Radiative decays of the heavy flavored baryons in light cone QCD sum rules

Radiative decays of the heavy flavored baryons in light cone QCD sum rules

T.M. Aliev,*,† K. Azizi,‡ and A. Ozpinecix

Physics Department, Middle East Technical University, 06531, Ankara, Turkey(Received 5 January 2009; revised manuscript received 28 February 2009; published 17 March 2009)

The transition magnetic dipole and electric quadrupole moments of the radiative decays of the sextet

heavy flavored spin 3=2 to the heavy spin 1=2 baryons are calculated within the light cone QCD sum rules

approach. Using the obtained results, the decay rate for these transitions are also computed and compared

with the existing predictions of the other approaches.

DOI: 10.1103/PhysRevD.79.056005 PACS numbers: 11.55.Hx, 13.30.Ce, 13.40.Em, 13.40.Gp

I. INTRODUCTION

In the last years, considerable experimental progress hasbeen made in the identification of new bottom baryonstates. The CDF Collaboration [1] reported the first obser-vation of the��

b and��b . After this discovery, the DO [2],

and CDF [3] Collaborations announced the observation ofthe �b� state (for a review see [4]). The BABARCollaboration discovered the ��

c state [5] and observedthe �c state in B decays [6]. Lastly, the BABAR andBELLE Collaborations observed �c state [7].

Heavy baryons containing c and b quarks have beensubject of the intensive theoretical studies (see [8] andreferences therein). A theoretical study of experimentalresults can give essential information about the structureof these baryons. In this sense, study of the electromagneticproperties of the heavy baryons receives special attention.One of the main static electromagnetic quantities of thebaryons is their magnetic moments. The magnetic mo-ments of the heavy baryons have been discussed withindifferent approaches in the literature (see [9] and referen-ces therein).

In the present work, we investigate the electromagneticdecays of the ground state heavy baryons containing singleheavy quark with total angular momentum J ¼ 3

2 to the

heavy baryons with J ¼ 12 in the framework of the light

cone QCD sum rules method. Note that, some of theconsidered decays have been studied within heavy hadronchiral perturbation theory [10], heavy quark and chiralsymmetries [11,12], 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 ofthe light decuplet baryons to the octet baryons have beenstudied in the framework of the light cone QCD sum rulesin [15].

The work is organized as follows. In Sec. II, the lightcone QCD sum rules for the form factors describing elec-tromagnetic transition of the heavy J ¼ 3

2 , to the heavy

baryons with J ¼ 12 are obtained. Section III encompasses

the numerical analysis of the transition magnetic dipoleand electric quadrupole moments as well as the radiativedecay rates. A comparison of our results for the total decaywidth with the existing predictions of the other approachesis also presented in Sec. III.

II. LIGHT CONE QCD SUM RULES FOR THEELECTROMAGNETIC FORM FACTORS OF THE

HEAVY FLAVORED BARYONS

We start this section with a few remarks about theclassification of the heavy baryons. Heavy baryons withsingle heavy quark belong to either SU(3) antisymmetric�3F or symmetric 6F flavor representations. For the baryonscontaining single heavy quark, in the mQ ! 1 limit, the

angular momentum of the light quarks is a good quantumnumber. The spin of the light diquark is either S ¼ 1 for 6For S ¼ 0 for �3F. The ground state will have angular mo-mentum l ¼ 0. Therefore, the spin of the ground state is1=2 for �3F representing the �Q and �Q baryons, while it

can be both 3=2 or 1=2 for 6F, corresponding to �Q, ��Q,

�0Q, �

�Q, �Q and ��

Q states, where � indicates spin 3=2

states.After this remark, let us calculate the electromagnetic

transition form factors of the heavy baryons. For this aim,consider the following correlation function, which is themain tool of the light cone QCD sum rules:

T�ðp; qÞ ¼ iZ

d4xeipxh0 j Tf�ðxÞ ��ð0Þg j 0i�; (1)

where � and �� are the generic interpolating quark cur-

rents of the heavy flavored baryons with J ¼ 1=2 and 3=2,respectively, and � means the external electromagneticfield. In this work we will discuss the following electro-magnetic transitions

��Q ! �Q�; ��

Q ! �Q�; ��Q ! �Q�; (2)

where Q ¼ c or b quark.In the QCD sum-rules approach, this correlation func-

tion is calculated in two different ways: from one side, it iscalculated in terms of the quarks and gluons interacting inQCD vacuum. In the phenomenological side, on the other

*Permanent address: Institute of Physics, Baku, Azerbaijan†[email protected]‡[email protected]@metu.edu.tr

PHYSICAL REVIEW D 79, 056005 (2009)

1550-7998=2009=79(5)=056005(10) 056005-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.79.056005

hand, it is saturated by a tower of hadrons with the samequantum numbers as the interpolating currents. The physi-cal quantities are determined by matching these two differ-ent representations of the correlation function.

The hadronic representation of the correlation functioncan be obtained inserting the complete set of states with thesame quantum numbers as the interpolating currents.

T�ðp; qÞ ¼ h0 j � j 2ðpÞip2 �m2

2

h2ðpÞ j 1ðpþ qÞi�

� h1ðpþqÞ j �� j 0iðpþ qÞ2 �m2

1

þ . . . ; (3)

where h1ðpþ qÞj and h2ðpÞj denote heavy spin 3=2 and1=2 states and m1 and m2 represent their masses, respec-tively, and q is the photon momentum. In the above equa-tion, the dots correspond to the contributions of the higherstates and continuum. For the calculation of the phenome-nological part, it follows from Eq. (3) that we need to knowthe matrix elements of the interpolating currents betweenthe vacuum and baryon states. They are defined as

h1ðpþ q; sÞ j ��ð0Þ j 0i ¼ �1 �u�ðpþ q; sÞ;h0 j �ð0Þ j 2ðp; s0Þi ¼ �2uðp; s0Þ;

