Analysis of gamma(*) Lambda -> Sigma(0) transition in QCD

Analysis of

_{! }

_{transition in QCD}

T. M. Aliev,1,*,†

K. Azizi,2,‡and M. Savc
1,§

1_{Physics Department, Middle East Technical University, 06531 Ankara, Turkey} 2

Physics Department, Faculty of Arts and Sciences, Dog˘us¸ University, Ac{badem-Kad{ko¨y, 34722 Istanbul, Turkey (Received 27 March 2013; published 20 May 2013)

The

 ! 0transition form factors are investigated within the light-cone QCD sum rules method. Using the most general form of the interpolating current of0baryon and the distribution amplitudes of  baryon we calculate the Q2_{dependence of the electromagnetic form factors. Our results are compared}

with the predictions of the covariant spectator quark model.

DOI:10.1103/PhysRevD.87.096013 PACS numbers: 11.55.Hx, 12.38.t, 13.40.Gp

I. INTRODUCTION

The investigation of electromagnetic form factors of hadrons plays a key role in understanding their internal structure. The form factors measured in experiments de-scribe the spatial distribution of charge and magnetization of hadrons [1], and indicate the deviation of hadron struc-ture from the pointlike particle. At present, the studies are mainly focused on the nucleon form factors. Recent ex-perimental and theoretical progress on this subject can be found in [1,2] and references therein.

The study of electromagnetic form factors of the ground state spin-1=2 baryons receives special interest. However, except the proton and neutron, the electromagnetic form factors of other members, the octet baryons, have not yet been measured. The main difficulty can be attributed to the unstable nature of the baryons containing a strange quark. From a theoretical point of view, the main problem is related to the fact that the formation of hadrons belongs to the non-perturbative region of QCD where the non-perturbative approach does not work. For this reason some nonperturbative ap-proaches are needed in order to calculate these form factors, and the QCD sum rules method is recognized to be the most predictive one among all other nonperturbative approaches. Another advantage of the QCD sum rules method is that it is based on the fundamental QCD Lagrangian.

The nucleon electromagnetic form factors are calculated in framework of the light-cone version QCD sum rules method for the Ioffe and general currents in [3,4]. The electromagnetic form factors of, , and  baryons are studied for the Chernyak-Zhitnisky and Ioffe currents in [5]. The electromagnetic form factors of octet baryons for the most general form of the interpolating currents are studied within the light-cone QCD sum rules method in [6]. It should be noted here that the electromagnetic form factors of nucleons and other members of octet baryons have already been studied in numerous works within the

framework of lattice calculations (see [7] and references therein), and relativistic constituent quark model [8].

In the present work, we study the electromagnetic tran-sition form factors of the

 ! 0 in the framework of the light-cone QCD sum rules method using the most gen-eral form of the interpolating current for the0baryon. This decay is studied in the framework of the nonrelativistic quark model and general QCD parametrization method [9], the covariant spectator quark model [10], chiral pertur-bative theory [11,12], chiral quark model [13], and Skyrme model [14]. The

 ! 0 transition is interesting in several respects: it is unique between two different baryons that belong to the same octet family even in an exact isospin symmetry case. The second interesting peculiarity of this transition is that having different initial and final baryons is contrary to the case observed in elastic scattering of the octet baryons. For these reasons, the electric charge form factor GEðQ2Þ at Q2¼ 0 should vanish. Hence, the value of

GEðQ2Þ is expected to be small in its dependence on Q2.

Therefore, investigation of the Q2 dependence of the form factors receives special interest. It should be noted that the magnetic moment for the

 ! 0 transition is investi-gated within the light-cone QCD sum rules method in [15]. The modern status of QCD and particularly the QCD sum rules for baryons is presented in great detail in [16].

The structure of this paper is organized as follows. In Sec. II, we derive sum rules for the form factors of the

 ! 0transition. In Sec.III, we present our numerical results and conclusions.

