rev. D 47 (1993) S. Tawfig, J. G. Koerner, P. J. O Donnel, Phys. Rev. D 63 (1) M. A. Ivanov, et. al, Phys. Rev. D 6 (1999) S. L. Zhu, Y. B. Dai, Phys. Rev. D 59 (1999) T. M. Aliev, A. Ozpineci, Nucl. Phys. B 73 (6) H. F. Joens, M. D. Scadron, Ann. Phys. 81(1973)1. 17 R. C. E. Devenish, T.S. Eisenschitz and J. G. Korner, Phys. Rev. D 14 (1976) A. Khodjamirian, D. Wyler, arxiv:hep-ph/ I. I. Balitsky, V. M. Braun, Nucl. Phys. B 311 (1989) 541. V. M. Braun, I. E. Filyanov, Z. Phys. C 48 (199) K. G. Chetyrkin, A. Khodjamirian, A. A. Pivovarov, Phys. Lett. B 661 (8)5; I. I. Balitsky, V. M. Braun, A. V. Kolesnichenko, Nucl. Phys. B 31 (1989) 59. P. Ball, V. M. Braun, N. Kivel, Nucl. Phys. B 649 (3) K. Azizi, M. Bayar, A. Ozpineci, arxiv: hep-ph. 4 V. M. Belyaev, B. L. Ioffe, JETP 56 (198) J. Rohrwild, JHEP 79 (7) 73. 7

29 6 V. M. Belyaev, I. I. Kogan, Yad. Fiz. 4 (1984)

30 8 G M (Σ --->Σ ) 4 s =6. GeV s =6.4 GeV cosθ Figure 1: The dependence of the magnetic dipole form factor G M for Σ Σ on cosθ for two fixed values of the continuum threshold s. Bare lines and lines with the circles correspond to the M = GeV and M = 5 GeV, respectively. 3 G M (Σ + --->Σ+ ) s =6. GeV s =6.4 GeV cosθ Figure : The same as Fig. 1, ut for Σ + Σ +. 9

31 6 G M (Σ + c --->Σ+ c ) 4 s =.7 GeV s =.9 GeV cosθ Figure 3: The same as Fig. 1, ut for Σ + c Σ + c and M = 6 GeV and M = 9 GeV. 1 G M (Σ ++ c --->Σ++ c ) s =.7 GeV s =.9 GeV cosθ Figure 4: The same as Fig. 3, ut for Σ ++ c Σ ++ c. 3

32 G M (Σ --->Λ ) s =6. GeV s =6.4 GeV cosθ Figure 5: The same as Fig. 1, ut for Σ Λ. 1 G M (Σ + c --->Λ+ c ) 5 s =.7 GeV s =.9 GeV cosθ Figure 6: The same as Fig. 3, ut for Σ + c Λ + c. 31

33 G M (Ξ --->Ξ ) s =6.3 GeV s =6.5 GeV cosθ Figure 7: The same as Fig. 1, ut for Ξ Ξ. 3 G M (Ξ --->Ξ ) 1 s =6.3 GeV s =6.5 GeV cosθ Figure 8: The same as Fig. 1, ut for Ξ Ξ. 3

34 1. G M (Ξ c --->Ξ c ) s =.8 GeV s =3. GeV cosθ Figure 9: The same as Fig. 3, ut for Ξ c Ξ c. 1 G M (Ξ + c --->Ξ+ c ) s =.8 GeV s =3. GeV cosθ Figure 1: The same as Fig. 3, ut for Ξ + c Ξ + c. 33

35 . G E (Σ --->Σ ).1 s =6. GeV s =6.4 GeV cosθ Figure 11: The dependence of the electric dipole form factor G E for Σ Σ on cosθ for two fixed values of the continuum threshold s. Bare lines and lines with the circles correspond to the M = GeV and M = 5 GeV, respectively..8 G E (Σ + --->Σ+ ).6.4. s =6. GeV s =6.4 GeV cosθ Figure 1: The same as Fig. 11, ut for Σ + Σ +. 34

36 G E (Σ + c --->Σ+ c ).6.4. s =.7 GeV s =.9 GeV cosθ Figure 13: The same as Fig. 11, ut for Σ + c M = 9 GeV. Σ + c and M = 6 GeV and.1 G E (Σ ++ c --->Σ++ c ).5 s =.7 GeV s =.9 GeV cosθ Figure 14: The same as Fig. 13, ut for Σ ++ c Σ ++ c. 35

37 G E (Σ --->Λ )..1 s =6. GeV s =6.4 GeV cosθ Figure 15: The same as Fig. 11, ut for Σ Λ..16 G E (Σ + c --->Λ+ c ) s =.7 GeV s =.9 GeV cosθ Figure 16: The same as Fig. 13, ut for Σ + c Λ + c. 36