(4)

where �1 and �2 are the residues of the heavy baryons,u�ðp; sÞ is the Rarita-Schwinger spinor and s and s0 are thepolarizations of the spin 3=2 and 1=2 states, respectively.The electromagnetic vertex h2ðpÞ j 1ðpþ qÞi� of the spin

3=2 to spin 1=2 transition is parametrized in terms of thethree form factors in the following way [16,17]

h2ðpÞ j 1ðpþqÞi�¼ e �uðp;s0ÞfG1ðq�"6 �"�q6 ÞþG2½ðP"Þq��ðPqÞ"��5

þG3½ðq"Þq��q2"��5gu�ðpþqÞ; (5)

where P ¼ pþðpþqÞ2 and "� is the photon polarization

vector. Since for the considered decays the photon is real,the terms proportional to G3 are exactly zero, and foranalysis of these decays, we need to know the values ofthe form factors G1ðq2Þ and G2ðq2Þ only at q2 ¼ 0. Fromthe experimental point of view, more convenient form

factors are magnetic dipole GM, electric quadrupole GE

and Coulomb quadrupole GC which are linear combina-tions of the form factors G1 and G2 (see [15]). At q

2 ¼ 0,these relations are

GM ¼�ð3m1 þm2ÞG1

m1

þ ðm1 �m2ÞG2

�m2

3;

GE ¼ ðm1 �m2Þ�G1

m1

þG2

�m2

3:

(6)

In order to obtain the explicit expressions of the corre-lation function from the phenomenological side, we alsoperform summation over spins of the spin 3=2 particlesusing

Xs

u�ðp; sÞ �u�ðp; sÞ ¼ ðp6 þmÞ2m

��g�� þ 1

3��

� 2p�p�

3m2� p�� p��

3m

�: (7)

In principle, the phenomenological part of the correlatorcan be obtained with the help of the Eqs. (3)–(7). As notedin [9,15], at this point two problems appear: 1) all Lorentzstructures are not independent, b) not only spin 3=2, butspin 1=2 states also give contributions to the correlationfunction. In other words the matrix element of the current�� between vacuum and spin 1=2 states is nonzero. In

general, this matrix element can be written in the followingway

h0 j ��ð0Þ j Bðp; s ¼ 1=2Þi¼ ðAp� þ B��Þuðp; s ¼ 1=2Þ: (8)

Imposing the condition �� ¼ 0, one can immediately

obtain that B ¼ � A4 m.

In order to remove the contribution of the spin 1=2 statesand deal with only independent structures in the correlationfunction, we will follow [9,15] and remove those contri-butions by ordering the Dirac matrices in a specific way.For this aim, we choose the ordering for Dirac matrices as"6 q6 p6 ��. Using this ordering for the correlator, we get

T� ¼ e�1�2

1

p2 �m22

1

ðpþ qÞ2 �m21

½½"�ðpqÞ � ð"pÞq��f�2G1m1 �G2m1m2 þG2ðpþ qÞ2

þ ½2G1 �G2ðm1 �m2Þ�p6 þm2G2q6 �G2q6 p6 g�5 þ ½q� 6�� "� 6��fG1ðp2 þm1m2Þ �G1ðm1 þm2Þp6 g�5

þ 2G1½6�ðpqÞ � q6 ð"pÞ�q��5 �G1 6�q6 fm2 þ p6 gq��5at the end or which are proportional toðpþ qÞ��: (9)

For determination of the form factors G1 and G2, we needtwo invariant structures. We obtained that among allstructures, the best convergence comes from the structures"6 p6 �5q� and q6 p6 �5ð"pÞq� for G1 and G2, respectively.To get the sum-rules expression for the form factorsG1 and G2, we will choose the same structures also

from the QCD side and match the correspondingcoefficients. We also would like to note that, the correla-tion function receives contributions from contact terms.But, the contact terms do not give contributions to thechosen structures (for a detailed discussion see e. g.[15,18]).

T.M. ALIEV, K. AZIZI, AND A. OZPINECI PHYSICAL REVIEW D 79, 056005 (2009)

056005-2

On QCD side, the correlation function can be calculatedusing the operator product expansion. For this aim, weneed the explicit expressions of the interpolating currentsof the heavy baryons with the angular momentums J ¼3=2 and J ¼ 1=2. The interpolating currents for the spinJ ¼ 3=2 baryons are written in such a way that the lightquarks should enter the expression of currents in symmet-ric way and the condition �� ¼ 0 should be satisfied.

The general form of the currents for spin J ¼ 3=2 baryonssatisfying both aforementioned conditions can be writtenas [9]

�� ¼ A�abcfðqaT1 C��qb2ÞQc þ ðqaT2 C��Q

bÞqc1þ ðQaTC��q

b1Þqc2g; (10)

where C is the charge conjugation operator and a, b and care color indices. The value of normalization factor A andquark fields q1 and q2 for corresponding heavy baryons isgiven in Table I (see [9]).

The general form of the interpolating currents for theheavy spin 1=2 baryons can be written in the following

form:

��Q¼ � 1ffiffiffi

2p �abcfðqaT1 CQbÞ�5q

c2 þ �ðqaT1 C�5Q

bÞqc2� ½ðQaTCqb2Þ�5q

c1 þ �ðQaTC�5q

b2Þqc1�g;

��Q;�Q¼ 1ffiffiffi

6p �abcf2½ðqaT1 Cqb2Þ�5Q

c þ �ðqaT1 C�5qb2ÞQc�

þ ðqaT1 CQbÞ�5qc2 þ �ðqaT1 C�5Q

bÞqc2þ ðQaTCqb2Þ�5q

c1 þ �ðQaTC�5q

b2Þqc1g; (11)

where � is an arbitrary parameter and � ¼ �1 corre-sponds to the Ioffe current. The quark fields q1 and q2for the corresponding heavy spin 1=2 baryons are as pre-sented in Table II.After performing all contractions of the quark fields in