II. SUM RULES FOR

_{! }0_TRANSITION

FORM FACTORS

The transition form factors for

 ! 0 are deter-mined by the matrix element of the electromagnetic current between the and 0 baryons. Using the conservation of the electromagnetic current, this matrix element can be determined in the following way:

h0_ðp0_Þjjel jðpÞi¼ u0ðp0Þ
F1ðQ2Þ 
 6qq q2   i mþm0q _F 2ðQ2Þ  uðpÞ; (1) *taliev@metu.edu.tr

†_{Permanent address: Institute of Physics, Baku, Azerbaijan.} ‡_{kazizi@dogus.edu.tr}

where q ¼ p  p0, Q2¼ q2 and ¼2i½
;
.

Here, F1ðQ2Þ and F2ðQ2Þ are the Dirac and Pauli type

form factors, respectively.

Experimentally, a more convenient set of the electromag-netic form factors are the Saches form factors defined as

GEðQ2Þ ¼ F1ðQ2Þ  Q2 ðmþ m0Þ2F2ðQ 2_Þ; GMðQ2Þ ¼ F1ðQ2Þ þ F2ðQ2Þ: (2) In order to calculate the form factors F1ðQ2Þ and F2ðQ2Þ

for the

 ! 0 transition, we consider the following correlation function:

ðp;qÞ ¼ i

Z

d4xeiqxh0jTf0ð0Þjel

ðxÞgjðpÞi; (3)

where T means the time ordering, jðpÞi is the  baryon state with four-momentum p, 0 _{is the interpolating}

current for the₀ baryon, i.e.,

0 ¼pffiffiffi2_"abcfðuaT_CsbÞ
_5dcþ ðdaT_CsbÞ
_5uc

þ ðuaT_C

5sbÞdcþ ðdaTC
5sbÞucg: (4)

Here C is the charge conjugation operator,  is an arbitrary parameter, and jelis the electromagnetic current defined as

jelðxÞ ¼ euuðxÞ
uðxÞ þ eddðxÞ
dðxÞ þ essðxÞ
sðxÞ:

(5) The correlation function can be calculated in terms of hadrons (phenomenological part) and in terms of quark and gluon degrees of freedom. Equating these two representa-tions of the correlation function (1) we get the sum rules for the form factors of

 ! 0 transition.

Saturating (1) with the hadronic states with the quantum numbers of0baryon and separating the ground state, for the phenomenological part, we get

ðp; qÞ ¼

h0j0j₀ðp0Þih₀ðp0Þjjel jðpÞi

m2_0 p02

þ    ; (6) where    denotes contribution of the higher states and continuum.

The matrix element h0j0j

0i is determined as

h0j0j

0i ¼ 0uðp 0_Þ;

where 0 is the residue of the0 baryon. Moreover, the

matrix element h₀jjel_jðpÞi is determined as is given in Eq. (1). Using these definitions, for the phenomenological part, we get ph ¼0ð6p 0_{þ m} 0Þ m2_0 p02
F1ðQ2Þ 
 6qq q2   i mþ m0q _F 2ðQ2Þ  uðpÞ: (7)

We see from Eq. (7) that there appears numerous structures in determining the transition form factors F1ðQ2Þ and

F2ðQ2Þ. For this aim we choose the structures p and

p6q, as a result of which, for the coefficients of the

selected structures, we get ð1Þ_¼20F1ðQ2Þ m2_0 p02 ; ð2Þ¼ 2 m0þ m 0F2ðQ2Þ m2_0 p02 : (8) As has already been noted, these form factors are de-scribed in terms of baryon distribution amplitudes (DAs). The  baryon matrix element of three-quark operator "abc_hua

ða1xÞdbða2xÞsc
ða3xÞjðpÞi is given in terms of 

baryon DAs. The definition of this matrix element in terms of DAs and expressions of these DAs can be found in [5].

In constructing sum rules for the transition form factors F1ðQ2Þ and F2ðQ2Þ, we need the expression for the

correlation function from the QCD side. This correlation function in QCD can be calculated for large negative p02 and q2¼ Q2 in terms of  baryon distribution ampli-tudes using the operator product expansion. Matching then the coefficients of the structures p and p6p in the

expressions of the correlation function in the phenomeno-logical and QCD sides, we get the sum rules for the transition form factors F1ðQ2Þ and F2ðQ2Þ of the

 !