38 G E (Ξ --->Ξ )..1 s =6.3 GeV s =6.5 GeV cosθ Figure 17: The same as Fig. 11, ut for Ξ Ξ..4 G E (Ξ --->Ξ ). s =6.3 GeV s =6.5 GeV cosθ Figure 18: The same as Fig. 11, ut for Ξ Ξ. 37

39 . G E (Ξ c --->Ξ c ).1 s =.8 GeV s =3. GeV cosθ Figure 19: The same as Fig. 13, ut for Ξ c Ξ c.. G E (Ξ + c --->Ξ+ c ) s =.8 GeV s =3. GeV cosθ Figure : The same as Fig. 13, ut for Ξ + c Ξ + c. 38

40 3 G M (Σ --->Σ ) 1 s =6. GeV s =6.4 GeV M (GeV ) Figure 1: The dependence of the magnetic dipole form factor G M for Σ Σ on the Borel mass parameter M for two fixed values of the continuum threshold s. Bare lines, lines with the circles and lines with the dimonds correspond to the Ioffe currents (β = 1), β = 5 and β =, respectively. 9 G M (Σ + --->Σ+ ) 6 3 s =6. GeV s =6.4SGeV M (GeV ) Figure : The same as Fig. 1, ut for Σ + Σ +. 39

41 4 G M (Σ + c --->Σ+ c ) s =.7 GeV s =.9 GeV M (GeV ) Figure 3: The same as Fig. 1, ut for Σ + c M = 9 GeV. Σ + c and M = 6 GeV and 8 G M (Σ ++ c --->Σ++ c ) 6 4 s =.7 GeV s =.9 GeV M (GeV ) Figure 4: The same as Fig. 3, ut for Σ ++ c Σ ++ c. 4

42 G M (Σ --->Λ ) s =6. GeV s =6.4 GeV M (GeV ) Figure 5: The same as Fig. 1, ut for Σ Λ. 8 G M (Σ + c --->Λ+ c ) 6 4 s =.7 GeV s =.9 GeV M (GeV ) Figure 6: The same as Fig. 3, ut for Σ + c Λ + c. 41

43 G M (Ξ --->Ξ ) s =6.3 GeV s =6.5 GeV M (GeV ) Figure 7: The same as Fig. 1, ut for Ξ Ξ. 3 G M (Ξ --->Ξ ) 1 s =6.3 GeV s =6.5 GeV M (GeV ) Figure 8: The same as Fig. 1, ut for Ξ Ξ. 4

44 1. G M (Ξ c --->Ξ c ) s =.8 GeV s =3. GeV M (GeV ) Figure 9: The same as Fig. 3, ut for Ξ c Ξ c. 1 G M (Ξ + c --->Ξ+ c ) s =.8 GeV s =3. GeV M (GeV ) Figure 3: The same as Fig. 3, ut for Ξ + c Ξ + c. 43

45 .3 G E (Σ --->Σ )..1 s =6. GeV s =6.4 GeV M (GeV ) Figure 31: The dependence of the electric dipole form factor G E for Σ Σ on the Borel mass parameter M for two fixed values of the continuum threshold s. Bare lines, lines with the circles and lines with the dimonds correspond to the Ioffe currents (β = 1), β = 5 and β =, respectively..8 G E (Σ + --->Σ+ ).6.4. s =6. GeV s =6.4 GeV M (GeV ) Figure 3: The same as Fig. 31, ut for Σ + Σ +. 44

46 .6 G E (Σ + c --->Σ+ c ).4. s =.7 GeV s =.9 GeV M (GeV ) Figure 33: The same as Fig. 31, ut for Σ + c M = 9 GeV. Σ + c and M = 6 GeV and.1 G E (Σ ++ c --->Σ++ c ).5 s =.7 GeV s =.9 GeV M (GeV ) Figure 34: The same as Fig. 33, ut for Σ ++ c Σ ++ c. 45

47 . G E (Σ --->Λ ) s =6. GeV s =6.4 GeV M (GeV ) Figure 35: The same as Fig. 31, ut for Σ Λ..16 G E (Σ + c --->Λ+ c ) s =.7 GeV s =.9 GeV M (GeV ) Figure 36: The same as Fig. 33, ut for Σ + c Λ + c. 46

48 .4 G E (Ξ --->Ξ ). s =6.3 GeV s =6.5 GeV M (GeV ) Figure 37: The same as Fig. 31, ut for Ξ Ξ..3 G E (Ξ --->Ξ )..1 s =6.3 GeV s =6.5 GeV M (GeV ) Figure 38: The same as Fig. 31, ut for Ξ Ξ. 47

49 . G E (Ξ c --->Ξ c ).1 s =.8 GeV s =3. GeV M (GeV ) Figure 39: The same as Fig. 33, ut for Ξ c Ξ c.. G E (Ξ + c --->Ξ+ c ).1 s =.8 GeV s =3. GeV M (GeV ) Figure 4: The same as Fig. 33, ut for Ξ + c Ξ + c. 48