Eq. (1), we get the following expression for the correlationfunctions responsible for the ��0

b ! �0b� and ��0

b ! �0b�

transitions in terms of the light and heavy quark propaga-tors

T��0

b!�0

b� ¼ iffiffiffi

3p �abc�a0b0c0

Zd4xeipxh�ðqÞ j f��5S

c0cd Tr½Sb0bb ��S

0a0au � þ �5S

c0cu Tr½Sb0bd ��S

0a0ab � þ �Tr½�5S

aa0u ��S

0bb0b �Scc0d

� �Tr½�5Saa0b ��S

0bb0d �Scc0u � �5S

c0ad ��S

0a0bu Sb

0cb þ �5S

c0bd ��S

0b0ab Sa

0cu � �5S

c0bu ��S

0b0ad Sa

0cb þ �5S

c0au ��S

0a0bb Sb

0cd

� �Sca0

d ��S0ab0u �5S

bc0b þ �Sc

0bd ��S

0b0ab �5S

a0cu � �Sc

0bu ��S

0b0ad �5S

a0cb þ �Sc

0au ��S

0a0bb �5S

b0cd j 0i; (12)

T��0

b!�0

b� ¼ i

3�abc�a0b0c0

Zd4xeipxh�ðqÞ j f�5S

c0cs Tr½Sb0bb ��S

0a0au � � 2�5S

c0cb Tr½Sb0as ��S

0a0bu � þ 2�Tr½�5S

a0bu ��S

0b0as �Sc0cb

� �Tr½�5Sa0au ��S

0b0bb �Sc0cs þ �5S

c0cu Tr½Sb0bs ��S

0a0ab � � �Tr½�5S

a0ab ��S

0b0bs �Sc0cu � �5S

c0bu ��S

0b0as Sa

0cb

þ 2�5Sc0ab ��S

0b0bs Sa

0cu � 2�5S

c0bb ��S

0a0au Sb

0cs þ �5S

c0as ��S

0a0bu Sb

0cb � �5S

c0bs ��S

0b0ab Sa

0cu þ 2�Sc

0ab ��S

0b0bs �5S

a0cu

� 2�Sc0bb ��S

0a0au �5S

b0cs þ �Sc

0as ��S

0a0bu �5S

b0cb � �Sc

0bs ��S

0b0ab �5S

a0cu þ �5S

c0au ��S

0a0bb Sb

0cs

� �Sc0bu ��S

0b0as �5S

a0cb þ �Sc

0au ��S

0a0bb �5S

b0cs j 0i; (13)

where S0 ¼ CSTC.The correlation functions for all other possible transitions can be obtained by the following replacements

TABLE I. The value of normalization factor A and quark fieldsq1 and q2 for the corresponding heavy spin 3=2 baryons.

Heavy spin 32 baryons A q1 q2

��þðþþÞbðcÞ 1=

ffiffiffi3

pu u

��0ðþÞbðcÞ

ffiffiffiffiffiffiffiffi2=3

pu d

��ð0ÞbðcÞ 1=

ffiffiffi3

pd d

��0ðþÞbðcÞ


ps u

��ð0ÞbðcÞ


ps d

��ð0ÞbðcÞ 1=

ffiffiffi3

ps s

TABLE II. The quark fields q1 and q2 for the correspondingheavy spin 1=2 baryons.

Heavy spin 12 baryons q1 q2

�þðþþÞbðcÞ u u

�0ðþÞbðcÞ u d

��ð0ÞbðcÞ d d

�0ðþÞbðcÞ u s

��ð0ÞbðcÞ d s

�0ðþÞbðcÞ u d

RADIATIVE DECAYS OF THE HEAVY FLAVORED . . . PHYSICAL REVIEW D 79, 056005 (2009)

056005-3

T��

b!��

b� ¼ T

��0b!�0

b� ðu ! dÞ; T

��þb

!�þb

� ¼ T��0

b!�0

b� ðd ! uÞ; T��þ

c !�þc

� ¼ T��0

b!�0

b� ðb ! cÞ;

T��0c !�0

c� ¼ T

��b

!��b

� ðb ! cÞ; T��þþc !�þþ

c� ¼ T

��þb

!�þb

� ðb ! cÞ; T��

b!��

b� ¼ T

��0b!�0

b� ðu ! dÞ;

T��0c !�0

c� ¼ T

��b

!��b

� ðb ! cÞ; T��þc !�þ

c� ¼ T

��0b!�0

b� ðb ! cÞ; T

��0b!�0

b� ¼ T

��0b!�0

b� ðs ! dÞ;

T��þc !�þ

c� ¼ T

��0b!�0

b� ðb ! cÞ:

(14)

In SU(2) symmetry limit, from Eq. (14) it follows that,

T��þðþþÞ

bðcÞ !�þðþþÞbðcÞ

� þ T��ð0Þ

bðcÞ !��ð0ÞbðcÞ

� ¼ 2T��0ðþÞ

bðcÞ !�0ðþÞbðcÞ

� : (15)

The correlators in Eqs. (12) and (13) get three differentcontributions: 1) Perturbative contributions, 2) Mixed con-tributions, i.e., the photon is radiated from short distanceand at least one of the quark pairs forms a condensate.3) Nonperturbative contributions, i.e., when photon is radi-ated at long distances. This contribution is described by thematrix element h�ðqÞ j �qðx1Þ�qðx2Þ j 0i which is parame-trized in terms of photon distribution amplitudes withdefinite twists.