0 _transition.

In order to enhance the ground state contribution and suppress the higher state contributions, it is necessary to perform Borel transformation on the theoretical and phe-nomenological parts of the correlation function. After the Borel transformation, we get the final expressions for the transition form factors F1ðQ2Þ and F2ðQ2Þ as

F1ðQ2Þ ¼ ffiffiffi 2 p 4 1 2_0e m2 0=M2
Z1 x0 dx   2ðxÞ x þ 4ðxÞ M2x2 6ðxÞ 2M4_x3  e  Q2 x M2xþ m2 x M2  þ 4ðx0Þ Q2þ m2_x2₀ 12x0 6ðx0Þ ðQ2_{þ m}2 x20ÞM2 þ 1 2 x2₀ ðQ2_{þ m}2 x20Þ  d dx0 6ðx0Þ x0ðQ2þ m2x20ÞM2  es0=M2  ; (9) F2ðQ2Þ ¼ ffiffiffi 2 p 4 m0þ m_ 20 e m2_0=M2
Z1 x0 dx   02ðxÞ x þ 0 4ðxÞ M2x2 0 6ðxÞ 2M4_x3  e  Q2 x M2xþ m2 x M2  þ 04ðx0Þ Q2þ m2_x2₀ 12x0 06ðx0Þ ðQ2_{þ m}2 x20ÞM2 þ 1 2 x20 ðQ2_{þ m}2 x20Þ  d dx0 06ðx0Þ x0ðQ2þ m2x20ÞM2  es0=M2  ; (10)

where 6ðxÞ¼4eum3ð1þÞxðm2x2þQ2Þ B6ðxÞþ4edm3ð1þÞxðm2x2þQ2Þ~~B6ðxÞ þ8esm2fm2msð1Þx2^^C6þð1þÞ½mxðm2x2þQ2Þ ^^B6msðQ2^^B6þ2m2x2^^B8ÞgðxÞ; 4ðxÞ¼eumf2m2x½2ð1Þ C6ð1þÞð2 B65 B8ÞðxÞþ½2ð1Þðm2x2ð D5 C4þ2 C5ÞQ2ð D2 C2ÞÞ þð1þÞðQ2_{ð3 B} 2þ7 B4Þþm2x2ð2 H12 E1 B2þ B410 B520 B7ÞÞðxÞ 2m2 x Zx 0 dx3½2ð1ÞV M 1 þ5ð1þÞT1Mðx;1xx3;x3Þg þedmf2m2x½2ð1Þ~~C6ð1þÞð2~~B65~~B8ÞðxÞþ½ð1Þð2m2x2ð ~D5þ ~C42 ~C5Þ þQ2_{ð ~D} 2þ ~C2ÞÞþð1þÞðQ2ð3 ~B2þ7 ~B4Þm2x2ð2 ~H12 ~E1þ ~B2 ~B4þ10 ~B5þ20 ~B7ÞÞðxÞ 2m2 x Zx 0 dx1½2ð1ÞV M 1 þ5ð1þÞT1Mðx1;x;1x1xÞg þ2esmf2mð1þÞ½mxð2 ^^B6 ^^B8Þms^^B6ðxÞþ½ð1Þð2ðm2x2^C5þQ2^C2Þ mmsxð2 ^C2 ^C4 ^C5ÞÞð1þÞðQ2ð ^B23 ^B4Þþm2x2ð ^B2 ^B4þ2 ^B5þ4 ^B7Þ 4mmsxð ^B4 ^B5ÞÞðxÞ2m2ð1þÞx Zx 0 dx1T M 1 ðx1;1x1x;xÞg; 2ðxÞ¼2eumf½ð1Þð D2þ C2Þð1þÞð B2 B4ÞðxÞþx Z x 0 dx3½ð1ÞðA3þ2V13V3Þ ð1þÞðP1þS15T1þ10T3Þðx;1xx3;x3Þgþ2edmf½ð1Þð ~D2 ~C2Þþð1þÞð ~B2 ~B4ÞðxÞ þxZx 0 dx1½ð1ÞðA32V1þ3V3Þð1þÞðP1þS1þ5T110T3Þðx1;x;1x1xÞg þ4esfm½ð1Þ ^C2ð1þÞð ^B2 ^B4ÞðxÞþ Zx 0 dx1fð1ÞðmxV3þmsV1Þ þð1þÞ½2mxT3ðmxþ2msÞT1gðx1;1x1x;xÞg; 0₆ðxÞ¼4eum2ð1þÞðm2x2þQ2Þ B6ðxÞ4edm2ð1þÞðm2x2þQ2Þ~~B6ðxÞ 8esm2fmmsð1Þx ^^C6þð1þÞ½ðm2x2þQ2Þ ^^B6þmmsxð ^^B62 ^^B8ÞgðxÞ; 04ðxÞ¼eum2fð1þÞ B6ðxÞþ2x½ð1Þð D2þ D5 C2 C4þ2 C5Þþð1þÞð H1 E12 B23 B45 B510 B7ÞðxÞ þ2ð1ÞZ x 0 dx3ðA M 1 V1MÞðx;1xx3;x3Þgþedm2fð1þÞ~~B6ðxÞþ2x½ð1Þð ~D2þ ~D5þ ~C2þ ~C42 ~C5Þ þð1þÞð ~H1 ~E1þ2 ~B2þ3 ~B4þ5 ~B5þ10 ~B7ÞðxÞþ2ð1Þ Zx 0 dx1ðA M 1 þVM1 Þðx1;x;1x1xÞg 2esmf5mð1þÞ ^^B6ðxÞþ2½ð1Þðmx ^C5ðmxþmsÞ ^C2Þð1þÞðmxð ^B4þ ^B5þ2 ^B7Þ msð ^B2þ ^B4ÞÞðxÞþ2mð1Þ Zx 0 dx1V M 1 ðx1;1x1x;xÞg; 02ðxÞ¼2euð1Þ Zx 0 dx3ðA1V1Þðx;1xx3;x3Þþ2edð1Þ Z x 0 dx1ðA1þV1Þðx1;x;1x1xÞ 4esð1Þ Zx 0 dx1V1ðx;1xx3;x3Þ; (11)