The results of the contributions when the photon inter-acts with the quarks perturbatively is obtained by replacingthe propagator of the quark that emits the photon by

Sab�� )�Z

d4ySfreeðx� yÞA6 SfreeðyÞ�ab

��: (16)

The free light and heavy quark propagators are defined as:

Sfreeq ¼ ix622x4

� mq

42x2;

SfreeQ ¼ m2Q

42

K1ðmQ

ffiffiffiffiffiffiffiffiffi�x2

pÞffiffiffiffiffiffiffiffiffi

�x2p � i

m2Qx6

42x2K2ðmQ

ffiffiffiffiffiffiffiffiffi�x2

pÞ; (17)

where Ki are Bessel functions. The nonperturbative con-tributions to the correlation function can be easily obtainedfrom Eq. (12) replacing one of the light quark propagatorsthat emits a photon by

Sab�� ! � 1

4�qa�jq

bð�jÞ��; (18)

where � is the full set of Dirac matrices �j ¼f1; �5; ��; i�5��; ��=

ffiffiffi2

p g and sum over index j is im-

plied. Remaining quark propagators are full propagatorsinvolving the perturbative as well as the nonperturbativecontributions.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 thegluon field strength tensor give contributions to the propa-gators and in [20], it was shown that terms with two gluonsas well as four quarks operators give negligible smallcontributions, so we neglect them. Taking into accountthis fact, the expressions for the heavy and light quarkpropagators are written as:

SQðxÞ ¼ SfreeQ ðxÞ � igsZ d4k

ð2Þ4 e�ikx

Z 1

0dv

�k6 þmQ

ðm2Q � k2Þ2 G

��ðvxÞ�� þ 1

m2Q � k2

vx�G��

�;

SqðxÞ ¼ Sfreeq ðxÞ � mq

42x2� h �qqi

12

�1� i

mq

4x6�� x2

192m2

0h �qqi�1� i

mq

6x6�

� igsZ 1

0du

�x6

162x2G��ðuxÞ�� ux�G��ðuxÞ�� i

42x2� i

mq

322G��

��

�ln

��x2�2

4

�þ 2�E

��; (19)

where � is the scale parameter and we choose it at facto-rization scale i.e., � ¼ ð0:5 GeV–1 GeVÞ (see [9,21]).

As we already noted, for the calculation of the non-perturbative contributions, the matrix elements h�ðqÞ j�q�iq j 0i are needed. These matrix elements are parame-trized in terms of the photon distribution amplitudes (DA’s)(see [22]).

Now, using the above equations and expressions of thematrix elements in terms of the photon DA’s, one can getthe correlation function from the QCD side. Separating thecoefficients of the structures "6 p6 �5q� and q6 p6 �5ð"pÞq�respectively for the form factors G1 and G2 from bothQCD and phenomenological representations and equating

them, we get sum rules for the form factors G1 and G2. Tosuppress the contributions of the higher states and contin-uum, Borel transformations with respect to the variables p2

and ðpþ qÞ2 are applied. For ��0b ! �0

b, the sum rules of

G1 and G2 are obtained as

G1 ¼ � 1

�1�2ðm1 þm2Þ em2

1=M2

1em22=M2

2 ½eq1�1 þ eq2�1

� ðq1 $ q2Þ þ eb�01�

G2 ¼ 1

�1�2

em21=M2

1em22=M2

2 ½eq1�2 þ eq2�2ðq1 $ q2Þþ eb�

02�; (20)


056005-4

where q1 ¼ u, q2 ¼ d. The functions �i½�0i� can be writ-

ten as:

�i½�0i� ¼

Z s0

m2Q

e�s=M2�iðsÞ½�0

iðsÞ�dsþ e�m2Q=M

2

�i½�0i�;

(21)

where, the explicit expressions for functions �1, �01, �2, �

02,

�1, �01, �2 and �0

2 are given in the Appendix.Similarly, for ��0

b ! �0b, we get

G1 ¼ � 1

�1�2ðm1 þm2Þ em2

1=M21em

22=M

22 ½eq1�1 � eq2�1

� ðq1 $ q2Þ þ eb�01�

G2 ¼ 1

�1�2

em21=M2

1em22=M2

2 ½eq1�2 � eq2�2ðq1 $ q2Þþ eb�

02�; (22)

where q1 ¼ u, q2 ¼ s. The functions�i½�0i� are given by:

�i½�0i� ¼

Z s0

m2Q

e�s=M2%iðsÞ½%0

iðsÞ�dsþ e�m2Q=M2

i½0i�;

(23)

where, the explicit expressions of the functions %1, %01, %2,

%02, 1,

01, 2 and

02 are also presented in the Appendix.

From Eqs. (20) and (22) it is clear that for the calculationof the transition magnetic dipole and electric quadrupolemoments, the expressions for the residues �1 and �2 areneeded. These residues are determined from analysis oftwo point sum rules (see [9,23,24]).

III. NUMERICAL ANALYSIS

This section is devoted to the numerical analysis of themagnetic dipole GM and electric quadrupole GE formfactors at q2 ¼ 0 as well as the calculation of the decayrates for the considered radiative transitions. The inputparameters used in the analysis of the sum rules are takento be: h �uuið1 GeVÞ ¼ h �ddið1 GeVÞ ¼ �ð0:243Þ3 GeV3,h �ssið1 GeVÞ ¼ 0:8h �uuið1 GeVÞ, m2

0ð1GeVÞ¼ ð0:8�0:2ÞGeV2 [25], � ¼ ð0:5–1Þ GeV, and f3� ¼�0:0039 GeV2 [22]. The value of the magnetic suscepti-bility was obtained in different papers as �ð1 GeVÞ ¼�3:15� 0:3 GeV�2 [22], �ð1 GeVÞ ¼ �ð2:85�0:5Þ GeV�2 [26], and �ð1 GeVÞ ¼ �4:4 GeV�2 [27].From sum rules for the magnetic dipole GM and electricquadrupole GE, it is clear that we also need to know theexplicit form of the photon DA’s. The expressions for theseDA’s are also given in [22].