where M2 is the Borel parameter and x0 is given as

x0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðQ2_{þ s} 0 m2Þ2þ 4m2Q2 q 2m2  :

Here, s0 is the continuum threshold. In the expressions of ð 0_Þ

i ðxÞ, the functions F ðxiÞ are defined as

F ðx1Þ ¼ Zx1 1 dx 0 1 Z1x0 1 0 dx3F ðx 0 1;1  x01 x3;x3Þ; Fðx1Þ ¼ Zx1 1 dx 0 1 Zx0₁ 1 dx 00 1 Z1x00 1 0 dx3F ðx 00 1;1  x001 x3;x3Þ; ~ F ðx2Þ ¼ Zx2 1 dx 0 2 Z1x0 2 0 dx1F ðx1;x 0 2;1  x1 x02Þ; ~~Fðx2Þ ¼ Zx2 1 dx 0 2 Zx0₂ 1 dx 00 2 Z1x00 2 0 dx1F ðx1;x 00 2;1  x1 x002Þ; ^F ðx3Þ ¼ Zx3 1 dx 0 3 Z1x0 3 0 dx1F ðx1;1  x1 x 0 3;x03Þ; ^^Fðx3Þ ¼ Zx3 1 dx 0 3 Zx0₃ 1 dx 00 3 Z1x00 3 0 dx1F ðx1;1  x1 x 00 3;x003Þ: (12)