The sum rules for the electromagnetic form factors alsocontain three auxiliary parameters: the Borel mass parame-ter M2, the continuum threshold s0 and the arbitrary pa-rameter � entering the expression of the interpolatingcurrents of the heavy spin 1=2 baryons. The measurablephysical quantities, like the magnetic dipole and electric

quadrupole moments, should be independent of them.Therefore, we look for regions for these auxiliary parame-ters such that in these regions the GM and GE are practi-cally independent of them. The working region for M2 arefound requiring that not only the contributions of thehigher states and continuum should be less than the groundstate contribution, but the highest power of 1=M2 be lessthan say 30% of the highest power of M2. These twoconditions are both satisfied in the region 15 GeV2 �M2 � 30 GeV2 and 6 GeV2 � M2 � 12 GeV2 for bary-ons containing b and c-quark, respectively. The workingregions for the continuum threshold s0 and the generalparameter � are obtained considering the fact that theresults of the physical quantities are approximately un-changed. Note that, we give the absolute values of theGM and GE, since it is not possible to calculate the signof the residue from the mass sum-rules. Hence, it is notpossible to predict the signs of the GM or GE. The relativesign of the GM and GE can only be predicted using theQCD sum rules. As an example, the dependence of themagnetic dipole GM and the electric quadropole GE formfactors at q2 ¼ 0 on cos , where � ¼ tan at two fixedvalues of the continuum threshold s0 and Borel masssquare M2 for ��

b ! ��b � transition are depicted in

Figs. 1 and 2. The dependence of the GM and GE oncos have following common behavior: these form factorsare equal zero at some finite values of cos and theyincrease or decrease near to the end points, i.e., cos ¼�1. These can be explained as follows. The combination�1�2ð�ÞGMðEÞ as well as �2ð�Þ are linear functions of the�. The analysis of the two point sum rules, where �2ð�Þ isdetermined, shows that at all values of �, the �2ð�Þ differsfrom zero, and its values near the end points are enoughsmall. From analysis of the combination of �1�2ð�ÞGMðEÞwith respect to the cos , we obtain that it has zeros at somefinite values of the cos and it is also small at the endpoints. Therefore, the ratio of the �1�2ð�ÞGMðEÞ and �2ð�Þhas zeros at some finite values of the cos and it canincrease or decrease at the end points. As a result, the

-1 -0.5 0 0.5 1cosθ

0

1

2

3

|GM

|(Ξ∗−

b--->

Ξ− b) s0=6.3

2GeV

2

s0=6.5

2GeV

2

FIG. 1. The dependence of the magnetic dipoleGM for��b !

��b on cos for two fixed values of the continuum threshold s0.

Bare lines and lines with the circles correspond to the M2 ¼20 GeV2 and M2 ¼ 25 GeV2, respectively.


056005-5

zero and end points and any region near them are notsuitable regions for �. Analysis of the sum rules showthat the ‘‘working region’’ of � is �0:5< cos < 0:5,where the GM and GE have weak dependency on cos .The Ioffe current which corresponds to cos ’ �0:71, isout of the working region of �. Note that, similar behaviorwas obtained in analysis of the radiative decays of the lightdecuplet to octet baryons (see [15]).

We also show as an example, the dependence of themagnetic dipole moment GM and the electric quadropolemoment GE onM2 for��

b ! ��b � at two fixed values of

the continuum threshold s0 and three values of the arbitraryparameter � in Figs. 3 and 4. From these figures, it followsthat the sum rules for GM and GE exhibit a very weakdependency with respect to the M2 in the working region.Here, we would like to make the following remark. In [28],it was obtained that the effective threshold dependsstrongly on both the Borel parameter M2 and the momen-tum transfer square q2 (in our case q2 ¼ 0). This leads thatin applying the QCD sum-rules technique even to simpli-fied but explicitly soluble systems, the entire sum-ruleapproach is plagued by a class of ‘‘minimum’’ systematicuncertainties. This implies that, the usual criteria adoptedin the sum rules about the stability of the results withrespect to the variation of the Borel parameter M2 does

not provide realistic error estimates, so the actual errorsturn out be large and the extracted results may be off theirtrue values by some uncertainties arising from differentsources. Following [28], we consider these systematic un-certainties about 15%. Our numerical calculations showthat the results depend on s0 weakly. We should also stressthat our results practically do not change considering threevalues of � which we presented at the beginning of thissection. Our final results on the absolute values of themagnetic dipole and electric quadrupole moments of theconsidered radiative transitions are presented in Table III.The quoted errors in Table III are coming from variation ofthe continuum threshold s0, arbitrary parameter�, errors ininput parameters as well as the systematic uncertainties.Few words about the GE are in order. From Eq. (6), itfollows that the GE is proportional to the mass differencem1 �m2 and these masses are close to each other.Therefore, reliability of predictions for GE are question-able and one can consider them as order of magnitude.At the end of this section, we would like to calculate the

decay rate of the considered radiative transitions in termsof the multipole moments GE and GM:

15 20 25 30M

2(GeV

2)

0

1

2

3

|GM

|(Ξ∗−

b--->

Ξ− b) s0=6.3

2GeV

2

s0=6.5

2GeV

2

FIG. 3. The dependence of the magnetic dipole GM for��b !

��b on the Borel mass parameter M2 for two fixed values of the

continuum threshold s0. Bare lines, lines with the circles andlines with the diamonds correspond to the Ioffe currents (� ¼�1), � ¼ 5 and � ¼ 1, respectively.

15 20 25 30M

2(GeV

2)

0

0.01

0.02

0.03

|GE|(Ξ

∗−b--

->Ξ− b) s

0=6.3

2GeV

2

s0=6.5

2GeV

2

FIG. 4. The same as Fig. 3, but for the electric quadrupole GE.

TABLE III. The results for the magnetic dipole moment jGMjand electric quadrupole moment jGEj for the correspondingradiative decays in units of their natural magneton.

jGMj jGEj��0

b ! �0b� 1:0� 0:4 0:005� 0:002

��b ! ��

b � 2:1� 0:9 0:016� 0:007��þ

b ! �þb � 4:2� 1:9 0:026� 0:011

��þc ! �þ

c � 1:2� 0:3 0:0140� 0:0035��0

c ! �0c� 0:5� 0:2 0:0030� 0:0012

��þþc ! �þþ

c � 2:8� 1:0 0:030� 0:012��0

b ! �0b� 8:5� 3:8 0:085� 0:038

��b ! ��

b � 0:9� 0:4 0:011� 0:005��þ

c ! �þc � 4:0� 1:8 0:075� 0:032

��0c ! �0

c� 0:45� 0:18 0:007� 0:003��0

b ! �0b� 7:3� 3:4 0:075� 0:032

��þc ! �þ

c � 3:8� 1:4 0:060� 0:025

-1 -0.5 0 0.5 1cosθ

0

0.02

0.04

|GE|(Ξ

∗−b--

->Ξ− b) s

0=6.3

2GeV

2

s0=6.5

2GeV

2

FIG. 2. The same as Fig. 1, but for the electric quadrupole GE.


056005-6

� ¼ 3�

32

ðm21 �m2

2Þ3m3

1m22

ðG2M þ 3G2

EÞ: (24)

The results for the decay rates are given in Table IV. Incomparison, we also present the predictions of the con-stituent quark model in SUð2NfÞ �Oð3Þ symmetry [12],

the relativistic three-quark model [13] and heavy quarkeffective theory (HQET) [14] in this Table.In summary, the transition magnetic dipole moment

GMðq2 ¼ 0Þ and electric quadrupole moment GEðq2 ¼ 0Þof the radiative decays of the sextet heavy flavored spin 3=2to the heavy spin 1=2 baryons were calculated within thelight cone QCD sum rules approach. Using the obtainedresults for the electromagnetic moments GM and GE, thedecay rate for these transitions were also calculated. Thecomparison of our results on these decay rates with thepredictions of the other approaches is also presented.

ACKNOWLEDGMENTS

Two of the authors (K. A. and A.O.), would like to thankTUBITAK, Turkish Scientific and Research Council, fortheir partial financial support through the project number106T333 and also through the PhD program (K.A.).

APPENDIX

In this appendix, we present the explicit expressions ofthe functions �1, �

01, �2, �

02, �1, �

01, �2, �

02, %1, %

01, %2, %

02,

1, 01, 2, and 0

2 entering to the sum rules for the formfactors G1 and G2 are given. These functions are:

ffiffiffi3

p�1ðsÞ ¼ hq1q1ihq2q2i

��

3�’�ðu0Þ

�þm2

0hq2q2i�� ð1þ�Þ144mQ

2ð1þ c 11Þ

�þ hq2q2i

��mQ

42ðc 10 � c 21Þ

�

þ hq1q1imQ

962

��3ð1þ�Þðc 10 � c 21ÞAðu0Þ þ 2

��2ðc 10 � c 21Þfð1þ 5�Þ�1 ��2 � 4�3 ��4 þ 6�5

þ 2�6 þ 3�7 � 2�8 ��ð�2 þ�4 � 2�5 � 2�6 � 7�7 þ 2�8Þg � 3ð3þ�Þð�1þ c 02 þ 2c 10 � 2c 21Þ� ðu0 � 1Þ�1 þ 2fð1þ 3�Þ�1 � ð3þ�Þð�2 ��4Þ þ ð1þ 3�Þ�7g ln

�s

m2Q

�þ 3ð1þ�Þm2

Q�

�2c 10 � c 20

þ c 31 � 2 ln

�s

m2Q

��’�ðu0Þ

��þm4

Qð�� 1Þ5124

�4c 30 � 5c 42 � 4c 52 þ ð4c 30 � 8c 41 þ 5c 42 þ 4c 52Þu0 þ 6

��2c 10 � c 20 � 2 ln

�s

m2Q

��ð1þ u0Þ

�þ f3�m

2Qð�� 1Þ

1922

�8

��ðc 20 � c 31Þð�0

1 ��02Þ þ

�c 10 � ln

�s

m2Q

��

� ð�03 þ�0

1 ��02Þ�þ 6ðc 20 � c 31Þc aðu0Þ þ ð3c 31 þ 4c 32 þ 3c 42Þðu0 � 1Þ

�4c vðu0Þ � dc aðu0Þ

du0

��; (A1)

ffiffiffi3

p�01ðsÞ ¼ m2

0ðhq1q1i þ hq2q2iÞ�5ð1þ 3�Þð1þ c 11Þ

1442mQ

�þ ðhq1q1i þ hq2q2iÞ

mQð1þ 3�Þ82

�c 10 � c 21 � ln

�s

m2Q

��

þm4Qð1� �Þ7684

�2ð�3c 20 � 12c 30 þ 18c 31 þ 8c 32 þ 18c 41Þ � 3ðc 42 þ 4c 52Þ þ f2ð3c 20 þ 6c 30

� 18c 31 � 8c 32Þ þ 3ð�8c 41 þ c 42 þ 4c 52Þgu0 � 12c 10ð7þ 5u0Þ � 72ðc 10 þ u0Þ ln�m2

Q

s

�

þ 12f7þ 2c 10ð�1þ u0Þ � u0g ln�s

m2Q

��; (A2)

TABLE IV. The results for the decay rates of the correspond-ing radiative transitions in KeV.