We also use the following shorthand notations for the combinations of the distribution amplitudes: B2 ¼ T1þ T2 2T3; B4 ¼ T1 T2 2T7; B5 ¼ T1þ T5þ 2T8; B6 ¼ 2T1 2T3 2T4þ 2T5þ 2T7þ 2T8; B7 ¼ T7 T8; B8 ¼ T1þ T2þ T5 T6þ 2T7þ 2T8; C2 ¼ V1 V2 V3; C4 ¼ 2V1þ V3þ V4þ 2V5; C5 ¼ V4 V3; C6 ¼ V1þ V2þ V3þ V4þ V5 V6; D2 ¼ A1þ A2 A3; D4 ¼ 2A1 A3 A4þ 2A5; D5 ¼ A3 A4; D6 ¼ A1 A2þ A3þ A4 A5þ A6; E1 ¼ S1 S2; H1 ¼ P2 P1: (13)

It follows from Eqs. (9) and (10) that in order to calculate the form factors F1ðQ2Þ and F2ðQ2Þ the residue of the 0baryon

is needed. The general form of the interpolating current for0 baryon leads to the following result for its residue [17]:

2_0eM 2

0=M2 ¼ 1

2564ð5 þ 2 þ 52ÞM6E2ðxÞ þ₃₂ms2M2E0ðxÞfð5 þ 2 þ 52Þhssi  6ð1 þ 2Þðhuui þ h ddiÞg

þ 1 24

m2₀

M2ð1  Þf6ð1 þ Þhssiðh uui þ h ddiÞ þ ð1 þ Þh uuih ddig þ 3ms 322m20ðhuui þ h ddiÞð1  2_Þ

E ln  M2 2   ms

1922m20f2ð5 þ 2 þ 52Þhssi  3ð1 þ 2Þðhuui þ h ddiÞg

 1

6ð1  Þf3ð1 þ Þhssiðh uui þ h ddiÞ þ ð1 þ Þh uuih ddig; (14)

EnðxÞ ¼ 1  ex

Xn k¼1

xk

k!

describes the continuum subtraction and x ¼ s0=M2. It

should be noted that the masses and residues of nucleons and other members of the octet baryons, for Ioffe current ( ¼ 1) within the QCD sum rules approach, were firstly calculated in [18,19].

III. NUMERICAL ANALYSIS OF THE SUM RULES FOR THE TRANSITION FORM FACTORS In order to perform numerical analysis of the transition form factors F1ðQ2Þ and F2ðQ2Þ within the light-cone QCD

sum rules, we need to know the explicit expressions of the DAs for the baryon, as well as the values of nonpertur-bative parameters entering into them. These input parame-ters for the baryon are calculated within the two-point QCD sum rules method in [5] which are given as

f ¼ ð6:0  0:3Þ  103 GeV2;

1 ¼ ð1:0  0:3Þ  102 GeV2;

j2j ¼ ð0:83  0:05Þ  102 GeV2;

j3j ¼ ð0:83  0:05Þ  102 GeV2:

(15)

Other input parameters used in the numerical analysis are huuið1 GeVÞ ¼ h ddið1 GeVÞ ¼ ð0:243  0:01Þ3 _GeV3_,

hssi ¼ 0:8h uui, m2

0ð1 GeVÞ ¼ ð0:8  0:2Þ GeV2[20], and

m0 ¼ 1:192 GeV.

Moreover, the sum rules for the transition form factors F1ðQ2Þ and F2ðQ2Þ involve the continuum threshold s0,

Borel parameter M2, and the arbitrary parameter  enter-ing to the expression for the interpolatenter-ing current of the₀ baryon. For the value of the continuum threshold we shall use s0 ¼ ð2:8  3:0Þ GeV2, which is obtained from the

mass sum rules analysis [15]. The Borel parameter M2 is the auxiliary parameter and physical quantities such as F1ðQ2Þ and F2ðQ2Þ should be interdependent of it. The

lower bound of the Borel mass is obtained from the con-dition that the higher states and continuum contributions should be less than 40% of the perturbative contribution, while the upper limit of M2 is determined by demanding that the light-cone expansion with increasing twist should be convergent. Numerical analysis shows that both con-ditions are satisfied when M2lies in the region1:3 GeV2  M2  2:0 GeV2. In our calculations, we fix the lower bound of Q2 to be Q2¼ 1:0 GeV2, since above this value of Q2 the higher twist contributions are suppressed. In order to guarantee the higher resonance and continuum contributions to be smaller than the spectral density con-tribution, we consider the upper bound of Q2 as Q2  8:0 GeV2_{. In Figs.1 and2, we depict the dependence of}