� (present work) � [12] � [13] � [14]

��0b ! �0

b� 0:028� 0:020 0.15 - 0.08

��b ! ��

b � 0:11� 0:076 - - 0.32

��þb ! �þ

b � 0:46� 0:28 - - 1.26

��þc ! �þ

c � 0:40� 0:22 0.22 0:14� 0:004 -

��0c ! �0

c� 0:08� 0:042 - - -

��þþc ! �þþ

c � 2:65� 1:60 - - -

��0b ! �0

b� 135� 85 - - -

��b ! ��

b � 1:50� 0:95 - - -

��þc ! �þ

c � 52� 32 - 54� 3 -

��0c ! �0

c� 0:66� 0:41 - 0:68� 0:04 -

��0b ! �0

b� 114� 62 251 - 344

��þc ! �þ

c � 130� 65 233 151� 4 -


056005-7

ffiffiffi3

p�2ðsÞ ¼ hq1q1i 1

12mQ2

�3ð3þ �Þð�1þ c 03 þ 2c 12 þ c 22Þðu0 � 1Þ�2 þ 2

��2ð1þ �Þ þ ð�� 1Þc 11

þ 2ð1þ 3�Þðc 12 þ c 22Þ � 3þ �

2ð�1þ 2c 12 þ c 22Þð�9 � �5Þ

��þm2

Qð�� 1Þ1284

½ð12c 31 þ 19c 32 þ c 33

þ 12c 42Þðu0 � 1Þu0� þf3�ð�� 1Þ

962½4fð1� c 02Þ�9 þ ð�3þ 3c 02 þ 2c 10 � 2c 21Þ�11 � ð�1þ c 02

þ 2c 10 � 2c 21Þ�10 þ ð�1þ 3c 02 � 2c 03Þðu0 � 1Þ�11g � ð�1þ 3c 02 � 2c 03Þðu0 � 1Þc aðu0Þ�; (A3)

ffiffiffi3

p�02ðsÞ ¼

m2Qð�� 1Þ1924

�ð�1þ u0Þu0

�12c 10 � 6c 20 þ 24c 31 þ 27c 32 þ c 33 þ 18c 42 � 12 ln

�s

m2Q

��; (A4)

ffiffiffi3

p�1 ¼ m2

0hq1q1ihq2q2i��m4

QAðu0Þ48M6

þ m2Q

432M4f12½�2 � 2�3 � �4 þ 2�5 þ �ð2�1 þ �2 � 2�3 þ �4 � 2�6 þ 2�8Þ

þ 3ðu0 � 1Þ�1� � �Aðu0Þg þ 5�

54�’�ðu0Þ þ 1

108M2fð1� u0Þð�� 5Þ�1 þ 9�m2

Q�’�ðu0g�þ hq1q1ihq2q2i

��m2

QAðu0Þ12M2

þ 1

36f�4½�2 � 2�3 � �4 þ 2�5 þ �ð2�1 þ �2 � 2�3 þ �4 � 2�6 þ 2�8Þ þ 3ðu0 � 1Þ�1�

þ 3�Aðu0Þg�þm2

0hq2q2i��mQ

162þ

��m3Q

24M4þ ð1þ �ÞmQ

216M2

�f3�c

aðu0Þ�þ hq2q2i

��

6f3�mQc

aðu0Þ�; (A5)

ffiffiffi3

p�01 ¼ m2

0hq1q1ihq2q2i��m2

Q

3M4

�þ hq1q1ihq2q2i

�2�

3

�; (A6)

ffiffiffi3

p�2 ¼ m2

0hq1q1ihq2q2i�� 1

54M4ðu0 � 1Þð�� 5Þ�2 þ

m2Q

18M6f3ðu0 � 1Þ�2 � �ð�5 � �9Þg

�

þ hq1q1ihq2q2i�

2

9M2f�3ðu0 � 1Þ�2 þ �ð�5 � �9Þg

�; (A7)

�02 ¼ 0; (A8)

%1ðsÞ ¼ hq1q1ihq2q2i�1þ 2�

9�’�ðu0Þ

�þm2

0hq2q2i� ð1þ �Þ144mQ

2ð1þ c 11Þ

�þ hq2q2i

��mQ

42ðc 21 � c 10Þ

�

þ hq1q1imQ

2882½3ð1þ 5�Þðc 10 � c 21ÞAðu0Þ þ 2ð�2ðc 10 � c 21Þfð�1þ 7�Þ�1 þ �2 � 4�3 þ �4 þ 2�5

� 2�6 � 3�7 þ 2�8 þ �ð5�2 � 8�3 þ 5�4 � 2�5 � 10�6 � 3�7 þ 10�8Þg þ 3ð�1þ �Þ� ð�1þ c 02 þ 2c 10 � 2c 21Þðu0 � 1Þ�1 þ 2ð�1þ �Þð�1 þ �2 � �4 þ �7Þ ln

�s

m2Q

�� 3ð1þ 5�Þm2

Q�

��2c 10 � c 20 þ c 31 � 2 ln

�s

m2Q

��’�ðu0Þ

��þm4

Qð�� 1Þ5124

�6c 20 � 4c 30 þ 5c 42 þ 4c 52 þ ð6c 20 � 4c 30

þ 8c 41 � 5c 42 � 4c 52Þu0 � 12

�c 10 � ln

�s

m2Q

��ð1þ u0Þ

�� f3�m

2Qð�� 1Þ

5762

�8

�ðc 20 � c 31Þð�0

1 � �02Þ

þ�c 10 � ln

�s

m2Q

��ð3�0

3 � �01 þ �0

2Þ�þ 18ðc 20 � c 31Þc aðu0Þ þ 3ð3c 31 þ 4c 32 þ 3c 42Þðu0 � 1Þ

��4c vðu0Þ � dc aðu0Þ

du0

��; (A9)


056005-8

%01ðsÞ ¼ m2

0ðhq1q1i � hq2q2iÞ�5ð�� 1Þð1þ c 11Þ

4322mQ

�þ ðhq1q1i � hq2q2iÞ

mQð�� 1Þ242

�c 10 � c 21 � ln

�s

m2Q

��

þm3Qmq2ð�� 1Þ1284

�6c 10 � c 20 þ c 31 � 2f3þ c 10g ln

�s

m2Q

��; (A10)