the magnetic and electric form factors GMðQ2Þ and GEðQ2Þ

on Q2 at s0 ¼ 3 GeV2, M2¼ 1:4 GeV2, and at several

fixed values of . From these figures we see that the

FIG. 1. The dependence of the magnetic form factor GMðQ2Þ

of the

 ! 0 transition on Q2 at s0¼ 3:0 GeV2,

M2¼ 1:4 GeV2, and at several fixed values of the arbitrary parameter .

FIG. 2. The same as Fig.1, but for the electric charge form factor GEðQ2Þ.

FIG. 3. The dependence of the magnetic form factor GM of

the

 ! 0 transition on cos  at Q2¼ 1:0 GeV2, s0¼ 3:0 GeV2, and at several fixed values of the Borel mass

parameter M2.

magnitude of GMðQ2Þ and GEðQ2Þ for negative (positive)

values of  are negative (positive). Only the  ¼ 1 case is exceptional and at this value of , GEðQ2Þ is positive

although its value is quite small and very sensitive to the values of the input parameters.

As has already been noted, the sum rules for the tran-sition form factors F1ðQ2Þ and F2ðQ2Þ contain also the

auxiliary parameter . For this reason we should find the ‘‘working region’’ of , where these form factors exhibit no dependence on it. For this aim we shall work with a two-step procedure. At the first stage we use the mass sum rules for the 0 baryon analysis of which leads to the domain 0:6  cos   0:3, where  ¼ tan  (see also [17]). Having this region forcos  obtained from mass sum rules, next we analyze the dependence of form factors on this parameter. Hence, we present the dependence of GMðQ2Þ

and GEðQ2Þ on cos  in Figs.3and4at several fixed values

of other auxiliary parameters. We see from these figures that the domain 0:2  cos   0:2 is the common region where the transition form factors are practically indepen-dent ofcos .

In order to compare our predictions on the Q2 depen-dence of the transition form factors with the existing ones

in the literature, we note that there are only four works [7,8,10,12] in which Q2 dependence of the

 ! 0 transition form factors are studied. In all other works, these form factors are studied only at the point Q2 ¼ 0. These form factors are studied up to Q2 ¼ 0:4 GeV2 in [12]. Unfortunately, the light-cone sum rules method is not applicable in the region Q2< 1 GeV2 and for this reason we cannot compare our results with the predictions of [12]. When we compare our results on GMðQ2Þ with those

given in [8] we see that, they are very close to the pre-diction of [8] in the working region of 0:2  cos  

0:2, while our results on GEðQ2Þ are larger compared to

those obtained in [8]. A comparison of our results on GMðQ2Þ with the ones calculated in [10] shows that our

predictions are smaller than theirs. However, the situation is contrary in the case of GEðQ2Þ, i.e., our results are larger

compared to the predictions given in [10]. Therefore, checking the predictions of different approaches on the study of the Q2 dependence of the form factors for the

 ! 0 transition receives special interest. Further improvements of our predictions on the transition form factors could be achieved by including the Oð_sÞ correc-tions to DAs, as well as considering possible future improvements of nonperturbative input parameters.

In conclusion, we studied the

 ! 0transition form factors within the light-cone QCD sum rules using the most general form of the interpolating current for the0baryon. We obtained the working regions for the Borel mass parameter and the arbitrary parameter  entering to the expressions of the interpolating current. We observed that the electric charge form factor GEðQ2Þ is quite small as

expected. We also compared our results on GEðQ2Þ and

GMðQ2Þ with the predictions existing in the literature. We

saw that our results on GMðQ2Þ are very close to those that

are calculated by the relativistic constituent quark model [8]. We further observed that our prediction on the mag-netic (electric charge) form factor is smaller (larger) com-pared to the results of the covariant spectator quark model. The Q2dependence of the transition form factors presented in this work can be very useful in choosing the right model.

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