%2ðsÞ ¼ hq1q1i 1��

36mQ2

�3ð�1þ c 03 þ 2c 12 þ c 22Þðu0 � 1Þ�2 þ 2c 11ð2�3 þ �5 � 2�7 � �9Þ

þ 2

�1

2ð�9 � �5Þ þ ð2c 12 þ c 22Þ

�2�3 þ 3

2ð�5 � �9Þ � 2�7

��þ�m2

Qð�� 1Þ1284

½ð12c 31 þ 19c 32 þ c 33 þ 12c 42Þ

� ðu0 � 1Þu0� �f3�ð�� 1Þ2882

½4fð�1þ c 02 þ 4c 10 � 4c 21Þ�9 � 3ð�1þ c 02 þ 2c 10 � 2c 21Þ�11 þ ð�1þ c 02

� 2c 10 þ 2c 21Þ�10 þ 3ð�1þ 3c 02 � 2c 03Þðu0 � 1Þ�11g � 3ð�1þ 3c 02 � 2c 03Þðu0 � 1Þc aðu0Þ�; (A11)

%02ðsÞ ¼ 0; (A12)

1 ¼ m20hq1q1ihq2q2i

�ð1þ 2�Þm4QAðu0Þ

144M6þ m2

Q

1296M4f12½ð1� �Þð�2 � 2�3Þ � 3ð1þ �Þ�4 þ 2ð2þ �Þ�5 þ 2ð1þ 2�Þ

� ð�6 þ �7 � �8Þ þ 3ð2þ �Þðu0 � 1Þ�1� þ ð1þ 2�ÞAðu0Þg � 5ð1þ 2�Þ162

�’�ðu0Þ þ 1

108M2fð�1þ u0Þð5

þ 3�Þ�1 � 3ð1þ 2�Þm2Q�’�ðu0g

�þ hq1q1ihq2q2i

��ð1þ 2�Þm2

QAðu0Þ36M2

þ 1

108f4½ð�� 1Þ�2 þ 2�3 þ 3�4

� 2ð2�5 þ �6 þ �7 � �8 � 3�1Þ � 6u0�1 � �ð2�3 � 3�4 þ 2�5 þ 4ð�6 þ �7 � �8Þ þ 3ðu0 � 1Þ�1Þ� � 3ð1

þ 2�ÞAðu0Þg�þm2

0hq2q2i��mQ

162�

��m3Q

24M4þ ð1þ �ÞmQ

216M2

�f3�c

aðu0Þ�þ hq2q2i

��

6f3�mQc

aðu0Þ�;

(A13)

01 ¼ m2

0hq1q1ihq2q2i5ð�� 1Þ

432

�m3Qmq2

M6�mQmq2

M4

�

� hq1q1ihq2q2i�ð�� 1ÞmQmq2

36M2

�; (A14)

2 ¼m20hq1q1ihq2q2i

�1

54M4ðu0 � 1Þð3�þ 5Þ�2

þ m2Q

54M6f3ð2þ�Þðu0 � 1Þ�2 � ð1þ 2�Þð�5 � �9Þg

�

þ hq1q1ihq2q2i� �2

27M2f3ð2þ�Þðu0 � 1Þ�2

� ð1þ 2�Þð�5 � �9Þg�; (A15)

02 ¼ 0: (A16)

Note that, in the above equations, the terms proportional toms are omitted because of their lengthy expressions, butthey have been taken into account in numerical calcula-tions. The contributions of the terms�hG2i are also calcu-lated, but their numerical values are very small andtherefore for simplicity in the expressions these terms are

also omitted. The functions entering Eqs. (A1)–(A16) aregiven as

�i ¼Z

D�i

Z 1

0dvfið�iÞ�ð�q þ v�g � u0Þ;

�0i ¼

ZD�i

Z 1

0dvgið�iÞ�0ð�q þ v�g � u0Þ;

�i ¼Z 1

u0

hiðuÞdu ði ¼ 1; 2; 11Þ;

�i ¼Z

D�i

Z 1

0d �vhið�iÞ ð�q þ v�g � u0Þ

ði ¼ 3� 10Þ; c nm ¼ ðs�m2QÞn

smðm2QÞn�m

;

(A17)

where, the measure D�i is defined as

ZD�i ¼

Z 1

0d� �q

Z 1

0d�q

Z 1

0d�g�ð1�� q ��q ��gÞ:

(A18)

Here, f1ð�iÞ ¼ Sð�iÞ, f2ð�iÞ ¼ ~Sð�iÞ, f3ð�iÞ ¼ v~Sð�iÞ,f4ð�iÞ ¼ h5ð�iÞ ¼ T 2ð�iÞ, f5ð�iÞ ¼ h6ð�iÞ ¼ vT 2ð�iÞ,f6ð�iÞ ¼ h8ð�iÞ ¼ vT 3ð�iÞ, f7ð�iÞ ¼ h9ð�iÞ ¼ T 4ð�iÞ,


056005-9

f8ð�iÞ ¼ h10ð�iÞ ¼ vT 4ð�iÞ, f9ð�iÞ ¼ Að�iÞ,f10ð�iÞ ¼ g3ð�iÞ ¼ vAð�iÞ, f11ð�iÞ ¼ g2ð�iÞ ¼vV ð�iÞ, g1ð�iÞ ¼ V ð�iÞ, h1ðuÞ ¼ h�ðuÞ, h2ðuÞ ¼ðu� u0Þh�ðuÞ, h3ð�iÞ ¼ T 1ð�iÞ, h4ð�iÞ ¼ vT 1ð�iÞ,h7ð�iÞ ¼ T 3ð�iÞ and h11ðuÞ ¼ c vðuÞ are functions interms of the photon distribution amplitudes. In the aboveequations, ’�ðuÞ is the leading twist 2, c vðuÞ, c aðuÞ, AandV are the twist 3 and h�ðuÞ, A, T i (i ¼ 1, 2, 3, 4) are

the twist 4 photon DA’s, respectively, and � is the magneticsusceptibility of the quarks. The Borel parameter M2 used

in the sum rules is defined as M2 ¼ M21M2

2

M21þM2

2

and u0 ¼M2

1

M21þM2

2

. Since the masses of the initial and final baryons

are very close to each other, we set M21 ¼ M2

2 ¼ 2M2 andu0 ¼ 1=2.

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Radiative decays of the heavy flavored baryons in light cone QCD sum rules